2015 - 2016 - Brevard Public Schoolselementarypgms.sp.brevardschools.org/Home... · INTRODUCTION TO...

A Teacher’s Guide to

Marching Ahead

with the

Mathematics Florida Standards

2015 - 2016Grade 3

Getting the Facts about Mathematics Florida Standards

Third Grade Table of Contents

I. Planning

Introduction to Pacing and Sequencing

Pacing and Sequencing Chart

Test Item Specifications

Operations and Properties Tables

II. Standards for Mathematical Practice

What Do Good Problem Solvers Do?

What Constitutes a Cognitively Demanding Task?

Key Ideas in Mathematics

Standards for Mathematical Practice Descriptions

Standards for Mathematical Practice – Student Behaviors

Standards for Mathematical Practice – Student Friendly Language

Standards for Mathematical Practice – Sample Questions for Teachers to Ask

Standards for Mathematical Practice in Action

Standards for Mathematical Practice in 3rd Grade

Standards for Mathematical Practice Posters

III. Getting to Know the Mathematics Florida Standards (MAFS)

Breaking the Code

MAFS by Grade Level at a Glance

Mathematics Florida Standards Changes

CCSS Domains, Clusters, and Critical Areas of Focus

Domain Progressions

Third Grade Domain/Cluster Descriptors and Clarifications

NOTE: While some of the documents in this section were written based on Common Core Standards, they still contain information that can be used with Mathematics Florida Standards (MAFS). The changes as listed on the chart titled Mathematics Florida Standards Changes must be considered when using these documents.

IV. Additional Resources

Addition and Subtraction Strategies

Basic Multiplication and Division Strategies

Four Corners and Rhombus Math Graphic Organizer

Depth of Knowledge Levels/ Cognitive Complexity of Mathematics Items

Planning

INTRODUCTION TO PACING AND SEQUENCING- GRADE 3

INSTRUCTION:

All instruction must be standards-based. The textbook is a resource and textbook

lessons must be carefully chosen andaligned with the standards targeted forinstruction.

It is critical that the Pacing and SequenceChart and the FSA Test ItemSpecifications are used for planning andimplementing lessons.

The entire pacing and sequencing chartshould be previewed in order to beginwith the end in mind and understand howthe mathematical concepts growthroughout the year.

CONNECTIONS BETWEEN THE DOMAINS:

Standards are not meant to be taught inisolation.

Each standard supports other standardsand will continue to be developedthroughout the year.

MAFS.3.MD.1.2 (volume) andMAFS.3.MD.1.1 (time)The content of these standards must bedeveloped within short segments ratherthan being taught by completing thetextbook topics day by day.

PROBLEM-SOLVING:

Emphasis should be on engagingstudents in deeper levels of thinking andanalyzing.

Students must have many opportunitiesto explore the content of the standardsthrough real-world problem-solving tasks.

Mathematical discourse must be anintegral part of instruction.

MEASUREMENT:

Hands-on opportunities for students to beengaged in measurement are critical.

Hands-on measurement tasks may betaught within the science and socialstudies curriculum

VOCABULARY:

Correct mathematical vocabulary MUSTbe used. For example, students areexpected to use terms such as addend,sum, product, and so on.

ALGORITHMS AND FORMULAS:

No formal algorithms or formulas areused in 3rd grade

POST-FSA IDEAS:

Students should continue to work oncritical areas within the grade levelstandards

Project-based lessons and activities areencouraged.

Possible resources to use are:

AIMS Solve It! Navigating Through Numbers and

Operations in Grades 3-5, NCTM EnVision Math Worldscapes Literature

Library The Super Source Series,

ETA/Cuisenaire Teaching Student-Centered

Mathematics, Vol.1, J.A.Van de Walleand L.H. Lovin

Good Questions for Math Teaching, byPeter Sullivan and Pat Lilburn

Third Grade Mathematics Florida Standards, Pacing and Sequencing Chart, page 1 of 18, Brevard Public Schools, 2015 – 2016

*Explanations and Examples were excerpted and modified from Arizona Department of Education, North Carolina Department of Public Instruction documents, and FSA Test Item Specifications.

Standards for Mathematical Practice

Make sense of problems and persevere in solving

them.

Reason abstractly and quantitatively.

Construct viable arguments and critique the reasoning of others.

Model with mathematics.

Use appropriate tools

strategically.

Attend to precision.

Look for and make use of

structure.

Look for and express regularity

in repeated reasoning.

First Nine Weeks

Mathematics Florida Standards (MAFS)

Explanations and Examples *

MAFS.3.MD.2.3:

Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step “how many more” and “how many less” problems using information presented in scaled bar graphs. For example, draw a bar graph in which each square in the bar graph might represent 5 pets.

Students should have opportunities reading and solving problems using scaled graphs before being asked to draw one. The

following graphs all use five as the scale interval, but students should experience different intervals to further develop their

understanding of scaled graphs and how they relate to number facts.

While exploring data concepts, students should pose a question, collect data, analyze data, and interpret data. Students should be graphing data that are relevant to their lives.

MAFS.3.MD.1.1:

Tell and write time to the nearest minute and

measure time intervals in minutes. Solve

word problems involving addition and

subtraction of time intervals in minutes, e.g.,

by representing the problem on a number

line diagram.

This standard calls for students to solve elapsed time, including word problems. Students could use clock models or number lines to solve. On a number line, students should be given the opportunity to determine the intervals and size of jumps on their number line. Students could use predetermined number lines (e.g., intervals every 5 minutes) or open number lines (intervals determined by students).

Example: James wakes up for school at 6:45 a.m. He is ready to leave for school after he showers, dresses, and eats breakfast. It takes him 5 minutes to get showered, 15 minutes to get dressed and 15 minutes to eat breakfast. What time will he be ready to leave for school?

MAFS.3.MD.1.2:

Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (L). Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units.

This standard asks for students to reason about the units of mass and volume. Students need multiple opportunities weighing

classroom objects and filling containers to help them develop a basic understanding of the size of a liter, and weight of a gram, and a

kilogram. Milliliters may also be used to show amounts that are less than a liter. Word problems should only be one-step and include

the same units. Students need practice reading scales of different intervals. Students should estimate before actually finding the actual

measurement.



MAFS.3.NBT.1.1:

Use place value understanding to round

whole numbers to the nearest 10 or 100.

Number sense and computational understanding are built on a firm understanding of place value. Place value understanding extends beyond an algorithm or procedure for rounding. The expectation is that students acquire a deep understanding of place value and number sense and are able to explain and reason about the answers they get when rounding. Students need various experiences using a number line and a hundreds chart as tools to support their work with rounding.

MAFS.3.NBT.1.2:

Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.

Fluency involves accuracy (correct answer), efficiency (a reasonable amount of steps), and flexibility (using strategies that demonstrate number sense). The word algorithm refers to a procedure or a series of steps. There are other algorithms other than the standard algorithm. Third grade students should have experiences other than the standard algorithm. The standard algorithm for addition and subtraction is not introduced until 4th grade (MAFS).

Problems should include both vertical and horizontal forms, including opportunities for students to apply the commutative and

associative properties. Students explain their thinking and show their work by using strategies, and verify that their answer is

reasonable.

Example:

There are 178 fourth graders and 225 fifth graders on the playground. What is the total number of students on the playground?

Student 1

100 + 200 = 300

70 + 20 = 90

8 + 5 = 13

300 + 90 + 13 = 403 students

Student 2

I added 2 to 178 to get 180. I

added 220 to get 400. I added

the 3 left over to get 403.

Student 3

I know that 75 plus 25 equals 100. I then added 1

hundred from 178 and 2 hundreds from 275. I had a

total of 4 hundreds and I had 3 more left to add. So I

have 4 hundreds plus 3 more which is 403.



MAFS.3.OA.4.9:

Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends.

NOTE: This standard should be

developed throughout the year as new

operations are introduced.

This standard calls for students to examine arithmetic patterns involving both addition and multiplication. Arithmetic patterns are

patterns that change by the same rate, such as adding the same number. For example, the series 2, 4, 6, 8, 10 is an arithmetic

pattern that increases by 2 between each term.

This standard also involves identifying patterns related to the properties of operations.

Examples:

• Even numbers are always divisible by 2. Even numbers can always be decomposed into 2 equal addends (14 = 7 + 7). This

builds on student work with doubles in K-2.

• Multiples of even numbers (2, 4, 6, and 8) are always even numbers.

• On a multiplication chart, the products in each row and column increase by the same amount (skip counting).

• On an addition chart, the sums in each row and column increase by the same amount.

MAFS.3.OA.1.1:

Interpret products of whole numbers, e.g., interpret 5 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 7.

Multiplication requires students to think in terms of groups of things or an equal amount of objects, rather than individual things. Students learn that the multiplication symbol ‘x’ means “groups of” and problems such as 6 x 8 refer to 6 groups of 8.

The terms factor and product, should be used when describing multiplication. (factor x factor = product)

MAFS.3.OA.1.2:

Interpret whole-number quotients of wholenumbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8.

Students recognize the operation of division in two different types of situations. One situation requires determining how many groups

(Jim purchased 48 pencils. Each package contained 6 pencils. How many packages did he buy?) and the other situation requires sharing or determining how many in each group (Jim purchased 48 pencils. They were divided equally into 8 packages. How many pencils were in each package).

The terms factor and product should be used for division (product ÷ factor = factor) in order to develop understanding of these as inverse operations.



MAFS.3.OA.2.6:

Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8.

Multiplication and division are inverse operations and that understanding can be used to find the unknown. Fact family triangles

demonstrate the inverse operations of multiplication and division by showing the two factors and how those factors relate to the product.

Examples:

3 x 5 = 15 5 x 3 = 15 (factor x factor = product)

15 ÷ 3 = 5 15 ÷ 5 = 3 (product ÷ factor = factor)

Students need opportunities to see that the answer to a division problem is one of the factors.





them.





strategically.



structure.



Second Nine Weeks



MAFS.3.OA.1.4:

Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations

8 ? = 48, 5 = ÷ 3, 6 6 = ?

Students explore inverse relationships between multiplication and division.

Students apply their understanding of the meaning of the equal sign as ”the same as” or ‘balances” and are able to interpret an equation

with an unknown.

or

Equations in the form of a x b = c and c = a x b should be used interchangeably, with the unknown in different positions.

Example:

Rachel has 3 bags of marbles. She has a total of 12 marbles. If each bag contains the same amount of marbles, how many marbles are in

each bag?

3 x □ = 12

12 ÷ 3 = □

3x 18 18 5 x 6 x 5

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MAFS.3.OA.1.3:

Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

Students use a variety of representations for creating and solving one-step word problems. They use multiplication and division of whole

numbers up to 10x10.

Word problems should be represented in multiple ways including but not limited to:

Equations: 3 x 4 = ?, 4 x 3 = ?, 12 ÷ 4 = ? and 12 ÷ 3 = ?

Array:

Equal groups

Repeated addition: 4 + 4 + 4 or repeated subtraction

Three equal jumps forward from 0 on the number line to 12 or three equal jumps backwards from 12 to 0



MAFS.3.OA.2.5:

Apply properties of operations as strategies to multiply and divide. Examples: If 6 4 = 24 is known, then 4

6 = 24 is also known. (Commutative property of multiplication)

3 5 2 can be found by 3 5 = 15, then

15 2 = 30, or by 5 2 = 10, then 3 10 =

30. (Associative property of multiplication)

Knowing that 8 5 = 40 and 8 2 = 16, one

can find 8 7 as 8 (5 + 2) = (8 5) + (8 2) = 40 + 16 = 56. (Distributive property)

Students represent equations using various objects, pictures, words and symbols in order to develop their understanding of properties.

They must apply these properties flexibly and fluently. Students need not use formal terms for these properties.

Models help build understanding of the commutative property:

3 x 6 = 6 x 3

is the same quantity as

4 x 3 = 3 x 4

An array explicitly demonstrates the concept of the commutative property.

4 rows of 3 or 4 x 3 3 rows of 4 or 3 x 4



MAFS.3.OA.2.5: (Continued) Apply properties of operations as strategies to multiply and divide.

Examples: If 6 4 = 24 is known, then 4 6 = 24 is also known. (Commutative property of multiplication)

3 5 2 can be found by 3 5 = 15, then

15 2 = 30, or by 5 2 = 10, then 3 10 =

30. (Associative property of multiplication)

Knowing that 8 5 = 40 and 8 2 = 16, one

can find 8 7 as 8 (5 + 2) = (8 5) + (8 2) = 40 + 16 = 56. (Distributive property)

Using the associative property, students are able to create simpler calculations that are easier to do mentally.

Students are introduced to the distributive property of multiplication over addition as a strategy for using products they know to solve

products they don’t know.

Example:

If students are asked to find the product of 7 x 8, they might decompose 7 into 5 and 2 and then multiply 5 x 8 and 2 x 8 to arriveat 40 + 16 or 56. Students should learn that they can decompose either of the factors. It is important to note that the students mayrecord their thinking in different ways.

MAFS.3.NBT.1.3:

Multiply one-digit whole numbers by multiples of 10 in the range 10 - 90

(e.g., 9 80, 5 60) using strategies based on place value and properties of operations.

This standard extends students’ work in multiplication by having them apply their understanding of place value.

Students use base ten blocks, diagrams, or hundreds charts to multiply one-digit numbers by multiples of 10 from 10-90.

Examples:

They can interpret 2 x 40 as 2 groups of 4 tens or 8 groups of 10.

They understand that 5 x 60 is 5 groups of 6 tens or 30 tens, and know that 30 tens is 300.

After developing this understanding they begin to recognize the patterns in multiplying one-digit numbers by multiples of 10.

40 + 16 = 56

3 x 5 x 2 = (3 x 5) x 2

= 15 x 2

= 30

3 x 5 x 2 = 3 x (5 x 2)

= 3 x 10

= 30



MAFS.3.OA.4.8:

Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

Examples:

Jerry earned 231 points at school last week. This week he earned 79 points. If he uses 60 points to earn free time on a computer, howmany points will he have left?

A student may use the number line above to describe his/her thinking,

“231 + 9 = 240 so now I need to add 70 more. 240, 250 (10 more), 260 (20 more), 270, 280, 290, 300, 310 (70 more). Now I need to count back 60. 310, 300 (back 10), 290 (back 20), 280, 270, 260, 250 (back 60).”

On Monday Mike ran 5 miles. On Tuesday he ran 6 miles. His goal is to run 30 miles by the end of the week. How many miles doesMike have left to run in order to meet his goal? Write an equation and find the solution. (5+6+b=30)

When students solve word problems, they use various estimation skills which include identifying when estimation is appropriate,

determining the level of accuracy needed, selecting the appropriate method of estimation, and verifying solutions or determining the

reasonableness of solutions.

Estimation strategies include, but are not limited to: using benchmark numbers that are easy to compute (students select close whole numbers to determine an estimate).

using friendly or compatible numbers (students seek to fit numbers together, e.g., rounding to factors and grouping numberstogether that have round sums like 100 or 1000).

Here are some typical estimation strategies for the problem:

On a vacation, your family travels 267 miles on the first day, 194 miles on the second day and 34 miles on the third day. About how many miles did they travel?

Student 1 I first thought about 267 and 34. I noticed that their sum is about 300. Then I knew that 194 is close to 200. When I put 300 and 200 together, I get 500.

Student 2 I first thought about 194. It is really close to 200. I also have 2 hundreds in 267. That gives me a total of 4 hundreds. Then I have 67 in 267 and the 34. When I put 67 and 34 together that is really close to 100. When I add that hundred to the 4 hundreds that I already had I end up with 500.

Student 3 I rounded 267 to 300. I rounded 194 to 200. I rounded 34 to 30. When I added 300, 200 and 30, I know my answer will be about 530.



MAFS.3.OA.3.7:

Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that

8 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.

Fluency involves accuracy (correct answer), efficiency (a reasonable amount of steps) and flexibility (using strategies that demonstrate number sense. Students demonstrate fluency with multiplication facts through 10 and the related division facts. Multiplying and dividing fluently refers to knowledge of procedures, knowledge of when and how to use them appropriately, and skill in performing them flexibly, accurately, and efficiently.

Strategies students may use to attain fluency include

• Multiplication Properties (Zero, Identity, Commutative, Associative, and Distributive

• Doubles (2s facts), Doubling twice (4s), Doubling three times (8s)

• Tens facts (relating to place value, 5 x 10 is 5 tens or 50)

• Five facts (half of tens)

• Skip counting (counting groups of __ and knowing how many groups have been counted)

• Square numbers (ex: 3 x 3)

• Nines (10 groups less one group, e.g., 9 x 3 is 10 groups of 3 minus one group of 3)

• Decomposing into known facts (6 x 7 is 6 x 6 plus one more group of 6)

Students should have exposure to multiplication and division problems presented in both vertical and horizontal forms.

MAFS.3.MD.3.5:

Recognize area as an attribute of plane figures and understand concepts of area measurement.

a. A square with side length 1 unit, called “a unit square,” is said to have “one square unit” of area, and can be used to measure area.

b. A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units.

Students explore the concept of covering a region with unit squares to measure area.

a. One unit square can be used as a measuring tool to find the area of a shape.

b. The unit squares cover the shape and then are added together to determine the area of the shape.

MAFS.3.MD.3.6:

Measure areas by counting unit squares (square cm, square m, square in., square ft., and improvised units).

Students should find the area by counting in metric, customary or non-standard square units. Using different sized graph paper, students can explore the area measured in square centimeters and square inches.



MAFS.3.MD.3.7:

Relate area to the operations of multiplication and addition. a. Find the area of a rectangle with whole-

number side lengths by tiling it, andshow that the area is the same aswould be found by multiplying the sidelengths.

b. Multiply side lengths to find areas ofrectangles with whole-number sidelengths in the context of solving realworld and mathematical problems, andrepresent whole-number products asrectangular areas in mathematicalreasoning.

c. Use tiling to show in a concrete casethat the area of a rectangle with whole-number side lengths a and b + c is thesum of a x b and a x c. Use areamodels to represent the distributiveproperty in mathematical reasoning.

d. Recognize area as additive. Find areasof rectilinear figures by decomposingthem into non-overlapping rectanglesand adding the areas of the non-overlapping parts, applying thistechnique to solve real world problems.

Students tile areas of rectangles, determine the area, record the length and width of the rectangle, investigate the patterns in the numbers,

and discover that the area is the length times the width. For shapes made of composite rectangles (rectilinear figures), teachers can use a

hands-on activity to have students physically cut the shape into component rectangles before finding the area of each rectangle.

Example:

Joe and John made a poster that was 4’ by 3’. Mary and Amir made a poster that was 4’ by 2’. They placed their posters on the wall side-by-side so that that there was no space between them. How much area will the two posters cover?

Students use pictures, words, and numbers to explain their understanding of the distributive property in this context.

A rectilinear figure is a polygon composed of squares and rectangles.

Students can decompose a rectilinear figure into different rectangles or squares. They find the area of the figure by adding the areas of

each of the quadrilaterals together.

4 x 3 + 4 x 2 = 20 square feet 4 (3 + 2) = 20 square feet

4 x 5 = 20 square feet

4’

3’ 2’

3”

4” 7”

8”

10”

12”

3”

12” 7”

8”





them.





strategically.



structure.



Third Nine Weeks



MAFS.3.G.1.2:

Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. For example, partition a shape into 4 parts with equal area, and

describe the area of each part as 4

1 of the

area of the shape.

Given a shape, students partition it into equal parts recognizing that these parts all have the same area. They identify the fractional name of each part and are able to partition a shape into parts with equal areas in several different ways.

MAFS.3.NF.1.1:

Understand a fraction b

1as the quantity

formed by 1 part when a whole is partitioned

into b equal parts; understand a fraction b

a

as the quantity formed by a parts of size b

1.

Fractional models in third grade include area (parts of a whole) and number lines. Denominators are limited to 2,3,4,6,and 8. Set models (parts of a group) are not introduced in Grade 3. Students should focus on the concept that a fraction is composed of many pieces of a unit fraction, which has a numerator of 1.

For example, the fraction 6

3 is composed of 3 pieces that each have a size of

6

1.

Some important concepts related to developing understanding of fractions include:

Fractional parts must be equal-sized

The number of equal parts tell how many make a whole

As the number of equal pieces in the whole increases, the size of the fractional pieces decrease

The size of the fractional part is relative to the whole

When a whole is cut into equal parts, the denominator represents the number of equal parts

The numerator of a fraction is the count of the number of equal parts

Students need many opportunities with concrete models to develop understanding of fractions. They also need many opportunities to solve word problems that require parts of a whole.

4

1

4

1

4

1

4

1 4

1

4

1

4

1

4

1



MAFS.3.NF.1.2:

Understand a fraction as a number on the number line; represent fractions on a number line diagram.

a. Represent a fraction b

1on a number

line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that

each part has size b

1and that the

endpoint of the part based at 0 locates

the number b

1on the number line.

b. Represent a fraction b

a on a number

line diagram by marking off a lengths

b

1from 0. Recognize that the resulting

interval has size b

a and that its

endpoint locates the number b

a on the

number line.

On a number line from 0 to 1, students can partition (divide) it into equal parts and recognize that each segmented part represents the same length.

Students label each fractional part based on how far it is from zero to the endpoint.

Denominators are limited to 2,3,4,6, and 8. Number lines may extend beyond 1.



MAFS.3.NF.1.3:

Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.

a. Understand two fractions as equivalent(equal) if they are the same size, or thesame point on a number line.

b. Recognize and generate simple

equivalent fractions, e.g.,2

1 =

4

2,

6

4 =

3

2. Explain why the fractions are

equivalent, e.g., by using a visual fraction model.

c. Express whole numbers as fractions,and recognize fractions that areequivalent to whole numbers.Examples: Express 3 in the form

3 =1

3; recognize that

1

6 = 6; locate

4

4

and 1 at the same point of a numberline diagram.

d. Compare two fractions with the samenumerator or the same denominator byreasoning about their size. Recognizethat comparisons are valid only whenthe two fractions refer to the samewhole. Record the results ofcomparisons with the symbols >, =, or<, and justify the conclusions, e.g., byusing a visual fraction model.

3.NF.1.3a and 3.NF.1.3b: These standards call for students to use visual fraction models (area models) and number lines (linear models)to explore the idea of equivalent fractions. Students should only explore equivalent fractions using models and reasoning, rather than using algorithms or procedures.

3.NF.1.3c: This standard includes writing whole numbers as fractions.

3.NF.1.3d: This standard involves comparing fractions with or without visual fraction models including number lines. An importantconcept when comparing fractions is to look at the size of the parts and the number of the parts. Experiences should encourage

students to reason about the size of pieces, the fact that 3

1 of a cake is larger than

4

1 of the same cake. Since the same cake (the

whole) is split into equal pieces, thirds are larger than fourths.

In this standard, students should also reason that comparisons are only valid if the wholes are identical. For example, 2

1 of a

large pizza is a different amount than 2

1 of a small pizza. Students should be given opportunities to discuss and reason about which

2

1 is larger and why?

Denominators are limited to 2,3,4,6, and 8.

Items may not use the term “simplify” or “lowest terms” in directives.



MAFS.3.MD.2.4:

Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units—whole numbers, halves, or quarters.

It’s important to review with students how to read and use a standard ruler including details about halves and quarter marks on the ruler.

Students should connect their understanding of fractions to measuring to one-half and one-quarter inch. Third graders need many

opportunities measuring the length of various objects in their environment.

Some important ideas related to measuring with a ruler are:

The starting point of where one places a ruler to begin measuring

Measuring is approximate. Items that students measure will not always measure exactly4

1,

2

1 or one whole inch. Students will

need to measure to the nearest 4

1or

2

1.

Making paper rulers and folding to find the half and quarter marks may help students develop a stronger understanding ofmeasuring length.

Students generate data by measuring and then creating a line plot to display their findings. An example of a line plot is shown below:

Objects Measured

Measurement in Inches



MAFS.3.MD.4.8:

Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters

Examples:

Students should be able to use geoboards, tiles, and graph paper to find the perimeter of a polygon. Students should also be able to find

the perimeter of all the possible rectangles that have a given perimeter (e.g., find the rectangles with a perimeter of 14 cm).

Students use geoboards, tiles, graph paper, or technology to find all the possible rectangles with a given area (e.g., find the rectangles that

have an area of 12 square units).

Example:

Same Perimeter/Different Area Same Area/Different Perimeter

Perimeter Length Width Area Area Length Width Perimeter

14 in. 4 in. 3 in. 12 sq. in 12 sq. in. 1 in. 12 in. 26 in.

14 in. 5 in. 2 in. 10 sq. in 12 sq. in. 2 in. 6 in. 16 in.

14 in. 6 in. 1 in. 6 sq. in 12 sq. in 3 in. 4 in. 14 in.




Charts can allow the students to identify the factors of 12, connect the results to the commutative property, and discuss the differences in

perimeter within the same area. This chart can also be used to investigate rectangles with the same perimeter.





them.





strategically.



structure.



Fourth Nine Weeks



MAFS.3.G.1.1: Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories.

Students recognize shapes that are and are not quadrilaterals by examining the properties of the geometric figures. They

conceptualize that a quadrilateral must be a closed figure with four straight sides and begin to notice characteristics of the angles and

the relationship between opposite sides. Students should be encouraged to provide details and use proper vocabulary when

describing the properties of quadrilaterals. They sort geometric figures and identify squares, rectangles, t rapezo id , parallelograms

and rhombuses as quadrilaterals.

Students should classify shapes by attributes and draw shapes that fit specific categories.

For example, parallelograms include: squares, rectangles, parallelograms and rhombuses. Also, the broad category quadrilaterals include all types of parallelograms, trapezoids and other four-sided figures.

*The terms parallel and perpendicular lines are not assessed until fourth grade.



THIRD GRADE – CRITICAL AREAS OF FOCUS

In Grade 3, instructional time should focus on four critical areas:

(1) developing understanding of multiplication and division and strategies for multiplication and division within 100;

(2) developing understanding of fractions, especially unit fractions (fractions with numerator 1);

(3) developing understanding of the structure of rectangular arrays and of area; and

(4) describing and analyzing two-dimensional shapes.

(1) Students develop an understanding of the meanings of multiplication and division of whole numbers through activities and problems involving equal-sized groups, arrays, and area models; multiplication is finding an unknown product, and division is finding an unknown factor in these situations. For equal-sized group situations, division can require finding the unknown number of groups or the unknown group size. Students use properties of operations to calculate products of whole numbers, using increasingly sophisticated strategies based on these properties to solve multiplication and division problems involving single-digit factors. By comparing a variety of solution strategies, students learn the relationship between multiplication and division.

(2) Students develop an understanding of fractions, beginning with unit fractions. Students view fractions in general as being built out of unit fractions, and they use fractions along with visual fraction models to represent parts of a whole. Students understand that the size of a

fractional part is relative to the size of the whole. For example, 2

1 of the paint in a small bucket could be less paint than

3

1of the paint in a

larger bucket, but 3

1 of a ribbon is longer than

5

1 of the same ribbon because when the ribbon is divided into 3 equal parts, the parts are

longer than when the ribbon is divided into 5 equal parts. Students are able to use fractions to represent numbers equal to, less than, and greater than one. They solve problems that involve comparing fractions by using visual fraction models and strategies based on noticing equal numerators or denominators.

(3) Students recognize area as an attribute of two-dimensional regions. They measure the area of a shape by finding the total number of same size units of area required to cover the shape without gaps or overlaps, a square with sides of unit length being the standard unit for measuring area. Students understand that rectangular arrays can be decomposed into identical rows or into identical columns. By decomposing rectangles into rectangular arrays of squares, students connect area to multiplication, and justify using multiplication to determine the area of a rectangle.

(4) Students describe, analyze, and compare properties of two-dimensional shapes. They compare and classify shapes by their sides and angles,

and connect these with definitions of shapes. Students also relate their fraction work to geometry by expressing the area of part of a shape as

a unit fraction of the whole.

DRAFT

Grade3Mathematics ItemSpecifications

Grade 3 Mathematics Item Specifications Florida Standards Assessments

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The draft Florida Standards Assessments (FSA) Test Item Specifications (Specifications) are based upon the Florida Standards and the Florida Course Descriptions as provided in CPALMs. The Specifications are a resource that defines the content and format of the test and test items for item writers and reviewers. Each grade‐level and course Specifications document indicates the alignment of items with the Florida Standards. It also serves to provide all stakeholders with information about the scope and function of the FSA. Item Specifications Definitions Also assesses refers to standard(s) closely related to the primary standard statement. Clarification statements explain what students are expected to do when responding to the question. Assessment limits define the range of content knowledge and degree of difficulty that should be assessed in the assessment items for the standard. Item types describe the characteristics of the question. Context defines types of stimulus materials that can be used in the assessment items.

Context – Allowable refers to items that may but are not required to have context.

Context – No context refers to items that should not have context.

Context – Required refers to items that must have context.


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Technology‐EnhancedItemDescriptions:TheFloridaStandardsAssessments(FSA)arecomposedoftestitemsthatincludetraditionalmultiple‐choiceitems,itemsthatrequirestudentstotypeorwritearesponse,andtechnology‐enhanceditems(TEI).Technology‐enhanceditemsarecomputer‐delivereditemsthatrequirestudentstointeractwithtestcontenttoselect,construct,and/orsupporttheiranswers.Currently,thereareninetypesofTEIsthatmayappearoncomputer‐basedassessmentsforFSAMathematics.ForstudentswithanIEPor504planthatspecifiesapaper‐basedaccommodation,TEIswillbemodifiedorreplacedwithtestitemsthatcanbescannedandscoredelectronically.

Forsamplesofeachoftheitemtypesdescribedbelow,seetheFSATrainingTests.

Technology‐EnhancedItemTypes–Mathematics

1. EditingTaskChoice–Thestudentclicksahighlightedwordorphrase,whichrevealsadrop‐downmenucontainingoptionsforcorrectinganerroraswellasthehighlightedwordorphraseasitisshowninthesentencetoindicatethatnocorrectionisneeded.Thestudentthenselectsthecorrectwordorphrasefromthedrop‐downmenu.Forpaper‐basedassessments,theitemismodifiedsothatitcanbescannedandscoredelectronically.Thestudentfillsinacircletoindicatethecorrectwordorphrase.

2. EditingTask–Thestudentclicksonahighlightedwordorphrasethatmaybeincorrect,whichrevealsatextbox.Thedirectionsinthetextboxdirectthestudenttoreplacethehighlightedwordorphrasewiththecorrectwordorphrase.Forpaper‐basedassessments,thisitemtypemaybereplacedwithanotheritemtypethatassessesthesamestandardandcanbescannedandscoredelectronically.

3. HotText–a. SelectableHotText–Excerptedsentencesfromthetextarepresented

inthisitemtype.Whenthestudenthoversovercertainwords,phrases,orsentences,theoptionshighlight.Thisindicatesthatthetextisselectable(“hot”).Thestudentcanthenclickonanoptiontoselectit.Forpaper‐basedassessments,a“selectable”hottextitemismodifiedsothatitcanbescannedandscoredelectronically.Inthisversion,thestudentfillsinacircletoindicateaselection.


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b.Drag‐and‐DropHotText–Certainnumbers,words,phrases,or

sentencesmaybedesignated“draggable”inthisitemtype.Whenthestudenthoversovertheseareas,thetexthighlights.Thestudentcanthenclickontheoption,holddownthemousebutton,anddragittoagraphicorotherformat.Forpaper‐basedassessments,drag‐and‐drophottextitemswillbereplacedwithanotheritemtypethatassessesthesamestandardandcanbescannedandscoredelectronically.

4. OpenResponse–Thestudentusesthekeyboardtoenteraresponseintoatextfield.Theseitemscanusuallybeansweredinasentenceortwo.Forpaper‐basedassessments,thisitemtypemaybereplacedwithanotheritemtypethatassessesthesamestandardandcanbescannedandscoredelectronically.

5. Multiselect–Thestudentisdirectedtoselectallofthecorrectanswersfromamonganumberofoptions.Theseitemsaredifferentfrommultiple‐choiceitems,whichallowthestudenttoselectonlyonecorrectanswer.Theseitemsappearintheonlineandpaper‐basedassessments.

6. GraphicResponseItemDisplay(GRID)–Thestudentselectsnumbers,words,phrases,orimagesandusesthedrag‐and‐dropfeaturetoplacethemintoagraphic.Thisitemtypemayalsorequirethestudenttousethepoint,line,orarrowtoolstocreatearesponseonagraph.Forpaper‐basedassessments,thisitemtypemaybereplacedwithanotheritemtypethatassessesthesamestandardandcanbescannedandscoredelectronically.

7. EquationEditor–Thestudentispresentedwithatoolbarthatincludesavarietyofmathematicalsymbolsthatcanbeusedtocreatearesponse.Responsesmaybeintheformofanumber,variable,expression,orequation,asappropriatetothetestitem.Forpaper‐basedassessments,thisitemtypemaybereplacedwithamodifiedversionoftheitemthatcanbescannedandscoredelectronicallyorreplacedwithanotheritemtypethatassessesthesamestandardandcanbescannedandscoredelectronically.

8. MatchingItem–Thestudentchecksaboxtoindicateifinformationfromacolumnheadermatchesinformationfromarow.Forpaper‐basedassessments,thisitemtypemaybereplacedwithanotheritemtypethatassessesthesamestandardandcanbescannedandscoredelectronically.

9.TableItem–Thestudenttypesnumericvaluesintoagiventable.Thestudentmaycompletetheentiretableorportionsofthetabledependingonwhatisbeingasked.Forpaper‐basedassessments,thisitemtypemaybereplacedwithanotheritemtypethatassessesthesamestandardandcanbescannedandscoredelectronically.


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MathematicalPractices:

TheMathematicalPracticesareapartofeachcoursedescriptionforGrades3‐8,Algebra1,Geometry,andAlgebra2.Thesepracticesareanimportantpartofthecurriculum.TheMathematicalPracticeswillbeassessedthroughout.

MAFS.K12.MP.1.1:

Makesenseofproblemsandpersevereinsolvingthem.

Mathematicallyproficientstudentsstartbyexplainingtothemselvesthemeaningofaproblemandlookingforentrypointstoitssolution.Theyanalyzegivens,constraints,relationships,andgoals.Theymakeconjecturesabouttheformandmeaningofthesolutionandplanasolutionpathwayratherthansimplyjumpingintoasolutionattempt.Theyconsideranalogousproblems,andtryspecialcasesandsimplerformsoftheoriginalprobleminordertogaininsightintoitssolution.Theymonitorandevaluatetheirprogressandchangecourseifnecessary.Olderstudentsmight,dependingonthecontextoftheproblem,transformalgebraicexpressionsorchangetheviewingwindowontheirgraphingcalculatortogettheinformationtheyneed.Mathematicallyproficientstudentscanexplaincorrespondencesbetweenequations,verbaldescriptions,tables,andgraphsordrawdiagramsofimportantfeaturesandrelationships,graphdata,andsearchforregularityortrends.Youngerstudentsmightrelyonusingconcreteobjectsorpicturestohelpconceptualizeandsolveaproblem.Mathematicallyproficientstudentschecktheiranswerstoproblemsusingadifferentmethod,andtheycontinuallyaskthemselves,“Doesthismakesense?”Theycanunderstandtheapproachesofotherstosolvingcomplexproblemsandidentifycorrespondencesbetweendifferentapproaches.

MAFS.K12.MP.2.1:

Reasonabstractlyandquantitatively.

Mathematicallyproficientstudentsmakesenseofquantitiesandtheirrelationshipsinproblemsituations.Theybringtwocomplementaryabilitiestobearonproblemsinvolvingquantitativerelationships:theabilitytodecontextualize—toabstractagivensituationandrepresentitsymbolicallyandmanipulatetherepresentingsymbolsasiftheyhavealifeoftheirown,withoutnecessarilyattendingtotheirreferents—andtheabilitytocontextualize,topauseasneededduringthemanipulationprocessinordertoprobeintothereferentsforthesymbolsinvolved.


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Quantitativereasoningentailshabitsofcreatingacoherentrepresentationoftheproblemathand;consideringtheunitsinvolved;attendingtothemeaningofquantities,notjusthowtocomputethem;andknowingandflexiblyusingdifferentpropertiesofoperationsandobjects.

MAFS.K12.MP.3.1:

Constructviableargumentsandcritiquethereasoningofothers.

Mathematicallyproficientstudentsunderstandandusestatedassumptions,definitions,andpreviouslyestablishedresultsinconstructingarguments.Theymakeconjecturesandbuildalogicalprogressionofstatementstoexplorethetruthoftheirconjectures.Theyareabletoanalyzesituationsbybreakingthemintocases,andcanrecognizeandusecounterexamples.Theyjustifytheirconclusions,communicatethemtoothers,andrespondtotheargumentsofothers.Theyreasoninductivelyaboutdata,makingplausibleargumentsthattakeintoaccountthecontextfromwhichthedataarose.Mathematicallyproficientstudentsarealsoabletocomparetheeffectivenessoftwoplausiblearguments,distinguishcorrectlogicorreasoningfromthatwhichisflawed,and—ifthereisaflawinanargument—explainwhatitis.Elementarystudentscanconstructargumentsusingconcretereferentssuchasobjects,drawings,diagrams,andactions.Suchargumentscanmakesenseandbecorrect,eventhoughtheyarenotgeneralizedormadeformaluntillatergrades.Later,studentslearntodeterminedomainstowhichanargumentapplies.Studentsatallgradescanlistenorreadtheargumentsofothers,decidewhethertheymakesense,andaskusefulquestionstoclarifyorimprovethearguments.

MAFS.K12.MP.4.1:

Modelwithmathematics.

Mathematicallyproficientstudentscanapplythemathematicstheyknowtosolveproblemsarisingineverydaylife,society,andtheworkplace.Inearlygrades,thismightbeassimpleaswritinganadditionequationtodescribeasituation.Inmiddlegrades,astudentmightapplyproportionalreasoningtoplanaschooleventoranalyzeaprobleminthecommunity.Byhighschool,astudentmightusegeometrytosolveadesignproblemoruseafunctiontodescribehowonequantityofinterestdependsonanother.Mathematicallyproficientstudentswhocanapplywhattheyknowarecomfortablemakingassumptionsandapproximationstosimplifyacomplicatedsituation,realizingthatthesemayneedrevisionlater.Theyareabletoidentifyimportantquantitiesinapracticalsituationandmaptheirrelationshipsusingsuchtoolsasdiagrams,two‐waytables,graphs,flowchartsandformulas.Theycananalyzethoserelationshipsmathematicallytodrawconclusions.Theyroutinelyinterprettheir


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mathematicalresultsinthecontextofthesituationandreflectonwhethertheresultsmakesense,possiblyimprovingthemodelifithasnotserveditspurpose.

MAFS.K12.MP.5.1:

Useappropriatetoolsstrategically.Mathematicallyproficientstudentsconsidertheavailabletoolswhensolvingamathematicalproblem.Thesetoolsmightincludepencilandpaper,concretemodels,aruler,aprotractor,acalculator,aspreadsheet,acomputeralgebrasystem,astatisticalpackage,ordynamicgeometrysoftware.Proficientstudentsaresufficientlyfamiliarwithtoolsappropriatefortheirgradeorcoursetomakesounddecisionsaboutwheneachofthesetoolsmightbehelpful,recognizingboththeinsighttobegainedandtheirlimitations.Forexample,mathematicallyproficienthighschoolstudentsanalyzegraphsoffunctionsandsolutionsgeneratedusingagraphingcalculator.Theydetectpossibleerrorsbystrategicallyusingestimationandothermathematicalknowledge.Whenmakingmathematicalmodels,theyknowthattechnologycanenablethemtovisualizetheresultsofvaryingassumptions,exploreconsequences,andcomparepredictionswithdata.Mathematicallyproficientstudentsatvariousgradelevelsareabletoidentifyrelevantexternalmathematicalresources,suchasdigitalcontentlocatedonawebsite,andusethemtoposeorsolveproblems.Theyareabletousetechnologicaltoolstoexploreanddeepentheirunderstandingofconcepts.

MAFS.K12.MP.6.1:

Attendtoprecision.

Mathematicallyproficientstudentstrytocommunicatepreciselytoothers.Theytrytousecleardefinitionsindiscussionwithothersandintheirownreasoning.Theystatethemeaningofthesymbolstheychoose,includingusingtheequalsignconsistentlyandappropriately.Theyarecarefulaboutspecifyingunitsofmeasure,andlabelingaxestoclarifythecorrespondencewithquantitiesinaproblem.Theycalculateaccuratelyandefficiently,expressnumericalanswerswithadegreeofprecisionappropriatefortheproblemcontext.Intheelementarygrades,studentsgivecarefullyformulatedexplanationstoeachother.Bythetimetheyreachhighschooltheyhavelearnedtoexamineclaimsandmakeexplicituseofdefinitions.


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MAFS.K12.MP.7.1:

Lookforandmakeuseofstructure.

Mathematicallyproficientstudentslookcloselytodiscernapatternorstructure.Youngstudents,forexample,mightnoticethatthreeandsevenmoreisthesameamountassevenandthreemore,ortheymaysortacollectionofshapesaccordingtohowmanysidestheshapeshave.Later,studentswillsee7×8equalsthewellremembered7×5+7×3,inpreparationforlearningaboutthedistributiveproperty.Intheexpressionx²+9x+14,olderstudentscanseethe14as2×7andthe9as2+7.Theyrecognizethesignificanceofanexistinglineinageometricfigureandcanusethestrategyofdrawinganauxiliarylineforsolvingproblems.Theyalsocanstepbackforanoverviewandshiftperspective.Theycanseecomplicatedthings,suchassomealgebraicexpressions,assingleobjectsorasbeingcomposedofseveralobjects.Forexample,theycansee5–3(x–y)²as5minusapositivenumbertimesasquareandusethattorealizethatitsvaluecannotbemorethan5foranyrealnumbersxandy.

MAFS.K12.MP.8.1:

Lookforandexpressregularityinrepeatedreasoning.

Mathematicallyproficientstudentsnoticeifcalculationsarerepeated,andlookbothforgeneralmethodsandforshortcuts.Upperelementarystudentsmightnoticewhendividing25by11thattheyarerepeatingthesamecalculationsoverandoveragain,andconcludetheyhavearepeatingdecimal.Bypayingattentiontothecalculationofslopeastheyrepeatedlycheckwhetherpointsareonthelinethrough(1,2)withslope3,middleschoolstudentsmightabstracttheequation(y–2)/(x–1)=3.Noticingtheregularityinthewaytermscancelwhenexpanding(x–1)(x+1),(x–1)(x²+x+1),and(x–1)(x³+x²+x+1)mightleadthemtothegeneralformulaforthesumofageometricseries.Astheyworktosolveaproblem,mathematicallyproficientstudentsmaintainoversightoftheprocess,whileattendingtothedetails.Theycontinuallyevaluatethereasonablenessoftheirintermediateresults.


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ReferenceSheets:•Referencesheetsandz‐tableswillbeavailableasonlinereferences(inapop‐upwindow).Apaperversionwillbeavailableforpaper‐basedtests.•ReferencesheetswithconversionswillbeprovidedforFSAMathematicsassessmentsinGrades4–8andEOCMathematicsassessments.•ThereisnoreferencesheetforGrade3.•ForGrades4,6,and7,Geometry,andAlgebra2,someformulaswillbeprovidedonthereferencesheet.•ForGrade5andAlgebra1,someformulasmaybeincludedwiththetestitemifneededtomeettheintentofthestandardbeingassessed.•ForGrade8,noformulaswillbeprovided;however,conversionswillbeavailableonareferencesheet.•ForAlgebra2,az‐tablewillbeavailable.

Grade Conversions SomeFormulas z‐table3 No No No4 OnReferenceSheet OnReferenceSheet No5 OnReferenceSheet WithItem No6 OnReferenceSheet OnReferenceSheet No7 OnReferenceSheet OnReferenceSheet No8 OnReferenceSheet No No

Algebra1 OnReferenceSheet WithItem NoAlgebra2 OnReferenceSheet OnReferenceSheet YesGeometry OnReferenceSheet OnReferenceSheet No


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Content Standard MAFS.3.OA Operations and Algebraic Thinking MAFS.3.OA.1 Represent and solve problems involving multiplication and division.

MAFS.3.OA.1.1 Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 × 7.

Assessment Limits Whole number factors may not exceed 10 x 10. Students may not be required to write an equation to represent a product of

whole numbers.

Calculator No

Item Types Equation Editor Multiple Choice Multiselect Open Response Table Item

Context Allowable

Sample Item Item Type

Tom told Mary he planted 4 x 5 flowers. How might Mary describe the arrangement of flowers in Tom’s rectangular‐shaped garden?

Open Response

Tom told Mary he planted 48 flowers in the rectangular‐shaped garden. Which sentence could Mary use to describe how the flowers were planted? A. Tom planted 24 rows of 24 flowers. B. Tom planted 4 rows of 24 flowers. C. Tom planted 40 rows of 8 flowers. D. Tom planted 8 rows of 6 flowers.

Multiple Choice

See Appendix for the practice test item aligned to this standard.


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Content Standard MAFS.3.OA Operations and Algebraic Thinking MAFS.3.OA.1 Represent and solve problems involving multiplication and division. MAFS.3.OA.1.2 Interpret whole‐number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8.

Assessment Limits Whole number quotients and divisors may not exceed 10. Items may not require students to write an equation to represent a quotient of

whole numbers.

Calculator No

Item Types Equation Editor GRID Multiple Choice Multiselect Open Response

Context Allowable


Heidi has 12 apples and 6 bags. She places an equal number of apples in each bag. Drag apples to show how many apples are in each bag.

GRID



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Content Standard MAFS.3.OA Operations and Algebraic Thinking MAFS.3.OA.1 Represent and solve problems involving multiplication and division. MAFS.3.OA.1.3 Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

Assessment Limits All values in items may not exceed whole number multiplication facts of 10 x 10 or the related division facts.

Items may not contain more than one unknown per equation. Items may not contain the words “times as much/many.”

Calculator No

Item Types Equation Editor GRID Multiple Choice Multiselect

Context Required


Craig has 72 grapes. He separates the grapes into 9 equal groups. How many grapes are in each group?

Equation Editor



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Content Standard MAFS.3.OA Operations and Algebraic Thinking MAFS.3.OA.1 Represent and solve problems involving multiplication and division. MAFS.3.OA.1.4 Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 × ? = 48, 5 = _ ÷ 3, 6 × 6 = ?

Assessment Limits All values in items may not exceed whole number multiplication facts of 10 x 10 or the related division facts.

Items must provide the equation. Students may not be required to create the equation.

Calculator No

Item Types Equation Editor Multiple Choice Multiselect

Context No context


A division problem is shown. 9 = ÷ 3 What is the value of the unknown number?

Equation Editor

What is the value of the unknown number in the equation 72 ÷ = 9 ? Equation Editor



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Content Standard MAFS.3.OA Operations and Algebraic Thinking MAFS.3.OA.2 Understand properties of multiplication and the relationship between multiplication and division. MAFS.3.OA.2.5 Apply properties of operations as strategies to multiply and divide. Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)

Assessment Limit All values in items may not exceed whole number multiplication facts of 10 x 10 or the related division facts.

Items may contain no more than two properties in an equation (e.g., a x (b + c) = (a x b) + (c x a)).

Calculator No

Item Types Equation Editor GRID Matching Item Multiple Choice Multiselect

Context No context


An equation is shown. 4 x 9 = 9 x What is the missing value? A. 4 B. 5 C. 9 D. 13

Multiple Choice

Drag numbers to the boxes to create a different expression that is equal to (3 + 4) + 5.

GRID


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Select all the expressions that could be used to find 6 x 10. □ 10 x 6 □ 6 x (2 x 5) □ 6 + (2 x 5) □ (6 x 2) x 5 □ (6 x 8) x (6 x 2)

Multiselect



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Content Standard MAFS.3.OA Operations and Algebraic Thinking MAFS.3.OA.2 Understand properties of multiplication and the relationship between multiplication and division. MAFS.3.OA.2.6 Understand division as an unknown‐factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8.


Calculator No


Context No context


Create a multiplication equation that could be used to solve 21 ÷ 3 = . Equation Editor



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Content Standard MAFS.3.OA Operations and Algebraic Thinking MAFS.3.OA.3 Multiply and divide within 100. MAFS.3.OA.3.7 Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one‐digit numbers.


Calculator No

Item Types Equation Editor Multiple Choice Multiselect Table Item

Context No context


Multiply: 8 x 2 Equation Editor



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Content Standard MAFS.3.OA Operations and Algebraic Thinking MAFS.3.OA.4 Solve problems involving the four operations, and identify and explain patterns in arithmetic. MAFS.3.OA.4.8 Solve two‐step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

Assessment Limits Adding and subtracting is limited to whole numbers within 1,000. All values in multiplication or division situations may not exceed whole number

multiplication facts of 10 x 10 or the related division facts. Students may not be required to perform rounding in isolation. Equations may be provided in items.

Calculator No

Item Types Editing Task Choice Equation Editor Hot Text Multiple Choice Multiselect Open Response

Context Required


A bookstore has 4 boxes of books. Each box contains 20 books. On Monday, the bookstore sold 16 books. How many books remain to be sold?

Equation Editor

On Monday, a bookstore sold 75 books. On Tuesday, the bookstore sold 125 books. The bookstore must sell 500 books by Friday. Create an equation that can be used to find how many more books, b, the bookstore must sell by Friday.

Equation Editor



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Content Standard MAFS.3.OA Operations and Algebraic Thinking MAFS.3.OA.4 Solve problems involving the four operations, and identify and explain patterns in arithmetic. MAFS.3.OA.4.9 Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends.

Assessment Limits Adding and subtracting is limited to whole numbers within 1,000. All values in items may not exceed whole number multiplication facts of 10 x 10

or the related division facts.

Calculator No

Item Types Editing Task Choice Equation Editor GRID Hot Text Multiple Choice Multiselect Table Item

Context No context




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Content Standard MAFS.3.NBT Number and Operations in Base Ten MAFS.3.NBT.1 Use place value understanding and properties of operations to perform multi‐digit arithmetic. MAFS.3.NBT.1.1 Use place value understanding to round whole numbers to the nearest 10 or 100.

Assessment Limit Items may contain whole numbers up to 1,000.

Calculator No

Item Types Equation Editor GRID Matching Item Multiple Choice Multiselect Table Item

Context No context


What value is 846 rounded to the nearest 100? Equation Editor

A. Round 846 to the nearest hundred. B. Round 846 to the nearest ten.

Equation Editor

Select all the numbers that will equal 800 when rounded to the nearest hundred. □ 739 □ 751 □ 792 □ 805 □ 850

Multiselect

An incomplete table is shown. Complete the table by filling in the missing original numbers with possible values.

Original Number

Rounded to Nearest Ten

100

150

190

Table Item

Plot points on the number line to represent all whole number values that round to 500 when rounded to the nearest hundred and to 450 when rounded to the nearest ten.

GRID



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Content Standard MAFS.3.NBT Number & Operations in Base Ten MAFS.3.NBT.1 Use place value understanding and properties of operations to perform multi‐digit arithmetic. MAFS.3.NBT.1.2 Fluently add and subtract within 1,000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.

Assessment Limits Addends and sums are less than or equal to 1,000. Minuends, subtrahends, and differences are less than or equal to 1,000. Items may not require students to name specific properties.

Calculator No

Item Types Equation Editor GRID Multiple Choice Multiselect Table Item

Context No context


What is the sum of 153, 121, and 178?

Equation Editor



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Content Standard MAFS.3.NBT Number & Operations in Base Ten MAFS.3.NBT.1 Use place value understanding and properties of operations to perform multi‐digit arithmetic. MAFS.3.NBT.1.3 Multiply one‐digit whole numbers by multiples of 10 in the range 10–90 (e.g., 9 × 80, 5 × 60) using strategies based on place value and properties of operations.

Assessment Limit Items may not require students to name specific properties.

Calculator No

Item Types Equation Editor Matching Item Multiple Choice Multiselect

Context Allowable


What is the product of 7 and 50? Equation Editor

Select all expressions that have a product of 320. □ 3 x 90 □ 4 x 80 □ 5 x 60 □ 8 x 40 □ 9 x 30

Multiselect

Mr. Engle has 10 tables in his classroom. There are 3 students at each table. Each student has 6 glue sticks. A. How many glue sticks are at each table? B. How many glue sticks do all of Mr. Engle’s students have combined?

Equation Editor



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Content Standard MAFS.3.NF Number and Operations — Fractions MAFS.3.NF.1 Develop understanding of fractions as numbers.

MAFS.3.NF.1.1 Understand a fraction as the quantity formed by 1 part when a

whole is partitioned into b equal parts; understand a fraction as the quantity

formed by a parts of size . Also Assesses: MAFS.3.G Geometry MAFS.3.G.1 Reason with shapes and their attributes. MAFS.3.G.1.2 Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. For example, partition a shape into 4

parts with equal area, and describe the area of each part as of the area of the shape.

Assessment Limits Denominators are limited to 2, 3, 4, 6, and 8. Items are limited to combining or putting together unit fractions rather than

formal addition or subtraction of fractions. Maintain concept of a whole as one entity that can be equally partitioned in

various ways when working with unit fractions. Fractions a/b can be fractions greater than 1. Items may not use the term “simplify” or “lowest terms” in directives. Items may not use number lines. Shapes may include: quadrilateral, equilateral triangle, isosceles triangle, regular

hexagon, regular octagon, and circle.

Calculator No


Context Allowable for 3.NF.1.1; no context for 3.G.1.2


24 | P a g e M a y 2 0 1 6


Each model shown has been shaded to represent a fraction. Which model shows

shaded? A. B. C. D.

Multiple Choice

Each model shown has been shaded to represent a fraction. Which model shows

shaded? A. B. C. D.

Multiple Choice

A figure is shown. Part of the figure is shaded. Which fraction of the total area of the figure does the shaded part represent?

Equation Editor

A figure is shown. Part of the figure is shaded.

Which fraction of the total area of the figure does the shaded part represent?

Equation Editor


25 | P a g e M a y 2 0 1 6


A half of a shape is shown.

Click squares to complete the whole shape.

GRID

A sixth of a shape is shown.

Click squares to complete the whole shape.

GRID

Each shape shown represents of a whole. Drag the shapes into the box to show .

GRID

Each shape shown represents of a whole.

How many shapes should be put together to make ?

Equation Editor

See Appendix for the practice test item aligned to a standard in this group.


26 | P a g e M a y 2 0 1 6

Content Standard MAFS.3.NF Number and Operations – Fractions MAFS.3.NF.1 Develop understanding of fractions as numbers. MAFS.3.NF.1.2 Understand a fraction as a number on the number line; represent fractions on a number line diagram.

MAFS.3.NF.1.2a Represent a fraction on a number line diagram by defining the

interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize

that each part has size and that the endpoint of the part based at 0 locates the

number on the number line.

MAFS.3.NF.1.2b Represent a fraction on a number line diagram by marking off a

lengths from 0. Recognize that the resulting interval has size and that its

endpoint locates the number on the number line.

Assessment Limits Denominators are limited to 2, 3, 4, 6, and 8. Number lines in MAFS.3.NF.1.2b items may extend beyond 1. Only whole number marks may be labeled on number lines.

Calculator No


Context No context


Which number line is divided into thirds? A.

B.

C. D.

Multiple Choice


27 | P a g e M a y 2 0 1 6


What fraction is represented by the total length marked on the number line shown?

Equation Editor

What fraction is represented by the length marked on the number line shown?

Equation Editor



28 | P a g e M a y 2 0 1 6

Content Standard MAFS.3.NF Number and Operations — Fractions MAFS.3.NF.1 Develop understanding of fractions as numbers. MAFS.3.NF.1.3 Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. MAFS.3.NF.1.3a Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.

MAFS.3.NF.1.3b Recognize and generate simple equivalent fractions, e.g., ,

. Explain why the fractions are equivalent, e.g., by using a visual fraction

model. MAFS.3.NF.1.3c Express whole numbers as fractions, and recognize fractions that

are equivalent to whole numbers. Examples: Express 3 in the form 3 = ;

recognize that = 6; locate and 1 at the same point of a number line diagram.

MAFS.3.NF.1.3d Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

Assessment Limits Denominators are limited to 2, 3, 4, 6, and 8. Fractions must reference the same whole entity that can be equally partitioned,

unless item is assessing MAFS.3.NF.1.3d. Items may not use the term “simplify” or “lowest terms” in directives. Visual models may include number lines and area models. Only whole number marks may be labeled on number lines.

Calculator No

Item Types Equation Editor GRID Matching Item Multiple Choice Multiselect Table Item

Context Allowable


29 | P a g e M a y 2 0 1 6


Jenni and Jimmy’s equal‐sized pizzas are each cut into 8 pieces. Jenni eats 2 slices of her pizza, and Jimmy eats 3 slices of his pizza.

Click on Jenni’s pizza to show how much she ate. Click on Jimmy’s pizza to show how much he ate.

Drag <, >, or = to the box to make a true statement.

GRID

Jenni’s and Jimmy’s equal‐sized pizzas are each cut into 8 slices. Jenni eats 2 slices of her pizza, and Jimmy eats 3 slices of his pizza.

Complete the comparison of Jenni’s pizza to Jimmy’s pizza.

GRID


30 | P a g e M a y 2 0 1 6


Mary has two models, each divided into equal‐sized sections. The first model has been shaded to represent a fraction. Click to shade sections on the second model to show a fraction equivalent to the one in the first model. Create a true comparison of the 2 fractions.

GRID



31 | P a g e M a y 2 0 1 6

Content Standard MAFS.3.MD Measurement and Data MAFS.3.MD.1 Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.

MAFS.3.MD.1.1 Tell and write time to the nearest minute and measure time intervals in minutes. Solve word problems involving addition and subtraction of time intervals in minutes, e.g., by representing the problem on a number line diagram.

Assessment Limits Clocks may be analog or digital.

Digital clocks may not be used for items that require telling or writing time in

isolation.

Calculator No


Context Allowable


Alex arrives at the grocery store at 5:15 p.m. He leaves the grocery store 75 minutes later. Place an arrow on the number line to show the time he left the grocery store.

GRID

Alex arrives at the grocery store at 5:17 p.m. He leaves at 5:59 p.m. How many minutes was he in the grocery store?

Equation Editor

Alex has chores every day. The length of time, in minutes, of each chore is shown. He starts at 9:00 a.m. Complete the table to show what time he will start and finish each chore.

Chore

Time Needed to Complete the Chore Start Time End Time

Watering flowers 12 minutes 9:00 :

Sweeping kitchen 7 minutes : :

Dusting all rooms 14 minutes : :

Table Item



32 | P a g e M a y 2 0 1 6

Content Standard MAFS.3.MD Measurement and Data MAFS.3.MD.1 Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects. MAFS.3.MD.1.2 Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l). Add, subtract, multiply, or divide to solve one‐step word problems involving masses or volumes that are given in the same units.

Assessment Limits Items may not contain compound units such as cubic centimeters (cm3) or finding the geometric volume of a container.

Items may not require multiplicative comparison (e.g., “times as much/many”). Unit conversions are not allowed.

Calculator No


Context Allowable


How many liters (L) of water are in the following container?

Equation Editor


33 | P a g e M a y 2 0 1 6


Gina and Maurice have same‐sized containers filled with different amounts of water, as shown.

Gina’s container has 4 liters (L) of water. About how much water, in liters (L), does Maurice’s container have?

Equation Editor

Gina and Maurice have the containers shown.

Gina does not know how much water is in her container. Maurice’s container is the same size as Gina’s container. About how much less water, in liters (L), does Gina have than Maurice?

Equation Editor



34 | P a g e M a y 2 0 1 6

Content Standard MAFS.3.MD Measurement and Data MAFS.3.MD.2 Represent and interpret data. MAFS.3.MD.2.3 Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one‐ and two‐step “how many more” and “how many less” problems using information presented in scaled bar graphs. For example, draw a bar graph in which each square in the bar graph might represent 5 pets.

Assessment Limits The number of data categories are six or fewer. Items must provide appropriate scale and/or key unless item is assessing that

feature. Only whole number marks may be labeled on number lines.

Calculator No


Context Required


John surveys his classmates about their favorite foods, as shown in the table.

Favorite Food

Hamburger 2

Salad 5

Pizza 8

Click on the graph to complete the bar graph.

GRID


35 | P a g e M a y 2 0 1 6


John surveys his classmates about their favorite foods, as shown in the bar graph.

How many more classmates prefer pizza over salad?

Equation Editor

John surveys his classmates about their favorite foods, as shown in the table.

Favorite Food

Hot Dogs 5

Pizza 9

Salad 6

Chicken 3

Fish 8

Click on the graph to create a bar graph that represents the data.

GRID



36 | P a g e M a y 2 0 1 6

Content Standard MAFS.3.MD Measurement and Data MAFS.3.MD.2 Represent and interpret data. MAFS.3.MD.2.4 Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units—whole numbers, halves, or quarters.

Assessment Limits Standard rulers may not be used; only special rulers that are marked off in halves or quarters are allowed.

Measurements are limited to inches.

Calculator No

Item Types Equation Editor GRID Matching Item Multiple Choice Multiselect

Context Allowable


A pencil is shown.

What is the length of the pencil to the nearest whole inch?

Equation Editor

A pencil is shown.

What is the length of the pencil to the nearest half inch?

Equation Editor

A pencil is shown.

What is the length of the pencil to the nearest quarter inch?

Equation Editor



37 | P a g e M a y 2 0 1 6

Content Standard MAFS.3.MD Measurement and Data MAFS.3.MD.3 Geometric measurement: understand concepts of area and relate area to multiplication and addition. MAFS.3.MD.3.5 Recognize area as an attribute of plane figures and understand concepts of area measurement. MAFS.3.MD.3.5a A square with side length 1 unit, called “a unit square,” is said to have “one square unit” of area, and can be used to measure area. MAFS.3.MD.3.5b A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units. Also Assesses: MAFS.3.MD.3.6 Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units).

Assessment Limits Items may include plane figures that can be covered by unit squares. Items may not include exponential notation for unit abbreviations (e.g., “cm2”).

Calculator No

Item Types Equation Editor Multiple Choice Multiselect

Context Allowable


Alex put the tiles shown on his floor.

What is the area, in square feet, of Alex’s floor?

Equation Editor


38 | P a g e M a y 2 0 1 6


The area of Alex’s floor is 30 square feet. Select all the floors that could be Alex’s.

□

□

□

□

□

Multiselect



39 | P a g e M a y 2 0 1 6

Content Standard MAFS.3.MD Measurement and Data MAFS.3.MD.3 Geometric measurement: understand concepts of area and relate area to multiplication and addition. MAFS.3.MD.3.7 Relate area to the operations of multiplication and addition. MAFS.3.MD.3.7a Find the area of a rectangle with whole‐number side lengths by tiling it, and show that the area is the same as would be found by multiplying the

side lengths. MAFS.3.MD.3.7b Multiply side lengths to find areas of rectangles with whole‐number side lengths in the context of solving real world and mathematical problems, and represent whole‐number products as rectangular areas in mathematical reasoning. MAFS.3.MD.3.7c Use tiling to show in a concrete case that the area of a rectangle with whole‐number side lengths a and b + c is the sum of a × b and a × c. Use area models to represent the distributive property in mathematical reasoning. MAFS.3.MD.3.7d Recognize area as additive. Find areas of rectilinear figures by decomposing them into non‐overlapping rectangles and adding the areas of the non‐overlapping parts, applying this technique to solve real world problems.

Assessment Limits Figures are limited to rectangles and shapes that can be decomposed into rectangles.

Dimensions of figures are limited to whole numbers. All values in items may not exceed whole number multiplication facts of 10 x 10.

Calculator No


Context Allowable


40 | P a g e M a y 2 0 1 6


A park is in the shape of the rectangle shown.

What is the area, in square miles, of the park?

Equation Editor

A park is shown.

What is the area, in square miles, of the park?

Equation Editor



41 | P a g e M a y 2 0 1 6

Content Standard MAFS.3.MD Measurement and Data MAFS.3.MD.4 Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures. MAFS.3.MD.4.8 Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.

Assessment Limits For items involving area, only polygons that can be tiled with square units are allowable.

Dimensions of figures are limited to whole numbers.

All values in items may not exceed whole number multiplication facts of 10 x 10.

Items are not required to have a graphic, but sufficient dimension information

must be given.

Calculator No


Context Required


Ben is planning a garden. Which measurement describes the perimeter of his garden? A. the length of fence he will need B. the amount of soil he will need C. the number of seeds he will buy D. the length of the garden multiplied by the width

Multiple Choice

Ben’s garden has a perimeter of 32 feet. Draw a rectangle that could represent the garden.

GRID

Ben has a rectangular garden with side lengths of 2 feet and 5 feet. What is the perimeter, in feet, of Ben’s garden?

Equation Editor

Ben wants to create a rectangular garden with an area less than 40 square feet. He has 30 feet of fencing. Draw a rectangle that could represent Ben’s garden.

GRID



42 | P a g e M a y 2 0 1 6

Content Standard MAFS.3.G Geometry MAFS.3.G.1 Reason with shapes and their attributes. MAFS.3.G.1.1 Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories.

Assessment Limits Shapes may include two‐dimensional shapes and the following quadrilaterals: rhombus, rectangle, square, parallelogram, and trapezoid.

Items may reference and/or rely on the following attributes: number of sides, number of angles, whether the shape has a right angle, whether the sides are the same length, and whether the sides are straight lines.

Items may not use the terms “parallel” or “perpendicular.” Items that include trapezoids must consider both the inclusive and exclusive

definitions. Items may not use the term "kite" but may include the figure.

Calculator No

Item Types Editing Task Choice GRID Hot Text Matching Item Multiple Choice Multiselect Open Response

Context No context

Sample Item Item Types

A square and a trapezoid are shown below. Which attributes do these shapes always have in common? □ number of sides □ side lengths □ angle measures □ right angles □ number of angles

Multiselect

Select the shapes that are always quadrilaterals and not rectangles. □ rhombus □ parallelogram □ triangle □ trapezoid □ square

Multiselect


43 | P a g e M a y 2 0 1 6


Draw a quadrilateral that is not a rectangle. GRID

What is the name of a shape that is a quadrilateral but not a rectangle? A. hexagon B. parallelogram C. square D. triangle

Multiple Choice



44 | P a g e M a y 2 0 1 6

Appendix A

The chart below contains information about the standard alignment for the items in the Grade 3

Mathematics FSA Computer‐Based Practice Test at http://fsassessments.org/students‐and‐

families/practice‐tests/.

Content Standard Item Type Computer‐Based Practice Test

Item Number

MAFS.3.OA.1.1 Table Item 10

MAFS.3.OA.1.2 Multiselect 4

MAFS.3.OA.1.3 Equation Editor 17

MAFS.3.OA.1.4 Multiple Choice 1


MAFS.3.OA.2.6 GRID 14

MAFS.3.OA.3.7 Table Item 6



MAFS.3.NBT.1.1 Matching Item 2

MAFS.3.NBT.1.2 Multiselect 15

MAFS.3.NBT.1.3 Equation Editor 22

MAFS.3.NF.1.1 GRID 19

MAFS.3.NF.1.2b GRID 5

MAFS.3.NF.1.3c Multiselect 9

MAFS.3.MD.1.1 Multiple Choice 13

MAFS.3.MD.1.2 Equation Editor 3

MAFS.3.MD.2.3 GRID 11


MAFS.3.MD.3.6 Multiple Choice 20

MAFS.3.MD.3.7d Equation Editor 8


MAFS.3.G.1.1 Open Response 7


45 | P a g e M a y 2 0 1 6

Appendix B: Revisions

Page(s) Revision Date

13 Item types and sample items revised. May 2016

14‐15 Assessment limits revised. May 2016

16 Sample item revised. May 2016

17 Sample items revised. May 2016

18 Item types revised. May 2016







28‐30 Assessment limits and sample items revised. May 2016

31 Item types and sample items revised. May 2016

32‐33 Assessment limits, item types, and sample items revised. May 2016

34‐35 Assessment limits and item types revised. May 2016


39‐40 Sample items revised. May 2016

41 Assessment limits revised. May 2016

42‐43 Assessment limits, item types, and sample items revised. May 2016

44 Appendix A added to show Practice Test information. May 2016

Common Operation Situations and Properties, page 1 of 3, Brevard Public Schools, 2015 – 2016

Table 1. Common addition and subtraction situations.6

Result Unknown Change Unknown Start Unknown

Add to

Take from

Put together/take apart2

Compare3

Two bunnies sat on the grass. Three more bunnies hopped there. How many bunnies are on the grass now?

2 + 3 = ?

Two bunnies were sitting on the grass. Some more bunnies hopped there. Then there were five bunnies. How many bunnies hopped over to the first two?

2 + ? = 5

Some bunnies were sitting on the grass. Three more bunnies hopped there. Then there were five bunnies. How many bunnies were on the grass before?

? + 3 = 5

Five apples were on the table. I ate two apples. How many apples are on the table now?

5 – 2 = ?

Five apples were on the table. I ate some apples. Then there were three apples. How many apples did I eat?

5 – ? = 3

Some apples were on the table. I ate two apples. Then there were three apples. How many apples were on the table before?

? – 2 = 3

Total Unknown Addend Unknown Both Addends

Unknown1

Three red apples and two green apples are on the table. How many apples are on the table?

3 + 2 = ?

Five apples are on the table. Three are red and the rest are green. How many apples are green?

3 + ? = 5, 5 – 3 = ?

Grandma has five flowers. How many can she put in her red vase and how many in her blue vase?

5 = 0 + 5 5 = 5 + 0 5 = 1 + 4 5 = 4 + 1 5 = 2 + 3 5 = 3 + 2

difference Unknown Bigger Unknown Smaller Unknown

Difference Unknown Bigger Unknown Smaller Unknown

(“How many more?” version): Lucy has two apples. Julie has five apples. How many more apples does Julie have than Lucy?

(“How man fewer?” version): Lucy has two apples. Julie has five apples. How many fewer apples does Lucy have than Julie?

2 + ? = 5 5 – 2 = ?

(Version with “more”): Julie has three more apples than Lucy. Lucy has two apples. How many apples does Julie have?

(Version with “fewer”): Lucy has 3 fewer apples than Julie. Lucy has two apples. How many apples does Julie have?

2 + 3 = ? 3 + 2 = ?

(Version with “more”): Julie has three more apples than Lucy. Julie has five apples. How many apples does Lucy have?

(Version with “fewer”): Lucy has 3 fewer apples than Julie. Julie has five apples. How many apples does Lucy have?

5 – 3 = ? ? + 3 = 5

1These take apart situations can be used to show all the decompositions of a given number. The associated equations, which have the total on the left of the equal sign, help children understand that the = sign does not always mean makes or results in but always does mean is the same number as. 2Either addend can be unknown, so there are three variations of these problem situations. Both Addends Unknown is a productive extension of this basic situation, especially for small numbers less than or equal to 10. 3For the Bigger Unknown or Smaller Unknown situations, one version directs the correct operation (the version using more for the bigger unknown and using less for the smaller unknown). The other versions are more difficult.

6Adapted from Box 2-4 of Mathematics Learning in Early Childhood, National Research Council (2009, pp. 32, 33).


Table 2. Common multiplication and division situations.7

Unknown Product

3 × 6 = ?

Group Size Unknown (“How many in each

group?” Division)

3 × ? = 18, and 18 ÷ 3 = ?

Number of Groups Unknown

(“How many groups?” Division)

? × 6 = 18, and 18 ÷ 6 = ?

Equal Groups

Arrays4, Area5

Compare3

There are 3 bags with 6 plums in each bag. How many plums are there in all?

Measurement example: You need 3 lengths of string, each 6 inches long. How much string will you need altogether?

If 18 plums are shared equally into 3 bags, then how many plums will be in each bag?

Measurement example: You have 18 inches of string, which you will cut into 3 equal pieces. How long will each piece of string be?

If 18 plums are to be packed 6 to a bag, then how many bags are needed?

Measurement example: You have 18 inches of string, which you will cut into pieces that are 6 inches long. How many pieces of string will you have?

There are 3 rows of apples with 6 apples in each row. How many apples are there?

Area example: What is the area of a 3 cm by 6 cm rectangle?

If 18 apples are arranged into 3 equal rows, how many apples will be in each row?

Area example: A rectangle has area 18 square centimeters. If one side is 3 cm long, how long is a side next to it?

If 18 apples are arranged into equal rows of 6 apples, how many rows will there be?

Area example: A rectangle has area 18 square centimeters. If one side is 6 cm long, how long is a side next to it?

A blue hat costs $6. A red hat costs 3 times as much as the blue hat. How much does the red hat cost?

Measurement example: A rubber band is 6 cm long. How long will the rubber band be when it is stretched to be 3 times as long?

A red hat costs $18 and that is 3 times as much as a blue hat costs. How much does a

blue hat cost?

Measurement example: A rubber band is stretched to be18 cm long and that is 3 times as long as it was at first. How long was the rubber band at first?

A red hat costs $18 and a blue hat costs $6. How many times as much does the red hat cost as the blue hat?

Measurement example: A rubber band was 6 cm long at first. Now it is stretched to be 18 cm long. How many times as long is the rubber band now as it was at first?

General a × b = ? a × ? = p, and p ÷ a = ? ? × b = p, and p ÷ b = ?

4The language in the array examples shows the easiest form of array problems. A harder form is to use the terms

rows and columns: The apples in the grocery window are in 3 rows and 6 columns. How many apples are in there?

Both forms are valuable. 5Area involves arrays of squares that have been pushed together so that there are no gaps or overlaps, so array

problems include these especially important measurement situations.

7The first examples in each cell are examples of discrete things. These are easier for students and should be given

before the measurement examples.


TABLE 3. THE PROPERTIES OF OPERATIONS. Here a, b and c stand for arbitrary numbers in a given number system. The properties of operations apply to the rational number system, the real number system, a nd the complex number system.

TABLE 4. THE PROPERTIES OF EQUALITY. Here a, b and c stand for arbitrary numbers in the rational, real, or complex number systems.

TABLE 5. THE PROPERTIES OF INEQUALITY. Here a, b and c stand for arbitrary numbers in the rational or real number systems.

Exactly one of the following is true: a < b, a = b, a > b.

If a > b and b > c then a > c.

If a > b, then b < a.

If a > b, then–a < –b.

If a > b, then a ± c > b ± c.

If a > b and c > 0, then a × c > b × c.

If a > b and c < 0, then a × c < b × c.

If a > b and c > 0, then a ÷ c > b ÷ c.

If a > b and c < 0, then a ÷ c < b ÷ c

Associative property of addition

Commutative property of addition

Additive identity property of 0

Existence of additive inverses

Associative property of multiplication

Commutative property of multiplication

Multiplicative identity property of 1

Existence of multiplicative inverses

Distributive property of multiplication over addition

(a + b) + c = a + (b + c)

a + b = b + a

a + 0 = 0 + a = a

For every a there exists –a so that a + (–a) = (–a) + a = 0

(a × b) × c = a × (b × c)

a × b = b × a

a × 1 = 1 × a = a

For every a ≠ 0 there exists 1/a so that a × 1/a = 1/a × a = 1

a × (b + c) = a × b + a × c

Reflexive property of equality

Symmetric property of equality

Transitive property of equality

Addition property of equality

Subtraction property of equality

Multiplication property of equality

Division property of equality

Substitution property of equality

a = a

If a = b, then b = a.

If a = b and b = c, then a = c.

If a = b, then a + c = b + c.

If a = b, then a – c = b – c.

If a = b, then a × c = b × c.

If a = b and c ≠ 0, then a ÷ c = b ÷ c.

If a = b, then b may be substituted for a in any expression containing a.

Standards for

Mathematical Practice

Look for

and create efficient

strategies

Use tools and

technology strategically

Do what makes sense and be

persistent

Use number sense when representing

a problem

What do good problem

solvers do?

Be precise with words,

numbers, and symbols

Use math to describe a

real situation or problem

Look for and use

patterns and connections

Make conjectures and prove or

disprove them

What Constitutes a Cognitively Demanding Task?

Lower-level demands (memorization) • Involve either reproducing previously learned facts, rules, formulas, or definitions or committing facts,

rules, formulas or definitions to memory. • Cannot be solved using procedures because a procedure does not exist or because the time frame in

which the task is being completed is too short to use a procedure • Are not ambiguous. Such tasks involve the exact reproduction of previously seen material, and what is to

be reproduced is clearly and directly stated. • Have no connection to the concepts or meaning that underlie the facts, rules, formulas, or definitions

being learned or reproduced. Lower-level demands (procedures without connections to meaning)

• Are algorithmic. Use of the procedure either is specifically called for or is evident from prior instruction, experience, or placement of the task.

• Require limited cognitive demand for successful completion. Little ambiguity exists about what needs to be done and how to do it.

• Have no connection to concepts or meaning that underlie the procedure being used. • Are focused on producing correct answers instead of on developing mathematical understanding. • Require no explanation or explanations that focus solely on describing the procedure that was used.

Higher-level demands (procedures with connections to meaning)

• Focus students’ attention on the use of procedures for the purpose of developing deeper levels of understanding of mathematical concepts and ideas.

• Suggest explicitly or implicitly pathways to follow that are broad general procedures that have close connections to underlying conceptual ideas as opposed to narrow algorithms that are opaque with respect to underlying concepts.

• Usually are represented in multiple ways, such as visual diagrams, manipulatives, symbols, and problem situations. Making connections among multiple representations helps develop meaning.

• Require some degree of cognitive effort. Although general procedures may be followed, they cannot be followed mindlessly. Students need to engage with conceptual ideas that underlie the procedures to complete the task successfully and that develop understanding.

Higher-level demands (doing mathematics)

• Require complex and non-algorithmic thinking - a predictable, well-rehearsed approach or pathway is not explicitly suggested by the task, task instructions, or a worked-out example.

• Require students to explore and understand the nature of mathematical concepts, processes, or relationships.

• Demand self-monitoring or self-regulation of one’s own cognitive processes. • Require students to access relevant knowledge and experiences and make appropriate use of them in

working through the task. • Require considerable cognitive effort and may involve some level of anxiety for the student because of

the unpredictable nature of the solution process required.

Arbaugh, F., & Brown, C.A. (2005). Analyzing mathematical tasks: a catalyst for change? Journal of Mathematics Teacher Education, 8, p. 530.

Key Ideas in the Mathematics Florida Standards (MAFS)

Focus: Greater focus on fewer topics

Focus deeply on the standards for mastery and the ability to transfer skills.

Focus deeply on the major work of each grade as follows:

In grades K-2: Concepts, skills, problem solving related to addition and subtraction.

In grades 3-5: Concepts, skills, and problem solving related to multiplication and division of

whole numbers and fractions.

In grade 6: Ratios and proportional relationships, and early algebraic expressions and

equations.

This focus will enable students to gain strong foundations, including a solid understanding of

concepts, and the ability to apply the math they know to solve problems inside and outside the

classroom.

Coherence: Linking topics and thinking across grades

Coherence is about making math make sense.

Mathematics is a coherent body of knowledge made up of interconnected concepts.

The standards are designed around coherent progressions from grade to grade.

Learning is carefully connected across grades so that students can build new understanding

onto foundations built in previous years.

Each standard is not a new event, but an extension of previous learning.

It is critical to think across grade levels and examine the progressions to see how major

content is developed across grades.

Rigor: Calls for a balance of tasks that require conceptual understanding,

procedural skills and fluency, and application of mathematics to solve problems

Rigor refers to deep, authentic command of mathematical concepts.

The following three aspects of rigor must be pursued with equal intensity to help students

meet the standards:

Conceptual understanding: The standards call for conceptual understanding of key

concepts. Students must be able to access concepts from a number of perspectives. This

will allow them to see math as more than a set of mnemonics or discrete procedures.

Procedural skills and fluency: The standards call for speed and accuracy in calculation with

a balance of practice and understanding. Students must practice simple calculations such

as single-digit multiplication with meaning, in order to have access to more complex

concepts and procedures.

Application: The standards call for students to have solid conceptual understanding and

procedural fluency. They are expected to apply their understanding and procedural skills in

mathematics to problem solving situations.

-Adapted from www.corestandards.org

Standards for Mathematical Practice The Standards for Mathematical Practice describe behaviors that all students will develop in the Common Core Standards. These practices rest on important “processes and proficiencies” including problem solving, reasoning and proof, communication, representation, and making connections. These practices will allow students to understand and apply mathematics with confidence.

1. Make sense of problems and persevere in solving them.• Find meaning in problems• Analyze, predict, and plan solution pathways• Verify answers• Ask them the question: “Does this make sense?”

2. Reason abstractly and quantitatively.• Make sense of quantities and their relationships in problems• Create coherent representations of problems

3. Construct viable arguments and critique the reasoning of others.• Understand and use information to construct arguments• Make and explore the truth of conjectures• Justify conclusions and respond to arguments of others

4. Model with mathematics.• Apply mathematics to problems in everyday life• Identify quantities in a practical situation• Interpret results in the context of the situation and reflect on

whether the results make sense

When given a problem, I can make a plan to solve it and check my answer.

I can use numbers and words to help me make sense of problems.

I can explain my thinking and consider the mathematical thinking of others.

I can recognize math in everyday life and use math I know to solve problems.

5. Use appropriate tools strategically.• Consider the available tools when solving problems• Be familiar with tools appropriate for their grade or course (pencil and paper, concrete models, ruler,

protractor, calculator, spreadsheet, computer programs, digital content located on a website, and othertechnological tools)

6. Be precise.• Communicate precisely to others• Use clear definitions, state the meaning of symbols and be careful

about specifying units of measure and labeling axes• Calculate accurately and efficiently

7. Look for and make use of structure.• Recognize patterns and structures• Step back for an overview and shift perspective• See complicated things as single objects or as being composed of several objects

8. Look for and identify ways to create shortcuts when doing problems.• When calculations are repeated, look for general methods, patterns and

shortcuts• Be able to evaluate whether an answer makes sense

I can use math tools to help me explore and understand math in my world.

I can be careful when I use math and clear when I share my ideas.

I can see and understand how numbers and shapes are put together as parts and wholes.

I can notice when calculations are repeated.

Standard for Mathematical Practice

Student Friendly Language

1. Make sense of problems andpersevere in solving them.

• I can try many times tounderstand and solve amath problem.

2. Reason abstractlyand quantitatively.

• I can think about the mathproblem in my head, first.

3. Construct viable argumentsand critique the reasoningof others.

• I can make a plan, called astrategy, to solve theproblem and discuss otherstudents’ strategies too.

4. Model with mathematics.• I can use math symbols and

numbers to solve theproblem.

5. Use appropriate toolsstrategically.

• I can use math tools,pictures, drawings, andobjects to solve the problem.

6. Attend to precision.• I can check to see if my

strategy and calculationsare correct.

7. Look for and make useof structure

• I can use what I alreadyknow about math to solvethe problem.

8. Look for and express regularityin repeated reasoning.

• I can use a strategy that Iused to solve another mathproblem.

Carroll County Public Schools, http://www.carrollk12.org/instruction/instruction/elementary/math/curriculum/common/default.asp

Florida State Standards Standards for Mathematical Practice Sample Questions for Teachers to Ask

Make sense of problems and persevere in solving them

Reason abstractly and quantitatively

Construct viable arguments and critique the reasoning of others Model with mathematics

Teachers ask: • What is this problem asking?• How could you start this

problem?• How could you make this

problem easier to solve?• How is ___’s way of solving

the problem like/different fromyours?

• Does your plan make sense?Why or why not?

• What tools/manipulativesmight help you?

• What are you having troublewith?

• How can you check this?

Teachers ask: • What does the number ____

represent in the problem? • How can you represent the

problem with symbols and numbers?

• Create a representation of theproblem.

Teachers ask: • How is your answer different

than _____’s? • How can you prove that your

answer is correct? • What math language will help

you prove your answer? • What examples could prove or

disprove your argument? • What do you think about

_____’s argument • What is wrong with ____’s

thinking? • What questions do you have

for ____? *it is important that the teacherimplements tasks that involve discourse and critiquing of reasoning

Teachers ask: • Write a number sentence to

describe this situation • What do you already know

about solving this problem? • What connections do you see?• Why do the results make

sense?• Is this working or do you need

to change your model?*It is important that the teacherposes tasks that involve real world situations

Use appropriate tools strategically Attend to precision Look for and make use of

structure Look for and express regularity

in repeated reasoning

Teachers ask: • How could you use

manipulatives or a drawing to show your thinking?

• Which tool/manipulative wouldbe best for this problem?

• What other resources couldhelp you solve this problem?

Teachers ask: • What does the word ____

mean? • Explain what you did to solve

the problem. • Compare your answer to

_____’s answer • What labels could you use?• How do you know your answer

is accurate?• Did you use the most efficient

way to solve the problem?

Teachers ask: • Why does this happen?• How is ____ related to ____?• Why is this important to the

problem?• What do you know about ____

that you can apply to thissituation?

• How can you use what youknow to explain why thisworks?

• What patterns do you see?*deductive reasoning (movingfrom general to specific)

Teachers ask: • What generalizations can you

make? • Can you find a shortcut to

solve the problem? How would your shortcut make the problem easier?

• How could this problem helpyou solve another problem?

*inductive reasoning (moving fromspecific to general)

Standards for Mathematical Practice in Action

Practice Sample Student Evidence Sample Teacher Actions

1. Make senseof problemsandpersevere insolving them

Display sense-making behaviors Show patience and listen to others Turn and talk for first steps and/or generate solution plan Analyze information in problems Use and recall multiple strategies Self-evaluate and redirect Assess reasonableness of process and answer

Provide open-ended problems Ask probing questions Probe student responses Promote and value discourse Promote collaboration Model and accept multiple approaches

2. Reasonabstractlyandquantitatively

Represent abstract and contextual situations symbolically Interpret problems logically in context Estimate for reasonableness Make connections including real life situations Create and use multiple representations Visualize problems Put symbolic problems into context

Model context to symbol and symbol to context Create problems such as “what word problem

will this equation solve?” Give real world situations Offer authentic performance tasks Place less emphasis on the answer Value invented strategies Think Aloud

3. Constructviableargumentsand critiquethereasoning ofothers

Questions others Use examples and non-examples Support beliefs and challenges with mathematical evidence Forms logical arguments with conjectures and counterexamples Use multiple representations for evidence Listen and respond to others well Uses precise mathematical vocabulary

Create a safe and collaborative environment Model respectful discourse behaviors “Find the error” problems Promote student to student discourse (do not

mediate discussion) Plan effective questions or Socratic formats Provide time and value discourse

4. Model withmathematics

Connect math (numbers and symbols) to real-life situations Symbolize real-world problems with math Make sense of mathematics Apply prior knowledge to solve problems Choose and apply representations, manipulatives and other

models to solve problems Use strategies to make problems simpler Use estimation and logic to check reasonableness of an answer

Model reasoning skills Provide meaningful, real world, authentic

performance-based tasks Make appropriate tools available Model various modeling techniques Accept and value multiple approaches and

representations

5. Useappropriatetoolsstrategically

Choose appropriate tool(s) for a given problem Use technology to deepen understanding Identify and locate resources Defend mathematically choice of tool

Provide a “toolbox” at all times with all available tools – students then choose as needed

Model tool use, especially technology for understanding

6. Attend toprecision

Communicate (oral and written) with precise vocabulary Carefully formulate questions and explanations (not retelling

steps) Decode and interpret meaning of symbols Pay attention to units, labeling, scale, etc. Calculate accurately and effectively Express answers within context when appropriate

Model problem solving strategies Give explicit and precise instruction Ask probing questions Use ELA strategies of decoding,

comprehending, and text-to-self connections for interpretation of symbolic and contextual math problems

Guided inquiry

7. Look for andmake use ofstructure

Look for, identify, and interpret patterns and structures Make connections to skills and strategies previously learned to

solve new problems and tasks Breakdown complex problems into simpler and more

manageable chunks Use multiple representations for quantities View complicated quantities as both a single object or a

composition of objects

Let students explore and explain patterns Use open-ended questioning Prompt students to make connections and

choose problems that foster connections Ask for multiple interpretations of quantities

8. Look for andexpressregularity inrepeatedreasoning

Design and state “shortcuts” Generate “rules” from repeated reasoning or practice (e.g.

integer operations) Evaluate the reasonableness of intermediate steps Make generalizations

Provide tasks that allow students to generalize Don’t teach steps or rules, but allow students to

explore and generalize in order to discover and formalize

Ask deliberate questions Create strategic and purposeful check-in points

Standards for Mathematical Practice, (from North Carolina Department of Education, http://www.ncpublicschools.org/), Third Grade, page 1 of 2, 2015 - 2016

STANDARDS FOR MATHEMATICAL PRACTICE IN THIRD GRADE The Standards for Mathematical Practice are expected to be integrated into every mathematics lesson for all students Grades K-12. Below are a few examples of how these Practices may be integrated into tasks that students complete.

Practice Explanation and Example

1. Make sense of problemsand persevere in solvingthem.

Mathematically proficient students in third grade know that doing mathematics involves solving problems and discussing how they solved them. Students make sense of a problem by determining and explaining its meaning and look for ways to solve it. Third grade students may use concrete objects or pictures to help themselves conceptualize and solve problems. They may check their thinking by asking themselves, “Does this make sense?”. They listen to the strategies of others and will try different approaches. They often will use another method to check their answers.

2. Reason abstractly andquantitatively.

Mathematically proficient students in third grade should recognize that a number represents a specific quantity. They connect the quantity to written symbols and create a logical representation of the problem at hand, considering both the appropriate units involved and the meaning of quantities.

3. Construct viablearguments and critiquethe reasoning of others.

Mathematically proficient students in third grade may construct arguments using concrete referents, such as objects, pictures, and drawings. They refine their mathematical communication skills as they participate in mathematical discussions involving questions like “How did you get that?” and “Why is that true?”. They explain their thinking to others and respond to others’ thinking.

4. Model with mathematics.

Mathematically proficient students in third grade experiment with representing problem situations in multiple ways including numbers, words (mathematical language), drawing pictures, using objects, acting out, making a chart, list, or graph, creating equations, etc. Students need opportunities to connect these different representations and explain the connections. They should be able to use all of these representations as needed. Third grade students should evaluate their results in the context of the situation and reflect on whether or not the results make sense.

5. Use appropriate toolsstrategically.

Mathematically proficient students in third grade consider the available tools (including estimation) when solving a mathematical problem and decide when certain tools might be helpful. For instance, they may use graph paper to find all the possible rectangles that have a given perimeter. They compile the possibilities into an organized list or a table and determine whether they have all the possible rectangles.

http://www.ncpublicschools.org/

Standards for Mathematical Practice, (from North Carolina Department of Education, http://www.ncpublicschools.org/), Third Grade, page 2 of 2, 2015- 2016

Practice Explanation and Example

6. Attend to precision.

Mathematically proficient students in third grade develop their mathematical communication skills. They try to use clear and precise language in their discussions with others and in their own reasoning. They are careful about specifying units of measure and state the meaning of the symbols they choose. For instance, when figuring out the area of a rectangle, they record their answers in square units.

7. Look for and make useof structure.

Mathematically proficient students in third grade look closely to discover a pattern or structure. For instance, students use properties of operations as strategies to multiply and divide (commutative and distributive properties).

8. Look for and expressregularity in repeatedreasoning.

Mathematically proficient students in third grade should notice repetitive actions in computation and look for more shortcut methods. For example, students may use the distributive property as a strategy for using products they know to solve products that they don’t know. If students are asked to find the product of 7 x 8, they might decompose 7 into 5 and 2 and then multiply 5 x 8 and 2 x 8 to arrive at 40 + 16 or 56. In addition, third grade students continually evaluate their work by asking themselves, “Does this make sense?”.

http://www.ncpublicschools.org/

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MAFS

Breaking the Code Mathematics Florida Standards

MAFS.5.OA.1.1

MAFS = Mathematics Florida Standards 5 = Fifth Grade OA = Operations and Algebraic Thinking

1 = Cluster – Write and interpret numerical expressions.

1 = Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.

K – 5 Domains

CC = Counting and Cardinality OA = Operations and Algebraic Thinking NBT = Number and Operations in Base Ten

MD = Measurement and Data G = Geometry

Subject/Standards Domain

Grade Level

Cluster

Standard

Third Grade Mathematics Florida Standards 2015 - 2016

Third Grade Mathematics Florida Standards, page 1 of 4, Brevard Public Schools, 2015 - 2016

Domain: OPERATIONS AND ALGEBRAIC THINKING

Cluster 1: Represent and solve problems involving multiplication and division.

MAFS.3.OA.1.1:

Interpret products of whole numbers, e.g., interpret 5 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 7.

MAFS.3.OA.1.2:

Interpret whole-number quotients of wholenumbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8.

MAFS.3.OA.1.3:

Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

MAFS.3.OA.1.4:

Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations

8 ? = 48, 5 = ÷ 3, 6 6 = ?

Cluster 2: Understand properties of multiplication and the relationship between multiplication and division.

MAFS.3.OA.2.5:

Apply properties of operations as strategies to multiply and divide. Examples: If 6 4 = 24 is known, then 4 6 = 24 is also

known. (Commutative property of multiplication) 3 5 2 can be found by 3 5 = 15, then 15 2 = 30, or by 5 2 = 10, then

3 10 = 30. (Associative property of multiplication.) Knowing that 8 5 = 40 and 8 2 = 16, one can find 8 7 as 8 (5 + 2)

= (8 5) + (8 2) = 40 + 16 = 56. (Distributive property.)

MAFS.3.OA.2.6:

Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8.

Cluster 3: Multiply and divide within 100.

MAFS.3.OA.3.7:

Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g.,

knowing that 8 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.

Cluster 4: Solve problems involving the four operations, and identify and explain patterns in arithmetic.

MAFS.3.OA.4.8:

Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.



MAFS.3.OA.4.9:

Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends.

Domain: NUMBER AND OPERATIONS IN BASE TEN

Cluster 1: Use place value understanding and properties of operations to perform multi-digit arithmetic.

MAFS.3.NBT.1.1:

Use place value understanding to round whole numbers to the nearest 10 or 100.

MAFS.3.NBT.1.2:

Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.

MAFS.3.NBT.1.3:

Multiply one-digit whole numbers by multiples of 10 in the range 10 - 90 (e.g., 9 80, 5 60) using strategies based on place value and properties of operations.

Domain: NUMBER AND OPERATIONS - FRACTIONS

Cluster 1: Develop understanding of fractions as numbers.

MAFS.3.NF.1.1:

Understand a fraction b

1 as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction

b

a as the quantity formed by a parts of size

b

1.

MAFS.3.NF.1.2: Understand a fraction as a number on the number line; represent fractions on a number line diagram.

a. Represent a fractionb

1on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b

equal parts. Recognize that each part has size b

1and that the endpoint of the part based at 0 locates the number

b

1on the

number line.

b. Represent a fractionb

a on a number line diagram by marking off a lengths

b

1from 0. Recognize that the resulting interval

has size b

a and that its endpoint locates the number

b

a on the number line.



MAFS.3.NF.1.3:

Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. a. Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.

b. Recognize and generate simple equivalent fractions, e.g., 2

1 =

4

2,

6

4 =

3

2. Explain why the fractions are equivalent, e.g.,

by using a visual fraction model. c. Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3

in the form 3 = 1

3; recognize that

1

6 = 6; locate

4

4 and 1 at the same point of a number line diagram.

d. Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

Domain: MEASUREMENT AND DATA

Cluster 1: Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.

MAFS.3.MD.1.1:

Tell and write time to the nearest minute and measure time intervals in minutes. Solve word problems involving addition and subtraction of time intervals in minutes, e.g., by representing the problem on a number line diagram.

MAFS.3.MD.1.2:

Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (L). Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units.

Cluster 2: Represent and interpret data.

MAFS.3.MD.2.3:

Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step “how many more” and “how many less” problems using information presented in scaled bar graphs. For example, draw a bar graph in which each square in the bar graph might represent 5 pets.

MAFS.3.MD.2.4:

Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units—whole numbers, halves, or quarters.

Cluster 3: Geometric measurement: understand concepts of area and relate area to multiplication and to addition.

MAFS.3.MD.3.5:

Recognize area as an attribute of plane figures and understand concepts of area measurement. a. A square with side length 1 unit, called “a unit square,” is said to have “one square unit” of area, and can be used to

measure area. b. A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units.

MAFS.3.MD.3.6:

Measure areas by counting unit squares (square cm, square m, square in., square ft., and improvised units).



MAFS.3.MD.3.7:

Relate area to the operations of multiplication and addition. a. Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be

found by multiplying the side lengths.b. Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real world and

mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning.c. Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of

a x b and a x c. Use area models to represent the distributive property in mathematical reasoning.d. Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and

adding the areas of the non-overlapping parts, applying this technique to solve real world problems.

Cluster 4: Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.

MAFS.3.MD.4.8:

Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.

Domain: GEOMETRY

Cluster 1: Reason with shapes and their attributes.

MAFS.3.G.1.1:

Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories.

MAFS.3.G.1.2:

Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. For example,

partition a shape into 4 parts with equal area, and describe the area of each part as 4

1 of the area of the shape.

3rd Grade


STANDARD CODE

REVISED/ DELETED/NEW

STANDARD

MACC.3.MD.1.2 PREVIOUS

Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l). Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem.

MAFS.3.MD.1.2 REVISED Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l). Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units.

4th Grade


STANDARD CODE


STANDARD


Use the four operations to solve word problems1 involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.

MAFS.4.MD.1.2 REVISED

Use the four operations to solve word problems1 involving distances, intervals of time, and money, including problems involving simple fractions or decimals.2 Represent fractional quantities of distance and intervals of time using linear models. (1See Table 2 Common Multiplication and Division Situations) (2Computational fluency with fractions and decimals is not the goal for students at this grade level.)

MAFS.4.OA.1.a NEW Determine whether an equation is true or false by using comparative relational thinking. For example, without adding 60 and 24, determine whether the equation 60 + 24 = 57 + 27 is true or false.

MAFS.4.OA.1.b NEW Determine the unknown whole number in an equation relating four whole numbers using comparative relational thinking. For example, solve 76 + 9 = n + 5 for n by arguing that nine is four more than five, so the unknown number must be four greater than 76.

MACC.4.OA.2.4 PREVIOUS

Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1–100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1–100 is prime or composite.

MAFS.4.OA.2.4 REVISED

Investigate factors and multiples. A. Find all factor pairs for a whole number in the range 1–100. B. Recognize that a whole number is a multiple of each of its factors. Determine whether a

given whole number in the range 1–100 is a multiple of a given one-digit number. C. Determine whether a given whole number in the range 1–100 is prime or composite.

5th Grade


STANDARD CODE


STANDARD

MACC.5.G.2.4 PREVIOUS Classify two-dimensional figures in a hierarchy based on properties.

MAFS.5.G.2.4 REVISED Classify and organize two-dimensional figures into Venn diagrams based on the attributes of the figures.

MACC.5.MD.1.1 PREVIOUS Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems.

MAFS.5.MD.1.1 REVISED Convert among different-sized standard measurement units (i.e., km, m, cm; kg, g; lb., oz.; l, ml; hr., min, sec) within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems.


Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume. a. Find the volume of a right rectangular prism with whole-number side lengths by packing it

with unit cubes, and show that the volume is the same as would be found by multiplying theedge lengths, equivalently by multiplying the height by the area of the base. Representthreefold whole-number products as volumes, e.g., to represent the associative property ofmultiplication.

b. Apply the formulas V = l × w × h and V = b × h for rectangular prisms to find volumes of rightrectangular prisms with whole- number edge lengths in the context of solving real world andmathematical problems.

c. Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts,applying this technique to solve real world problems.

MAFS.5.MD.3.5 REVISED

Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume. a. Find the volume of a right rectangular prism with whole-number side lengths by packing it

with unit cubes, and show that the volume is the same as would be found by multiplying theedge lengths, equivalently by multiplying the height by the area of the base. Representthreefold whole-number products as volumes, e.g., to represent the associative property ofmultiplication.

b. Apply the formulas V = l × w × h and V = B × h for rectangular prisms to find volumes ofright rectangular prisms with whole- number edge lengths in the context of solving real worldand mathematical problems.

c. Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts,applying this technique to solve real world problems.

6th Grade


STANDARD CODE


STANDARD

MACC.6.RP.1.3 PREVIOUS

Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

a. Make tables of equivalent ratios relating quantities with whole- number measurements, findmissing values in the tables, and plot the pairs of values on the coordinate plane. Use tablesto compare ratios.

b. Solve unit rate problems including those involving unit pricing and constant speed. Forexample, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could bemowed in 35 hours? At what rate were lawns being mowed?

c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 timesthe quantity); solve problems involving finding the whole, given a part and the percent.

d. Use ratio reasoning to convert measurement units; manipulate and transform unitsappropriately when multiplying or dividing quantities.

MAFS.6.RP.1.3 REVISED

Use ratio and rate reasoning to solve real-world and mathematical problems1, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

a. Make tables of equivalent ratios relating quantities with whole- number measurements, findmissing values in the tables, and plot the pairs of values on the coordinate plane. Use tablesto compare ratios.

b. Solve unit rate problems including those involving unit pricing and constant speed. Forexample, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could bemowed in 35 hours? At what rate were lawns being mowed?

c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 timesthe quantity); solve problems involving finding the whole, given a part and the percent.

d. Use ratio reasoning to convert measurement units; manipulate and transform unitsappropriately when multiplying or dividing quantities.

e. Understand the concept of Pi as the ratio of the circumference of a circle to its diameter.

(1See Table 2 Common Multiplication and Division Situations)

Domain Progression, Brevard Public Schools, 2013-2014 Page 1 of 22

DOMAIN PROGRESSION OPERATIONS AND ALGEBRAIC THINKING

Third Grade Fourth Grade Fifth Grade

Represent and solve problems involving multiplication and division. 3.OA.1.1: Interpret products of whole numbers (e.g.,

interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each).

For example, describe a context in which a total number of objects can be expressed as 5 × 7.

3.OA.1.2: Interpret whole number quotients of wholenumbers (e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each).

For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8.

3.OA.1.3: Use multiplication and division within 100 tosolve word problems in situations involving equal groups, arrays, and measurement quantities (e.g., by using drawings and equations with a symbol for the unknown number to represent the problem).

Use the four operations with whole numbers to solve problems.

4.OA.1.1: Interpret a multiplication equation as acomparison (e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5). Represent verbal statements of multiplicative comparisons as multiplication equations.

4.OA.1.2: Multiply or divide to solve word problemsinvolving multiplicative comparison (e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison).

4.OA.1.3: Solve multi-step word problems posed withwhole numbers and having whole number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

Write and interpret numerical expressions.

5.OA.1.1: Use parenthesis, brackets, or braces innumerical expressions, and evaluate expressions with these symbols.

5.OA.1.2: Write simple expressions that recordcalculations with numbers, and interpret numerical expressions without evaluating them.

For example, express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18,932 + 921) is three times as large as 18,932 + 921, without having to calculate the indicated sum or product.

Analyze patterns and relationships. 5.OA.2.3: Generate two numerical patterns using two

given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane.

For example, given the rule “Add 3” and the starting number 0, and given the rule “Add 6” and the starting number 0, generate terms in the resulting sequences and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.



3.OA.1.4: Determine the unknown whole number in amultiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 × ? = 48, 5 = ☐ ÷ 3, 6 × 6 = ?.

Understand properties of multiplication and the relationship between multiplication and division.

3.OA.2.5: Apply properties of operations as strategiesto multiply and divide.

Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (commutative property of multiplication)

3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (associative property of multiplication)

Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (distributive property)

3.OA.2.6: Understand division as an unknown-factorproblem.

For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8.

Gain familiarity with factors and multiples.

4.OA.2.4: Find all factor pairs for a whole number inthe range 1-100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1-100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1-100 is prime or composite.

Generate and analyze problems.

4.OA.2.5: Generate a number or shape pattern thatfollows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself.

For example, given the rule “Add 3” and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate this way.



Multiply and divide within 100.

3.OA.3.7: Fluently multiply and divide within 100, usingstrategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations.

By the end of Grade 3, know from memory all products of two one-digit numbers.

Solve problems involving the four operations, and identify and explain patterns in arithmetic.

3.OA.4.8: Solve two-step word problems using the fouroperations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

3.OA.4.9: Identify arithmetic patterns (includingpatterns in the addition table or multiplication table), and explain them using properties of operations.

For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends.


NUMBERS AND OPERATIONS BASE IN TEN

Third Grade Fourth Grade Fifth Grade Use place value understanding and properties of operations to perform multi-digit arithmetic.

3.NBT.1.1: Use place value understanding to roundwhole numbers to the nearest 10 or 100.

3.NBT.1.2: Fluently add and subtract within 1,000using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.

3.NBT.1.3: Multiply one-digit whole numbers bymultiples of 10 in the range 10-90 (e.g., 9 × 80, 5 × 60) using strategies based on place value and properties of operations.

Generalize place value understanding for multi-digit whole numbers. 4.NBT.1.1: Recognize that in a multi-digit whole

number, a digit in one place represents ten times what it represents in the place to its right.

For example, recognize that 700 ÷ 7 = 10 by applying concepts of place value and division.

4.NBT.1.2: Read and write multi-digit whole numbersusing base ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

4.NBT.1.3: Use place value understanding to roundmulti-digit whole numbers to any place.

Use place value understanding and properties of operations to perform multi-digit arithmetic.

4.NBT.2.4: Fluently add and subtract multi-digit wholenumbers using the standard algorithm.

4.NBT.2.5: Multiply a whole number of up to four digitsby a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

Understand the place value system.

5.NBT.1.1: Recognize that in a multi-digit number, adigit in one place represents 10 times as much as it represents in the place to its right and

101 of what it represents in the place to

its left.

5.NBT.1.2: Explain patterns in the number of zeros ofthe product when multiplying a number of powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole number exponents to denote powers of 10.

5.NBT.1.3: Read, write, and compare decimals tothousandths.

a. Read and write decimals to thousandthsusing base-ten numerals, number names, and expanded form (e.g., 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 ×

101 + 9 ×

1001 + 2 ×

000,11 ).

b. Compare two decimals to thousandthsbased on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

5.NBT.1.4: Use place value understanding to rounddecimals to any place.



4.NBT.2.6: Find whole number quotients andremainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

Perform operations with multi-digit whole numbers and with decimals to hundredths.

5.NBT.2.5: Fluently multiply multi-digit whole numbersusing the standard algorithm.

5.NBT.2.6: Find whole number quotients of wholenumbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

5.NBT.2.7: Add, subtract, multiply, and divide decimalsto hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction. Relate the strategy to a written method, and explain the reasoning used.


NUMBER AND OPERATIONS - FRACTIONS


Develop understanding of fractions as numbers.

3.NF.1.1: Understand a fraction b1 as the quantity

formed by 1 part when a whole is partitioned into b equal parts; understand a fraction

ba

as the quantity formed by a parts of size b1 .

3.NF.1.2: Understand a fraction as a number on thenumber line; represent fractions on a number line diagram.

a. Represent a fraction b1 on a number line

diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size

b1 and the endpoint of the part

based at 0 locates the number b1 on the

number line. b. Represent a fraction

ba on a number line

diagram by marking off a lengths b1 from

0. Recognize that the resulting intervalhas size

ba on the number line.

Extend understanding of fractional equivalence and ordering.

4.NF.1.1: Explain why a fraction ba is equivalent to a

fraction bnan

×× by using visual fraction

models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.

4.NF.1.2: Compare two fractions with differentnumerators and different denominators (e.g., by creating common numerators and denominators, or by comparing to a benchmark fraction such as

21 ). Recognize

that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions (e.g., by using the visual fraction model).

Extend understanding of fractional equivalence and ordering.

4.NF.2.3: Understand a fraction ba with 𝑎 > 1 as a sum

of fractions b1 .

Use equivalent fractions as a strategy to add and subtract fractions.

5.NF.1.1: Add and subtract fractions with unlikedenominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators.

For example: 32 +

45 =

128 +

1215 =

1223 . (In general,

ba +

dc =

bdbcad )( + ).

5.NF.1.2: Solve word problems involving addition andsubtraction of fractions referring to the same whole, including cases of unlike denominators (e.g., by using visual fraction models or equations to represent the problem). Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers.

For example, recognize an incorrect result

52 +

21 =

73 , by observing that

73 <

21 .



3.NF.1.3: Explain equivalence of fractions in specialcases, and compare fractions by reasoning about their size.

a. Understand two fractions as equivalent (equal)if they are the same size, or the same point on a number line.

b. Recognize and generate simple equivalentfractions (e.g.,

21 =

42 ,

64 =

32 ). Explain

why the fractions are equivalent (e.g., by using a visual fraction model).

c. Express whole numbers as fractions, andrecognize fractions that are equivalent to whole numbers.

For example: Express 3 in the form 3 = 13 ; recognize

that 16 = 6; locate

44 and 1 at the same point on a

number line diagram.

d. Compare two fractions with the samenumerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions (e.g., by using a visual fraction model).

a. Understand addition and subtraction offractions as joining and separating parts referring to the same whole.

b. Decompose a fraction into a sum of fractionswith the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions (e.g., by using a visual fraction model).

For example: 83 =

81 +

81 +

81 ;

83 =

81 +

82 ;

2 81 = 1 + 1 +

81 =

88 +

88 +

81

c. Add and subtract mixed numbers with likedenominators (e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction).

d. Solve word problems involving addition andsubtraction of fractions referring to the same whole and having like denominators (e.g., by using visual fraction models and equations to represent the problem).

4.NF.2.4: Apply and extend previous understandings ofmultiplication to multiply a fraction by a whole number.

a. Understand a fraction ba as a multiple of

b1 .

Use equivalent fractions as a strategy to add and subtract fractions.

5.NF.2.3: Interpret a fraction as division of the numerator by

the denominator (ba = 𝑎 ÷ b). Solve word

problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers (e.g., by using visual fraction models or equations to represent the problem).

For example, interpret 43 as the result of dividing

3 by 4, noticing that 43 multiplied by 4 equals 3, and that

when 3 wholes are shared equally among 4 people, each

person has a share of size 43 . If 9 people want to share a

50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?

5.NF.2.4: Apply and extend previous understandings ofmultiplication to multiply a fraction or whole number by a fraction.

a. Interpret the productba × q as a parts of a partition

of q into b equal parts; equivalently, as the result of a sequence of operations 𝑎 × q ÷ b.

For example, use a visual fraction model to show

32 × 4 =

38 , and create a story context for this equation. Do

the same with 32 ×

54 =

158 . (In general,

ba ×

dc =

dbac .)



For example: use a visual fraction model to represent

45 as the product 5 ×

41 , recording the conclusion by

the equation 45 = 5 ×

41 .

b. Understand a multiple of ba as a multiple of

b1 ,

and use this understanding to multiply a fraction by a whole number.

For example: use a visual fraction model to express 3 ×

52 as 6 ×

51 , recognizing this product as

56 . (In

general, n × ba =

ban × ).

c. Solve word problems involving multiplication ofa fraction by a whole number (e.g., by using visual fraction models and equations to represent the problem).

For example: if each person at a party will eat 83 of a

pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?

b. Find the area of a rectangle with fractional sidelengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.

5.NF.2.5: Interpret multiplication as scaling (resizing),by:

a. Comparing the size of a product to the size ofone factor on the basis of the size of the other factor, without performing the indicated multiplication.

b. Explaining why multiplying a given number by afraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence

ba =

bnan

×× to the effect of

multiplying ba by 1.



Understand decimal notation for fractions, and compare decimal fractions.

4.NF.3.5: Express a fraction with denominator 10 as anequivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100. For example: express

103 as

10030 , and add

103 +

1004 =

10034 .

4.NF.3.6: Use decimal notation for fractions withdenominators 10 or 100.

For example: rewrite 0.62 as 10062 ; describe a length

as 0.62 meters; locate 0.62 on a number line diagram.

4.NF.3.7: Compare two decimals to hundredths byreasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions (e.g., by using a visual model).

5.NF.2.6: Solve real world problems involvingmultiplication of fractions and mixed numbers (e.g., by using visual fraction models or equations to represent the problem).

5.NF.2.7: Apply and extend previous understandings ofdivision to divide unit fractions by whole numbers and whole numbers by unit fractions.

a. Interpret division of a unit fraction by a non-zerowhole number, and compute such quotients.

For example, create a story context for 31 ÷ 4, and use

a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that

31 ÷ 4 =

121 because

121 × 4 =

31 .

b. Solve real-world problems involving division ofunit fractions by non-zero whole numbers and division of whole numbers by unit fractions (e.g., by using visual fraction models and equations to represent the problem).

For example, how much chocolate will each person get if 3 people share

21 lb. of chocolate equally? How

many 31 cup servings are in 2 cups of raisins?


MEASUREMENT AND DATA


Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.

3.MD.1.1: Tell and write time to the nearest minute andmeasure time intervals in minutes. Solve word problems involving addition and subtraction of time intervals in minutes (e.g., by representing the problem on a number line diagram).

3.MD.1.2: Measure and estimate liquid volumes andmasses of objects using standard units of grams (g), kilograms (kg), and liters (L). Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units (e.g., by using drawings such as a beaker with a measurement scale to represent the problem).

Represent and interpret data.

3.MD.2.3: Draw a scaled picture graph and a scaledbar graph to represent a data set with several categories. Solve one- and two-step “how many more” and “how many less” problems using information presented in scaled bar graphs.

For example, draw a bar graph in which each square in the bar graph might represent 5 pets.

Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.

4.MD.1.1: Know relative sizes of measurement unitswithin one system of units including km, m, cm; kg, g; lb., oz.; L, mL; hr., min., and sec. Within a single system of measurement, express measurements in a larger unit. Record measurement equivalents in a two-column table.

For example, know that 1 ft. is 12 times as long as 1 in. Express the length of a 4 ft. snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1,12), (2,24), (3,36),…

4.MD.1.2: Use the four operations to solve wordproblems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.

Convert like measurement units within a given measurement system.

5.MD.1.1: Convert among different sized standardmeasurement units within a given measurement system (e.g., convert 5 cm to 0.05 m) and use these conversions in solving multi-step, real-world problems.

Represent and interpret data.

5.MD.2.2: Make a line plot to display a data set ofmeasurements in fractions of a unit (

21 ,

41 ,

81 ). Use operations on fractions for

this grade to solve problems involving information presented in line plots.

For example, given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally.



3.MD.2.4: Generate measurement data by measuringlengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units - whole numbers, halves, or quarters.

Geometric measurement: understand concepts of area and relate area to multiplication and to addition.

3.MD.3.5: Recognize area as an attribute of planefigures and understand concepts of area management.

a. A square with side length 1 unit, called “a unitsquare,” is said to have “one square unit” of area,and can be used to measure area.

b. A plane figure that can be covered without gapsor overlaps by n unit squares is said to have anarea of n square units.

3.MD.3.6: Measure areas by counting unit squares(square cm, square m, square in., square ft., and improvised units).

3.MD.3.7: Relate area to the operations ofmultiplication and addition.

a. Find the area of a rectangle with whole numberside lengths by tiling it, and show that the area isthe same as would be found by multiplying theside lengths.

4.MD.1.3: Apply the area and perimeter formulas forrectangles in real world and mathematical problems.

For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor.

Represent and interpret data. 4.MD.2.4: Make a line plot to display a data set of

measurements in fractions of a unit (

21 ,

41 ,

81 ). Solve problems involving addition

and subtraction of fractions by using information presented in line plots.

For example, from a line plot, find and interpret the difference in length between the longest and shortest specimens in an insect collection.

Geometric measurement: understand concepts of angles and measure angles.

4.MD.3.5: Recognize angles as geometric shapes thatare formed wherever two rays share a common endpoint, and understand concepts of angle measurement.

a An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through

3601 of a

circle is called a “one-degree angle,” and can be used to measure angles.

Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.

5.MD.3.3: Recognize volume as an attribute of solidfigures and understand concepts of volume measurement.

a. A cube with side length 1 unit, called a “unitcube,” is said to have “one cubic unit” of volume,and can be used to measure volume.

b. A solid figure which can be packed without gapsor overlaps using n unit cubes is said to have avolume of n cubic units.

5.MD.3.4: Measure volumes by counting unit cubes,using cubic cm, cubic in., cubic ft., and improvised units.

5.MD.3.5: Relate volume to the operations ofmultiplication and addition and solve real- world and mathematical problems involving volume.

a. Find the volume of a right-rectangular prism withwhole number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole number products as volumes (e.g., to represent the associative property of multiplication).



b. Multiply side lengths to find areas of rectangleswith whole number side lengths in the context ofsolving real world and mathematical problems,and represent whole number products asrectangular areas in mathematical reasoning.

c. Use tiling to show in a concrete case that thearea of a rectangle with whole number sidelengths a and b + c is the sum of a × b and a × c.Use area models to represent the distributiveproperty in mathematical reasoning.

d. Recognize area as additive. Find areas ofrectilinear figures by decomposing them into non-overlapping rectangles and adding the areas ofthe non-overlapping parts, applying thistechnique to solve real world problems.

Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.

3.MD.4.8: Solve real world and mathematical problemsinvolving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.

b. An angle that turns through n one-degree anglesis said to have an angle measurement of ndegrees.

4.MD.3.6: Measure angles in whole number degreesusing a protractor. Sketch angles of a specified measure.

4.MD.3.7: Recognize angle measure as additive. Whenan angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems (e.g., by using an equation with a symbol for the unknown angle measure).

b. Apply the formulas V = l × w × h andV = B × h for rectangular prisms to findvolumes of right-rectangular prisms withwhole number edge lengths in the context ofsolving real world and mathematicalproblems.

c. Recognize volume as additive. Find volumesof solid figures composed of two non- overlapping right-rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems.


GEOMETRY

Third Grade Fourth Grade Fifth Grade Sixth Grade

Reason with shapes and their attributes.

3.G.1.1: Understand that shapes indifferent categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories.

3.G.1.2: Partition shapes into parts withequal areas. Express the area of each part as a unit fraction of the whole.

For example, partition a shape into 4 parts with equal area, and describe the area of each part as

41 of the area of

the shape.

Draw and identify lines and angles, and classify shapes by properties of their lines and angles.

4.G.1.1: Draw points, lines, linesegments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.

4.G.1.2: Classify two-dimensionalfigures based on the presence or absence of parallel or perpendicular lines, or presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles.

4.G.1.3: Recognize a line of symmetryfor a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.

Graph points on the coordinate plane to solve real-world and mathematical problems.

5.G.1.1: Use a pair of perpendicularnumber lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x-axis and x-coordinate, y-axis and y-coordinate).

5.G.1.2: Represent real-world andmathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.

Solve real-world and mathematical problems involving area, surface area, and volume.

6.G.1.1: Find the area of right triangles,other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

6.G.1.2: Find the volume of a right-rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l× w× h and V = B× h to find volumes of right- rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.


Third Grade Fourth Grade Fifth Grade Sixth Grade

Classify two-dimensional figures into categories based on their properties.

5.G.2.3: Understand that attributesbelonging to a category of two-dimensional figures also belong to all subcategories of that category.

For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.

5.G.2.4: Classify two-dimensionalfigures in a hierarchy based on properties.

6.G.1.3: Draw polygons in thecoordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.

6.G.1.4: Represent three-dimensionalfigures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.


RATIOS AND PROPORTIONAL RELATIONSHIPS

Sixth Grade Seventh Grade Understand ratio concepts and use ratio reasoning to solve problems. 6.RP.1.1: Understand the concept of a ratio and use ratio language to describe a

ratio relationship between two quantities. For example, the ratio of wings to beaks in the bird house at the zoo was 2:1 because for every 2 wings there was 1 beak. For every vote candidate A received, candidate C received nearly three votes.

6.RP.1.2: Understand the concept of a unit rate ba associated with a ratio 𝑎:b with

b ≠ 0, and use rate language in the context of a ratio relationship. For example, this recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is

43 cup of flour for each cup of sugar. We paid $75 for 15 hamburgers, which is a

rate of $5 per hamburger.

6.RP.1.3: Use ratio and rate reasoning to solve real-world and mathematicalproblems (e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations).

a. Make tables of equivalent ratios relating quantities with whole numbermeasurements, finding missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.

b. Solve unit rate problems including those involving unit pricing and constantspeed.

For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?

c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantitymeans

10030 times the quantity); solve problems involving finding the

whole, given a part and the percent. d. Use ratio reasoning to convert measurement units; manipulate and

transform units appropriately when multiplying or dividing quantities.

Analyze proportional relationships and use them to solve real-world and mathematical problems. 7.RP.1.1: Compute unit rates associated with ratios of fractions, including ratios of

lengths, areas, and other quantities measured in like or different units. For example, if a person walks

21 mile in each

41 hour, compute the unit rate as

the complex fraction 1214 miles per hour, equivalently 2 miles per hour.

7.RP.1.2: Recognize and represent proportional relationships between quantities.a. Decide whether two quantities are in a proportional relationship (e.g., by

testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin).

b. Identify the constraint of proportionality (unit rate) in tables, graphs,equations, diagrams, and verbal descriptions of proportional relationships.

c. Represent proportional relationships by equations.For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed t = pn.

d. Explain what a point (x, y) on the graph of a proportional relationshipmeans in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate

7.RP.1.3: Use proportional relationships to solve multi-step ratio and percentproblems.

Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error


THE NUMBER SYSTEM

Sixth Grade Seventh Grade Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

6.NS.1.1: Interpret and compute quotients of fractions, and solve world problemsinvolving division of fractions by fractions (e.g., by using visual fraction models and equations to represent the problem).

For example, create a story context for 32 ÷

43 and use a visual fraction model to

show the quotient; use the relationship between multiplication and division to explain that

32 ÷

43 =

98 because

43 of

98 is

32 (In general

ba ÷

dc =

bcad .) How

much chocolate will each person get if 3 people share 21 lb. of chocolate equally?

How many 43 cup servings are in

32 cup of yogurt? How wide is a rectangular strip

of land with a length of 43 mile and an area of

21 square mile?

Compute fluently with multi-digit numbers and find common factors and multiples.

6.NS.2.2: Fluently divide multi-digit numbers using the standard algorithm.

6. NS.2.3: Fluently add, subtract, multiply, and divide multi-digit decimals using thestandard algorithm for each operation.

6.NS.2.4: Find the greatest common factor of two whole numbers less than or equalto 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1-100 with a common factor as a multiple of a sum of two whole numbers with no common factor.

For example, express 36 + 8 as 4(9 + 2).

Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. 7.NS.1.1: Apply and extend previous understandings of addition and subtraction to

add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.

a. Describe situations in which opposite quantities combine to make 0.For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged.

b. Understand p + q as the number located a distance |q| from p, in thepositive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts.

c. Understand subtraction of rational numbers as adding the additive inverse,p - q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.

d. Apply properties of operations as strategies to add and subtract rationalnumbers.

7.NS.1.2. Apply and extend previous understandings of multiplication and divisionand of fractions to multiply and divide rational numbers.

a. Understand that multiplication is extended from fractions to rationalnumbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1)=1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.


Sixth Grade Seventh Grade

Apply and extend previous understandings of numbers to the system of rational numbers. 6.NS.3.5: Understand that positive and negative numbers are used together to

describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.

6.NS.3.6: Understand a rational number as a point on the number line. Extend

number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.

a. Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself (e.g., -(-3) = 3) and that 0 is its own opposite.

b. Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.

c. Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.

6.NS.3.7: Understand ordering and absolute value of rational numbers.

a. Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram.

b. Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers is a rational number. If p and q are integers, then –

qp =

qp− =

qp−

. Interpret quotients of rational

numbers by describing real-world contexts.

c. Apply properties of operations as strategies to multiply and divide rational numbers.

d. Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates to 0s or eventually repeats.

7.NS.1.3. Solve real-world and mathematical problems involving the four operations with rational numbers.


Sixth Grade Seventh Grade

For example, interpret -3 > -7 as a statement that -3 is located to the right of -7 on a number line oriented from left to right.

b. Write, interpret, and explain statements of order for rational numbers in real-world contexts.

For example, write -3°C > -7°C to express the fact that -3°C is warmer than -7°C. c. Understand the absolute value of a rational number as its distance from 0

on the number line; interpret absolute value as magnitude for a positive or negative quantity in real-world situation.

For example, for an account balance of -30 dollars, write |-30| = 30 to describe the size of debt in dollars.

d. Distinguish comparisons of absolute value from statements about order.For example, recognize that an account balance less than -30 dollars represents a debt greater than 30 dollars.

6.NS.3.8: Solve real-world and mathematical problems by graphing points in all fourquadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.


EXPRESSIONS AND EQUATIONS

Sixth Grade Seventh Grade Apply and extend previous understandings of arithmetic to algebraic expressions.

6.EE.1.1: Write and evaluate numerical expressions involving whole numberexponents.

6.EE.1.2: Write, read, and evaluate expressions in which letters stand fornumbers.

a. Write expressions that record operations with numbers and with lettersstanding for numbers.

For example, express the calculation “Subtract y from 5” as 5 – y. b. Identify parts of an expression using mathematical terms (sum, term,

product, factor, quotient, coefficient); view one or more parts of an expression as a single entity.

For example, describe the expression 2(8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms.

c. Evaluate expressions at specific values of their variables. Includeexpressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations).

For example, use the formulas V = s3 and A = 6s2 to find the volume and surface area of a cube with sides of length s =

21 .

6.EE.1.3: Apply the properties of operations to generate equivalent expressions.For example, apply the distributive property to the expression 3(2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6(4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.

Use properties of operations to generate equivalent expressions.

7.EE.1.1: Apply properties of operations as strategies to add, subtract, factor, andexpand linear expressions with rational coefficients.

7.EE.1.2: Understand that rewriting an expression in different forms in a problemcontext can shed light on the problem and how the quantities in it are related.

For example, a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply by 1.05”.

Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

7.EE.2.3: Solve multi-step real-life and mathematical problems posed with positiveand negative rational numbers in any form (whole numbers, fractions, and decimals) using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies.

For example: If a woman making $25 an hour gets a 10% raise, she will make an additional

101 of her salary an hour, or $2.50, for a new salary of $27.50. If you want

to place a towel bar 9 43 inches long in the center of a door that is 27

21 inches

wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.

7.EE.2.4: Use variables to represent quantities in a real-world or mathematicalproblem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.


Sixth Grade Seventh Grade 6.EE.1.4: Identify when two expressions are equivalent (e.g., when the two

expressions name the same number regardless of which value is substituted into them).

For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for.

Reason about and solve one-variable equations and inequalities. 6.EE.2.5: Understand solving an equation or inequality as a process of answering a

question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.

6.EE.2.6: Use variables to represent numbers and write expressions when solving areal-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

6.EE.2.7: Solve real-world and mathematical problems by writing and solvingequations of the form x + p = q and px = q for cases in which p, q and x are all non-negative rational numbers.

6.EE.2.8: Write an inequality of the form x > c or x < c to represent a constraint orcondition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.

Represent and analyze quantitative relationships between dependent and independent variables. 6.EE.3.9: Use variables to represent two quantities in a real-world problem that change

in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and the independent variables using graphs and tables, and relate these to the equation.

For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.

a. Solve word problems leading to equations of the formpx + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.

For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?

b. Solve word problems leading to inequalities of the formpx + x > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem.

For example, as a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be a least $100. Write an inequality for the number of sales you need to make and describe the solutions.


STATISTICS AND PROBABILITY

Sixth Grade Seventh Grade Develop understanding of statistical variability.

6.SP.1.1: Recognize a statistical question as one that anticipates variability in thedata related to the question and accounts for it in the answers.

For example, “How old am I?” is not a statistical question, but “How old are the students in my school?” is a statistical question because one anticipates variability in students’ ages.

6.SP.1.2: Understand that a set of data collected to answer a statistical questionhas a distribution which can be described by its center, spread, and overall shape.

6.SP.1.3: Recognize that a measure of center for a numerical data set summarizesall of its values with a single number, while a measure of variation describes how its values vary with a single number.

Summarize and describe distributions.

6.SP.2.4: Display numerical data in plots on a number line, including dot plots,histograms, and box plots.

6.SP.2.5: Summarize numerical data sets in relation to their context, such as by:a. Reporting the number of observations.b. Describing the nature of the attribute under investigation, including how it

was measured and its units of measurement. c. Giving quantitative measures of center (median and/or mean) and

variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.

d. Relating the choice of measures of center and variability to the shape ofthe data distribution and the context in which the data were gathered.

Use random sampling to draw inferences about a population.

7.SP.1.1: Understand that statistics can be used to gain information about apopulation by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.

7.SP.1.2: Use data from a random sample to draw inferences about a populationwith an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions.

For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be.

Draw informal comparative inferences about two populations.

7.SP.2.3: Informally assess the degree of visual overlap of two numerical datadistributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability.

For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable.

7.SP.2.4: Use measures of center and measures of variability for numerical datafrom random samples to draw informal comparative inferences about two populations.

For example, decide whether the words in a chapter of a seventh grade science book are generally longer than the words in a chapter of a fourth grade science book.


Sixth Grade Seventh Grade Investigate chance processes and develop, use, and evaluate probability models. 7.SP.3.5: Understand that the probability of a chance event is a number between 0 and 1

that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability

around 21 indicates an event that is neither unlikely nor likely, and a probability

near 1 indicates a likely event. 7.SP.3.6: Approximate the probability of a chance event a probability around by collecting

data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability.

For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times. 7.SP.3.7: Develop a probability model and use it to find probabilities of events. Compare

probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.

a. Develop a uniform probability model by assigning equal probability to all outcomes,and use the model to determine probabilities of events.

For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected.

b. Develop a probability model (which may not be uniform) by observing frequencies indata generated from a chance process.

For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny appear to be equally likely passed on the observed frequencies?

7.SP.3.8: Find probabilities of compound events using organized lists, tables, treediagrams, and simulation.

a. Understand that, just as with simple events, the probability of a compound event is thefraction of outcomes in the sample space for which the compound event occurs.

b. Represent sample spaces for compound events using methods such as organized lists,tables, and tree diagrams. For an event described in everyday language (e.g., “rollingdouble sixes”) identify the outcomes in the sample space which compose the event.

c. Design and use a simulation to generate frequencies for compound events.For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?

THIRD GRADE DOMAIN/CLUSTER DESCRIPTORS AND CLARIFICATION

Third Grade Domain and Cluster Descriptors and Clarifications (from Ohio Department of Education, www.ode.state.oh.us/), page 1 of 11, 2013-2014

Domain: Operations and Algebraic Thinking Cluster: Represent and solve problems involving multiplication and

division.

In Grade 2, students found the total number of objects using rectangular arrays, such as a 5 × 5, and wrote equations to represent the sum. This strategy is a foundation for multiplication because students should make a connection between repeated addition and multiplication.

Students need to experience problem-solving involving equal groups (whole unknown or size of group is unknown) and multiplicative comparison (unknown product, group size unknown, or number of groups unknown) as shown in Table 2 of the Common Core State Standards for Mathematics, page 89. No attempt should be made to teach the abstract structure of these problems.

Encourage students to solve these problems in different ways to show the same idea and be able to explain their thinking verbally and in written expression. Allowing students to present several different strategies provides the opportunity for them to compare strategies. Sets of counters, number lines to skip count and relate to multiplication, and arrays/area models will aid students in solving problems involving multiplication and division. Allow students to model problems using these tools.

Provide a variety of contexts and tasks so that students will have more opportunity to develop and use thinking strategies to support and reinforce learning of basic multiplication and division facts. Have students create multiplication problem situations in which they interpret the product of whole numbers as the total number of objects in a group and write as an expression. Also, have students create division problem situations in which they interpret the quotient of whole numbers as the number of shares.

Students can use known multiplication facts to determine the unknown fact in a multiplication or division problem. Have them write a multiplication or division equation and the related multiplication or division equation. For example, to determine the unknown whole number in 27 ÷ = 3, students should use knowledge of the related multiplication fact of 3 × 9 = 27. They should ask themselves questions such as, “How many 3s are in 27?” or “3 times what number is 27?”. Have them justify their thinking with models or drawings.

Common Misconceptions

Students also think that 3 ÷ 15 = 5 and 15 ÷ 3 = 5 are the same equations. The use of models is essential in helping students eliminate this understanding.

http://www.ode.state.oh.us/



Domain: Operations and Algebraic Thinking Cluster: Understand properties of multiplication and the relationship

between multiplication and division.

Students need to apply properties of operations (commutative, associative, and distributive) as strategies to multiply and divide. Applying the concept involved is more important than students knowing the name of the property. Understanding the commutative property of multiplication is developed through the use of models as basic multiplication facts are learned. For example, the result of multiplying 3 × 5 (15) is the same as the result of multiplying 5 × 3 (15).

To find the product of three numbers, students can use what they know about the product of two of the factors and multiply this by the third factor. For example, to multiply 5 × 7 × 2, students know that 5 × 2 is 10. Then, they can use mental math to find the product of 10 × 7 (70). Allow students to use their own strategies and share with the class when applying the associative property of multiplication.

Splitting arrays can help students understand the distributive property. They can use a known fact to learn other facts that may cause difficulty. For example, students can split a 6 × 9 array into 6 groups of 5 and 6 groups of 4; then, add the sums of the groups. The 6 groups of 5 is 30 and the 6 groups of 4 is 24. Students can write 6 x 9 as (6 × 5) + ( 6 × 4). Students’ understanding of the part/whole relationships is critical in understanding the connection between multiplication and division.




Domain: Operations and Algebraic Thinking Cluster: Multiply and divide within 100.

Students need to understand the part/whole relationships in order to understand the connection between multiplication and division. They need to develop efficient strategies that lead to the big ideas of multiplication and division. These big ideas include understanding the properties of operations, such as the commutative and associative properties of multiplication and the distributive property. The naming of the property is not necessary at this stage of learning.

In Grade 2, students found the total number of objects using rectangular arrays, such as a 5 × 5, and wrote equations to represent the sum. This is called unitizing, and it requires students to count groups, not just objects. They see the whole as a number of groups of a number of objects. This strategy is a foundation for multiplication in that students should make a connection between repeated addition and multiplication.

As students create arrays for multiplication using objects or drawing on graph paper, they may discover that three groups of four and four groups of three yield the same results. They should observe that the arrays stay the same, although how they are viewed changes. Provide numerous situations for students to develop this understanding.

To develop an understanding of the distributive property, students need to decompose the whole into groups. Arrays can be used to develop this understanding. To find the product of 3 × 9, students can decompose 9 into the sum of 4 and 5 and find 3 × (4 + 5). The distributive property is the basis for the standard multiplication algorithm that students can use to fluently multiply multi-digit whole numbers in Grade 5.

Once students have an understanding of multiplication using efficient strategies, they should make the connection to division. Using various strategies to solve different contextual problems that use the same two one-digit whole numbers requiring multiplication allows for students to commit to memory all products of two one-digit numbers.




Domain: Operations and Algebraic Thinking Cluster: Solve problems involving the four operations, and identify and

explain patterns in arithmetic.

Students gain a full understanding of which operation to use in any given situation through contextual problems. Number skills and concepts are developed as students solve problems. Problems should be presented on a regular basis as students work with numbers and computations. Researchers and mathematics educators advise against providing “key words” for students to look for in problem situations because they can be misleading. Students should use various strategies to solve problems. Students should analyze the structure of the problem to make sense of it. They should think through the problem and the meaning of the answer before attempting to solve it.

Encourage students to represent the problem situation in a drawing or with counters or blocks. Students should determine the reasonableness of the solution to all problems using mental computations and estimation strategies. Students can use base ten blocks on centimeter grid paper to construct rectangular arrays to represent problems.

Students are to identify arithmetic patterns and explain those using properties of operations. They can explore patterns by determining likenesses, differences, and changes. Use patterns in addition and multiplication tables.




Domain: Number and Operations in Base Ten Cluster: Use place value understanding and properties of operations to

perform multi-digit arithmetic.

Prior to implementing rules for rounding, students need to have opportunities to investigate place value. A strong understanding of place value is essential for the developed number sense and the subsequent work that involves rounding numbers. Building on previous understandings of the place value of digits in multi-digit numbers, place value is used to round whole numbers. Dependence on learning rules can be eliminated with strategies such as the use of a number line to determine which multiple of 10 or of 100 a number is nearest (5 or more rounds up, less than 5 rounds down). As students’ understanding of place value increases, the strategies for rounding are valuable for estimating, justifying, and predicting the reasonableness of solutions in problem-solving.

Strategies used to add and subtract two-digit numbers are now applied to fluently add and subtract whole numbers within 1,000. These strategies should be discussed so that students can make comparisons and move toward efficient methods. Number sense and computational understanding is built on a firm understanding of place value. Understanding what each number in a multiplication expression represents is important. Multiplication problems need to be modeled with pictures, diagrams, or concrete materials to help students understand what the factors and products represent. The effect of multiplying numbers needs to be examined and understood.

The use of area models is important in understanding the properties of operations of multiplication and the relationship of the factors and its product. Composing and decomposing area models is useful in the development and understanding of the distributive property in multiplication. Continue to use manipulatives like hundreds charts and place-value charts.


The use of terms like “round up” and “round down” may confuse students. For example, the number 37 would round to 40 or they say it “rounds up”. The digit in the tens place is changed from 3 to 4 (rounds up). This misconception is what causes the problem when applied to rounding down. The number 32 should be rounded (down) to 30, but using the logic mentioned for rounding up, some students may look at the digit in the tens place and take it to the previous number, resulting in the incorrect value of 20. To remedy this misconception, students need to use a number line to visualize the placement of the number and/or ask questions such as: “What tens are 32 between and which one is it closer to?”. Developing the understanding of what the answer choices are before rounding can alleviate much of the misconception and confusion related to rounding.




Domain: Number and Operations – Fractions Cluster: Develop understanding of fractions as numbers.

This is the initial experience students will have with fractions and is best done over time. Students need many opportunities to discuss fractional parts using concrete models to develop familiarity and understanding of fractions. Expectations in this domain are limited to fractions with denominators 2, 3, 4, 6 and 8.

Understanding that a fraction is a quantity formed by part of a whole is essential to number sense with fractions. Fractional parts are the building blocks for all fraction concepts. Students need to relate dividing a shape into equal parts and representing this relationship on a number line, where the equal parts are between two whole numbers. Help students plot fractions on a number line by using the meaning of the fraction.

As students counted with whole numbers, they should also count with fractions. Counting equal-sized parts helps students determine the number of parts it takes to make a whole and recognize fractions that are equivalent to whole numbers. Students need to know how big a particular fraction is and can easily recognize which of two fractions is larger. The fractions must refer to parts of the same whole. Benchmarks such as 1

2 and 1 are also useful in

comparing fractions.

Equivalent fractions can be recognized and generated using fraction models. Students should use different models and decide when to use a particular model. Make transparencies to show how equivalent fractions measure up on the number line.

Venn diagrams are useful in helping students organize and compare fractions to determine the relative size of the fractions, such as more than 1

2 , exactly 1

2, or less than 1

2 . Fraction bars

showing the same sized whole can also be used as models to compare fractions. Students are to write the results of the comparisons with the symbols >, =, or <, and justify the conclusions with a model.


The idea that the smaller the denominator, the smaller the piece or part of the set, or the larger the denominator, the larger the piece or part of the set, is based on the comparison that in whole numbers, the smaller a number, the less it is, or the larger a number, the more it is. The use of different models, such as fraction bars and number lines, allows students to compare unit fractions to reason about their sizes. Students think all shapes can be divided the same way. Present shapes other than circles, squares, or rectangles to prevent students from overgeneralizing that all shapes can be divided the same way.




Domain: Measurement and Data Cluster: Represent and interpret data.

Representation of a data set is extended from picture graphs and bar graphs with single-unit scales to scaled picture graphs and scaled bar graphs. Intervals for the graphs should relate to multiplication and division with 100 (product is 100 or less and numbers used in division are 100 or less). In picture graphs, use values for the icons in which students are having difficulty with multiplication facts. For example, represents 7 people. If there are three , students should use known facts to determine that the three icons represents 21 people. The intervals on the vertical scale in bar graphs should not exceed 100.

Students are to draw picture graphs in which a symbol or picture represents more than one object. Bar graphs are drawn with intervals greater than one. Ask questions that require students to compare quantities and use mathematical concepts and skills.

Students are to measure lengths using rulers marked with halves and fourths of an inch and record the data on a line plot. The horizontal scale of the line plot is marked off in whole numbers, halves, or fourths. Students can create rulers with appropriate markings and use the ruler to create the line plots.


Although intervals on a bar graph are not in single units, students count each square as one. To avoid this error, have students include tick marks between each interval. Students should begin each scale with 0. They should think of skip-counting when determining the value of a bar since the scale is not in single units.




Domain: Measurement and Data Cluster: Solve problems involving measurement and estimation of

intervals of time, liquid volumes, and masses of objects.

Learning to tell time has much to do with learning to read a dial-type instrument and little with time measurement. Students have experience in telling and writing time from analog and digital clocks to the hour and half hour in Grade 1 and to the nearest five minutes, using a.m. and p.m. in Grade 2. Now students will tell and write time to the nearest minute and measure time intervals in minutes. Provide analog clocks that allow students to move the minute hand. Students need experience representing time from a digital clock to an analog clock and vice versa.

Provide word problems involving addition and subtraction of time intervals in minutes. Have students represent the problem on a number line. Students should relate using the number line with subtraction from Grade 2.

Provide opportunities for students to use appropriate tools to measure and estimate liquid volumes in liters only and masses of objects in grams and kilograms. Students need practice in reading the scales on measuring tools since the markings may not always be in intervals of one. The scales may be marked in intervals of two, five, or ten.




Domain: Measurement and Data Cluster: Geometric measurement: Understand concepts of area and relate

area to multiplication and to addition.

Students can cover rectangular shapes with tiles and count the number of units (tiles) to begin developing the idea that area is a measure of covering. Area describes the size of an object that is two-dimensional. The formulas should not be introduced before students discover the meaning of area. The area of a rectangle can be determined by having students lay out unit squares and count how many square units it takes to completely cover the rectangle without overlaps or gaps. Students need to develop the meaning for computing the area of a rectangle. A connection needs to be made between the number of squares it takes to cover the rectangle and the dimensions of the rectangle. Ask questions such as:

• What does the length of a rectangle describe about the squares covering it?• What does the width of a rectangle describe about the squares covering it?

The concept of multiplication can be related to the area of rectangles using arrays. Students need to discover that the length of one dimension of a rectangle tells how many squares are in each row of an array and the length of the other dimension of the rectangle tells how many squares are in each column. Ask questions about the dimensions if students do not make these discoveries. For example:

• How do the squares covering a rectangle compare to an array?• How is multiplication used to count the number of objects in an array?

Students should also make the connection of the area of a rectangle to the area model used to represent multiplication. This connection justifies the formula for the area of a rectangle.

Common Misconceptions Students may confuse perimeter and area when they measure the sides of a rectangle and then multiply. Pose problems situations that require students to explain whether they are to find the perimeter or area.




Domain: Measurement and Data Cluster: Geometric measurement: recognize perimeter as an attribute of

plane figures and distinguish between linear and area measures.

Geoboards can be used to find the perimeter of rectangles. Provide students with different perimeters and have them create the rectangles on the geoboards. Have students share their rectangles with the class. Have discussions about how different rectangles can have the same perimeter with different side lengths.

Students experienced measuring lengths of inches and centimeters in Grade 2. They have also related addition to length and writing equations with a symbol for the unknown to represent a problem. Once students know how to find the perimeter of a rectangle, they can find the perimeter of rectangular-shaped objects in their environment. They can use appropriate measuring tools to find lengths of rectangular-shaped objects in the classroom. Present problem situations involving perimeter, such as finding the amount of fencing needed to enclose a rectangular shaped park, or how much ribbon is needed to decorate the edges of a picture frame. Also present problem situations in which the perimeter and two or three of the side lengths are known, requiring students to find the unknown side length.

Students need to know when a problem situation requires them to know that the solution relates to the perimeter or the area. They should have experience with understanding area concepts when they recognize it as an attribute of plane figures. They also discovered that when plane figures are covered without gaps by n unit squares, the area of the figure is n square units.

Students need to explore how measurements are affected when one attribute to be measured is held constant and the other is changed. Using square tiles, students can discover that the area of rectangles may be the same, but the perimeter of the rectangles varies. Geoboards can also be used to explore this same concept.

Common Misconceptions Students think that when they are presented with a drawing of a rectangle with only two of the side lengths shown or a problem situation with only two of the side lengths provided, these are the only dimensions they should add to find the perimeter. Encourage students to include the appropriate dimensions on the other sides of the rectangle. With problem situations, encourage students to make a drawing to represent the situation in order to find the perimeter.




Domain: Geometry Cluster: Reason with shapes and their attributes.

In earlier grades, students have experiences with informal reasoning about particular shapes through sorting and classifying using their geometric attributes. Students have built and drawn shapes given the number of faces, number of angles, and number of sides. The focus now is on identifying and describing properties of two-dimensional shapes in more precise ways using properties that are shared rather than the appearances of individual shapes. These properties allow for generalizations of all shapes that fit a particular classification.

Development in focusing on the identification and description of shapes’ properties should include examples and non-examples, as well as examples and non-examples drawn by students of shapes in a particular category. For example, students could start with identifying shapes with right angles. An explanation as to why the remaining shapes do not fit this category should be discussed. Students should determine common characteristics of the remaining shapes.

In Grade 2, students partitioned rectangles into two, three, or four equal shares, recognizing that the equal shares need not have the same shape. They described the shares using words such as, halves, thirds, half of, a third of, etc., and described the whole as two halves, three thirds, or four fourths. In Grade 4, students will partition shapes into parts with equal areas (the spaces in the whole of the shape). These equal areas need to be expressed as unit fractions of the whole shape (e.g., describe each part of a shape partitioned into four parts as

41 of the area of the shape).


Additional Resources

Addition and Subtraction Strategies, page 1, 2015 - 2016

ADDITION AND SUBTRACTION STRATEGIES

The development of strategies for addition and subtraction is a critical area in the Common Core State

Standards. By using and comparing a variety of solution strategies students build their understanding

of the relationship between addition and subtraction.

*The following information regarding addition and subtraction strategies has been adapted from:

Van de Walle, J.A., & Lovin, L.H. (2006). Teaching Student-Centered Mathematics, Volume I. Boston:

Pearson. See chapters three and four of this book for further clarification of addition and subtraction

strategies.

Addition Strategies Subtraction Strategies

Zero Think-Addition

One More/Two More Build Up Through Ten

Doubles Back Down Through Ten

Near Doubles Invented

Sums of Ten

Make Ten

Ten Plus

Invented

Commutative Property

Associative Property

SPECIAL NOTES:

Basic facts for addition are combinations of numbers where both addends are less than 10.

Subtraction facts correspond to the addition facts.

Fluency involves accuracy (correct answer), efficiency (a reasonable amount of steps that does

not include counting), and flexibility (using strategies that demonstrate number sense).

Every child, including ESE children, can master the basic facts with efficient mental tools.

Steps to Mastery:

1) Children must develop an understanding of number relationships and the operations.

2) Children need to develop efficient strategies for fact retrieval.

3) Teachers need to provide practice of selection of strategies once they have been developed.

Children who do not learn mental strategies will continue to count on their fingers since they

have no other strategies to solve basic addition and subtraction problems.

AVOID PREMATURE DRILL: if a child does not know a fact and is given a timed test; the

child will revert to counting.

Downplay counting on as a strategy because children often get confused as to why they can

count for some problems but not others. It is used as a crutch where other strategies would be

more efficient.

Many of the strategies apply to more than one fact. Therefore, students need to choose the one

that works best for them through discussion and justification.

Encourage discussion so students can justify and defend their method. This allows the students

to hear other methods that might lead to the development of a more effective strategy.


Zero One addend is always zero

The sum of any addend and zero is the original addend.

There are 19 facts where zero is one of the addends

Be sure to show 0 + 6 and 6 + 0

Children assume that addition sentences result in a larger number

Note: This may seem easy; however, students over generalize that an addition sentence always equals a larger

sum.

One/Two More One addend is 1 or 2

36 facts

Students are ready for these activities when they can identify 1 or 2 more without counting

Doubles The two addends are the same 0 + 0, 1 + 1, 2 + 2, etc.

There are 10 doubles facts

These facts will be anchors for other facts (such as 4 + 4 = 8 so 4 + 5 = 9 , see Near-Doubles)

Near-Doubles All combinations where one addend is more than the other

Note: Some children will double the smaller fact and add up 6 + 6 = 12 so 6 + 7 = 13. Others will double the

greater fact and subtract one 7 + 7 = 14 so 7 + 6 = 13

*Be sure students are exposed to both so they can decide which is better for them.

Sums of Ten The two addends equal the sum of ten

These facts will be anchors for other facts (such as 9 + 1=10, so 9 + 4 becomes 10 + 3)

Ten Plus One addend is 10, 10 + 4, 4 + 10

Children need to recognize that a set of ten and a set of 4 total 14 without counting.

* This is not an appropriate place for the term 1 ten as regrouping for first graders. The term 1 set of ten not a 1 in

the tens place should be used to meet the needs of the early first grade student.

Make-Ten These facts all have 8 or 9 as one of the addends

Children use 10 as a way to “bridge” to get the sum 6 + 8. Start with 8; decompose the 6 into 4 + 2 add the

2 to 8 and get a sum of 10. 10 and the remaining 4 equals 14 so 6 + 8 = 14.

Commutative Property The order of the addends does not change the sum

2 + 5 = 5 + 2

Associative Property The sum is the same regardless of the grouping of the addends.

2 + (6 + 4) = 2 + 10 = 12

NOTE: Counting on is not a sophisticated strategy. Children coming from Kindergarten are expected to

recognize small sets of numbers but may count. Children in first and second grade are expected to take the

next step by creating and using more sophisticated strategies such as the ones listed below.

Addition Strategies (Continued)


Addition Strategies

Remaining 4 Facts 3 + 5 3 + 6 4 + 7 5 + 7

The children have learned or discovered strategies to solve the 4 strategies above. Now encourage the students to

apply and choose a strategy that will work for them.

7 + 4 decompose the 4 into 3 + 1 to make ten, add 1 more

7 + 5 decompose the 5 to make 3 + 2, therefore making a ten creating a fact they know (7 + 3 = 10), then

add 2 more

Invented Students create and/or apply any of the above strategies to other equations.

Students will create ways to solve problems that are not noted above.

Encourage students to create other ways to solve problems other than counting.

Invented strategies are number-oriented, flexible, and constructed by students.

7+5

Make Ten 7 + 3 = 10

+2 12

Invented (using what I

know)

7 + 7 = 14 7 + 5 = 12

Invented (applying a

near double)

7 + 6 = 13 7 + 5 = 12

Circle Map


Subtraction Strategies

Think-Addition

The student understands subtraction as an unknown addend problem.

This strategy works best for sums less than 10 because 64 % of the 100 subtraction facts

fall into this category, for example: 9 – 4 (think 4 + 5 = 9)

Such facts as 7 – 2 would go along well with 2 more, now think 2 less along with

2 + 5 = 7, so 7 – 2 = 5

Build Up Through Ten

This group includes all the facts where the part is either 8 or 9

Start with the 8 or 9 and ask how much to ten and then build up

Back Down Through 10

It is most useful for facts where one digit is close to the number it is being subtracted

from

14 - 6, remove six from a ten frame and then two more to get the eight

Known as decomposing a number leading to a ten in Common Core

Invented

Students will create ways to solve problems that are not noted above

Encourage students to create other ways to solve problems other than counting

Circle Map

14-8

8+6

14 - 8 (4+4)

14 - 4=10-4=6

8+7=15 8+6=14

8+8=16 -2 14 8-2=6

8+2=10 +4 14

Think addition

Back Down Through 10

Invented (using a fact I know)

Invented

Build Up Through 10


MACC.2.NBT.2

Use place value understanding and properties of operations to add and subtract.

The standard algorithm is introduced and taught in fourth grade: 4.NBT.4.

Second and third grade students are encouraged to invent strategies when solvingmulti digit addition and subtraction problems for the following reasons. Place value concepts are enhanced.

Students make fewer errors as they are focused on the number and numberrelationships.

Less reteaching is necessary as they are inventing for themselves what makessense.

Mental computation and estimation are enhanced.

Flexible thinking of number leads to strategies and this thinking is often fasterthan standard algorithms.

Strategies serve students just as well as traditional algorithms on tests (includingFCAT 2.0).

Students who look at the meaning of numbers and use what they know to solveproblems know and use more mathematics than those that follow a procedure.

Samples of Invented Strategies for Addition

Place Value: 352 + 675

300 + 50 + 2

600 + 70 + 5

900 + 120 + 7

1,027 = 1,000 + 20 + 7

Friendly Tens: 352 + 675

327 + 700 = 1,027

Compensate: 352 + 675

350 is easier to add to

650

1,000

Now I pick up my 25 + 2 = 27 1,000 + 27 = 1,027

Adding Hundreds or Ones first: 352 + 675

300 + 600 = 900

50 + 70 = 120

2 + 5 = 7

1,027


Samples of Invented Strategies for Subtraction

Place Value: 675 – 352 600 + 70 + 5 300 + 50 + 2 323 = 300 + 20 + 3

Add up: 675 – 352

352 + 8 = 360 360 + 40 = 400 400 + 275 = 675

323

Students may extend 275 + 5 (5+ 3 = 8) to get 280 + 20 (20 + 20 = 40) to get 300 +23 = 323

See Chapter 6 strategies for whole-number computation in Van de Walle, J.A., & Lovin, L.H. (2006). Teaching Student-Centered Mathematics, Volume I, Boston: Pearson .

Basic Multiplication and Division Fact Strategies, page 1 of 3, adapted from Van de Walle, J. A., Lovin, L. H, Karp, K. S, & Bay-Williams, J. M. (2014). Teaching Student-Centered Mathematics, Volume II. Boston; Pearson.

Strategies for BASIC Multiplication and Division Facts

The development of strategies for multiplication and division is a critical area in the Mathematics Florida Standards (MAFS). By using and comparing a variety of solution strategies students build their understanding of the relationship between multiplication and division.

The following information regarding multiplication and division strategies has been adapted from: Van de Walle, J.A., Lovin, L.H, Karp, K.S, & Bay-Williams, J.M. (2014). Teaching Student-Centered Mathematics, Volume II. Boston; Pearson. See chapters eight, nine, and eleven of this book for further clarification of multiplication and division strategies and.

Multiplication Strategies Division Strategies

Doubles Think Multiplication and then Apply a Known Multiplication Fact Fives

Zeros and Ones

Nifty Nines

Using Known Facts to Derive Other Facts

SPECIAL NOTES:

The use of a problem-based approach and a focus on reasoning strategies are critical todeveloping mastery of the multiplication and related division facts. Thus, story problems should beused to develop reasoning strategies for basic fact mastery.

BASIC Multiplication Fact Strategies

Doubles

These are facts with 2 as a factor and are equivalent to the addition doubles, so students shouldalready know these.

Students need experiences to help them realize that 2 x 8 is the same as double 8 (8 + 8).

Fives

These are facts that have 5 as a first or second factor.

Mastery development ideas:

Skip count by fives: 0, 5, 10, 15, 20 . . .

Connect counting by fives with arrays that have 5 dotsFor example three rows is 3 x 5

Connect to counting minutes on the clock.


Zeros and Ones

These are facts that have at least one factor that is either 0 or 1.

While these facts seem easy, they can be confusing to students because of rules for addition. Forexample, when zero is added to a number, it does not change the number (8 + 0 = 8). However,8 x 0 = 0. Adding 1 to a number results in the next number, or one more (8 + 1 = 9), but a numbermultiplied by one does not change the number (8 x 1 = 8).

The use of rules that are strictly procedural, such as “anything times zero is zero” should be avoided.

Nifty Nines

Facts with factors of 9 may be among the easiest to learn because of reasoning strategies andpatterns. 9 x 8 is the same as 10 x 8 less one set of 8, or 80 – 8 = 72 The tens digit is always one less than the other factor (the factor other than 9) and the sum of the

digits in the product is always 9. Therefore, for the fact 9 x 8, the tens digit is 7 and since the two digits in the product must add to 9 the ones digit is 2 and the product is 72.

Patterns are not rules without reasons. Students should be challenged to understand why they work.

Using Known Facts to Derive Other Facts

Reasoning Strategies:

Double and Double Again This applies to all facts with a factor of 4. For example, 4 x 6 is the same as 2 x 6 doubled. Note that for some facts such as 4 x 8, doubling

the product may result in a difficult addition problem. For 4 x 8, a student knows 2 x 8 is 16, and then doubles 16. Doubling 16 is a difficult addition and simply adding 16 + 16 defeats the purpose of efficient reasoning. Students should use effective and efficient addition strategies such as, “I know 15 + 15 is 30 and 16 + 16 is 2 more, or 32.”

Double and One More This works with facts that have 3 as one factor. For example, 3 x 6 is 2 x 6 and 6 more

(12 + 6 = 18). Note that 3 x 8 and 3 x 9 result in challenging mental additions.

Half then Double This applies to all facts with one even factor. For example, 6 x 8; half of 6 eights is 3 eights,

3 times 8 is 24, double 24 is 48.

Close Fact strategy This involves adding one more set to a known fact. For example, think of 6 x 8 as 6 eights. Five

eights is close and results in 40. Six eights is one more eight, or 48. Using 5 x 8 to figure out 6 x 8, the language “6 groups of eight” or “6 eights” can help students

remember to add 8 more not 6 more. The Close Fact strategy can be used with any multiplication fact. It reinforces students’ number

sense and relationships between numbers.


BASIC Division Fact Strategies

Reasoning

Mastery of basic division facts is dependent on the inverse relationship of multiplication anddivision. For example, to solve 48 ÷ 6, we might naturally ask ourselves, “Six times what is 48?”The reasoning strategy is to (1) think multiplication, and then (2) apply a known multiplicationfact.

Near facts: 60 ÷ 8; mentally review a short sequence of multiplication facts comparing eachproduct to 60: 6 x 8 = 48 (too low), 7 x 8 = 56 (close), 8 x 8 = 64 (too high). It must be 7, so thatis 56 with 4 left over.

NOTE: Division with remainders if much more prevalent in the real world than basic division facts that have no remainders. Students should be able to solve these near fact problems with reasonable speed.

Four-Corners-and-a-Rhombus Math Graphic Organizer

What do you already know? Brainstorm ways to solve this problem.

Try two ways to solve the problem here. List words and phrases you need to include in your communication write up.

What do you need to

find out?

Cognitive Complexity of Mathematics Items

Low Complexity This category relies heavily on the recall and recognition of previously learned concepts and principles. Items typically specify what the student is to do, which is often to carry out come procedure that can be performed mechanically. It is not left to the student to come up with an original method or solution. The list below illustrates some, but not all, of the demands that items in the low complexity category might make:

• Recall or recognize a fact, term, or property.• Identify appropriate units or tools for common measurements.• Compute a sum, difference, product, or quotient.• Recognize or construct an equivalent representation.• Perform a specified operation or procedure.• Evaluate a variable expression, given specific values for the variables.• Solve a one-step problem.• Retrieve information from a graph, table, or figure.• Perform a single-unit conversion.

Moderate Complexity Items in the moderate complexity category involve more flexibility of thinking and choice among alternatives than do those in the low complexity category. They require a response that goes beyond the habitual, is not specified, and ordinarily has more than a single step. The student is expected to decide what to do, using informal methods of reasoning and problem solving strategies, and to bring together skill and knowledge from various domains. The list below illustrates some, but not all, of the demands that items of moderate complexity might make.

• Solve a problem requiring multiple operations.• Solve a problem involving spatial visualization and/or reasoning.• Retrieve information from a graph, table, or figure and use it to solve a problem.• Compare figures or statements.• Determine a reasonable estimate.• Extend an algebraic or geometric pattern.• Provide a justification for steps in a solution process.• Formulate a routine problem, given data and conditions.• Represent a situation mathematically in more than one way.• Select and/or use different representations, depending on situation and purpose.

High Complexity High complexity items make heavy demands on student thinking. Students must engage in more abstract reasoning, planning, analysis, judgment, and creative thought. The item requires that the student think in an abstract and sophisticated way. The list below illustrates some, but not all, of the demands that items in the high complexity category might make:

• Perform a procedure having multiple steps and multiple decision points.• Describe how different representations can be used for different purposes.• Solve a non-routine problem (as determined by grade-level appropriateness).• Analyze similarities and differences between procedures and concepts.• Generalize an algebraic or geometric pattern.• Formulate an original problem, given a situation.• Solve a problem in more than one way.• Explain and justify a solution to a problem.• Describe, compare, and contrast solution methods.• Formulate a mathematical model for a complex situation.• Analyze or produce a deductive argument.• Provide a mathematical justification.

NOTE: The complexity of an item is generally NOT dependent on the multiple-choice distractors. The options may affect the difficulty of the item, not the complexity of the item.

2015 - 2016 - Brevard Public Schoolselementarypgms.sp.brevardschools.org/Home... · INTRODUCTION TO...

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