BOSTON PUBLIC SCHOOLS, 2013-2014 Grade 4 Mathematics Scope...

Boston Public Schools Mathematics Department Grade 4 Scope and Sequence, 2013-2014 Last updated 07.26.13

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BOSTON PUBLIC SCHOOLS, 2013-2014 Grade 4 Mathematics Scope and Sequence Guide

I. Introduction: In grade 4, instructional time should focus on three critical areas: (1) developing understanding and fluency with multi-digit multiplication, and developing understanding of dividing to find quotients involving multi-digit dividends; (2) developing an understanding of fraction equivalence, addition and subtraction of fractions with like denominators, and multiplication of fractions by whole numbers; (3) understanding that geometric figures can be analyzed and classified based on their properties, such as having parallel sides, perpendicular sides, particular angle measures, and symmetry. (MCF 2011; p.43) By the end of the year, students should be proficient with the grade 4 Mathematical Content and Practice Standards.

II. Essential Questions for the Year: 1. How does finding the common characteristics among similar problems create more efficient problem solvers? 2. How do mathematical operations relate to each other and extend and apply through all domains of number? 3. How does becoming more fluent in number and operations help student become better mathematicians? 4. How does building fractions from unit fractions extend previous understandings of operations on whole numbers to operations with fractions? 5. How does analyzing, comparing, building and classifying 2-dimensional shapes deepen understanding of properties of 2- dimensional shapes?

III. Strengthening Fluency with the Curriculum Resources:

The Massachusetts Curriculum Framework for Mathematics (MCF 2011) names standards for fluency with single-digit combinations in addition, subtraction, multiplication and division at different grade levels. “The word fluent is used in the Standards to mean ‘fast and accurate’. Fluency in each grade involves a mixture of just knowing some answers, knowing some answers from patterns (e.g., “adding 0 yields the same number”), and knowing some answers from the use of strategies. It is important to push sensitively and encouragingly toward fluency of the designated numbers at each grade level, recognizing that fluency will be a mixture of these kinds of thinking, which may differ across students. As should be clear from the foregoing, this is not a matter of instilling facts divorced from their meanings, but rather as an outcome of a multi-year process that heavily involves the interplay of practice and reasoning.” (excerpt from K, Counting and Cardinality; K–5, Operations and Algebraic


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Thinking CCSS Progression) The Framework defines procedural fluency with multi-digit numbers, including decimals, and fraction as “skill in carrying out procedures flexibly, accurately, efficiently, and appropriately” (MCF 2011, p.15). Our materials provide many opportunities for students to engage in the interplay of practice and reasoning and to strengthen their procedural fluency including: 1. Primary Curriculum Materials 2. First in Math 3. Number Talks Number Talks The only Ten Minute Math Routine included in this year’s Scope and Sequence is Number Talks. Kindergarten through fifth grade teachers will facilitate Number Talks with all students at least three days a week. Number Talks are designed to support proficiency with grade level fluency standards. The goal of Number Talks is for students to compute accurately, efficiently, and flexibly. In addition to developing efficient computation strategies, Number Talks encourages students to make sense of mathematics, communicate mathematically, and reason and prove solutions. The key components of successful Number Talks: I. A safe and accepting classroom environment and mathematical community II. Classroom discussions (PROTOCOL)

1. Teacher provides the problem. 2. Teacher provides students opportunity to solve problem mentally. 3. Students show a visual cue when they are ready with a solution. Students signal if they have solved it in more than one way

too. A quiet form of acknowledgement allows time for students to think, while the process continues to challenge those that already have an answer. This is why we encourage the use of a “thumbs up” close to the body as opposed to hands up.

4. Teacher calls for answers. S/he collects all answers- correct and incorrect- and records answers. 5. Students share strategies and justifications with peers.

III. The teacher’s role as a “facilitator, questioner, listener, and learner” IV. Use of mental math to increase efficiency and knowledge of number relationships V. Purposeful computation problems that support mathematical goals in number and operations


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The BPS math office has provided resources to schools, on mybps.org and through copies of the Number Talks book, to implement the Number Talks routine. However, these are only meant to be resources. The purpose of Number Talks is for each teacher to use the protocol to address the needs of his/her students. “Crafting problems that guide students to focus on mathematical relationships is an essential part of number talks that is used to build mathematical understanding and knowledge. The teacher’s goals and purposes for the number talk should determine the numbers and operations that are chosen. Careful planning before the number talk is necessary to design ‘just right’ problems for students. “ (Number Talks by Sherry Parrish, Math Solutions 2010, p.14) Teachers are encouraged to design their own Number Talks, based upon informal and formal assessment data. For example, at the beginning of 4th grade, teachers should consider revisiting ideas about addition and subtraction from previous grades to be ready to begin the Addition, Subtraction, and Number System Unit of Study in the coming months. “As you begin to implement number talks in your classroom, start with small numbers that are age and grade-level appropriate. Using small numbers serves two purposes: 1) students can focus on the nuances of the strategy, instead of on the magnitude of the numbers, and 2) students are able to build confidence in their mathematical abilities.” (Number Talks by Sherry Parrish, Math Solutions 2010, p.183) Later in the year, Number Talks can be used to revisit operations with decimals and fractions. Areas to consider when selecting Number Talk Problems (Number Talks by Sherry Parrish, Math Solutions 2010, p. 373): 1. Overgeneralizations. When students are investigating which strategies work with different operations, they often over generalize, and try to apply their generalizations to all operations. An example is when students are convinced that compensation works with addition and then assume it will also work with subtraction, multiplication, or division. 2. Inefficient strategies. Sometimes students become more focused on a specific strategy and ignore efficiency. If you have given them a problem that lends itself to using landmark numbers or compensation, such as 1999 + 1999, yet the majority of your students solve this either with the standard U.S. algorithm or by breaking it apart by place value, you would want to craft problems to address this issue. 3. Evidence from exit cards. Exit cards are an excellent way to keep a pulse on students’ understanding and use of strategies. If students struggle with a specific type of problem or operation on their exit cards, this would guide the types of problems and strategies for the next day’s number talk.

IV. Standards for Mathematical Practice in Grade 4:

The Common Core State Standards for Mathematical Practice are practices 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. These examples taken from North Carolina Public Schools unpacked math standards document for 4th grade, see below. These can also be found at http://www.dpi.state.nc.us/docs/acre/standards/common-core-tools/unpacking/math/4th.pdf.


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• MP1 - Make sense and persevere in solving them. Mathematically proficient students in grade 4 know that doing mathematics involves solving problems and discussing how they solved them. Students explain to themselves the meaning of a problem and look for ways to solve it. Fourth graders may use concrete objects or pictures to help them 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. • MP2 - Reason abstractly and quantitatively. Mathematically proficient fourth grade students 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. They extend this understanding from whole numbers to their work with fractions and decimals. Students write simple expressions, record calculations with numbers, and represent or round numbers using place value concepts. • MP3 - Construct viable arguments and critique the reasoning of others. In fourth grade mathematically proficient students may construct arguments using concrete referents, such as objects, pictures, and drawings. They explain their thinking and make connections between models and equations. 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. • MP4 - Model with mathematics. Mathematically proficient fourth grade students experiment with representing problem situations in multiple ways including numbers, words (mathematical language), drawing pictures, using objects, making a chart, list, or graph, creating equations, etc. Students need opportunities to connect the different representations and explain the connections. They should be able to use all of these representations as needed. Fourth graders should evaluate their results in the context of the situation and reflect on whether the results make sense. • MP5 - Use appropriate tools strategically. Mathematically proficient fourth grader students 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 or a number line to represent and compare decimals and protractors to measure angles. They use other measurement tools to understand the relative size of units within a system and express measurements given in larger units in terms of smaller units.


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• MP6 - Attend to precision. As fourth grader students 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, including using the equal sign consistently and appropriately. For instance, they use appropriate labels when creating a line plot. • MP7 - Look for and make use of structure. In fourth grade, mathematically proficient students look closely to discover a pattern or structure. For instance, students use properties of operations to explain calculations (partial products model). They relate representations of counting problems such as tree diagrams and arrays to the multiplication principle of counting. They generate number or shape patterns that follow a given rule. • MP8 - Look for and express regularity in repeated reasoning. Students in fourth grade should notice repetitive actions in computation to make generalizations. Students use models to explain calculations and understand how algorithms work. They also use models to examine patterns and generate their own algorithms. For example, students use visual fraction models to write equivalent fractions.

V. Appendix:

Throughout the scope and sequence, references are made to the appendix. These are resources you will use in addition to the Investigations curriculum materials. These resources consist of a variety of kinds of materials: former TMM or other materials created by the BPS elementary math office, materials created by BPS teachers who were part of the workgroup that created this scope and sequence guide, and links to materials available on such sites as Illustrativemathematics.org and NCTM Illuminations.

VI. Abbreviations Used in this Scope and Sequence Guide:

MCF 2011 is the Massachusetts Curriculum Framework for Mathematics, 2011. CCSS is the Common Core State Standards. The CCSS Guide is the Investigations and the Common Core State Standards booklet that helps align our curriculum resources to the new MCF standards. The lessons for these sessions end in a letter, A or B. The Progressions documents are documents that the authors of the CCSS for mathematics created to help teachers and school districts understand the depth and breadth of the new CCSS standards for mathematics and how they develop over time. SAB is the Investigations Student Activity Book. Please note: This year grade 4 will be using some grade 5 units from Investigations. Also Investigations, Unit 1 from grade 4, Factors, Multiples, and Arrays has been permanently moved to grade 3 because the work of the unit is grade 3 MCF standards.


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UNIT OF STUDY 1: MULTIPLICATION AND DIVISION Primary Curricular Resources: Grade 4, Investigations, Unit 3: Multiple Towers and Division Stories and Grade 5, Investigations, Unit 1: Number Puzzles and Multiple Towers Estimated Instructional Time: 44 days End of Unit Assessment: November 7th September 6th - November 8th

Overarching Questions:

! What is the relationship between multiplication and division? ! How does knowing the factors help you fluently solve multiplication and

division problems? ! What are different models of and models for multiplication and division?

How do models and representations help you to understand multiplication and division? How do these models and representations connect to equations?

! What are efficient methods for finding products and quotients? ! How is multiplicative comparison different from additive comparison?

Standards for Mathematical Practice Focus MP1 Make sense of problems and persevere in solving them: persevere in solving problems. MP2 Reason abstractly and quantitatively: write simple expressions, record calculations with numbers, and represent and round numbers using place value concepts. Engage in active discussions about problem solving strategies. MP3 Construct viable arguments and critique the reasoning of others: practice academic language, explain thinking to others and respond to others’ thinking, and engage in active discussions justifying solutions through problem solving strategies. MP7 Look for and make use of structure: students use what they know to solve problems they don’t know.

Instructional Notes: • Many lessons in this unit are omitted because the new standards place that work in Grade 3. • The unit is a review of factors, which students learned in grade 3, and introducing understanding of square, prime, and composite numbers. • Also new to this unit are multiplicative compare problems. • The unit continues to build the understanding of the Distributive Property. For example 8!6=(4!6)+(4!6)=(5!6)+(3!6) and the accompanying array, open array, and

area representations. Read Unit 1 Teacher Edition pages 115-116 to be familiar with different representations of multiplication. Read TE pages 117-119 to be familiar with the various levels of understandings of arrays.

• Dimensions of arrays should be drawn as accurately as possible as shown in lesson 2.2 in Grade 5 Unit 1. (It should not be a box divided into four equal parts because this does not support the distributive property, which is what this visual representation is highlighting.) The array model should not be called a box. Explicitly connect this work to area and use of area formula as well. This concept will be revisited in Unit of Study 4, Geometry, Geometric Measurement, and Geometry as well as in Unit of Study 5 Multiplication and Division later in the year.

• The standards expect students to represent problems using an equation, representing the unknown with a symbol or a letter, as they did in grade 3. • New to Grade 4 last year was Multiplicative Comparison. “Consider two diving boards, one 40 feet high, the other 8 feet high. Students in earlier grades learned to

compare these heights in an additive sense—‘This one is 32 feet higher than that one’—by solving additive Compare problems (2.OA.1) and using addition and subtraction to solve word problems involving length (2.MD.5). Students in Grade 4 learn to compare these quantities multiplicatively as well: ‘This one is 5 times as high as that one.’ (4.OA.1, 4.OA.2, 4.MD.1, 4.MD.2). In an additive comparison, the underlying question is what amount would be added to one quantity in order to result in the other. In a multiplicative comparison, the underlying question is what factor would multiply one quantity in order to result in the other.” (K, Counting and Cardinality; K–5, Operations and Algebraic Thinking Progression p. 29)


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• See p 184 in MCF 2011 for examples of prime versus composite numbers; one is neither prime nor composite. A prime number is a number greater than 1 that has only two factors. Composite numbers have more than 2 factors.

• MA.4.5a Students should be working developing fluency in multiplication and division up to 12 x 12; students should have ample time with Number Talks, class work, and homework to practice. Teachers and students should keep track of multiplication combinations in which students are fluent, and continue to practice until proficiency is reached.

Concepts developed in this unit • Apply their understanding of models and representations for multiplication

(equal-sized groups, arrays, area models), place value, and properties of operations, in particular the distributive property, as they develop, discuss, and use efficient, accurate, and generalizable methods to compute products of multi-digit whole numbers.

• Extend their concept of multiplication to make multiplicative comparisons (4.OA.1)

• Gain familiarity of factors and multiples. • Apply understanding of models to compute products of multi-digit whole

numbers. • Massachusetts Standard MA.4.NBT.5a requires students know multiplication

and division facts through 12 x 12. • Students become fluent in multiplication and division facts through their

understanding of array models, area models, and equal sized groups not by rote memorization.

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

• Solve division problems in which remainders must be interpreted. • Illustrate and explain multiplication and division calculations by using

equations, with a symbol for the unknown, rectangular arrays, and area models.

• Develop, strengthen, and become fluent with multiplication facts up to 12 x 12.

Prior knowledge expected • 3.OA.3 Use multiplication and division within 100 to solve word problems in

situations involving equal groups, arrays, and measurement quantities • 3.OA.5 Apply properties of operations as strategies to multiply and divide.

Examples: If 6 x 4 = 24 is known, then 4 x 6 = 24 is also known. (Commutative property of multiplication.) 3 x 5 x 2 can be found by 3 x 5 = 15 then 15 x 2 = 30, or by 5 x 2 = 10 then 3 x 10 = 30. (Associative property of multiplication.) Knowing that 8 x 5 = 40 and 8 x 2 = 16, one can find 8 x 7 as 8 x (5 + 2) = (8 x 5) + (8 x 2) = 40 + 16 = 56. (Distributive property.)

• 3.OA.6 Understand division as an unknown-factor problem. • 3.OA.2 Interpret whole-number quotients of whole numbers, e.g., interpret 56

x 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. Interpret whole-number quotients of whole numbers.

• 3.OA.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 x ? = 48, 5

= ! ÷ 3,, 6 x 6 = ?.

• 3.OA.7 Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 x 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.

Learning Outcomes Operations and Algebraic Thinking 4.OA Use the four operations with whole numbers to solve problems. 1. Interpret a multiplication equation as a comparison, 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. 2. Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to


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represent the problem, distinguishing multiplicative comparison from additive comparison.[1] 3. Solve multi-step word problems posed with whole 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. Gain familiarity with factors and multiples. 4. 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. Number and Operations in Base Ten 4.NBT Use place value understanding and properties of operations to perform multi-digit arithmetic. 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 ÷ 70 = 10 by applying concepts of place value and division. 5. Multiply a whole number of up to four digits by 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. MA.5a. Know multiplication facts and related division facts through 12 x 12. 6. Find whole-number quotients and remainders 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. Measurement and Data 4.MD Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit. 3. Apply the area and perimeter formulas for rectangles in real-world and mathematical problems. Area, connected to arrays, area models, and multiplication will be the focus in this unit. Perimeter will be studied in the addition and subtraction unit. Students will revisit both area and perimeter in the geometry unit.

MA 2011 Framework Citation

After completing each investigation, students will be able to:

Days Primary Curriculum Resource

4.NBT.5 4.MD.3 4.MA.5a.

• Solve, discuss and record multiplication problems with 2-digit numbers and consider ways to break apart problems in order to make them easier to solve.

• Represent the breaking apart of multiplication problems by fitting smaller arrays together to construct a larger array.

• Illustrate and explain the calculation by using equations, rectangular arrays and/or area models.

• Create story problems representing area. • Represent problems using equations with a letter standing

for the unknown quantity.

Academic language: multiplication, array, unmarked array, area model, dimension,

5 days Gr4 U3 Multiple Towers and Division Stories U3 1.1 Solving Multiplication Problems As students work on breaking up the arrays, make explicit connection to finding the total area of the rectangle. U3 1.2 Making Big Arrays -- use recording sheet U3 1.3 Small Array/Big Array -- use recording sheet U3 1.4 Small Array/Big Array cont. Have students write in area next to product on SAB page 11. Highlight that they are breaking up the arrays to find the total area.


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dimensions, rows, columns, factor, factor pairs, missing factor, multiple, product, partial product , prime number, square number, composite number, even, odd, half, double, estimate, estimation, representation, parenthesis

U3 1.5 Assessment: Solving 18x7 Students should be connecting the array to the problem. Look for students who are using the representation as a means to solve versus students who are marking the array one way and solving the problem another way. As part of this assessment students write a word problem to go with 18 x 7. Students can do this on the back of the page. Students’ problem should reflect multiplication, number of groups and number in a group.

4.NBT.1 4.NBT.5 4.OA.4 4.MA.5a

! Use an array and/or area model to represent multiplication ! Determine whether one number is a factor or multiple of another number

! Identify prime, square, composite, even and odd numbers ! Identify and develop strategies for learning multiplication combinations not yet known fluently ! Use known multiplication combinations to find equivalent multiplication combinations and multiplication combinations related by place value ! Apply/extend vocabulary skills/concepts through number puzzles Academic language: even, odd, half, double, estimate, estimation, multiples, multiple towers, 10th multiple, 20th multiple, 25th multiple

7 days

Gr 5 Unit 1: Number Puzzles and Multiple Towers

U1: 1.1 Building and using Arrays Use Activity 1, TMM (for 10-15 minutes)as an opportunity to review the use of parentheses and equations. See Olivia’s work on the top of page 31 of the Teacher Guide. Make sure students know why we use parentheses and how to write an equation. U1: 1.2 Identifying Properties of Numbers Activity 2: When referring to “unmarked arrays” continue to include the idea of area. Any array that is made with 24 tiles, and each tile is 1 square unit, has an area of 24 square units. U1: 1.3 What Numbers Have Which Properties? U1: 1.4 Multiplying with More Than Two Numbers U1: 1.5 Assessment: Number Puzzles and Finding Factors U1: 1.6 Number Puzzles and Finding Factors, continued U1: 1.7 OMIT Prime Factorization: This is not a grade 4 or grade 5 standard. Instead of U1: 1.7continue with Number Puzzles or do a task from Illustrative Math site: http://www.illustrativemathematics.org/illustrations/1493 http://www.illustrativemathematics.org/illustrations/1484 Use the comment sections of the task for guidance on the full group lesson summary.

4.NBT.1 4.NBT.5 4.OA.4

• Explain patterns when multiplying by multiples of 10 • Estimate products. • Efficiently solve 2-digit by 2-digit problems.

7 days

Gr5 U1: Number Puzzles and Multiple Towers U1: 2.1 Naming Multiplication Strategies


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4.MA.5a

• Use multiples of 10 to multiply efficiently. • Break up multiplication problems efficiently. • Use clear and concise notation. • Represent problems using equations with a letter standing

for the unknown quantity. • Name multiplication strategies and represent them using

arrays. • Develop flexibility in solving and representing

multiplication problems. • Use precision to accurately represent array models. • Develop flexibility in solving and representing multiplication

problems. Academic language: Use symbols: is greater than, >; is less than, <; is equal to, =; is approximately equal to, ( ) multiplication equation (e.g., 4 x 3 = 12), multiplication expression (e.g., 4 x 3)

Include an open array when solving 35 x 28. Emphasize precision of drawing arrays to support students’ understanding of the distributive property. Do not use a “box” or use the word “box” or “box method” to describe the array/area model representation. U1: 2.2 Comparing Representations U1 2.3 Which Product is Greater? The purpose of the following optional task is for students to solve a multi-step multiplication problem in a context that involves area. In addition, the numbers were chosen to determine if students have a common misconception related to multiplication: http://www.illustrativemathematics.org/illustrations/876 See commentary to support the full group summary discussion. U1: 2.4 Multiplication Cluster Problems Use distributive property to write equations for cluster problems as appropriate. Multiplying by ten: use language “ten times greater,” not “adding zero.” Note: “We can calculate 6 x 700 by calculating 6 x 7 and then shifting the result to the left two places (by placing two zeros at the end to show that these are hundreds) because 6 groups of 7 hundred is 6 x 7 hundreds, which is 42 hundreds, or 4,200.” K-5, Number and Operations Base Ten, Progressions, p 13. U1: 2.5 Multiplication Cluster Problems, continued U1: 2.6 How Do I Start? Develop flexibility in multiplicative reasoning. U1: 2.7 Assessment: What is the Answer? As part of this embedded assessment students write a word problem to go with 32 x 26. Students can do this on the back of the page. Students’ problems should reflect multiplication, number of groups and number in a group.


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4.OA.1 4.OA.2 4.OA.3 4.MA.5a.

• Solve and model two different kinds of division situations ((Massachusetts Curriculum Framework 2011 page 184).

• Discuss strategies for solving division problems that involve making groups of the divisor.

• Discuss division problems with remainders and focus on how the remainder affects the solution in each problem situation.

• Develop strategies for division by solving division problems and finding the missing dimension on an array when given the numbers of squares in the array and one dimension.

• Solve and represent pairs of related multiplication and division problems.

• Create and solve division story problems involving area, equal groups, and compare.

• Represent problems using equations with a letter standing for the unknown quantity.

Academic language: inverse relationship, division, divisor, dividend, quotient, remainder, missing factor

5 days Gr4 Unit 3: Multiple Towers and Division Stories U3 2.1: Looking at Division U3 2.2: Division with Remainders U3 2.3: Division Stories U3 2.4: Strategies for Division U3 2.5: Related Multiplication and Division Problems Note: Students should become familiar with the Partial Products notation for solving division problems. This is sometimes referred to as Dividing Down.


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4.OA.1 4.OA.2 4.OA.3 4.MA.5a.

• Solve word problems that involve multiplicative comparison. • Draw pictures, create equations and story problems that

explain multiplicative comparison. • Identify and verbalize what quantity is being multiplied and

by how many times. • Represent problems using equations with a letter standing

for the unknown quantity. Academic language: multiplicative comparison, for multiplicative comparison: ____ times as many as

6 days Day 1 Multiplicative Comparison Common Core 1.6A: Multiplicative Comparison (CCSS Guide, p CC3) Days 2-6 Use the problems created in the packet: Multiplicative Compare Problems, which includes extensive Teacher Notes. Also see: http://www.illustrativemathematics.org/illustrations/263 Refer to page 184 in MCF 2011 to see examples and/or the Progression document: Counting and Cardinality; K–5, Operations and Algebraic Thinking Progression p. 29. Examples: Unknown Product: A blue scarf costs $3. A red scarf costs 6 times as much. How much does the red scarf cost? (3 x 6 = p). Group Size Unknown: A book costs $18. That is 3 times more than a DVD. How much does a DVD cost? (18 ÷ p = 3 or 3 x p = 18). Number of Groups Unknown: A red scarf costs $18. A blue scarf costs $6. How many times as much does the red scarf cost compared to the blue scarf? (18 ÷ 6 = p or 6 x p = 18).

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


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4.OA.2 4.OA.4 4.OA.5 4.NBT.1 4.NBT.5 4.MA.5a.

• Generate an ordered series of multiples of 2-digit numbers and examine patterns and relationships in their lists of multiples.

• Identify landmark multiples, such as the 10th and 20th multiple of a number.

• Solve and represent related problems that involve multiple of 10 (e.g., 6 x 4 and 6 x 40) and describe the relationship between the products.

• Solve 2-digit multiplication problems and examine the mathematical relationship that underlies the pattern that they see when a number is multiplied by a multiple of 10.

• Demonstrate knowledge of multiplication combinations to 12 x 12.

Academic language: multiple, factor, 10th multiple, 20th multiple, 25th multiple, variable (letter standing for an unknown quantity)

4 days Gr4 Unit 3: Multiple Towers and Division Stories U3 3.1: Building Multiple Towers U3 3.2: Multiplying Groups of 10 U3 3.3: Multiplying 2-Digit Numbers U3 3.4: Multiplication Combinations

4.OA.2 4.OA.4 4.OA.5 4.NBT.1 4.NBT.5

• Examine what happens when one or both factors in a multiplication expression are doubled or halved, including the use of the open array/area model to see why and how this strategy works.

• Find ways to break apart the problems in order to use

5 days Gr4 Unit 3: Multiple Towers and Division Stories U3 4.1: Doubles and Halves: Omit Activity 1 and start with Activity 2. U3 4.2: Multiplication Cluster Problems


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4.MA.5a. familiar number relationships. • Develop and practice strategies for solving multiplication

problems with 2-digit numbers. • Continue to develop and practice strategies for solving

multiplication problems with 2-digit numbers. • Represent problems using equations with a letter standing

for the unknown quantity. Academic language: doubling, doubled, halving, halved, variable (letter standing for an unknown quantity)

U3 4.3: Strategies for Multiplication U3 4.4: Strategies for Multiplication, cont.

1 day BPS End of Unit Assessment


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UNIT OF STUDY 2: FRACTIONS AND DECIMALS Primary Curricular Resource: Grade 4, Investigations, Unit 6: Fraction Cards and Decimal Squares Fraction and Decimal Squares Packet Estimated Instructional Time: 35 days End of Unit Assessment: January 13th November 12th – January 14th


! How do you know if two fractions are equivalent? How can you use reasoning to compare fractions? (See Teacher Note, pp 151-152: Strategies for Comparing Fractions).

! How can you use what you know about equivalent fractions and relationships between fractions to compare and order fractions?

! What is the same and what is different when you add and subtract with fractions when compared to adding and subtracting whole numbers?

! What is the same and what is different when you add and subtract with decimals when compared to adding and subtracting whole numbers?

! What is the same and what is different when you multiply a whole number by a fraction when compared to multiplying two whole numbers?

! How can you use what you know about fractions to help you compare decimals?

! What do you learn about fractions (and decimals) through different visual models?

Standards for Mathematical Practice Focus MP 3 Construct viable arguments and critique the reasoning of others. Explain thinking and make connections between models and equations. Explain thinking to others and respond to others’ thinking. Constructively critique the strategies and reasoning of their classmates.

MP 5 Use appropriate tools strategically. Organize thinking in a diagram and use diagram to make sense of mathematics.

MP 6 Attend to precision. Link common language to academic language and use clear and precise language in their discussions with others and in their own reasoning.

MP 8 Look for and express regularity in repeated reasoning. Students in fourth grade should notice repetitive actions in computation to make generalizations. Students use models to explain calculations and understand how algorithms work. They also use models to examine patterns and generate their own algorithms. For example, students use visual fraction models to write equivalent fractions.

Instructional Notes: Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, and 12.

Materials to prepare This unit includes additional resources, entitled Fractions and Decimal Squares. This collection of resources will be posted to mybps.org Elementary Math, Grade 4 page. The resources include: directions and helpful hints to make fraction cards, number lines, recording sheets for games, pre-made fraction cards, and exit tickets. Rather than using the 4 x 6 arrays for the first investigation, use the blank rectangles found in the Fraction and Decimal Squares packet posted on mybps.org. Prepare needed materials in advance.

Fraction Cards


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Through the act of partitioning rectangles, students develop an understanding of fractions and relationships among fractions. Some students may have difficulty folding or partitioning rectangles into equal parts. Thirds can be particularly difficult. Creating Fraction Cards allow students time to work through this problem for themselves and to help each other with the actual folding or partitioning. Making fractional parts equal is a focus throughout the unit, and students will have several opportunities to make sure that fractional parts are the same size. Fraction Cards also provide an opportunity for students to strengthen understanding of relationships between fractions and fraction equivalency.

Why certain lessons are omitted Adding and subtracting with unlike denominators: Note the footnote on 4.NF.5: “Students who can generate equivalent fractions can develop strategies for adding fractions with unlike denominators in general. But addition and subtraction with unlike denominators in general is not a requirement at this grade.” For this reason sessions: 1.5, 1.6, and 1.7 are omitted.

Fractions in lowest terms. There is no explicit requirement in the Standards about simplifying fractions or putting fractions into lowest terms. What there is instead is an important progression of concepts and skills relating to fraction equivalence (e.g., 4.NF.4). Observe that putting a fraction into lowest terms is a special case of generating equivalent fractions. (Of course, generating equivalent fractions can go “either way” – starting from 4/12, we might generate an equivalent fraction 1/3, or we might generate an equivalent fraction 40/120.) While the standards don’t make an explicit demand that answers to fraction problems be put in lowest terms, teachers are of course free to impose that requirement if they wish. If students have a good understanding of fraction equivalence, and fluency with multiplication and division and knowledge of the times table, then putting fractions into lowest terms is presumably not too much of a problem. But in any case, 4/12 and 1/3 are equally correct ways to express 4 ! 1/12 as a fraction. (J. Zimba, New Twists on an Old Standard) Language for fractions is also more precise. We should be saying “fractions greater than one whole” (not improper fractions), “fractions equal to one whole,” and “fractions less than one whole.” Comparing and Equivalency Both standards for equivalency and comparing fractions references using a visual model. When creating visual models for comparing and equivalency, recognize that comparisons and equivalencies are valid only when the two fractions refer to the same whole.


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Representations: Value vs. location

When you and students represent a fraction on the number line, it is important to represent the length. See picture below. The distance between 0 and 5/3 represents the value 5/3.


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Concepts developed in this unit

! Students develop understanding of fraction equivalence and operations with fractions. They recognize that two different fractions can be equal (e.g., 15/9 = 5/3), and they develop methods for generating and recognizing equivalent fractions.

! Students extend previous understandings about how fractions are built from unit fractions, composing fractions from unit fractions, decomposing fractions into unit fractions, and using the meaning of fractions and the meaning of multiplication to multiply a fraction by a whole number. (MCF 2011 page 43)

Prior knowledge expected 3.NF.1 Students understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b. (3.NF.1). That is, students view fractions as being built out of unit fractions, and they use fractions along with visual fraction models to represent parts of a whole (number line and area model). 3.NF.2. Understand a fraction as a number on the number line; represent fractions on a number line diagram. a. Represent a fraction 1/b 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 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line. b. Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line. 3.NF.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., 1/2 = 2/4, 4/6 = 2/3. 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 = 3/1; recognize that 6/1 = 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.




3.NF.1 3.NF.2a 3.NF.2b

Note: These are grade 3 standards. This should be a review. • Identify numerator and denominator.

2 days

Days 1-2 U6 1.1: Fractions of an Area: Halves, Fourths, and Eighths Rather than using the 4 x 6 arrays, use the blank rectangles found in the Fraction and Decimal Squares packet posted on mybps.org. Lesson can be


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3.NF.3a 3.NF.3b 3.NF.3c

• Use correct fraction notation for the parts they identify.

• Represent fractions less than 1 in an area model. Academic language: fraction, numerator, denominator, equivalent, equivalence, half, halves, fourth, fourths, third, thirds, sixth, sixths

taught as structured, just refer to rectangle rather than 4x6 arrays. See additional teaching notes for this lesson in the Fraction and Decimal Squares packet posted on mybps.org. U6 1.2: Fractions of an Area: Thirds and Sixths Continue to use the same rectangle sheet 1a. During the discussion at the end of the lesson (“How are Thirds and Sixths related?”), continue to refer to the visual model to highlight equivalency. Ask students, “Looking at 1/3 = 2/6, what happens to the number of pieces and the size of the pieces?” This conversation on equivalency is going to be the first of many in this unit. Equivalency is a critical area of focus in grade 4. OMIT U6 1.3: Fractions of Groups of Things OMIT U6 1.4: Same Parts, Different Wholes OMIT U6 1.5: Assessment: Identifying and Comparing Fractions OMIT U6 1.6: Combinations That Equal 1 OMIT U6 1.7: Adding Fractions

4.NF.1 4.NF.3a 4.NF.3b 4.NF.3d 4.NF.4a

• Represent fractions less than 1, equal to 1, and greater than 1 in an area model and on a number line.

• Describe the relationship between fractions in a visual model.

• Identify equivalent fractions and use a visual model to justify equivalence.

• Decompose a fraction into a sum of fractions with the same denominator in more than one way with an equation.

Academic language: equation, number line, partition, location (on the number line), length (on the number line), interval (on the number line), number of equal parts, size of equal parts

5 days Day 1 U6 2.1 and 2.2: Fraction Cards Modification of launch p. 69 Teacher’s Guide. Use blank rectangles for the launch. 1. “Let’s say that these rectangles are sandwiches, and I ate 3/2 of a sandwich. How could you show 3/2 of a sandwich?” 2. Ask students how they would represent 3/2. 3. Give students time to represent the number. 4. Bring class together and share different representations. Use the conversation on p. 70 as a guide. 5. Ask students, “Is there another way to write 3/2?” Encourage students to justify their thinking and reasoning with the representation. 6. Ask students how you might write 3/2 or 1 1⁄2 as an addition equation. Record equations. (For example:1⁄2 + 1⁄2 + 1⁄2 = 3/2; 1+1⁄2 = 1 1⁄2) Making Fraction Cards " See p. 71 in Teacher’s Guide for a suggestion on how to launch making fraction cards. "Please note: In addition to representing the fraction in an area model, students will decompose the fraction into the sum of fractions with the same denominator. Decomposing the fraction into a variety of sums helps students understand the fractional amount. This can be done as shown on the fraction card examples as a string of equivalent expressions or as separate equations.


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" On the first day students will use fraction squares (M16). See sample cards below.

Day 2 Introduce the number line. " Ask students how they would show 1 1⁄2 on the number line. Encourage students to use what they know from the previous day’s discussion to help them with this task. "Students will paste the number line on the back of the card and represent the fraction on the number line too. See sample cards below. " Students will continue to record equations for each fraction.

Note: Not every student has to make every fraction card. All students should have experience creating some fractions less than one and some fractions greater than one using an area model and the number line. All students should have experience creating equations for their representations. Days 3-5 Fraction Card Workshop


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1. Continue making Fraction Cards, with area models and number lines. 2. Introduce Adding and Subtracting Fraction Story Problems found in the Fraction and Decimal Squares packet posted on mybps.org. As students continue to work on the fraction cards, introduce addition and subtraction story problems as a part of workshop. Encourage students to use what they know about fractions to add and to subtract. Students should use visual fraction models (number line or area model) and equations to represent the problems. This packet should be printed before the lesson. Note: Adding Fractions and Subtracting Fractions: Do not use Resource Masters C29, C30. Instead use the word problems in the grade 4 Fractions and Decimal Squares packet posted on mybps.org. Students should be using strategies such as adding by place -- add the whole numbers first, then add the fractions, adding on -- start with the subtrahend and add on to get to the minuend, maintain the difference by making an easier problem to solve, for example 3 # - 1 $, may be solved by creating an equivalent equation: (3 # + #) - (1 $ + #) = 3 % - 2. Note: Students are not required to know the words subtrahend and minuend, although it makes discussing a subtraction problem easier when one knows how to refer to the two numbers when describing a strategy to solve a subtraction problem. In subtraction the minuend is the number you subtract from and the subtrahend is the number being subtracted. POSSIBLE SHARE CONVERSATIONS During the 5 days Given the needs of your students, use the share at the end of each of the five days (the end of the lesson) as an opportunity to highlight mathematical ideas you notice your students thinking about, struggling with, connections between representations, and begin to make generalizations. Here are some possible ideas for the share: 1. Which fractions were easy to make? Why? Which were challenging? Why? What suggestions do you have for your friends as we continue to work on the cards tomorrow? 2. What helped you write equations for your fraction cards? Let’s look at Derek’s card for 3⁄8. Does 3⁄8=2/8+1⁄8? Does 3⁄8=1⁄8+1⁄8+1⁄8? Is there a way we can use multiplication to write 3⁄8= 1⁄8 + 1⁄8 + 1⁄8? Where do you see each equation in the diagram? 3. Does 1 3⁄4 = 1 + 3/4? How do you know? What is another way we can write 1 3⁄4? 4. Does 3⁄4 = 6/8? Where can we see that in the rectangle? Where do you see that in the number line? What is happening to the number of pieces and the size of each piece in both visual models? Do you think this is true for other equivalent fractions? 5. What do you notice about all of these fractions? 5/3, 9/4, 5/2, 8/6, 9/6, 5/4, 6/3, 3/2, 4/2? Is there another way to write these fractions as a mixed number?


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6. What do you notice about 3/3 and 8/8? Are they equivalent? How do you know? How can you prove that with the representations you created? Are there other fractions that are equal to 3/3 and 8/8? What do we notice about all of these fractions and what can we say is true about all of these fractions? For intervention and extension suggestions see page CC50 in the Gr 4 CCSS Guide book.

4.NF 1 4.NF 2

• Compare fractions by reasoning about fraction equivalencies and other fraction relationships they know. (For example, comparing to benchmark fractions, creating common denominators, and/or numerators.)

• Record comparisons using <, >, or = • Justify conclusion/ easoning with a visual model • Recognize fractions equivalent to 0,1, and 2 • Compare fractions to 1⁄2 • Order fractions using benchmarks (landmarks) and

equivalencies and reasoning about a fraction’s relationship to benchmarks (landmarks)

Academic language: benchmark, landmark, is greater than, is less than, is equal or equivalent (use symbols), number of equal parts, size of equal parts, equivalent fraction Students often refer separately to the size of the equal parts and the number of equal parts. However, they must think about and examine both the size and the number of equal parts together, as a relationship. For

example, we might hear students say that

!

56

is greater

than

!

68

because sixths are larger equal parts than

eighths, but we must also consider the number of equal

parts. In this example the student would be correct,

!

56

6 days

Days 1-2 U6 2.3: Capture Fractions (2 days and you will revisit Capture Fractions in session 2.5) Introduce game. Use session 2.3 as a guide. " When playing Capture Fractions it might help to use the pre-made cards on mybps.org, Elementary Math, grade 4. " It is important to move students away from just relying on the visual representation when comparing fractions. The 4th grade standards require reasoning and connecting the reasoning to a visual model. (For example when comparing " and ! they should notice that " is less than % while ! is only " away from 1 whole.) Students use the recording sheet included in the Fractions and Decimal Squares packet posted on mybps.org. " Before class ends, ask students to select one pair fractions from their recording sheet and respond to the exit ticket task individually. See attached exit ticket in the Fractions and Decimal Squares packet posted on mybps.org. Use the data collected from the exit ticket to inform your next day’s session. Strategies for comparing fractions. As the students discuss these strategies, create Anchor Charts to post in the classroom so students can refer to them as they play Capture Fractions. See pp. 151-156 and pp. 165-168 in Teacher Guide. 1. When the numerators are the same, compare denominators. The fraction with the larger denominator (the smallest sized pieces) is less than the fraction with the smaller denominator (the larger sized pieces). 2. When the denominators are the same, compare numerators. The fraction with the larger numerator (more pieces) is larger. 3. Compare both fractions to a benchmark such as % or 1 whole. First example, if you have two fractions that are less than % and you look at the missing piece to get %, the fraction with the smaller missing piece is larger than the other one. Second example, if two fractions are missing the same number of pieces less than 1, the fraction with smaller pieces is bigger. Third example, if a fraction has a numerator that is less than the denominator, it is less than one. If a fraction has a numerator that is more than the denominator, it is greater than one. The fraction greater than 1 is the larger fraction.


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is greater than

!

68

, however the student’s reasoning is

not correct.

(Day 2 of Capture Fractions) HIGHLIGHT EQUIVALENCY Day 3 Finding Equivalent Fractions: Refer to the packet entitled, Fractions and Decimal Squares packet posted on mybps.org. See Finding Equivalent Fractions in the packet. Day 4 U6 2.4: Comparing Fractions to Landmarks Follow session 2.4 as a guide. Use this task as an opportunity to reinforce the following comparing strategies you noted as students played Capture Fractions 1. Compare to a benchmark (1/2, 1, 2) 2. Use fraction equivalence Day 5 U6 2.5: Fractions on a Number Line (Ordering fractions) Follow session 2.5 as a guide. Important note - Although this lesson is named fractions on the number line, it is more about ordering fractions. Exact precision is not the goal. Students should be able to place fractions relative to the approximate location. The expectation of precision has to do with the precision of language students use to justify why they placed the fraction where they did. For example, “3/8 is less than ! because 3/8 + 1/8 = !. 3/8 is greater than " because "+ 1/8= 3/8. So, 3/8 is 1/8 away from both " and !. I know that 3/8 is halfway between " and !.” - When students play Capture Fraction, the goal is for the majority of students should be reasoning which fraction is larger and not using a visual model to determine which fraction is larger. Students who do reason should be able to connect their reasoning back to a visual model. Day 6 U6 2.6: Assessment: Comparing Fractions: See the Comparing Fractions Assessment in the in the Fractions and Decimal Squares packet posted on mybps.org. Follow Session 2.6 as a guide

4.NF.1 4.NF.3a 4.NF.3b 4.NF.3c 4.NF.3d

• Add and subtract fractions by using what they know about fractions, the relationship between addition and subtraction, and the properties of the operations

• Solve word problems using equations and visual

4 days

Days 1-2 Gr 4 CCSS Guide U6 2.7A: Adding and Subtracting Mixed Numbers Encourage discussion about the connection between a line plot and a number line.


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4.MD.4

models • Represent solution on a visual model • Apply and extend what students know about

adding and subtracting with whole number to adding and subtracting fractions.

Academic language: The new standards use more precise language. For instance “improper fractions” and “proper fractions” are no longer used. We now use: [the fraction is] greater than one, less than one, equal or equivalent to one. • Create a line plot using a set of data that has

fractional units. • Answer interpretive questions using information

from a line plot, such as what is the difference between the longest and shortest data points?

• Create and answer word problems with line plots. Academic language: line plot, data points, equation, equivalent, variable (using a letter for an unknown), [fraction that is] greater than one, [fraction that is] less than one, [fraction that is] equivalent or equal to one

Using data from a line plot to add and subtract fractions will be the focus on day 3 and possibly day 4. Given the needs of your students, use the share (the end of each lesson) as an opportunity to highlight mathematical ideas you notice your students thinking about, struggling with, connections between representations, and highlight ways students are applying and extending previous understandings of operations on whole numbers.

1. Examine different strategies students used for comparing the length of the Giant Swallowtail and the Tiger (question #2). When examining strategies, include representations as part of the examination to ground the conversation and highlight the properties of the operations and the relationship between addition and subtraction.

For example:

3 $ + ? = 5 # (Adding up) 5 #- 3 $ = 5 # - 3 – # - % (Keeping on number whole and subtracting back in parts) 5 # (+ #) – 3 $ (+ #)= 5 %- 4 (Creating an equivalent problem)

2. When combining fractions, you might notice that some students add 1 5/8 + 2 7/8 and get the sums 3 12/8, 4 4/8 and 4 1/2. If this is the case, this might be an interesting thing to bring to the whole group share to highlight equivalency. For example: Does 1 5/8 + 2 7/8 = 3 12/8? Or 1 5/8 + 2 7/8 = 4 4/8? Or 1 5/8 + 2 7/8 = 4 %? How does the representation support your reasoning and logic? Does 3 12/8 = 4 4/8 = 4 %? Day 3 Data on a Number Line Adding and Subtracting Mixed Numbers See problems in Fraction and Decimal Square packet posted on the elementary grade 4 page of mybps.org. Day 4 Workshop Students can work on: Center 1. Adding and Subtracting Mixed Numbers using Information on Line Plots Center 2. Working with equivalent fractions or Play Fraction Compare


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Center 3. Solving fraction word problems See supporting work in the in the Fractions and Decimal Squares packet posted on mybps.org.

4.NF.4a 4.NF.4b 4.NF.4c

• Keep track of the number of groups and the size of each group in a visual model.

• Record a multiplication equation that represents the problem.

• Apply and extend what students know about multiplying with whole number to multiplying with a fraction.

Academic language: equation, equivalent, variable (using a letter for an unknown), [fraction that is] greater than one, [fraction that is] less than one, [fraction that is] equivalent or equal to one,

3 days Days 1-2 U6 3A.1: Multiplying Whole Numbers and Fractions Investigations and the Common Core Follow Session 3A.1 as a guide Omit Gr 4 CCSS Guide U6 3A.2 Day 3 Gr 4 CCSS Guide U6 3A.3: Investigations and the Common Core Assessment: Multiplying with Fractions Follow Session 3A.3 as a guide The following problems do not reflect a Grade 4 Standard. Ignore these problems or reinterpret the multiplication problem in the following: C40: #3, reinterpret #4 as 4 groups of 7/10, reinterpret #6 as 12 groups of 5/6 Assessment #1 Daily Practice C42: #2 Daily Practice C45: #5, #6 Daily Practice C46: #1

4.NF.5 4.NF.6 4.NF.7 4.MD.2

• Read and write tenths and hundredths. • Represent decimals on a 10 x 10 grid and a

number line. • Express decimals as equivalent fractions with 10 or

100 as a denominator. • Express fractions with denominators 10 or 100 in

decimal notation. • Compare two decimal numbers by reasoning

about their size and justify conclusion by a visual model.

• Use decimal equivalence to combine decimals. • Solve word problems and represent solution on the

number line. Academic language: equation, variable (using a letter for an unknown), tenths, hundredths, benchmark, landmark, is greater than, is less than, is equal or equivalent (use symbols), number line, intervals [on the number line, i.e., number

10 days

Days 1-2 U6 3.1 Representing Decimals Follow Session 3.1 as a guide " In addition to the Decimal grids as a visual model use the posted number lines on mybps.org as a model too. " As students work on the activity Decimals on the 10 x 10 square, students should also represent the decimal on the number line. It is not a grade 4 expectation for students to understand thousandths. Examining thousandths provides an opportunity to illuminate the structure of our number system and the powers of 10. Students will continue to explore these ideas in Grade 5. " In addition to writing the decimal that they are representing under each representation students should record decimals as equivalent fractions with 10 or 100 as a denominator. Optional Task from Illustrative Mathematics: http://www.illustrativemathematics.org/illustrations/103


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line is intervals of tenths], reading decimal numbers correctly: 0.25 is twenty-five hundredths NOT “zero point two five”

Day 3 U6 3.2: Comparing Decimals Follow Session 3.2 as a guide. Use Comparing Decimal recording sheet on mybps.org. Below are some items to pay attention to. " As students are playing, ensure that students are reading decimals. For example, 0.25 is twenty-five hundredths NOT “zero point two five”. " Students may use the visual models in different ways. Some students may need the visual model to determine which decimal is larger. If this is true, support the student in beginning to reason about the meaning of the decimal numbers. Others may reason about the size of each decimal to determine which decimal is larger. If this is true, encourage the student to justify this reasoning on a number line or 10 x 10 grid. Days 4-5 U6 3.3: Representing and Combining Decimals Follow Session 3.3 as a guide. Below are some suggestions to make the lesson even stronger. " After posing the question, “Is the following statement true: 3/10, 0.3 and 30/100 are equal.” Make sure that you look at both visual models: the number line and the 10 x 10 grid. As you look at both visual models, use the following questions to examine decimal equivalency. 1. What happens to the number of pieces and the size of each piece? 2. What is true about the area (10x10 grid) or length (number line) of each representation? Why do you think that is true? 3. In what ways is this similar to equivalent fractions? " Just like when students played Capture Fractions, use the attached recording sheet. Attached also is an exit ticket. The data collected from the exit ticket will give you valuable information to inform your next lesson.

Day 6 U6 3.4: Estimating and Adding Miles and Tenths of a Mile Follow Session 3.4 as a guide


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" Students should use visual models to help them combine decimals. " You will have to extend the number line diagram. " When recording equations for, record equations both as decimals and fractions. Days 7-8 U6 3.5: Comparing and Combining Decimals Follow Session 3.5 as a guide Day 9-10 U6 3.6: Comparing and Combining Decimals Follow Session 3.6 as a guide



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UNIT OF STUDY 3: ADDITION, SUBTRACTION, AND THE NUMBER SYSTEM Primary Curricular Resources: Grade 4, Investigations Unit 5: Landmarks and Large Numbers and Grade 5, Investigations Unit 3: Thousands of Miles, Thousands of Seats Strengthening Computational Fluency in Addition and Subtraction: Consolidating Strategies Estimated Instructional Time: 25 days End of Unit Assessment: February 26th January 15th - February 27th

Overarching Questions: • How does the value of a digit change as it moves one place to the left or one

place to the right? (Be explicit and use language: ten times greater, ten times less)

• How/why are landmark numbers useful? • How can place value be useful when adding or subtracting? • When adding or subtracting by a multiple of 10, 100 or 1,000 what digits

change and why? • How are adding and subtracting using place value strategies related to using

the standard algorithm for adding and subtracting? • When using an algorithm to add or subtract, how does composing and

decomposing base-ten units help you solve the problem? • How does expanded notation help you understand place value? • What is the nth number in any given pattern? • How can understanding of place value help round multi-digit numbers to any

place? • How does using representations, such as a number line, help you understand

addition and subtraction? • When finding the perimeter of a shape, why do we add the dimensions of the

shape?

Standards for Mathematical Practice Focus MP2 Reason abstractly and quantitatively. Represent situations by de-contextualizing tasks into numbers and symbols (story problem --> equation) MP5 Use appropriate tools strategically. Gain experience competence with a specific tool to recognize the differential power it offers. Consider the available tools (including estimation) when solving a mathematical problem and decide when certain tools might be helpful. MP7 Look for and make use of structure. Students use what they know to solve problems they don’t know.

Instructional Notes:

! Rounding is an important strategy for estimation. Students will round as they estimate throughout the year.

! Have students write the number in expanded form after they practice saying or writing the number to a partner.

! Compare two multi-digit numbers based on meanings of the digits in each place (place value), using >, =, or < symbols to record the results of the comparison, including larger numbers.

! Be explicit with visual representations that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right and ten times less than the digit to its left. For example 700 ÷ 70 = 10 by applying concepts of place value and division.

! Students need to practice writing numbers in expanded form (e.g. 3765 as 3000 + 700 + 60 + 5 and (1000 x 3) + (100 x 7) + (10 x 6) + (1 x 5)


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! Grade 4 expectations in this domain are limited to whole numbers less than or equal to 1,000,000.

! In Investigation 3, Working with Numbers to 10,000, use this as an opportunity to continue work with the US Standard Algorithm for addition

! Number patterns will be covered during this unit. Shape patterns will be covered in the geometry unit.

! We will be using Grade 5 Unit 3, Thousand of Miles, Thousands of Seats for an assessment (session 1.2 ) of numbers to 10,000 as well as lessons on addition and subtraction (session 3.2, 3.3 and 3.4).

! Grade 5 will no longer be teaching from Unit 3, Thousand of Miles, Thousands of Seats. Feel free to use any activity pages during this grade 4 unit of study.

! Consolidating Computational Fluency in Addition and Subtraction can be found on mybps.com, Elementary Mathematics, Grade 4.

Concepts developed in this unit: • In any multi-digit whole number, recognizes a digit in one place represents 10

times as much as it represents in the place to its right. • Reads, writes, and compares multi-digit whole numbers using base-10

numerals, number names in expanded form and inequality symbols (>, <, =) up to 1,000,000.

• Rounds any number to any place and chooses appropriate context given a rounded number.

• Represents addition and subtraction on a number line. • Represents the problem using equations with a letter standing for the unknown

quantity. • Fluently (accurately and quickly) adds and subtracts multi-digit whole

numbers (up to and including 4-digit numbers) using the standard algorithm, by composing and decomposing base ten units. Expanded notation supports this work. (See session 2.4 in grade 5 unit, Thousands of Miles, Thousands of Seats)

Prior knowledge expected 3.NBT.1 Use place value understanding to round whole numbers to the nearest 10 or 100. 3.NBT.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. From Grade 2: 2.NBT.3 Read and write numbers to 1000 using base-ten numerals, number names, and expanded form. 2.NBT.4 Compare two three-digit numbers based on meanings of the hundreds, tens and ones digits, using greater than, equal to, and less than symbols to record the results of the comparisons. 2.NBT.7 Add and subtract within 1000, 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. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds. 2.NBT.8 Mentally add 10 or 100 to a given number 100–900, and mentally subtract 10 or 100 from a given number 100–900.

Learning Outcomes Number and Operations in Base Ten 4.NBT Generalize place value understanding for multi-digit whole numbers.

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 ÷ 70 = 10 by applying concepts of place value and division.

2. Read and write multi-digit whole numbers using 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.

3. Use place value understanding to round any multi-digit whole number to any place. Use place value understanding and properties of operations to perform multi-digit arithmetic. 4. Fluently add and subtract multi-digit whole numbers using the standard algorithm.

Generate and analyze patterns. 5. Generate a number [or shape] pattern that follows 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 in this way.


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After completing each investigation, students will be able to: Days Primary Curriculum Resource

4.NBT.1 4.NBT.2 4.NBT.3 4.OA.3 4.MD.3

! Students have at least one efficient and accurate addition strategy from grade 3, that they can explain.

! Students use their chosen strategy efficiently and accurately combining larger “chunks” of numbers, and using place value, with a minimum number of steps.

! Students can name the place value of the digits in a number and the value of the digit in the number.

! Mentally solve starter problems or break apart numbers.

! Follow through one of the starts to solve the final problem.

! Round numbers to the nearest ten and nearest hundred.

! Write numbers to 1,000 in expanded form.

! Use >, =, and < symbols to compare numbers to 1,000.

! Be able to describe the steps to solve the problems using academic language.

Academic language: place value, hundreds place, tens place, ones place, rounding, round to the nearest 10, rounding to the nearest 100, expanded form, less than, greater than (use symbols), variable (using a letter for an unknown).

5 days

Gr4 U5: Landmarks and Large Numbers OMIT: U5 Investigation 1 This is grade 3 work. May use the session 1.5 Assessment: Numbers to 1,000 as an optional pre-assessment. Use a number such as 600 or 699 to see if students understand larger numbers surrounding century numbers. This should be a fifteen-minute pre-assessment. Optional: use any Homework or Daily Practice Pages from session 1 as homework in the next few days. Optional Quiz: See Grade 4, Differentiation and Intervention Guide, Unit 5, p 43. This could also be used as a pre-assessment or homework. Days 1-5 U5 2.1: Solving Addition Problems U5 2.2: Addition Strategies Students need to be proficient with the US Standard algorithm by the end of grade 4 so focus on place value strategies in preparation for the work with the US standard algorithm. OMIT: U5 2.3: Starter Problems Students need to be proficient with the US Standard algorithm by the end of grade 4. U5 2.4 Studying the U.S. Algorithm for Addition U5 Gr 4 CCSS Guide 1.5A Place Value Understanding Optional Quiz: See Grade 4, Differentiation and Intervention Guide, Unit 5, p 62. Day 5: Problem Solving


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Solve multi-operation problems that include perimeter and measurement problems. These are posted to mybps.org. The following task is ordering larger whole numbers: http://www.illustrativemathematics.org/illustrations/459 The task includes three-digit numbers with large hundreds digits and four-digit numbers with small thousands digits so that students must infer the presence of a 0 in the thousands place in order to compare. It also includes numbers with strategically placed zeros and an unusual request to order them from greatest to least in addition to the more traditional least to greatest.

4.NBT.1 4.NBT.2 4.NBT.3

! Students read, write, and sequence numbers to 10,000 and 100,000.

! Students understand the place value relationships between 10, 100, 1,000, and 10,000.

! Adding and subtracting multiples of 100 and 1,000.

! Finding the difference between a number and 10,000.

! Finding combinations of 3-digit numbers that add to 1,000.

! Using story contexts and representations, such as number lines, to explain and justify solutions to subtraction problems.

Academic language: value of the digit, thousand, thousands place, ten thousand, ten thousand place, hundred thousand, hundred thousand place, million, millions place, multiples of 100, multiples of 1,000, variable (using a letter for an unknown), greater than, less than (use symbols).

4 days Gr5 U3: Thousands of Miles, Thousands of Seats

1.1 Working with the 10,000 Chart

You will need the 10,000 chart from the Grade 5 curriculum. If one is not

available, you will need to create it using the hundreds sheets. Instead of

students making individual books, this lesson will be done with the whole

class using the large chart. Post the chart so that students can refer back to

place value strategies. 1.2 Assessment: Numbers on the 10,000 chart Assess all students and record information on Assessment Checklist. Note: Use the first half of the Assessment Checklist, Numbers to 100,000 to record students’ response. This is in Resource Masters, Unit 3 M3. You will use the second half of the checklist when you assess students’ ability to read, write and sequence numbers to 100,000 in Session 3.3. 1.3 How Many Steps to 10,000? In notating problems, use a letter for a variable whenever possible. Students are may solve some of these problems by adding up. For instance, if students are adding up you might write: 1,025 + N = 10,000. If other students are using subtraction, you might write: 10,000 - Z = 1,025. typically we do not use X as a letter for a variable in this grade level because we use this to notate multiplication. Gr 4 CCSS Guide U5 3.6A Larger Place Values Rounding Task: Use this for an exit task or a quiz or homework http://www.illustrativemathematics.org/illustrations/745 Use the comment section of this task to help the summary in the whole group discussion.


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4.NBT.1 4.NBT.2 Preparing for 4.NBT.4 4.OA.3

! Solving addition and subtraction problems with large numbers by focusing on the place value of the digits.

! Finding the difference between a number and 10,000.

! Finding combinations of 3-digit numbers that add to 1,000. Academic language: value of the digit, million, ten million, hundred million, billion, ten billion, hundred billion, trillion

2 days Gr5 U3: Thousands of Miles, Thousands of Seats 1.4 Adding and Subtracting Large Numbers Omit Activity 1: TMM Introducing Estimation and Number Sense: Closest Estimate 1.5 Adding and Subtracting Large Numbers

4.NBT.2 4.NBT.4 4.OA.3

! Solve whole number addition and subtraction problems accurately and efficiently using the U.S. standard algorithm.

! Understanding the meaning and notation of the U.S. Standard Algorithm for subtraction.

! Using clear and concise notation for recording addition and subtraction strategies.

Academic language: expanded form, standard form, value of the digit, all place value names

3 days Gr 4 CCSS Guide 4.4A Studying the U.S. Standard Algorithm for Subtraction Read Math Note 2 and 3 on p CC41. When writing the numbers of a problem out, call student’s attention to what expanded form means. That is, say, 200 + 80 + 3 or 200 + 80 + 3 is the expanded form for the number 283. When we write 283 in standard form, it is 283. Optional Quiz: See Grade 4, Differentiation and Intervention Guide, Unit 5, p 66 and 70. Gr5 Unit 3: Thousands of Miles, Thousands of Seats Omit 2.1 Naming Subtraction Strategies Omit 2.2 Practicing Subtraction Omit 2.3 Subtraction Starter problems Alternative strategies for subtraction are part of the mental math strategies in Number Talks. The goal for grade 4 is for students to understand and use the US standard algorithm accurately and efficiently. 2.4 Studying the U.S. Algorithm Help students make the connection between subtracting by place and the US Algorithm. 2.5 Assessment: Subtraction Problems Consolidating Strategies.... Assessment, select appropriate problems to meet your students’ needs from Strengthening Computational Fluency in Addition and Subtraction: Consolidating Strategies. Grade 4: Strengthening the Transition to the MCF 2011 EXAMPLES 1-6 are addition.

4.NBT.2 ! Solve addition and subtraction problems with large numbers by focusing on place value of the digits

5 days Gr5 Unit 3: Thousands of Miles, Thousands of Seats


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4.NBT.4 ! Adding and subtracting multiples of 100 and 1,000 ! Solve addition and subtraction problems with large

numbers by focusing on the place value of the digits and using the U.S. standard algorithm

! Interpreting and solving multi-step problems ! Solving whole-number addition and subtraction

problems efficiently ! Using clear and concise notation for recording addition

and subtraction strategies and the U.S. standard algorithm

! Interpreting and solving multi-step problems ! Reading, writing, and sequencing numbers to 10,000 and

100,000 ! Use standard US algorithm to add and subtract. Academic language: expanded form, standard form, value of the digit, all place value names

3.1 Assessment: Division Facts and Close to 7,500 Do Activity 1 and 2: Close to 7,500 and Comparing Strategies Omit Division Facts. Assessment on Division Facts could be used for homework. 3.2 Stadium Data Students may use any strategy and also need to use the US standard algorithm to continue to gain familiarity and become proficient with the US standard algorithm. 3.3 Assessment: Numbers to 100,000 and Rock On! Students may use any strategy and must also use the US standard algorithm to continue to become proficient with the US standard algorithm. 3.4 Rock On! Students may use any strategy and also need to use the U.S. standard algorithm to continue to gain familiarity and become proficient with the U.S. standard algorithm. One extra day to continue to work on becoming proficient with US standard algorithms for addition and subtraction.



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UNIT OF STUDY 4: GEOMETRY, GEOMETRIC MEASUREMENT, AND MEASUREMENT Primary Curricular Resources: Grade 4, Investigations, Unit 4: Size, Shape, and Symmetry Grade 5, Investigations, Unit 5: Measuring Polygons Estimated Instructional Time: 33 days End of Unit Assessment: April 16th February 28th – April 17th

Overarching Questions: • What are different geometric attributes we can use to classify two-

dimensional shapes? • What can you use about angles measures you know to help you find angle

measures you don’t know? • Describe how the types of quadrilaterals you have learned about are different

and how they are the same. • What strategies help you figure out the measurement of an angle? • What is an angle? • What is the building block/pattern unit of this pattern? • What is perimeter? How do you find the perimeter of a shape? • What is area? How do you find the area of a shape? • Perimeter is measured in linear units; area is measured in square units. How are

these two measures different? • How can you use the four operations to solve real world problems involving

various measurements? • How can you express the same measurement with different units within the

same system of units? • Given 2 dimensions, can students distinguish how to use them to apply area

and perimeter formula in rectangles and real-world situations?

Standards for Mathematical Practice Focus MP3 Construct viable arguments and critique the reasoning of others. Throughout this unit there are ongoing opportunities for students to express and defend mathematical arguments (Guess My Rule). MP5 Use appropriate tools strategically. Students use mathematical tools to foster mathematical understanding (Power Polygons, angle rulers, protractors) MP6 Attend to precision. Mathematically proficient students communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning (Guess My Rule) MP4 Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. MP6 Attend to precision. Mathematically proficient students are careful about specifying units of measure. MP7 Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure.

Instructional Notes: • Omit Investigation 1 because 4th grade students are not required to measure objects. • Omit Investigation 4 because it does not fit the grade 4 standards. • Throughout teaching Investigation 3 teach angle measurement within the context of a circle. The angle ruler helps students understand angles in the context of a circle. Refer to standard 4.MD.5 See Note about angle measurement and labeling angles. • TMM Quick Images: 2-D - Students visualize and analyze images of 2-D geometric figures. After briefly viewing an image of a 2-D design, students draw it from the mental image they formed during their brief viewing. Focus for this should be on parallel lines, perpendicular lines, or right triangles in the images. (Implementing Investigations Grade 4 pp. 28 - 30)


35

• For “Guess My Rule,” students should use parallel and perpendicular lines and right triangles as rules. • When students are creating angles in Investigation 3 make sure they are writing equations. See teaching notes on pp. CC13 - CC14 in the Investigations and Common Core Standards book. • Refer to Dialogue Box - Building Angles p.167 • Refer to Teacher Note - Classification of Quadrilaterals p 155 • This builds towards fifth grade students being able to classify two dimensional figures in a hierarchy based on properties (For example, scalene, isosceles, etc.) Students in grade 4 begin using the four operations to solve real world problems involving measurement quantities such as liquid volume, mass and time with whole numbers, fractions and decimals. • Teacher will select Workshop materials from BPS TMM Unit and Unit Conversion and Measuring Attributes • LogoPaths is an optional possibility to provide students with the opportunity to determine the perimeter of completed figures using known and unknown dimensions. (Optional) U4 1.3 Activity 2, 1.3. Math Workshop 3B, 1.4 Math Workshop 1B, 1.5 Math Workshop 1B (4.MD.3) Teacher Note: Metric and U.S. Standard Measures p 149 Student Math Handbook Perimeter p.104-105 Area p 114

Concepts developed in this unit: • Students describe, analyze, compare, and classify two-dimensional shapes by

their properties including explicit use of angle sizes, and the related geometric properties of perpendicularity and parallelism.

• Students use side length to classify triangles as equilateral, equiangular, isosceles, or scalene; and can use angle size to classify them as acute, right, or obtuse.

• Students learn to cross-classify, for example, naming a shape as a right isosceles triangle.

• Through building, drawing, and analyzing two-dimensional shapes, students deepen their understanding of properties of two-dimensional shapes and the use of them to solve problems involving symmetry.

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

• Selecting area or perimeter formula to solve problems. • Students learn to use and apply area and perimeter formula to solve real-

world and mathematical problems. • Students use appropriate measuring tools for linear measurement. • Students understand the concepts of angle and measure angles. • Students understand that an angle that turns through 1/360 of a circle is called

a “one-degree” angle and degrees are the unit to measure angles.

Prior knowledge expected 2.G.3 Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape. 3.MD.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. 3.MD.6 Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units). 3.MD.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 ´ b and a ´ d. Use area models to represent the distributive property in mathematical reasoning. e. Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-


36

overlapping parts, applying this technique to solve real-world problems. 3.G.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. 3.G.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 areas and describe the area of each part as " of the area of the shape. 3.NF.1 Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.

Learning Outcomes Generate and analyze patterns 4.OA 5. Generate a number or shape pattern that follows 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 in this way. In this unit of study the focus will be shapes. Measurement and Data 4.MD Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.

1. Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller 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), ...

2. Use the four operations to solve word problems 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.

3. Apply the area and perimeter formulas for rectangles 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.

Geometric Measurement: Understand concepts of angle and measure angles. 5. Recognize angles as geometric shapes that are 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 the circular arc between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a “one-degree angle,” and can be used to measure angles.

b. An angle that turns through n one-degree angles is said to have an angle measure of n degrees. 6. Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure. 7. Recognize angle measure as additive. When an 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. Geometry 4.G Draw and identify lines and angles, and classify shapes by properties of their lines and angles.


37

1. Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures. 2. Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size.

Recognize right triangles as a category, and identify right triangles. 3. Recognize a line of symmetry for 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.




4.MD.3 4.NBT.4 4.NF.3a

• Review what linear measurement is and the tools used to measure length.

• Find objects that equal several measurement units (1 centimeter, 1 inch, 1 foot, 1 yard, 1 meter) and then use these benchmarks to estimate lengths inside and outside their classroom.

• Use measurement tools to measure the lengths they previously estimated with benchmarks.

• Measure the perimeter of objects in the classroom.

• Apply the perimeter formula for rectangles in real-world and mathematical problems.

• Solve problems involving area and perimeter

• Apply the area and perimeter formula

• Explain why the formulas work Academic language: length, linear measurement, perimeter, inch, foot, yard, centimeter, meter, benchmark, area, volume

5 days Gr4 U4 Size, Shape, and Symmetry Days 1-5 Day 1 U4 1.1 Measurement Benchmarks Just do Activity 1. Omit Activity 2 and 3 Day 2 U4 1.2 Measurement Tools Day 3 U4 1.3 Assessment: How Long is Our Classroom? This lesson will focus on perimeter. Do Measuring Perimeter Do SAB p 14 for Perimeter Problems (Daily Practice) Omit 3B if you do not have computers to do LogoPaths. Omit 3C OMIT U4 1.4 and 1.5 Days 4 and 5 ! While students are expected to use formulas to calculate area and perimeter of rectangles, they need to understand and be able to communicate their understanding of why the formulas work. ! Formula for area is l x w and the answer will always be in square units. ! Formula for perimeter can be 2l + 2w OR 2(l + w) and the answer will always


38

be in linear units. For supporting work: use SAB pp 67 and 68 and p R32, Perimeters of Shapes from Differentiation and Intervention Guide SAB p 74 has students find area and perimeter of rectangles Quiz from Differentiation and Intervention Guide R31 may be used here.

4.G.1 • Categorize shapes as polygons or not polygons based on their attributes.

• Classify polygons by attribute, including number of sides, length of sides, and size of angles

Academic language: quadrilateral (four sided, closed figure), square, rectangle, trapezoid, rhombus, parallelogram, triangle, equilateral triangle, hexagon, concave polygon, interior angles, convex polygon, angles, vertex, vertices. Note: Students may use this academic language from grade 3. They will have additional opportunities to use geometric academic language throughout this unit of study. Teachers also should continue to use geometric academic language throughout this unit.

1 day Day 6 U4 2.1: Is it a Polygon? Refer to p 59 Ongoing Assessment: Observing Students at Work OMIT U4 2.2 Making Polygons

4.G.1 4.G.2

! Identify parallel and perpendicular lines and line segments (a line goes on indefinitely, a line segment is a piece of a line)

! Identify right angles, acute angles, and obtuse angles

! Identify right triangles Academic language: point, line, ray, parallel (parallel lines, parallel line segments, parallel sides) perpendicular (perpendicular -- forming right angles) intersecting, intersecting lines, intersecting line segments, angles, angle size, 90º angle, right angle, less than 90º angle, acute angle, greater than 90º angle, obtuse angle, Note: A ray is a part of a line that has one endpoint and extends indefinitely in one direction. An angle is two rays that share an endpoint.

1 day Day 7 Gr 4 CCSS Guide U4 2.3A: Identifying Geometric Figures Also refer to Differentiation and Intervention Guide for Practice and Intervention, p 48-49.

4.G.1 4.G.2

• Recognize number of sides as a descriptor of various polygons.

3 days Days 8-10


39

• Classify polygons by attribute, including number of sides, length of sides, and size of angles.

• Combine polygons to make new polygons. • Develop vocabulary to describe attributes and

properties of quadrilaterals. • Develop vocabulary to describe attributes and

properties of quadrilaterals. • Understand relationship between squares and

rectangles. Academic language: attribute [of a polygon], property [of a polygon], characteristic [of a polygon], sides, parallel (parallel lines, parallel line segments, parallel sides), a pair of parallel sides, two pairs of parallel sides, perpendicular (perpendicular lines form 90º angles) intersecting lines, intersecting line segments, angles, angle size, 90º angle, right angle, less than 90º angle, acute angle, greater than 90º angle, obtuse angle

U4 2.3: Sorting Polygons (Guess My Rule?) Include Activity 3 - Introducing Names for Polygons with SAB p. 20 from 2.2. Don’t do 3B part of workshop U4 2.4: Sorting Quadrilaterals See teaching notes on parallel and perpendicular lines in Investigations and the Gr 4 CCSS Guide, p CC13. U4 2.5: Assessment: What is a Quadrilateral? See teaching note: Parallel and perpendicular lines in Investigations and the Gr 4 CCSS Guide p CC13. Put constraint on embedded assessment that one shape should have pair of parallel sides and another shape that has two pairs of parallel sides. The language of “a pair of parallel sides” and “two pairs of parallel sides” is important for students to understand and distinguish. Students may also see: “at least one pair of parallel sides” which means the polygon could have one or more pairs of parallel sides.

4.MD.7 4. MD.5a 4. MD.5b 4.NBT.4

! Identify a right angle as 90° (90 degrees). ! Use acute angles to build right angles. ! Using known angles to find the measure of other

angles. ! Use known angles to find the measure of other

angles. Academic language: angle, degrees, parallel lines, perpendicular lines, parallel line segment, perpendicular line segment, right triangles, acute angles, obtuse angles, right angles,

3 days Day 11-13 U4 3.1: Making Right Angles See Teaching Note Equations for Making Right Angles in Gr 4 CCSS Guide, pp CC13. Write addition equations. TMM: Quick Images: 2-D Focus: Ask students about presence of parallel lines, perpendicular lines, right triangles and where they are? Task from Illustrative Mathematics that may be used as additional work or homework: http://www.illustrativemathematics.org/illustrations/1273 U4 3.2: More or Less Than 90 Degrees? See Teaching Note Equations for How Many Degrees? in CCSS pg. CC14 (Write addition equations) TMM: Quick Images: 2-D Focus: Ask students about presence of parallel lines, perpendicular lines, right triangles and where they are? U4 3.3: Assessment: Building Angles See Teaching Note Addition and Subtraction Equations and Equations for Angles That Make 120 Degrees in CCSS pg. 14 TMM: Quick Images: 2-D Focus: Ask students about presence of parallel lines, perpendicular lines, right triangles and where they are?


40

4.G.1 4.G.2 4.MD.7 4. MD.5a 4. MD.5b

! Use a protractor to accurately measure angles ! Understand the relationship between the degree

measure of an angle and circular arcs. Academic language: angle, degrees, parallel lines, perpendicular lines, parallel line segment, perpendicular line segment, right triangles, acute angles, obtuse angles, right angles, angle ruler, protractor, degrees, 360 degrees in a circle

1 day Day 14 U4 3.4A: Lines and Angles TMM: Quick Images: 2-D Focus: Ask students about presence of parallel lines, perpendicular lines, right triangles, acute angles, obtuse angles, and where they are? The angle ruler is designed to help students see the 360 degree rotation that forms angle measures.

4.G.1 4.G.2 4.MD.7 4. MD.5a 4.MD.5b 4.MD.6 4.NBT.4

• Use a protractor to accurately measure angles • Understand the relationship between the degree

measure of an angle and circular arcs. • Label and name angles with an angle symbol and

three letters Academic language: angle ruler, protractor, degrees, 360 degrees in a circle Naming Angles: Angles are named with the angle symbol and three points. The vertex must be the middle point in the angle name. Names for the angles formed by these intersecting rays:

!

"1:

!

"WXZ or

!

"ZXW

!

"2:

!

"YXZ or

!

"ZXY

Names for the angles formed by pentagon ABCDE and diagonal AD: Since only one angle is at each vertex B, C, and E, the single letter of the vertex may be used to name the angle.

!

"1:

!

"EAD or

!

"DAE

1 day Day 15 Measuring Angles Teacher Note: • Now that students have been introduced to the protractor have them go back to previous lesson in this investigation and measure angles with protractors. • Students could also measure power polygon pieces. • Students could also create their own angles with power polygon pieces and then use the protractor to measure those angles. • Guess My Rule cards will be blown up and students can measure angles on these shapes with protractors. • See mybps.org on the elementary math grade 4 page for real world problems involving angle measurement. The following task has notation that is important for grade 4 students to learn. This task labels angles with an angle symbol and three letters, as described in the Instructional Notes. http://www.illustrativemathematics.org/illustrations/1168

!"

#"

$"

%"&"

#$%$

&$

''$

($)$*$

'"("

)" *"

+"

$

,"


41

!

"2:

!

"DAB or

!

"BAD

!

"3:

!

"B or

!

"ABC or

!

"CBA 4: C or BCD or DCB 5: ADC or CDA 6: ADE or EDA 7: E or AED or DEA 1 + 2: EAB or BAE 5 + 6: EDC or CDE

• Identify attributes of polygons and sort polygons by attributes

• Describing triangles by the sizes of their angles and the lengths of their sides

• Using attributes to describe and compare quadrilaterals including parallelograms, rectangles, and rhombuses, and squares

• Identifying attributes of polygons Academic language: right triangle, obtuse triangle, acute triangle, scalene triangle, isosceles triangle, equilateral triangle, quadrilateral, parallel (sides, lines, line segments), trapezoid

3 days Grade 5 U5 Measuring Polygons You will be using the grade 5 Unit 5 book, Measuring Polygons for this set of lessons. Note: You will need Gr4/5 Shape Cards posted on mybps.org elementary math grade 4 page. Students or teacher should prepare these cards before math class. Student pages will also be posted to mybps.org. Days 16-18 1.1 Triangles 1.2 Quadrilaterals Omit Activity 3 if you do not have access to computers to do LogoPaths. 1.3 Relationships Among Quadrilaterals In the math workshop for Activity 1, omit LogoPaths if you do not have a computer.

4.G.3

• Recognize a line of symmetry for a two-dimensional figure as a line across the figure.

• Fold along a line of symmetry into matching parts. • Identify line-symmetric figures and draw lines of

symmetry. Academic language: symmetry, line symmetry, mirror symmetry, (line symmetry and mirror symmetry are the same), line(s) of symmetry, diagonal

3 days Days 19-21 Symmetry Use symmetry materials posted on the elementary page of grade 4 mybps.org and the tasks below. http://www.illustrativemathematics.org/illustrations/676 http://www.illustrativemathematics.org/illustrations/1060 http://www.illustrativemathematics.org/illustrations/1059 http://www.illustrativemathematics.org/illustrations/1058 OMIT All of Investigation 4


42

4.MD.1 4.MD.2 4.NBT.2 4.NBT.4

• Converting measurements in larger units to smaller units

• Making tables of equivalent measurements • Using four operations to solve word problems

involving measurements • Convert measurements in larger units to smaller units • Make tables of equivalent measurements • Apply area and perimeter formula Academic language: weight, mass, capacity, pounds, ounces

5 days Day 22-26 Gr 4 CCSS Guide U7 3.5A Measurement Equivalents Day 6 Follow lesson Gr 4 CCSS Guide U7 3.5A Day 7 Workshop Use Unit and Unit Conversion and Measuring Attributes from BPS TMM Center 1: Measuring Attributes Center 2: Unit and Unit Conversions Center 3: More work with Area and Perimeter (solving problems with missing dimensions) Day 8 and 9 Gr 4 CCSS Guide U7 3.5B Problem Solving Involving Measurements Two days are given for 3.5B Day 9 Workshop: Select materials from this unit to differentiate for students’ needs. Include multi-operation word problems with area and perimeter problems. These are posted on the elementary grade 4 page on mybps.org. Workshop Use Unit and Unit Conversion and Measuring Attributes from BPS TMM Center 1: Measuring Attributes Center 2: Unit and Unit Conversions Center 3: More work with Area and Perimeter (solving problems with missing dimensions) OMIT ALL OF INVESTIGATION 4



43

UNIT OF STUDY 5: CONSOLIDATING FLUENCY WITH MULTIPLICATION AND DIVISION Primary Curricular Resources: Grade 4, Investigations, Unit 8, How Many Packages? How Many Groups? Grade 5 Strengthening Computational Fluency Multiplication and Division Estimated Instructional Time: 33 days End of Unit Assessment: June 16th April 28th –June 19th


! How are the operations of multiplication and division related? ! How can estimating help you make sense of a problem? ! What representations help you make sense of multiplication and division

problems? How are these representations different? How are these representations the same?

! How do I choose efficient strategies for multiplication and division based on the numbers in the problem?

Standards for Mathematical Practice Focus MP2 Reason abstractly and quantitatively. 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 and use appropriate label. MP3 Construct viable arguments and critique the reasoning of others. Students explain their thinking to others and respond to others’ thinking. MP7 Look for and make use of structure. Look closely to discover a pattern or structure. For instance, students use properties of operations to explain calculations (partial products model). MP8 Look for and express regularity in repeated reasoning. Students notice repetitive actions in computation to make generalizations.

Instructional Notes:

! Students are expected to fluently illustrate and explain the calculation using equations, rectangular arrays, and/or area models. ! Students are to make good choices about efficient strategies for multiplying and dividing based on the numbers in the problem. ! Whenever possible write equations using a variable. Alternate the placement of the variable so that it is not always the product or the quotient, for

instance a division problem can be expressed as a missing factor problem using a variable, when appropriate. ! Refer to Instructional Note about “Using the Distributive Property” on page 16 in Unit 8. ! Omit Investigation 3 because the grade 4 MCF 2011 expectation is only 1-digit divisors.

Concepts developed in this unit • Students apply their understanding of models for multiplication, place

value, and properties of operations as they develop, discuss, and use efficient, accurate, and generalizable procedures to multiply up to four

Prior knowledge expected • 3.OA.3 Use multiplication and division within 100 to solve word problems in

situations involving equal groups, arrays, and measurement quantities • 3.OA.5 Apply properties of operations as strategies to multiply and divide.


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digits by one-digit whole numbers and two-digit by two-digit whole numbers.

• Students apply their understanding of models for division, place value, properties of operations, and the relationship of division to multiplication as they develop, discuss, and use efficient, accurate, and generalizable procedures to find quotients involving multi-digit dividends with up to four-digit dividends.

• Students select and accurately apply appropriate methods to estimate and mentally calculate quotients, and interpret remainders based upon the context. (MA frameworks 2011 page 43)

Examples: If 6 x 4 = 24 is known, then 4 x 6 = 24 is also known. (Commutative property of multiplication.) 3 x 5 x 2 can be found by 3 x 5 = 15 then 15 x 2 = 30, or by 5 x 2 = 10 then 3 x 10 = 30. (Associative property of multiplication.) Knowing that 8 x 5 = 40 and 8 x 2 = 16, one can find 8 x 7 as 8 x (5 + 2) = (8 x 5) + (8 x 2) = 40 + 16 = 56. (Distributive property.)

• 3.OA.6 Understand division as an unknown-factor problem. • 3.OA.2 Interpret whole-number quotients of whole numbers, e.g., interpret 56

x 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. Interpret whole-number quotients of whole numbers.

• 3.OA.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 x ? = 48, 5

= ! ÷ 3,, 6 x 6 = ?.

• 3.OA.7 Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 x 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.

• Connect to Multiple Towers and Division Stories • Students multiply and divide within 100. • Students understand relationship between multiplication and division. • Students understand meaning of multiplication and division. • Students model multiplication and division though equal sized groups, arrays,

area models. • Students can find the unknown factor which either represents the number of

groups or the number in a group. • Students can multiply one digit number by a multiple of ten. • Students know from memory all products of two one-digit numbers.

Learning Outcomes Operations and Algebraic Thinking 4.OA Use the four operations with whole numbers to solve problems.

3. Solve multi-step word problems posed with whole 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.

Gain familiarity with factors and multiples.


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4. 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.

Number and Operations in Base Ten 4.NBT Generalize place value understanding for multi-digit whole numbers

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÷ 70 = 10 by applying concepts of place value and division.

Use place value understanding and properties of operations to perform multi-digit arithmetic. 5. Multiply a whole number of up to four digits by 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. MA.5a. Know multiplication facts and related division facts through 12 x 12.

6. Find whole-number quotients and remainders 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.

Measurement and Data 4.MD Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit. 2. Use the four operations to solve word problems 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.




4.OA.3 4.MA.5a.

• Estimate the product of up to a four digit by a one digit number and a two digit by a two digit number

• Determine if estimate will be greater than or less than the actual

1 day U8 1.1: Making Estimates See Math Note on page 131 for “What is a Close Estimate” and page 33 about “Underestimates and Overestimates” Be sure to include examples of multiplication problems involving one-digit by four-digit numbers.

4.OA.3 4.NBT.5 4.OA.4 4.NBT.1 4.MA.5a.

! Determine if estimate should be greater or less than the actual and use this information when determining reasonableness of actual answer

! Compare estimate to actual answer for reasonableness

! Write story problem to match equation and identify the descriptor for the answer to a word problem, labeling it correctly

! Use efficient strategies based on place value and the

4 days U8 1.2: Breaking Numbers Apart See Teacher Note on page 113 “Multiplication Strategies” and on page 115 about “Breaking Numbers Apart to Solve Two Digit Multiplication Problems” Day 3 U8 1.3: Multiplication Cluster Problems Day 4


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properties of operations ! Fluently and efficiently model multiplication using

place value, arrays and/or area models and write equations that match the visual model

! Solve multiplication word problems using estimation, modeling with place value and/or array/area and writing equations that match the model

! Explain how they are breaking a problem apart in ways that make it easier to solve it

! Use knowledge of multiplication combinations to find factors of larger numbers.

U8 1.4 Assessment Solving Multiplication Problems As students are solving both the Multiplication Cluster Problems and the Pencil Problems on SAB pp. 11, 12, 14 and 15, ask them to explain how they are breaking apart in ways that make it easier for them to solve it and how they are keeping track of the steps. Day 5 U8 1.5: Solving 2-Digit Problems

4.OA.4 4.NBT.5 4.MD.2 4.MA.5a.

• Identify and explain which factor to change in order to compute mentally

• Use mental math to solve two digit by two digit problem and connect equations and models to the mental math computations

• Efficiently and fluenty solve multiplication word problems using estimation, modeling with place value and/or array/area and write equations that match the model

• Keep track of steps in their solution • Be able to connect numbers in solution to context of

work problem.

6 days U8 2.1: Making an Easier Problem U8 2.2: How Did You Start? U8 2.3: How Did you Start? continued U8 2.4: Practice Multiplication U8 2.4 A: Multiplying 4-Digit by 1 Digit Numbers U8 2.5: Assessment: 34 x 68. Use mid-unit assessment in Consolidating Fluency with Multiplication and Division packet posted to mybps.org on the elementary math grade 4 page.

4.NBT.6 4.MA.5a.

• Estimate the quotient of up to a four digit by a one digit number, including word problems

• Determine if estimate will be greater than or less than the actual

• Efficiently and fluently Illustrate solution of division using arrays and/or area models and write equations that match the model

• Efficiently and fluently model division by thinking about missing factor, using a multiplication model

• Use knowledge of multiples of ten to help solve the problem

• Determine how to interpret the remainder in the context of the word problem

6 days OMIT U8 3.1 OMIT U8 3.2 OMIT U8 3.3 OMIT U8 3.4 OMIT U8 3.5 Strengthening Computational Fluency Division 3-digit dividends U8 3.5A Dividing 4-Digit Numbers OMIT U8 3.6 Strengthening Computational Fluency Division 4-digit dividends


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4.OA.3 4.MD.2 4.MA.5a.

• Fluently and efficiently solve multi-step word problems posed with whole 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.

3 days Solve multi-operation problems. Use problems from Consolidating Fluency with Multiplication and Division packet posted to mybps.org on the elementary math grade 4 page.

9 days

BOSTON PUBLIC SCHOOLS, 2013-2014 Grade 4 Mathematics Scope...

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