Draft Unit Plan aProblems Overview: Teacher...

Draft Unit Plan 6.RP.1-3: Understand Ratio Concepts and Use Ratio Reasoning to Solve Problems

Draft December 2011 of 1Maryland Common Core State Curriculum Unit for Grade 6 Mathematics

Overview: The overview statement is intended to provide a summary of major themes in this unit.

This is the first time that students have formally studied ratio and proportion. The study of ratios and proportional reasoning extends students�’ work in measurement and multiplication and division in elementary grades. Ratios and proportional reasoning are the foundation for further study in mathematics, science and are useful in everyday life. Students will be working with ratios, rates, unit rates and percents to solve situations in daily life.

Teacher Notes: The information in this component provides additional insights which will help the educator in the planning process for the

unit. Ratios are used in geometry and in algebra when students study similar figures. The term ratio is used when units are the same and the term rate is used when units are different in the ratio. The CCSS uses ratio to

encompass both situations. The goal in sixth grade is to study ratios while leaving proportional studies for grade seven.

Enduring Understandings: Enduring understandings go beyond discrete facts or skills. They focus on larger concepts, principles, or

processes. They are transferable and apply to new situations within or beyond the subject.

At the completion of the unit on ratio concepts, the student will understand that: A ratio is a multiplicative comparison of two quantities. A rate is a special ratio with a denominator of one that compares different types of measures.

Essential Question(s): A questions is essential when it stimulates multi-layered inquiry, provokes deep thought and lively discussion, requires students to consider alternatives and justify their reasoning, encourages re-thinking of big ideas, makes meaningful connections with prior learning, and provides students with opportunities to apply problem-solving skills to authentic situations.

Is it important to know how to solve for unit rates? What is the connection between a ratio and a fraction? How are ratios used in the real world? How is a ratio or rate used to compare two quantities or values? Where can examples of ratios and rates be found? How can I model and represent rates and ratios? What are similarities and differences between fractions and ratios?


Draft Maryland Common Core State Curriculum Unit for Grade 6 Mathematics December 2011 of 2

Priority Clusters in Grade 6: According to the Partnership for the Assessment of Readiness for College and Careers (PARCC), some clusters require greater emphasis than others. The table below shows PARCC’s relative emphasis for each cluster. Prioritization does not imply neglect or exclusion of material. Clear priorities are intended to ensure that the relative importance of content is properly attended to. Note that the prioritization is in terms of cluster headings.

Key: Major Clusters Supporting Clusters Additional Clusters Ratios and Proportional Reasoning

Understand ratio concepts and use ratio reasoning to solve problems.

The Number System

Apply and extend previous understandings of multiplication and division to divide fractions by fractions. Compute fluently with multi-digit numbers and find common factors and multiples. Apply and extend previous understandings of numbers to the system of rational numbers.

Expressions and Equations

Apply and extend previous understandings of arithmetic to algebraic expressions.

Reason about and solve one-variable equations and inequalities.

Represent and analyze quantitative relationships between dependent and independent variables. Geometry

Solve real-world and mathematical problems involving area, surface area, and volume. Statistics and Probability

Develop understanding of statistical variability. Summarize and describe distributions.



Focus Standards (Listed as Examples of Opportunities for In-Depth Focus in the PARCC Content Framework document): According to the Partnership for the Assessment of Readiness for College and Careers (PARCC), this component highlights some individual standards that play an important role in the content of this unit. Educators may choose to give the indicated mathematics an especially in-depth treatment, as measured for example by the number of days; the quality of classroom activities for exploration and reasoning; the amount of student practice; and the rigor of expectations for depth of understanding or mastery of skills.

6.RP.3 When students work toward meeting this standard, they use a range of reasoning and representations to analyze proportional relationships.

Possible Student Outcomes:

The following list provides outcomes that describe the knowledge and skills that students should understand and be able to do when the unit is completed. The outcomes are often components of more broadly-worded standards and sometimes address knowledge and skills necessarily related to the standards. The lists of outcomes are not exhaustive, and the outcomes should not supplant the standards themselves. Rather, they are designed to help teachers delve deeply into the standards and augment as necessary, providing added focus and clarity for lesson planning purposes. This list is not intended to imply any particular scope or sequence. The student will be able to: Use ratios, rates and percent in a wide variety of contexts. Use tables, graphs, tape diagrams and double number line diagrams to represent equivalent ratios.

Progressions from Common Core State Standards in Mathematics: For an in-depth discussion of the overarching, “big picture” perspective on student learning of content related to this unit, see:

The Common Core Standards Writing Team (10 September 2011). Progressions for the Common Core State Standards in Mathematics (draft), accessed at: http://commoncoretools.files.wordpress.com/2011/09/ccss_progression_rp_67_2011_11_121.pdf

Vertical Curriculum Alignment: Vertical curriculum alignment provides two pieces of information: A description of prior learning that should support the learning of the concepts in this unit A description of how the concepts studied in this unit will support the learning of additional mathematics

http://commoncoretools.files.wordpress.com/2011/09/ccss_progression_rp_67_2011_11_121.pdf


Draft for 2011Maryland Common Core State Curriculum Unit Grade 6 Mathematics December of 4

Key Advances from Previous Grades: Between grade 5 and grade 6, students grow in their ability to analyze ratios, rates, unit rates and percents. Students use their knowledge of decimals and fractions to: o apply their skills with multiplication, division, and fractions to contribute to their study of ratios and unit rates (6.RP). o extend their work in measurement and in multiplication and division o work with percents

Additional Mathematics: Students will use ratios, rates, unit rates and percent skills:

o in grade 7 when working with proportional relationships and probability o in geometry and in algebra when studying similar figures and slopes of lines

Possible Organization of Unit Standards: This table identifies additional grade-level standards within a given cluster that support the

over-arching unit standards from within the same cluster. The table also provides instructional connections to grade-level standards from outside the cluster.

Overarching Unit Standards Supporting Standards within the

Cluster Instructional Connections outside

the Cluster 6.RP.1: Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.

6.RP.2: Understand the concept of a unit rate associated with a ratio a:b with b 0 (b not equal to zero), and use rate language in the context of a ratio relationship.

6.RP.3: 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,

6.RP.3a: Make tables of equivalent ratios

relating quantities with whole-number measurements, find missing values in the tables, and plot the

6.NS.8: Solve real-world and mathematical

problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates



or equations. pairs of values on the coordinate plane. Use tables to compare ratios.

and absolute value to find distances between points with the same first coordinate or the same second coordinate.

6.RP.3b: Solve unit rate problems including

those involving unit pricing and constant speed.

6.EE.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 independent variables using graphs and tables, and relate these to the equation.

6.RP.3c: Find a percent of a quantity as a

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

6.RP.3d: Use ratio reasoning to convert

measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.



Connections to the Standards for Mathematical Practice: This section provides samples of how the learning experiences for this unit support the development of the proficiencies described in the Standards for Mathematical Practice. The statements provided offer a few examples of connections between the Standards for Mathematical Practice in the content standards of this unit. The list is not exhaustive and will hopefully prompt further reflection and discussion. In this unit, educators should consider implementing learning experiences which provide opportunities for students to: 1. Make sense of problems and persevere in solving them.

Analyze a problem and depict a good way to solve the problem. Consider the best way to solve a problem. Can interpret the meaning of their answer to a given problem.

2. Reason abstractly and quantitatively

Consider the ideas that ratios can be represented in more than one way. Does your answer connect to the question?

3. Construct Viable Arguments and critique the reasoning of others.

Justify the process of working with a ratio to answer a question. Justify an argument using a table, graph, tape diagram or double line graph.

4. Model with Mathematics

Draw a diagram that represents ratio. Analyze an authentic problem and use a nonverbal representation of the problem. Use appropriate manipulatives.

5. Use appropriate tools strategically

Use virtual media and visual models to explore ratio and rate problems.

6. Attend to precision Demonstrate their understanding of the mathematical processes required to solve a problem by communicating all of the steps in

solving the problem. Label appropriately. Use the correct mathematics vocabulary when discussing problems.

Draft Unit Plan atio Concepts and Use Ratio Reasoning to Solve Problems 6.RP.1-3: Understand R


7. Look for and make use of structure. Look at a table or graph and recognize the relationship that is represented in each. Compare, reflect and discuss multiple solution methods.

8. Look for and express regularity in reasoning

Pay special attention to details and continually evaluate the reasonableness of their answers. Use of mathematical principles will help you in solving the problem.

Content Standards with Essential Skills and Clarifications: The Content Standards and Essential Skills and Knowledge statements shown in this section come directly from the Maryland State Common Core Curriculum Frameworks. Clarifications were added as needed. Please note that only the standards or portions of standards that needed further explanation have supporting statements. Educators should be cautioned against perceiving this as a checklist. All information added is intended to help the reader gain a better understanding of the standards.

Standard Essential Skills and Knowledge Clarification

6.RP.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.�”

Knowledge of ratio as a comparison of any two quantities

�• Knowledge of a ratio is not always a comparison of part-to-whole; Can be part-to-part or whole-to-whole

6.RP.1: ratio: Ratio is a comparison of two quantities or measures. Ratios can be expressed in the form ( ), a to b, or a:b. Ratios can be expressed as comparisons of a part to a whole, one part of a whole to another part of the same whole, or measures of two different types which is called a rate. Example: Part-to-whole would be the ratio of boys to the whole class. Part-to-part would be the ratio of boys to girls. Rate would be the ratio of miles per gallon to miles per hour.

6.RP.2: Understand the concept of a unit rate associated with a ratio a:b with b 0 (b not equal to zero), and

use rate language in the context

�• Knowledge that a unit rate emphasizes finding an equivalent ratio with a denominator of 1.

6.RP.2: unit rate: A ratio where the denominator is 1 unit. Example: If

15 bars of soap cost $6.75,one bar would cost $.45. =



of a ratio relationship. For example, "This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar." "We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger." (Expectations for unit rates in this grade are limited to non-complex fractions.)

6.RP.3: 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.

6a: Make tables of equivalent

ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.

�• Knowledge of multiplicative recursive patterns

�• Ability to use multiplicative relationships to extend an initial ratio to equivalent ratios; When working backward, use the inverse operation (division).

Ability to recognize a linear relationship appears when the pairs are plotted on the coordinate plane

6.RP.3: tape diagrams: Tape diagrams are linear models used to represent data and help students organize their thinking. Example: Casey read 7 more books than Jamie. If Casey read 16 books, how many books did Jamie read?

Number of books Jamie read

differenceNumber of books Casey read

Jamie + 7 = Casey

Casey �– 7 = Jamie

16 �– 7 = 9

So Jamie read 9 books.



6b: Solve unit rate problems including those involving unit pricing and constant speed. 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?

�• Ability to use division to determine unit rate

6c: Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole given a part and the percent.

�• Ability to introduce percent as a special rate where a part is compared to a whole and the whole always has a value of 100

�• Ability to solve problems using equivalent ratios. (NOTE: Proportions are not introduced until Grade 7.) This is developing proportional reasoning without formal proportions.

6.RP.6c: percent: is another name for hundredths Ratio of a number

to 100 with a percent sign. Per hundred Example: 75% = = .75

6d: Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.

�• Ability to expand ratio reasoning to units of measurement



Fluency Expectations/Recommendations: This section highlights individual standards that set expectations for fluency, or that otherwise represent culminating masteries. These standards highlight the need to provide sufficient supports and opportunities for practice to help students meet these expectations. Fluency is not meant to come at the expense of understanding, but is an outcome of a progression of learning and sufficient thoughtful practice. It is important to provide the conceptual building blocks that develop understanding in tandem with skill along the way to fluency; the roots of this conceptual understanding often extend one or more grades earlier in the standards than the grade when fluency is finally expected. PARCC has no fluency expectations related to ratio and proportion.

Evidence of Student Learning: The Partnership for Assessment of Readiness for College and Careers (PARCC) has awarded the Dana Center a grant to develop the information for this component. This information will be provided at a later date. The Dana Center, located at the University of Texas in Austin, encourages high academic standards in mathematics by working in partnership with local, state, and national education entities. Educators at the Center collaborate with their partners to help school systems nurture students' intellectual passions. The Center advocates for every student leaving school prepared for success in postsecondary education and in the contemporary workplace.

Common Misconceptions: This list includes general misunderstandings and issues that frequently hinder student mastery of concepts

regarding multiplication and division of fractions.

Students may: confuse mathematical terms such as ratio, rate, unit rate and percent. not understand how to set up a table or graph. not understand the difference between an additive relationship and a multiplicative relationship.

Interdisciplinary Connections: Interdisciplinary connections fall into a number of related categories: Literacy standards within the Maryland Common Core State Curriculum Science, Technology, Engineering, and Mathematics standards Instructional connections to mathematics that will be established by local school systems, and will reflect their specific grade-level

coursework in other content areas, such as English language arts, reading, science, social studies, world languages, physical education, and fine arts, among others.



Model Lesson Plan:

MSDE Mathematics Lesson Plan

Background Information Content/Grade Level

Ratios and Proportional Relationships/Grade 6

Unit

Understand ratio concepts and use ratio reasoning to solve problems.

Essential Questions/Enduring Understandings Addressed in the Lesson

Essential Questions Is a ratio is a multiplicative comparison of two quantities? What is the connection between a ratio and a fraction? How are ratios used in the real world? How is a ratio or rate used to comp re ta wo quantities or values?

Enduring Understandings Knowledge of a ratio is not always a comparison of part-to-whole; can be part-to-part or whole-to-

whole. Knowledge of the difference between a ratio and a fraction.

Standards Addressed in This Lesson 6.RP.1 language to describe a ratio relationship between

Understand the concept of a ratio and use ratio two quantities.

Lesson Topic iting Ratios

Defining and Wr

Relevance/Connections Students’ work with ratios and proportional relationships (6.RP) can be combined with their work in

representing quantitative relationships in real world problems that change in relationship to one another (6.EE.9). Formal use of proportions will be covered in grade 7.

Student Outcomes Students will:

now thK ree ways to write a ratio, 9:10, 9 to 10, Develop a definition of ratio

Draft Unit Plan .RP.1-3: Underst d nd Use Ratio Reasoning to Solve Problems 6 an Ratio Concepts a


Use ratio language to describe a relationship between two quantities Express ratios as part-to-part relationships

ps Express ratios as part-to-whole relationshi

Prior Knowledge Needed to Support This Learning

Knowledge of a fraction as a part-to-whole relationship. Knowledge of multiplication and division of fractions.

Method for Determining Student s a g of fractional part to whole relationship. Readiness for the Lesson

U e w m-up to determine student understandinr

Learning Experience

Component Which Standards of Mathematical Practice Details does this component address? Warm Up

NCTM Ratio, Proportions

/Drill

ee S& Proportional Reasoning pg.

1. Which is another way of expressing ?

A. + + +

B. +

C. 1

D. All of the above

2. Model answer choice A by

shading the fraction bars below.

3. How would your m del look different if you shaded a o

Make sense of problems and persevere in solving them

Reason abstractly and quantitatively Model with mathematics

structure Look for and make use of


Draft Maryland Common Core State Curriculum 6 Mathe 2011 of 13

Learning Experience

Component Which Standards of Mathematical Practice does this component address? Details

model for answer choice B?

.

4

Unit for Grade matics December

What is the value for the shad d part of the f ction bars above?

5.n #4?

e ra

What are all the possible ways of expressing the model

iven ig

Motivation

ote to Teacher: Give students a picture of an assortment of ports balls (see sample below).

Ns

Make sense of problems and persevere in solving them

Reason abstractly and quantitatively Model with mathematics



Learning Experience

Component tails Which Standards of Mathematical Practice does this component address? De

Given the values 5 and 7, discuss how these values are represented by the balls in the picture.

Activity 1 UDL Components

ltiple Means of Representation

on

eForm

Divide class into pairs or small groups; Give each group a

that compare each color to the total cubes. (Examples: red to total; blue

er:

h -

Mu

Multiple MeansAction and Exp

for ress

Multiple Means for Engagement

i

K y Questions ative Assessment

Summary

scoop of rainbow cubes. Use the cubes to write fractions

to total; etc.; all comparing color/part to total/whole.) Use labels wh n wre iting these fractions. Students should understand terminology: “part-to-whole.” How many red cubes do you have? Have many yellow cubes do you have? Can we compare red cubes to yellow cubes by writing a fraction? (AnswNo, because the denominator of a fraction represents a whole. This is a part-to-part relationship; one form of a ratio, which can be written using fraction notation. In the same way, a fraction (part-to-whole) is also a ratio.) Students should understand terminology: “part-to-part.” Define ratio (a comparison of two quantities or measures) & the 3 ways to write it. (Example: 2/3, 2:3, 2 to 3). W ith your partner, continue to write other ratios witthe rainbow cubes. (Students should include part-topart and part-to-whole.)

Use appropriate tools strategically

Model with mathematics Construct viable arguments and critique the

reasoning of others



Learning Experience

Component Which Standards of Mathematical Practice does this component address? Details

Pra c

:3, 2 t tio language. Ask students to nses

eir ratios and ) d in those

cti e writing ratios, including part-to-part and part-to-whole and use of all 3 methods to write a ratio (Example: 2/3,

o 3). Practice using ra2write ratios using objects in the classroom. Sample respocould include, but are not limited to:

Males to females Chairs to desks (Note: This may be a 1:1

relationship) ro p each studenG u students and have t (1) share th

(2 explain the type of relationship representeratios. Students should be able to defend their thinking.

Activity 2 UDL Components Multiple Means of

presentation

eForm

wer.

fRe

Multiple Means or Action and Express

Multiple Means for ion

Engagement K y Questions

ative Assessment Summary

On a 10-question quiz a student answers 8 questions correctly. What is the ratio of the number of questions nswered correctly to the total number questions? Allen says a

the answer is 10 to 8. Is he correct? Justify your ans Use additional scenarios to help students realize the significance of the order in which ratios are written given a cenario. Samples may include but are not limited to: s

Wins to losses Cost to number of items purchased

Construct viable arguments and critique the reasoning of others

Clo

eac r s of personal nte s

sure

Hav st th a partner or in a small group. Give e udents work wih g oup a photograph of something that ire t, e.g. from National Geographic or any other i

magazine. (Ask students to bring in a photo or a poster that they like.) Ask them to create ratios of comparisons based on

Make sense of problems and persevere in solving them: Identify and execute appropriate strategies to solve the problem.



Learning Experience

Component Details Which Standards of Mathematical Practice does this component address?

objects in the photographs. Identify each ratio as paror part-to-whole. Have groups exchange their photographs and ratios to share for correctness.

t-to-part

Supporting Information Interventions/Enrichments

Special Education/Learners

Common Misconceptions by students: Fractions and ratios are the same. A ratio is Students may no

Struggling

Enrichment/Gifted and Talented

ELL

just another name for a fraction. of two amounts. They will likely focus on just one

The transition for these students will be in that they need to take into account

t see ratios as a comparisonquantity.

two quantities wheapplying ratio reasoning.

Students may want to use additive properties when reasoning with ratios. However, equivalency and

n

reasoning with ratios must include multiplicative properties. 2 wins to 3 losses is NOT the same as 3 losses to 2 wins ( not equal to ).

ssible extension for enrichment: PoHave students discover the value of pi. Give students a number of circles, lids, bowls, plates, Frisbees, etc. Ask them to measure the diameter and measure the distance around the circles (see if any student calls it

that’s what it’s called.) Compare the measurement of the diameter to circumference. Or, you can tell themthe measurement to the circumference in the form of a ratio. Tell students to average all of the ratios that they found. Have a discussion about how this is equivalent to pi.

Materials

Rainbow Cubes, Picture of assortment of sports balls, Photographs or posters



Technology Optional: Interactive white board

Resources (must be available to all stakeholders) s

“Teaching Student Centered Mathematics; Grades 5-8” by Van de Walle “Developing Essential Under “Good Questions for Math Teaching

tanding of Ratios, Proportions, & Proportional Reasoning” by NCTM ; Why Ask Them and What to Ask; G

Schuster and Nancy Canavan Anderson rades 5-8” by Lainie

Draft Unit Plannd Use Ratio Reasoning to Solve Problems

6.RP.1-3: Understand Ratio Concepts a

Lesson Seeds:


seeds have been written particularly for the unit, with mind. The suggested activities are not ive, or sequential; they simply demonstrate how specific udents learn the

andards. They are designed to generate evidence of student teachers ideas for developing s.

MSDE Mathematics Lesson Seed Organiz

andard that can be used to build a lesson.

The lesson intended to be prescriptive, exhaustskills described in the sttheir own activitie

Lesson Seeds : The lesson seeds are ideas for the domain/st

specific standards in content can be used to help st

understanding and give

er

Lesson seeds are not meant to be all-inclusive, nor are they substitutes for instruction.

Domain: Ratio and Proportional Relationship

Standard: 6.RP.3

Use ratio and rate reasoning to solv bout tables of equivalent ratios, tape diagrams, e real-world and mathematical problems, e.g., by reasoning a



double number line diagrams, or equations.

b. Solve unit rate problems including those involving unit pricing and constant speed.

Purpose/Big Idea: Finding the best unit rate for wireless phone use.

Materials: Rate sheet for three companies

Activity:

Students are looking for the best cell phone plan for their families to buy for them. They looked on line and found these phone companies published these rates per m

Horizon Horizon EZZ EZZ Rocket Rocket

onth.

Talking Only Talking & Texting Talking Talking & Texting Talking Talking & Texting

450 minutes 39.99 59.99 69.99 89.99

700 minutes 69.99 99.99

900 minutes 59.99 79.99 89.99 109.99

1400 minutes 89.99 119.99

2000 minutes 99.99 129.99

Unlimited 119.99 149.99 69.99 89.99 99.99 119.99

ation do you need to compare these prices? What inform



W o to find it rate for eacWhich plan would you choose and why?

hat would you d the un h plan?

Essential Guiding Quest w is a ratio o ed to ions: Ho r rate us compare two quantities or values?

Sample Assessment Items: The items included in this component will be aligned to the standards in the unit and will include:

PARCC prototype items

nterventions/Enrichments: (Standard-specific modules that focus on student interventions/enrichments and on professional cluded later, as available from the vendor(s) producing the modules.)

Voca epts: This section of the Unit Plan is divided into two parts. Part I contains vocabulary and terminology ster which is the focus of this unit plan. Part II contains vocabulary and terminology from standards

outside of the focus cluster. These “outside standards” provide important instructional connections to the focus cluster.

ratio: : Ratio is a comparison of two quantities or measures. Ratios can be expressed in the form (

Items purchased from vendors

PARCC public release items Maryland Public release items

Idevelopment for teachers will in

bulary/Terminology/Concfrom standards that comprise the clu

Part I �– Focus Cluster: Understand Ratio Concepts and Use Ratio Reasoning to Solve Problems

), a to b, or a:b.

Ratios can be expressed as comparisons of:

1. part to a whole, one part of a whole to another part of the same whole. Part to whole would be the ratio of boys to the whole class. measures

of two different types which is called a rate.

2. part to part would be the ratio of boys to girls in a class.



Measures of two different types are miles per hour.

unit rate: A ratio where the denominator is 1 unit 75,one bar would cost

called a rate. Rate would be the ratio of miles per gallon to

. Example: If 15 bars of soap cost $6. $.45. =

e diagrams: Tape diagrams are linear models used to represent data and help students organize their thinking. Example: Casey read 7more books than Jamie. If Casey has read 16 books, how many books did Jamie read?

tap

ercent: is another name for hundredths Ratio of a number to 100 with a percent sign. Per hundred Example: 75% =

Number of books Casey read Jamie + 7 = Casey

16 �– 7 = 9

So Jamie read 9 books.

Casey �– 7 = Jamie

Number of books Jamie read difference

p = .75

Connections outside the F

Part II �– Instructional ocus Cluster

dependent variable: In a function of two variables, one variable is dependent and the oth ependent. In the equation y = 3x + 4, y is er is indthe dependent variable. Its value If the v 2, thendepends on the value of x. alue of x= the value of y is y = 3 2 + 4 or y = 10.

independent variable: In a function of two variables, one variables is dependent and the other independent. In the equation y = 3x + 4, x is the independent variable. The value of y depends on the value of x. If the value of x is 1, then the value of y is y = 3 1 + 4 or y = 7.

Resources: This section contains links to materials that are intended to support content instruction in this unit.

http://ime.math.arizona.edu/progressions/#products

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