boston public schools, 2013-2014 grade 5...

Boston Public Schools Mathematics Department Grade 5 Scope and Sequence, 2013-2014 Last updated 08.26.13

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BOSTON PUBLIC SCHOOLS, 2013-2014 Grade 5 Mathematics Scope and Sequence Guide " I. Introduction: In grade 5, instructional time should focus on three critical areas: (1) developing fluency with addition and subtraction of fractions, and developing understanding of the multiplication of fractions and of division of fractions in limited cases (unit fractions divided by whole numbers and whole numbers divided by unit fractions); (2) extending division to 2-digit divisors, integrating decimal fractions into the place value system and developing understanding of operations with decimals to hundredths, and developing fluency with whole number and decimal operations; and (3) developing understanding of volume. (MCF 2011; p. 48) By the end of the year, students should be proficient with the grade 5 Content and Practice Standards.

II. Essential Questions for the Year: 1. What is the relationship between whole number operations and decimal/fraction operations? What happens to the answer (sum, difference, product, quotient) with whole numbers? In comparison, what happens to the answer (sum, difference, product, quotient) with decimals and fractions? Why do you think this happens? 2. How does the position of a digit in a multi-digit number determine its value and its relationship to other digits in the number? 3. How does volume relate to addition and multiplication? 4. How can 2-D shapes be classified in a hierarchy?

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


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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 Thinking CCSS Progression) The Framework defines procedural fluency with multi-digit numbers, including decimals, and fractions 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 Guide 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. (Quiet form of acknowledgement allows time for students to think, while the process continues to challenge those that already have an answer.) 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


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V. Purposeful computation problems that support mathematical goals in number and operations (summarized from Number Talks by Sherry Parrish, Math Solutions 2010) The BPS Mathematics Department has provided resources 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 own 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 5th grade, teachers might want to initially revisit some ideas about addition and subtraction from previous grades. Later in the year, Number Talks can be used to revisit operations with decimals and fractions. “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) Areas to consider when selecting Number Talk Problems (Number Talks by Sherry Parrish, Math Solutions 2010, p. 373): 1. Over-generalizations. When students are investigating which strategies work with different operations, they often over-generalize, and try to apply their generalizations to all operations. 2. Inefficient strategies. Sometimes students become more focused on a specific strategy and ignore efficiency. 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.


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IV. Standards for Mathematical Practice in 5th grade: 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. We adapted materials from North Carolina Department of Public Instruction Instructional Support Tools, Grade 5, below. MP1 - Make sense of problems and persevere in solving them. Mathematically proficient students in grade 5 should solve problems by applying their understanding of operations with whole numbers, decimals, and fractions including mixed numbers. They solve problems related to volume and measurement conversions. Students seek the meaning of a problem and look for efficient ways to represent and solve it. They may check their thinking by asking themselves, “What is the most efficient way to solve the problem?”, “Does this make sense?”, and “Can I solve the problem in a different way?” MP2 - Reason abstractly and quantitatively. Mathematically proficient students in grade 5 should recognize that a number represents a specific quantity. They connect quantities 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 that record calculations with numbers and represent or round numbers using place value concepts. MP3 - Construct viable arguments and critique the reasoning of others. In fifth grade, mathematically proficient students may construct arguments using concrete referents, such as objects, pictures, and drawings. They explain calculations based upon models and properties of operations and rules that generate patterns. They demonstrate and explain the relationship between volume and multiplication. 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 students in grade 5 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. Fifth graders should evaluate their results in the context of the situation and whether the results make sense. They also evaluate the utility of models to determine which models are most useful and efficient to solve problems.


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MP5 - Use appropriate tools strategically. Mathematically proficient fifth graders consider the available tools (including estimation) when solving a mathematical problem and decide when certain tools might be helpful. For instance, they may use unit cubes to fill a rectangular prism and then use a ruler to measure the dimensions. They use graph paper to accurately create graphs and solve problems or make predictions from real world data. MP6 - Attend to precision. Mathematically proficient students in grade 5 continue to refine their mathematical communication skills by using clear and precise language in their discussions with others and in their own reasoning. Students use appropriate terminology when referring to expressions, fractions, geometric figures, and coordinate grids. They are careful about specifying units of measure and state the meaning of the symbols they choose. For instance, when figuring out the volume of a rectangular prism they record their answers in cubic units. MP7 - Look for and make use of structure. In fifth grade, mathematically proficient students look closely to discover a pattern or structure. For instance, students use properties of operations as strategies to add, subtract, multiply and divide with whole numbers, fractions, and decimals. They examine numerical patterns and relate them to a rule or a graphical representation. MP8 - Look for and express regularity in repeated reasoning. Mathematically proficient fifth graders use repeated reasoning to understand algorithms and make generalizations about patterns. Students connect place value and their prior work with operations to understand algorithms to fluently multiply multi-digit numbers and perform all operations with decimals to hundredths. Students explore operations with fractions with visual models and begin to formulate generalizations.

V. Appendix: Throughout the scope and sequence, references are made to the appendix. These are resources you will use in addition to the Investigations and Connected Math 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 which created this scope and sequence, and links to materials available on such sites as Illustrativemathematics.org and Illuminations (NCTM).


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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 Volume with Whole Numbers Primary Curricular Resources: Investigations: Unit 1 Number Puzzles and Multiple Towers Investigations: Unit 2 Prisms and Pyramids Investigations: Unit 7 How Many People? How Many Teams? Estimated Instructional Time: 22 days This unit starts on the first day of school and has 22 days of instruction plus two assessment days for Terra Nova, Supera, WIDA, and Predictive Sept. 4 - Oct. 7

Overarching Questions: ) How can I solve multiplication problems

using clear and concise notation?

) What is the relationship between area and volume?

) What is the relationship between volume and multiplication

Standards for Mathematical Practice Focus MP 1: Make sense of problems and persevere in solving them. Students must persist in solving multi-step, problems. MP 3: Construct viable arguments and critique the reasoning of others. At the beginning of the year, establish the expectation that students will be defending and challenging their conjectures regarding mathematical problems and procedures. MP 4: Model with mathematics. Students must be able to illustrate and explain their calculations with concrete models to solve real world problems. MP 6: Attend to precision. As students record and explain their equations and strategies, they use clear, concise notation and accurate representations.

Instructional Notes: ! Students must be able to multiply fluently using the U.S. standard algorithm.

! Calculating “fluently” means that students can compute flexibly, accurately, efficiently, and appropriately. Previously taught strategies remain important.

! Use work with volume to continue to develop fluency with multiplication.


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Concepts developed in this unit

! Algorithm for Multiplication

! Writing expressions/equations

! Application and use of the distributive property.

! Understanding attributes of volume

! Relating Volume to addition and multiplication in real world contexts

Prior knowledge expected:

) 4.NBT.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.

) 4.MD.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.

Learning Outcomes 5.NBT.5 Fluently multiply multi-digit whole numbers using the standard algorithm. 5.MD.3 Recognize volume as an attribute of solid figures and understand concepts of volume measurement. a. A cube with side length 1 unit, called a “unit cube,” is said to have “one cubic unit” of volume, and can be used to measure volume. b. A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units. 5.MD.4 Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units. 5.MD.5 Relate volume to the operations of multiplication and addition and solve real-world and mathematical problems involving volume. a. Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication. b. Apply the formulas V = l ! w ! h and V = b ! h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real-world and mathematical problems. c. Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real-world problems. Through intentional discussions, this work can be connected to the following standards, which will be revisited throughout the year: 5.NBT.2: Explain patterns in the number of zeros of the product when multiplying a number by powers of 10. 5.OA.1: Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. 5.OA.2: Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “add 8 and 7, then multiply by 2” as 2 ! (8 + 7). Recognize that 3 ! (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.


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MA 2011 Framework Citation

After completing each investigation, students will be able to:

Days Primary Curriculum Resource

MP3 Build a math community by using the first weeks of school to focus on constructing viable arguments by using math talk to encourage students to explain their thinking.

Use Classroom Discussions by Suzanne H. Chapin et al. as a resource. Practice using revoicing and wait time during whole group discussions and Number Talks. (Examples: You mean...? You’re trying to say that. . .?)

5.NBT.5

) Name multiplication strategies and represent them using arrays/area model

) Develop flexibility in solving and representing multiplication problems

) Throughout Investigations Unit 1, use academic language such as: expression, equation, factor, multiple, partial products, marked/unmarked array, area model, representation, equivalent problem, doubling, halving

Day 1 Investigations Unit 1: Number Puzzles and Multiple Towers Lesson 2.1 One of the strategies that might be presented and named by students is the standard algorithm. Teachers may include this as one of the multiplication strategies. However, the standard algorithm for multiplication should be not be explicitly taught on this day.

5.NBT.5

) Precisely and accurately represent array/area models

) Develop flexibility in solving and representing multiplication problems.

Day 2 Investigations Unit 1: Lesson 2.2 OMIT Activity 1 The use of partial products is an important step in developing the standard algorithm for multiplication. This is an important link to make for students on day 6.

5.NBT.5 5.OA.2 5.NBT.2

) Explain patterns when multiplying by multiples of 10

) Estimate products

Day 3 Investigations Unit 1: Lesson 2.3 Discussion in Activity 1 is extremely important to the NBT 2 standard, specifically, “Why is it incorrect to use the language “add a zero” when multiplying by a power of ten? For additional practice problems, use Differentiation and Intervention Guide (D&I), page R5. Reference Algebra Note on p. 87 of Teacher Manual to bring in the associative property and multiplying by the powers of 10.

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

! Develop arguments about how to generate equivalent expressions in multiplication

! Represent equivalent expressions in multiplication

Day 4 and 5

Investigations Unit 7: How Many People? How Many Teams? Lesson 1.1 * Students are still working toward multiplication fluency, which means being flexible, accurate, efficient, and appropriate when


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! Use story contexts and representations to support explanations of relationship between equivalent expressions

Throughout Investigations Unit 7, use academic language such as: algorithm, factor, multiple, expression, partial products, marked/unmarked array, area model, representation, equivalent problem, doubling, halving

solving problems. * Use correct notation for distributive property (OA 1) * Discuss how students know expressions are equivalent without solving (OA 2) *Investigations Unit 1: use pages from Lesson 2.4: SAB 36, 37, 38 as homework Part II Investigations Unit 7: Lesson 1.2 * Students should develop a conjecture regarding a pattern when creating an equivalent problem by the end of the lesson.

5.NBT.5 ) Recognize patterns completing the standard algorithm ) Use representations to model partial products ) Analyze the standard algorithm and the area model

Day 6 Introduce the standard algorithm for multiplication (appendix): Students solve three-digit by one-digit multiplication problems using partial products and the standard algorithm. Do pages in appendix as practice. Students discuss patterns they see in the algorithm, and the similarities between the strategies. Students try it out for themselves on practice pages. Have students do the homework on SAB p.19.

5.NBT.5 5.NBT.2

) Describe and compare strategies used to solve multi-digit multiplication problems

) Understand and apply the standard algorithm for multiplication

Day 7 Investigations Unit 7: Lesson 2.3 Standard algorithm for 2-digit by 2-digit and 3-digit by 2-digit numbers. For additional practice problems, use Differentiation and Intervention Guide, pages R55-56 Starter Problems SAB 41-42 (43 HW); 44-45. (Cut out starter problems; use standard algorithm to solve.)

5.NBT.5 5.NBT.2

) Describe and compare strategies used to solve multi-digit multiplication problems

) Understand and apply the standard algorithm for multiplication

Day 8 Investigations Unit 7: Lesson 2.4 For HW or additional practice, use Differentiation and Intervention Guide (D&I), page R6 and/or R57. (Ignore instructions and have students practice the standard algorithm.) Additional practice and homework pages to use: From Unit 7 Lessons 2.1 and 2.2: Student activity book pages 13, 16, 17, 18, 19. (Use these pages throughout the unit. Students should use the algorithm to solve problems.)

MP3, MP6 Build a math community that focuses on constructing viable arguments, critiquing the reasoning of others, and attending to precision.

On day eight, practice whole class discussions during the launch and/or the share at the end of the workshop. Remember to move the discussion slowly and use many repetitions. Encourage


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contributions from various students. Ensure that all students are focused and respectful.

5.MD.3a,b 5.MD.4 5.MD.5a,b

) Develop a strategy for determining the volume of rectangular prisms

) Recognize a solid figure can be packed without gaps or overlaps using unit cubes

) Find the volume of a right rectangular prism with whole number side lengths by packing it with unit cubes and show that the volume is the same as would be found by multiplying edge lengths

Throughout Investigations Unit 2, use academic language such as: dimensions, volume, layers, net, rectangular prism, cube, cubic centimeters, cubic inches, cubic feet, linear, length, width, height, area, base

Day 9 Investigations Unit 2: Prisms and Pyramids Lesson 1.1 * Omit TMM Activity 1 Quick images. This will be used on day 15 of unit.

5.MD.3a,b 5.MD.4 5.MD.5a,b

) Develop a strategy for determining the volume of rectangular prisms


Day 10 Investigations Unit 2 Lesson 1.2 For HW or additional practice, use Differentiation and Intervention Guide (D&I), page R11.

5.MD.3a,b 5.MD.4 5.MD.5a,b

) Apply the formulas V= l x w x h and V = b x h to find the volume of rectangular prisms

) Explore how the dimensions of a rectangular prism change when the volume is changed (doubled, halved, or tripled)

Day 11 Investigations Unit 2 Lesson 1.3

5.MD.3a,b 5.MD.4 5.MD.5a,b


) Explore how the dimensions of a rectangular prism change when the volume is changed (double, halved, or tripled)

) Solve real world problems by fitting rectangular packages in rectangular boxes

Day 12 Investigations Unit 2 Lesson 1.4

5.MD.3a,b ) Apply the formulas V= l x w x h and V = b x h to find the Day 13 Investigations Unit 2: Lesson 1.5 Students can complete page R12 in Differentiation and


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5.MD.4 5.MD.5a,b

volume of rectangular prisms

) Explore how the dimensions of a rectangular prism change when the volume is changed (double, halved, or tripled).

) Solve real world problems by organizing rectangular packages into rectangular boxes

) Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying edge lengths

Intervention Guide (D&I) guide for extra practice.

5.MD.3a,b 5.MD.4 5.MD.5a,b


) Explore how the dimensions of a rectangular prism change when the volume is changed (double, halved, or tripled).

) Solve real world problems by organizing rectangular packages to fit in rectangular boxes

) Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying edge lengths

Day 14 Investigations Unit 2: Lesson 1.6 For additional practice on volume: http://www.illustrativemathematics.org/illustrations/1308

5.MD.3a,b 5.MD.4 5.MD .5a,b,c


) Recognize volume as additive by finding the volume of a solid composed of two rectangular prisms

Day 15 Intro to non-overlapping prisms Select pages from Quick Images T25-T29 to introduce the concept of volume of solids composed from two rectangular prisms. Have students build them and find the volume of the two non-overlapping prisms.

MP 3 Build a math community that focuses on constructing viable arguments and critiquing the reasoning of others by practicing partner talk.

Consider using sentence frames to help support math language. (Example: I think that….because...)

5.MD.5c


) Recognize volume as additive (finding the volume of a solid composed of two rectangular prisms)

Day 16 and 17

Investigations Unit 2: CCSS Unit 2 1.5A Because of the difficulty level of this lesson, teachers should plan two instructional days to complete this lesson.


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5.M.D.4

) Recognize a cube with side length 1 unit, called a “unit cube,” is said to have “one cubic unit” of volume, and can be used to measure volume


) Measure volume by counting unit cubes using cubic cm, cubic in, cubic ft, and improvised units

Day 18 Investigations Unit 2 Combine Lesson 2.1 and Lesson 2.2 (OMIT Activity 3 of 2.2) Ask: How many cubic meters will fit inside our classroom? What unit of measure would you rather use (cubic meter, cubic feet, cubic inch, cubic centimeter)?

5.MD 5a,b,c

) Recognize a cube with side length 1 unit, called a “unit cube,” is said to have “one cubic unit” of volume, and can be used to measure volume

) Apply the formulas V= l x w x h and V = b x h to find the volume of rectangular prisms in the context of solving real-world and mathematical problems.

) Measure volume by counting unit cubes using cubic cm, cubic in, cubic ft, and improvised units

Day 19- 21

Real World Volume Problems See appendix lesson plans and practice pages Start project on Day 20: Project involves building non-overlapping rectangular prisms Use Differentiation and Intervention Guide (D&I), pages R13-15, for review work

Day 22 BPS Assessment


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Unit of Study 2: Division with Whole Numbers/Order of Operations Primary Curricular Resources: Investigations: Unit 1 Number Puzzles and Multiple Towers Investigations: Unit 7 How Many People? How Many Teams? Order of Operations (Appendix) Estimated Instructional Time: 20 days (and an additional day for the interim assessment) Oct. 8 - Nov.6

Overarching Questions: ) What is the relationship between multiplication and division?

) How can I represent my division calculations?

) How does the story context affect the remainder in a division problem?

) How does the order of operations affect the value of a given expression?

) How can I use parentheses/brackets/braces in numerical expressions?

Standards for Mathematical Practice Focus MP 1: Make sense of problems and persevere in solving them. Especially in the field day problems, students must persist in solving multi-step, multiple operations problems. MP 3: Construct viable arguments and critique the reasoning of others. At the beginning of the year, establish the expectation that students will be defending and challenging their conjectures regarding mathematical problems and procedures. MP 4: Model with mathematics. Students must be able to illustrate and explain their calculations with concrete models to solve real world problems. MP 6: Attend to precision. As students record and explain their equations and strategies, they use clear, concise notation and accurate representations.

Instructional Notes: ! Students continue to develop fluency with division. Note that the division algorithm is taught in 6th grade, and students are still developing efficient

strategies in grade 5. They must be able to illustrate their calculations with a rectangular array and/or area model.

! This year’s fifth graders may not have had extensive experience with the order of operations as named in the 3rd grade standard below under prior knowledge expected.

! Order of Operations: Expressions have depth of no greater than two, e.g., 3x[5+(8÷2)] is acceptable, but 3x{5+[8÷{4-2}]} is not. (PARCC)


) Developing fluency with division strategies

) Fractions as division

! Order of operations

! Translating words into expressions

Prior knowledge expected: 4.NBT.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.


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3.OA.8 Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies, including rounding.[1] [1] This standard is limited to problems posed with whole numbers and having whole number answers; students should know how to perform operations in the conventional order when there are no parentheses to specify a particular order (Order of Operations).

Learning Outcomes 5.NBT.6 Find whole number quotients with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. 5.NF.3 Interpret a fraction as division of the numerator by the denominator (a/b = a÷b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret ! as the result of dividing 3 by 4, noting that ! multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size !. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does the answer lie? 5.OA.1: Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols 5.OA.2: Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “add 8 and 7, then multiply by 2” as 2 ! (8 + 7). Recognize that 3 ! (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product




MP3 Build a math community focused on constructing viable arguments and critiquing the reasoning of others by supporting mathematical arguments.

Day 1 Have students use the following steps to develop a math argument:

1. State your claim. 2. Defend your claim by giving evidence. 3. Agree with a claim by adding more information or

another strategy. 4. Challenge a claim by giving another point of view.


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5.NBT.6

) Represent a division problem with a picture or diagram

) Create a story context for a division expression.

) Describe and compare strategies used to solve division problems

Throughout Investigations Units 1 and 7, use academic language such as: dividend, quotient, divisor, remainder, notation, factor, multiple

Day 1 Investigations Unit 1: Number Puzzles and Multiple Towers Lesson 3.1 *Focus on representations and meaning of partial products/quotients. Use arrays to model division. *Focus on division strategies: divide down.

5.NBT.6

) Use clear and concise notation.

) Fluently solve division problems with a 2-digit divisor

Day 2 Investigations Unit 1: Combine Lesson 3.2 and 3.3 (as you see fit for your students) Lesson 3.2 & Lesson 3.3 * Focus on division strategies: divide down * Use pages R59-60 from the D&I Guide for additional practice Note: Omit Lesson 3.4: SAB pages can be used as homework for days 2 - 4

5.NBT.6



) Solve multi-step word problems

) Use clear and concise notation

Day 3 Investigations Unit 1: Lesson 3.5 * Focus on link between cluster problems and dividing down Students can complete page R9 from the D&I Guide for additional practice or homework.

5.NBT.6





Day 4 Investigations Unit 1: Lesson 3.6 (M30 could be used at end) Omit Lesson 3.7 and 3.8 (use as optional work)

5.NBT.6 5.OA.2

) Generate and represent equivalent expressions in division

) Compare equivalent multiplication expressions to equivalent division expressions

Day 5 Investigations Unit 7: How Many People? How Many Teams? Lesson 1.4 *Students need to have understanding of relative size of divisor, quotient, and dividend: if two “parts” of equation change, how is other affected? (Connection to 5.OA.2: interpreting expressions without evaluating.)


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5.NF.3 ) Students will make sense of a division problem as a fraction

) Students will relate the story problem to the fractional answer

Day 6 and 7

Teacher Note: Fractions as Division (Appendix) The following activity can also be used along with the pages provided in the appendix: http://www.illustrativemathematics.org/illustrations/292

5.NBT.6 ) Describe and compare strategies used to solve division problems




Day 8 Investigations Unit 7: Lesson 3.1 * Students should think about what happens to the remainder. * Students should share their ideas about the remainder according to their story context. * Highlight contexts in which the remainder can be expressed as a fraction and have students think about potential fraction contexts for the remainder.





Day 9 and 10

Investigations Unit 7: Lesson 3.2 and Lesson 3.3 (Teachers decide which pages are most helpful for their students’ needs) * Use pages R59-60 from the D&I Guide for additional practice





Day 11 and12

Investigations Unit 7: Combine Lesson 3.4 and Lesson 3.5: Refining Division Strategies Use D&I Guide,page R58, for extra practice on day 12.





Day 13 Investigations Unit 7: Combine Lesson 3.6 and Lesson 3.7 Optional Assessment from 3.7 (assess students on use of “divide down” strategy) Students should solve problems on SAB pages 47-48 and you could end the day with the 3.7 Assessment Use SAB 49-51 for hw


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5.OA.1 5.OA.2

! Understand order of operations for addition, subtraction, multiplication, and division ! Use parentheses, brackets, or braces in numerical expressions ! Write simple expressions that record calculations with numbers

Day 14 - 15

Order of Operations (appendix)

5.NBT.5 5.NBT.6 5.OA.1 5.OA.2

) Solve multi-step word problems,

) Solve 2-digit by 2-digit or 3-digit multiplication problems fluently


) Use all four operations to solve problems.

) Describe and compare strategies to solve division problems

Day 16 - 19

Investigations Unit 7: Lesson 4.1, 4.2, 4.3, 4.4 * These problems represent an important part of the Massachusetts Curriculum Framework: estimating answers, real life contexts, choosing the operations, creating representations, efficiency, writing expressions with parentheses, and communicating solutions clearly and concisely. ** Note - Optional worksheet available to use that gives students more space for calculations and organization ** Use M31 as an assessment at end of lesson 4.4 ** Use D&I Guide, pages R61-63 as homework or group



!+!

Unit of Study 3: Place Value System/Addition and Subtraction with Decimals Primary Curricular Resources: Investigations Unit 6 Decimal Grids and Number Lines Expanded Notation and Powers of Ten (appendix) Investigations Unit 1 CCSS guide Estimated Instructional Time: 24 days (and an additional day for the interim assessment) Nov. 7 - Dec. 16

! Overarching Questions: ! How does place value change the value of a digit? ! How does the value change as you move “left” in place value? As

you move “right”? ! How do I round a decimal to the nearest place value given? ) Why are 0.5 and 0.50 equivalent?

) Given a decimal, can I name a decimal that is less? More?

) Given two decimals, can I name a decimal that is in between?

Standards for Mathematical Practice Focus MP 2 Reason abstractly and quantitatively Students will consider the units, involved attending to the meaning of quantities, not just how to compute them. MP 6 Attend to precision Students give carefully formulated claims and reasoning to each other. MP 7 Look for and make use of structure Students will use the pattern of place value to generate numbers in expanded form.

Instructional Notes:

• The cluster of standards, Understand the Place Value System, are integrated in the lessons below. Students will continue to develop their understanding of the place value system with decimals throughout the year. The “explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10” part of NBT.2 will be explicitly addressed again in Unit of Study #5.

) Emphasize the number line, visual representations (tenths, hundredths, and thousandths grids), base ten blocks.

) Enlarge tenths, hundredths, thousandths grids. (Zoom 220%)

) Grade 5 standards include operations (add/subtract) with decimals ONLY to the hundredths.

) Grade 5 standards include comparing two decimals to thousandths using < > = symbols. (Emphasize concrete models.)

) Expanded notation matching game sets A, B and C scaffold learning from whole numbers to exponents to decimals.


! The “position” of a digit affects its value

) Represent decimals as parts of an area and on a number line

) Interpret the meaning of digits in a decimal number

) Compare decimals to landmarks 0, ", 1

) Estimate sums/differences of decimal

Prior knowledge expected 4.NBT.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. 4.NBT.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. 4.NBT.3 Use place value understanding to round multi-digit whole numbers to any place.


#,!

numbers

) Addition and subtraction of decimals to the hundredths through reasoning about place value, equivalents, and representations

) Application of these skills in addition and subtraction contexts

4.NF.5 Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100. For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100. 4.NF.6 Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100 ; describe a length as 0.62 meters; locate 0.62 on a number line diagram. 4.NF.7 Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model.

Learning Outcomes 5.NBT.1 Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left. 5.NBT.2. Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10. 5.NBT.3. Read, write, and compare decimals to thousandths. a. Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., 347.392 = 3 ! 100 + 4 ! 10 + 7 ! 1 + 3 ! (1/10) + 9 ! (1/100) + 2 ! (1/1000). b. Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons. 5.NBT.4 Use place value understanding to round decimals to any place. 5.NBT.7 Add and subtract to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. 5.OA.1. Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. 5.OA.2. Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “Add 8 and 7, then multiply by 2” as 2 ! (8 + 7). Recognize that 3 ! (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.


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

5.NBT.1 5.NBT.2 5.NBT.3a

) Begin to recognize that the value of a digit represents 10 times as much as in the place to the right and 1/10 as much as in the place to the left

) Recognize the patterns of zeros in a product when multiplying a number by powers of 10

) Read and write decimals to the thousandths using base-ten numerals and expanded notation

Day 1 Expanded Notation and Powers of 10 (appendix.)


#!!

Throughout first part of unit, use academic language such as: place value, expanded and standard notation, powers of 10, exponents, and exponential.

5.NBT.1 5.NBT.2

) Understand powers of 10 expressed with exponents

) Use whole number exponents to denote powers of 10

Day 2 Expanded Notation and Powers of 10 (appendix)





) Read and write decimals to the thousandths using base- ten numerals and expanded notations

Day 3 Expanded Notation and Powers of 10 (appendix) Use Digit It game to practice concepts in this unit



) Read and write decimals to the thousandths using base-ten numerals and expanded notations

Day 4 Expanded Notation and Powers of Ten (appendix) ** Play Set A Matching Game (appendix)






Day 5 Expanded Notation and Powers of Ten (appendix) ** Play Sets A and B Matching Game (appendix)

5.NBT.1 5.NBT.3a,b

) Represent decimals as parts of an area. ) Read and write decimal values in tenths, hundredths,

and thousandths Throughout second part of unit: use academic language such as: fraction, decimal, tenths, hundredths, thousandths, grid

Day 6 Investigations Unit 6 Decimals on Grids and Number Lines Lesson 1.1 * Use pages R19 from the D&I Guide for quiz on place value

5.NBT.1 5.NBT.3a,b

) Represent decimals as parts of an area. ) Read and write decimals to the thousandths using base-

Day 7 Investigations Unit 6 Lesson 1.2 * When doing activity 2, Connect this to expanded notation


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ten numerals and expanded notations

(7 x 10,000 + 9 x 1,000 + 3 x 100 + 2 x 1 + 1 x 1 + 4 x 1/10 (or 0.10) + 5 x 1/100 (or 0.01) * Be explicit about how place value changes the value of the digit (10 times the size of the digit to right; 1/10 the size of the digit to the left: NBT 1)

5.NBT.1 NBT.3a,b

) Order decimals and justify their order through reasoning about decimal representations, equivalents, and relationships

Day 8 Investigations Unit 6: Lesson 1.3 Activity 3: ordering tenths and hundredths - “Decimals in Between”

5.NBT. 3a,b ) Order decimals and justify their order through reasoning about decimal representations, equivalents, and relationships

Day 9

Investigations Unit 6: Lesson 1.4 (whole lesson) Play decimal in between Discussion about comparing decimals

5.NBT.3a,b ) Order decimals and justify their order through reasoning about decimal representations, equivalents, and relationships

) Represent decimals as part of an area


Day 10

Investigations Unit 6: Lesson 1.5 - workshop Activity 1, 2a, and 2b Skip assessment or use as homework **Play Matching Game (Set C) - (appendix)

5.NBT.4 ) Round decimals to the nearest tenth and hundredth Day 11

Investigations Unit 6: Lesson 1.5A CCSS Day 11: Focus on use of number line as a model for rounding decimals Use page in appendix (rounding on the number line) ** Students can play Matching Game sets A, B & C

5.NBT.4 ) Round decimals to the nearest tenth and hundredth Day 12- Day 13

Investigations Unit 6: Lesson 1.5A CCSS Day 12: Do page c64 with continued focus on number line Continue practice with estimation and rounding decimals Day 13: Do page c65 Continue practice with rounding decimals and estimation of decimals

5.NBT.7 ) Represent decimals as a part of an area

) Estimate sums of decimal numbers

) Use representations to add tenths, hundredths, and thousandths.

Day 14

Investigations Unit 6 Lesson 2.1- Fill Two Addition of decimals using representations OMIT Lesson 2.2 (use as optional student activity book pages)


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5.NBT.7 ) Use representations to add tenths and hundredths

Day 15

Investigations Unit 6 Combine Lesson 2.3 and 2.4 Chart strategies for adding decimals (Omit Lesson 2.4 Activity 3 Decimal Compare - play if have time) OMIT 2.5 - 2.8 but optional SAB pages for practice/hw * Use pages R20-21 from the D&I Guide for additional practice in whole number addition.

5.NBT.7 ) Use representations to subtract tenths and hundredths

) Subtract decimals to the hundredths through reasoning about place value, equivalents, and representations

Days 16 and 17

Investigations Unit 6 CCSS 2.5A Decimal problems – subtraction *Instructional note- use number line and grids for representation of subtraction Day 16: Students solve problems and make posters. End with students presenting work and have discussion on naming subtraction strategies (create anchor chart) Some strategies that may be discussed: breaking one number apart and subtracting it in parts, adding up, making an equivalent problem, U.S. standard algorithm. Day 17: Begin with reviewing names of strategies; students practice subtracting using class strategies (appendix) * Use pages R22-27 from the D&I Guide for additional practice with whole number subtraction and addition.

5.NBT.4 5.NBT.7

) Estimate sums or differences with decimals

Day 18

CMP Bits and Pieces III Lesson 1.1 Even though students have not done the fraction to decimal benchmark work in Bits I, they can connect their understanding of money to the number line representation in About How Much?

5.NBT.7 ) Develop place value understanding of decimal addition and subtraction

Day 19

CMP Bits and Pieces III Lesson 1.2 Use number line or decimal grids for adding and subtracting with decimals


Day 20

Appendix: extra practice pages Bits III: ACE pages 13 - 14 (use problems to thousandths)


Day 21

Review (appendix) See appendix for extra work/review on decimals (4 pages)

5.OA.1 5.OA.2 ! Understand order of operations for addition,

subtraction, multiplication, and division

! Use parentheses, brackets, or braces in numerical

Day 22 and 23

Order of Operations/Numerical Expressions (appendix)

The following Illustrative Math activity can also be used to generate a discussion around the use of parentheses:


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expressions ! Write simple expressions that record calculations with numbers

http://www.illustrativemathematics.org/illustrations/555 OPTIONAL: This is a link to an activity created by the Ohio DOE to teach Common Core Standard 5.OA 1 and 2. Use if needed: http://ims.ode.state.oh.us/ODE/IMS/Lessons/Content/CMA_LP_S01_BE_L05_I09_01.pdf

Day 24

BPS Assessment


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Unit of Study 4: Addition and Subtraction with Fractions Primary Curricular Resources: Investigations Unit 4 What’s that Portion CMP Bits and Pieces I and Bits and Pieces II Estimated Instructional Time: 21 days Dec. 17 - Jan. 27

Overarching Questions: ) How could you prove that fractions are equivalent?

) What visual models help with the understanding of adding and subtracting fractions?

) What are the strategies for adding and subtracting fractions?

Standards for Mathematical Practice Focus MP2 Reason abstractly and quantitatively Students will connect quantities to written symbols and create a logical representation of the problem at hand, extending their understanding from whole numbers to fractions. MP3 Construct viable arguments and critique the reasoning of others When solving fraction addition and addition problems with unlike denominators, students will justify their own thinking, and critique the reasoning of others. MP4 Model with Mathematics Students will be able to solve problems arising in everyday life that involve fractions with unlike denominators, by identifying the important quantities and analyzing the mathematical relationship between the quantities.

Instructional Notes: ) Some of the work in Bits and Pieces I focuses on 4th grade standards, however, these lessons are important for scaffolding to the addition and

subtraction lessons that follow. ) Some of the work also builds the foundation for multiplication with fractions ) It is suggested that you do a reading lesson to introduce students to the set-up of the CMP book. ) it is recommended that you reinforce the concept of addition and subtraction of fractions using a number line, helping students to make a connection

to their work with addition and subtraction of whole numbers.


! Visual models for addition and subtraction of fractions

! Fluency with addition and

Prior knowledge expected: 4.NF.1 Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x b) by using visual fraction models, with attention to how the numbers and sizes of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. 4.NF.2 Compare two fractions with different numerators and different denominators, e.g., by creating common


#'!

subtraction of fractions and mixed numbers

! Application of adding and subtracting fractions and mixed numbers

denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. 4.NF.3 Understand a fraction a/b with a > 1 as a sum of fractions 1/b. a. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. b. Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ;21/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8. c. Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction. d. Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem.

Learning Outcomes 5.NF.1 Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such as way as to produce an equivalent sum or difference of fractions with like denominators 5.NF.2 Solve word problems involving addition and subtraction of fractions referring to the same whole including cases of unlike denominators, e.g. by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers.



Supports future work with 5.NF.4

Throughout Bits and Pieces I, use academic language such as: equivalent fractions, mixed numbers, fractions greater than 1 (improper fraction), unit fraction, benchmark number, common denominator

) Analyze a visual model for fractional parts

) Compare fractional parts with the same whole

Day 1 CMP Bits and Pieces I Lesson 1.1 and 1.2



) Compare fractional parts with the same whole

Day 2 CMP Bits and Pieces I Lesson 1.3


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) Compare fractional parts with different-sized wholes

) Identify fractional parts on a given visual model

Day 3 CMP Bits and Pieces I Lesson 1.4

4.NF.1 4.NF.2 Supports 5.NF.1


) Compare two fractions with different numerators and denominators

) Use benchmark fractions to compare fractions

Days 4 and 5

CMP Bits and Pieces I Combine Lessons 2.3, 2.4, 2.5 Pace lessons according to your students’ needs. * Use page R32 from the D&I Guide for additional practice ordering fractions

5.NF.1

) Add fractions with unlike denominators on a visual representation of a clock by replacing them with equivalent fractions

Throughout the rest of the unit, use academic language such as: area model, mixed number, unit fraction, number line, location, equivalent, greater than/less than, approximately equal to, sum, difference, equation, expression, clock model, clockwise rotation, counter clockwise rotation

Day 6 Investigations Unit 4 What’s That Portion Lesson 3.1

5.NF.1 ) Add fractions with unlike denominators; play “Roll around the Clock” by renaming with equivalent fractions

Day 7 Investigations Unit 4 Lesson 3.2 * Use pages R33-34 from the D&I Guide for additional practice adding and comparing fractions

5.NF.1 5.NF.2

) Add and subtract fractions with unlike denominators, including in story contexts

Day 8 Investigations Unit 4 Lesson 3.3 * Use pages R35-36 from the D&I Guide for additional practice adding fractions

5.NF.1 ) Use equivalent fractions to partition the fraction track board game, also order fractions and justify their order through reasoning about fraction representations, equivalence, and relationships

Day 9 Investigations Unit 4 Combine Lessons 3.4 & 3.5

5.NF.1 ) Add and subtract fractions by using a number line

Day 10 Investigations Unit 4 Lesson 3.6 Fraction Track Game

5.NF.1 ) Add and subtract fractions using a number line ) Add and subtract fractions through reasoning about

Days 11-12

Investigations Unit 4 Lesson 3.7 & 3.8 Math workshop


#*!

fraction equivalence and relationships OMIT Lesson 3.9 but use SAB pages 67 - 68 for practice as a part of workshop

5.NF.1 ) Add and subtract fractions within a story context Day 13 Investigations Unit 4 Lesson 3.10 Skip assessment; Complete daily practice pages 69-70

5.NF.2 ) Use benchmarks and other strategies to estimate the reasonableness of results of operations with fractions

Day 14 CMP Bits and Pieces II Lesson 1.1 Use Investigations fractions cards for this game. *Possible extension questions to use throughout Investigation 1 are #43-46 on page 14

5.NF.2 ) Use benchmarks and other strategies to estimate the reasonableness of results of operations with fractions within a story context

Day 15 CMP Bits and Pieces II Lesson 1.2

5.NF.2 ) Use benchmarks and other strategies to estimate the reasonableness of results of operations with fractions within a story context

Day 16 CMP Bits and Pieces II ACE pages 10- 11

5.NF.2 ) Add and subtract fractions within a story context Day 17

Bits and Pieces II Launch: Applications Connections Extensions Flower Garden (page 24) Copy lab sheet 2 ACE Exercise 1 page 124 in teacher’s edition Wrap-up discussion should focus on students justifying their thinking for parts d, and e. Using this as a launch will help prepare students for the next lesson, which is more of a challenge.


Bits and Pieces II Lesson 2.1 pages 17, 18, top of 19 Copy lab sheet 2.1 page 123 in teacher’s edition


Bits and Pieces II Lesson 2.2

5.NF.1 5.NF.2

) Add and subtract fractions within and without a story context

Day 20

Bits and Pieces II ACE pages 25- 26



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Unit of Study 5: Multiplication and Division with Decimals/Unit Conversions Primary Curricular Resources: Investigations Unit 6: Decimals on Grids and Number Lines CMP Bits and Pieces III Estimated Instructional Time: 23 days (and an additional two days for the interim and predictive assessments) Jan. 28 - March 10

Overarching Questions: ) What are the patterns you notice when multiplying or dividing by

the powers of ten? (10x greater; 1/10 the size) ) What representations can I use to demonstrate my

understanding? ) Is my answer reasonable? Why?

) What is the relationship between multiplying and dividing with whole numbers to multiplying and dividing with decimals?

) What is the relationship between converting units of measure within the metric system to multiplying and dividing decimals?

Standards for Mathematical Practice MP 2: Reason abstractly and quantitatively. Students create generalizations to explain the pattern for multiplying and dividing decimals. MP 6: Attend to precision. As students record and explain their equations and strategies, they try to use clear, concise notation and accurate representations MP 8: Look for and express regularity in repeated reasoning. Students will look for general methods and for shortcuts when doing repeated calculations.

Instructional Notes: ) In this unit, students will continue to develop the place value ideas in NBT.1 and NBT.2, begun in unit #3. There is particular focus on the “explain

patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10” part of NBT.2. ) 5th grade students are expected to multiply with decimals to get products in the tenths, hundredths, and thousandths only, and to divide in problems

involving tenths and/or hundredths. - Meets standard: 1.9 x 2.03 (tenths by hundredths) and 0.2 x 0.5 (tenths by tenths) - Does not meet standard: 1.34 x 0.12 (hundredths by hundredths results in a product with a place value beyond the 5th grade standard)


) Use representations and reasoning to multiply and divide whole numbers by powers of 10 & explain how and why it changes the value of the number and placement of the decimal point

) Develop strategies for multiplication and division of decimals to the hundredths

) Estimating products and quotients of decimal numbers ) Write a rule for multiplying decimal numbers

) Convert units within the metric system

) Relate the conversion of units within the metric system to division

Prior knowledge expected: 4.MD.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), … 4.MD.2 Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including


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and multiplication with decimals 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.

Learning Outcomes 5.NBT.1 Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left. 5.NBT.2. Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole number exponents to denote powers of 10. 5.NBT.3 Read, write, and compare decimals to thousandths. a. Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., 347.392 = 3 !100 + 4 ! 10 + 7 ! 1 + 3 ! (1/10) + 9 ! (1/100) + 2 ! (1/1000). 5.NBT.4 Use place value understanding to round decimals to any place. 5.NBT.7 Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/ or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. 5.MD.1 Convert among different-sized standard measurement units within a given measurement system (e.g. convert 5 cm to 0.05m) and use these conversions in solving multi-step, real world problems.



5.NBT.1 5.NBT. 2 5.NBT.3a 5.NBT.4 5.NBT.7

) Use representations and reasoning to multiply whole number by powers of 10, including 1.0, 0.1, and .01

) Explain the patterns in the placement of the decimal point when a decimal is multiplied by a power of 10

Throughout unit, use academic language such as: decimal, product, quotient, dividend, divisor, tenths, hundredths, estimation, reasonable, equation, factor, exponent, powers of ten, area model

Day 1 Investigations Unit 6 CCSS Lesson 3A.1 *Emphasize the language of “10 x the product”, relationship between 0.01 and 0.1

5.NBT.1 5.NBT. 2 5.NBT.3a 5.NBT.4

) Estimate products of decimal numbers ) Multiply decimals to hundredths through reasoning

about place value and multiplication ) Use number line to represent multiplication of

Day 2 Investigations Unit 6 Lesson CCSS 3A.2 *Students should have time to explore the use of the number line as a model for multiplying decimals


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5.NBT.7

decimals




about place value and multiplication ) Write a rule for multiplying decimal numbers


Day 3 Investigations Unit 6 Lesson CCSS 3A.3




) Use an area model for multiplying fractions


Day 4 Investigations Unit 6 CCSS Lesson 3A.4 * Use the teacher note on using the grid as an area model for multiplying decimals (tenths by tenths and tenths by hundredths) (appendix) * Students should create representations for the work from lesson 3A.4 page 76 and 77 (pick and choose problems to emphasize for discussion) *Discuss relationship between representations (What is happening with the product when you multiply decimals by decimals?)





Day 5 and Day 6

Bits and Pieces III: Investigation 2 (Decimal Times) ) Use pages 23, 24, 25, 28 and 29. See instructional note

below for choosing appropriate problems. ) Do not have students complete all pages. ) Continue practicing multiplication with decimals as well as

estimating products When using the Bits and Pieces practice problems only assign problems that meet the 5th grade standard (no products beyond thousandths):

Meets standard: 1.9 x 2.03 (tenths by hundredths) and 0.2 x 0.5 (tenths by tenths) Does not meet standard: 1.34 x 0.12 (hundredths by hundredths results in a product with place value beyond the 5th grade standard)

Some of these problems provide opportunities to estimate products. Have students round to the ones or tenths place when rounding.


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5.NBT.1 5.NBT. 2 5.NBT.6 5.NBT.7

) Use representations and reasoning to divide whole numbers by powers of 10 (i.e., 1.0, 0.1, 0.01)

) Explain the patterns in the placement of the decimal point when a decimal is divided by a power of 10

Days 7 and 8

Investigations Unit 6 Lesson CCSS 3A.5 ) See Teacher Note on representing division strategies for

decimal numbers on the hundredths grid and number line (appendix)

) DAY 7: Students solve problems using grids. In the CCSS guide, lesson should focus on pages CC125 and CC126. Students should solve teacher-created problems using 10 by 10 grids. Students should show their work using the following page: http://illuminations.nctm.org/Lessons/Percent/Percent-AS-10x10Grids.pdf

) Sample problems could include: 2 ÷ 0.1, 2 ÷ 0.2, 1 ÷ 0.5, 3 ÷ 0.6, etc.

) DAY 8: Students describe and make sense of patterns and placement of the decimal point. Follow the CCSS teacher guide pages CC128 and CC129.

) See appendix page - Division with Decimals for additional work on the placement of the decimal point and patterns when dividing by a power of 10. Use this sheet to create an anchor chart. Include student’s observations about the patterns they notice.

5.NBT.1 5.NBT. 2 5.NBT.7

) Estimate quotients of decimal numbers ) Divide decimals to the hundredths through reasoning

about place value and division

) Develop and explain division decimal strategies

) Explain the patterns in the placement of the decimal point when a decimal is divided by a power of 10

Days 9, 10, and 11

Investigations Unit 6 Lesson CCSS 3A.6 Day 9: Focus Lesson on page CC131 and CC132 Use number line representation (appendix for practice pages). Day 10: Focus lesson on CC133 Do page 82- 83 and have discussion about reasonable answers and practice division strategies. Day 11: Choose two problems from page 82 - 83 Students will create a poster with a word problem, solutions and a representation. During discussion, name strategies and create an anchor chart of strategies for the classroom. Students continue working on page 83, using more than one strategy to solve each problem. TEACHER NOTE: ) The following division strategies should be developed and

discussed over the 3 days: multiplying up, dividing down, creating an equivalent expression with whole numbers by multiplying by 10 or 100, and using reasoning to place the decimal point.


$$!

) Equivalent Expression: for example 4.8 ÷ 0.6 = 48 ÷ 6 (multiplying both the dividend and the divisor by 10)

) In addition to the problems from 3A.6, teachers should also use pages C98, C99, and C100 from the CCSS book in Unit 7 (How Many People? How Many Teams?)

5.NBT.7 ) Multiply and divide decimals to the hundredths through reasoning about place value and multiplication/division

Day 12 Investigations Unit 6 Lesson CCSS 3A.7 Students continue practicing strategies for dividing decimals

5.NBT.7 ) Multiply and divide decimals to the hundredths through reasoning about place value and multiplication/division

) Solve real world problems and explain reasoning for operation used to solve the problem

) Create representations for real world contexts

Days 13 and 14

Multiply or Divide? (appendix) Students solve real-world word problems using both multiplication and division with decimals. Students will practice identifying the correct operation for solving the given problem. Illustrative math activity: http://www.illustrativemathematics.org/illustrations/1293

5.NBT.4 5.NBT.7 5.MD.1

! Convert among different-sized units of measure within the metric system

! Use unit conversions to solve real-world problems

Day 15 Unit Conversions (appendix) Session 1.1: Metric Conversions: Length

5.NBT.4 5.NBT.7 5.MD.1



Day 16 Unit Conversions (appendix) Session 1.2: Metric Conversions: Liquids

5.NBT.4 5.NBT.7 5.MD.1



Day 17 Unit Conversions (appendix) Session 1.3: Metric Conversions: Mass

5.NBT.4 5.NBT.7 5.MD.1



Day 18 Unit Conversions (appendix) Session 1.4: Metric Conversions on a number line

5.NBT.4 5.NBT.7 5.MD.1



Day 19 Unit Conversions (appendix) 1.5: Metric Conversions using a table

5.MD.1 ! Convert among different-sized units of measure within the customary system


Day 20 Unit Conversions (appendix) 1.6: Time and Customary Measurement


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5.MD.1 ) Convert among different-sized units of measure within the customary system

) Use unit conversions to solve real-world problems

Day 21 Unit Conversions (appendix) 1.7: Customary Measurements

5.NBT.4 5.NBT.7 5.MD.1

) Convert among different-sized units of measure within the metric system

) Use unit conversions to solve real-world problems

Day 22 Metric Olympics: Choose activities that best meet student needs and are most feasible given your classroom and materials, etc. These are fun, hands-on activities for students. http://www.macombscience.org/uploads/5/8/3/4/583452/minimetricolympics.pdf Illustrative Math Options http://www.illustrativemathematics.org/illustrations/878



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Unit of Study 6: Multiplication and Division with Fractions Primary Curricular Resources: CMP Bits and Pieces II (Investigation 3) Investigations Unit 4 Common Core State Standards Guide Estimated Instructional Time: 25 days (and an additional two days for the ELA MCAS) March 11 - April 17

Overarching Questions:

) How does multiplication and division of fractions relate to whole numbers?

) How can you use a representation to model multiplication and division of fractions?

) What are real-world applications for multiplying and dividing fractions?

Standards for Mathematical Practice Focus MP 2 Reason abstractly and quantitatively. Students will be able to reason about the quantity of the product of a multiplication expression when one factor is greater than or less than one whole. MP 3 Construct viable arguments and critique the reasoning of others. Students will be able to critique the reasoning of the solutions of others as well as make connections to their own reasoning when multiplying or dividing fractions. MP 4 Model with mathematics. Students will be able to solve problems arising in everyday life that involve fractional lengths of sides to find area.

Instructional Notes: ! Students need to be fluent with multiplication of fractions; division with fractions continues in 6th grade. ! Use visual models for multiplication and division of fractions. ! Students need to develop generalizations about multiplying by a whole number, a mixed number, and a fraction. (See 5.NF.5 below). This will also

support the development of their estimation skills. Throughout the unit, ask students to explain what happens to the product when multiplying by the different kinds of numbers listed above. Connect this work to the work in Unit 3 with decimal multiplication and the understanding that was developed about the relationship of the digits in the place value system (i.e. that a digit represents 1/10 of the digit to its left.) Also, connect this to work students did in 4th grade with multiplicative comparison (See 4.OA.1 below). Note that multiplying by n/n is equivalent to multiplying by 1.

! When students are solving problems in CCSS lessons with “sections of land”, and when using the “brownie pan” model in CMP, be explicit about the fractional lengths of the dimensions. Use the language “square units” for the area and “units” (of length) for the dimensions (For example, the brownie pan model in Bits could represent 1 square unit) to meet the expectations of 5.NF.4.b. Refer back to the representations students used when multiplying decimals.


! Visual models to represent multiplication and division of fractions

! The relationship between the numerator and denominator

Prior knowledge expected: 4.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. a. Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 ! (1/4), recording the conclusion by the equation 5/4 = 5 ! (1/4). b. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 ! (2/5) as 6 ! (1/5), recognizing this


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! The relationship of multiplication and division of whole numbers to fractions

! Develop algorithm for multiplication of fractions and mixed numbers

product as 6/5. (In general, n ! (a/b ) = (n ! a)/b.) c. Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie? 4.OA.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.

Learning Outcomes 5.NF.3 Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie? 5.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. a. Interpret the product (a/b) ! q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a ! q ÷ b. b. Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas. 5.NF.5 Interpret multiplication as scaling (resizing), by: a. Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication. b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n!a)/(n!b) to the effect of multiplying a/b by 1. 5.NF.6 Solve real-world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem. 5.NF.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.




5.NF.4 5.NF.5

Throughout the unit, use academic language such as: fraction bar, fractional part, product, “of” refers to multiplication, expression/equation, model, estimation,

Day 1 Investigations Unit 4 CCSS Lesson 4A.1


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equivalence, area model, array, line plot, range ) Connect previous understandings of multiplication

of whole numbers to multiplication of a whole number by a fraction

) Interpret the product by using a visual fraction model.

5.NF.4 5.NF.5 5.NF.6

) Connect previous understandings of multiplication of whole numbers to multiplication of a whole number by a fraction


) Interpret multiplication as scaling (resizing) i.e., compare the size of a product to the size of one factor.

Day 2 Investigations Unit 4 CCSS Lesson 4 A.2 Discussion on CC36 addresses ideas in 5.NF.5 (See Instructional Note above.)

5.NF.4 5.NF.5 5.NF.6

) Connect previous understandings of multiplication of whole numbers to multiplication of a whole number by a mixed number

) Interpret the product by using a visual fraction model

) Solve real world problems involving mixed numbers multiplied by a whole number

) Interpret multiplication as scaling (resizing) i.e., compare the size of a product to the size of one factor.

Day 3 Investigations Unit 4 CCSS Lesson 4 A.3 *Performance Task: Students create a story problem and representation for 17 x 2 3/4 or 17 x 2 1/2 (depending on student needs)

5.NF.4 5.NF.5 5.NF.6

) Connect previous understandings of multiplication of whole numbers to multiplication of a fraction by a fraction

) Interpret the product by using a visual fraction model

) Solve real world problems involving multiplication of a fraction by a fraction

) Interpret multiplication as scaling (resizing) i.e. compare the size of a product to the size of one factor.

Day 4 Investigations Unit 4 CCSS Lesson 4 A.4 *OMIT LESSON 4A.5. Students will explore visual representations of multiplying fractions for more exploration in CMP. Do not teach rule. If you need additional scaffolding developing the algorithm- use this lesson after day 12. *Use pages 83 - 85 for optional in class or homework.


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5.NF.4 5.NF.5 5.NF.6

) Solve real world problems involving multiplication of a fraction by a fraction.


Day 5 Investigations Unit 4 CCSS Lesson 4 A.6 See appendix for additional work

5.NF.4 5.NF.5 5.NF.6


) Find area of a rectangle with fractional side lengths.


Day 6 CMP Bits and Pieces II Lesson 3.1 * Note: Focus on taking a “part of a part” in questions A to C; Discuss estimation strategies used in Part D

5.NF.4 5.NF.5 5.NF.6




Day 7 CMP Bits and Pieces II Lesson 3.2 Encourage students to look for patterns as they complete work in parts A to C. * Focus discussion on part D (p. 35 of Student Book) (Generalization regarding multiplication of whole numbers and multiplication of fractions) What is the relationship to the whole? Can you give an example to prove your thinking? *** STUDENTS MAY NOT RESOLVE THE QUESTION! Ideas will be explored in next two lessons.

5.NF.4 5.NF.5 5.NF.6

) Solve real world problems involving multiplication of fraction by a fraction.


Day 8 Floating Day Follow up with students who are still struggling to generalize patterns regarding multiplication of fractions.

5.NF.4 5.NF.5 5.NF.6

) Connect previous understandings of multiplication of whole numbers to multiplication of a mixed number by a fraction

) Solve real world problems involving mixed numbers multiplied by a fraction

Day 9 Investigations Unit 4 CCSS Lesson 4 A.7 *Performance Task: Students create story problems for page with a representation for page 90 #3-4 and page 91 #7-8

5.NF.4 5.NF.5 5.NF.6



Day 10 CMP Bits and Pieces II: Lesson 3.3 *Focus on how estimation helps to find the accurate answer.


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) Solve real world problems involving mixed numbers multiplied by a fraction

5.NF.4 5.NF.5 5.NF.6


) Interpret multiplication as scaling (resizing) i.e. compare the size of a product to the size of one factor

Day 11 CMP Bits and Pieces II: Lesson 3.4 * Focus discussion on parts B and D. In B, students are using diagrams to represent multiplication. The discussion in part D continues to develop the understanding of multiplication as scaling (resizing). * Continue to link diagrams to equations.

5.NF.4 5.NF.5 5.NF.6

) Connect previous understandings of multiplication of whole numbers to multiplication of a mixed number or fraction by a fraction

) Develop an algorithm for multiplying fractions

Day 12 CMP Bits and Pieces II: Lesson 3.5 * Use CCSS Lesson 4 A.5 for additional scaffolding if needed for students to develop algorithm for multiplying fractions

5.NF.4 5.NF.5 5.NF.6

) Analyze real-world problems involving fractions

Day 13 Floating day to finish up CMP work Students could also work on the following illustrative math activity about when to multiply fractions. They are asked to analyze a variety of math word problems and decide whether the problem is a fraction multiplication problem. This would be a good opportunity for students to practice Math Practice 3 (Construct Viable Arguments). http://www.illustrativemathematics.org/illustrations/609

5.NF.7 ) Connect previous understandings of division to divide by whole number by a unit fraction

Day 14 and Day 15

Investigations Unit 4 CCSS Lesson 4 A. 8 * Students will need additional paper to encourage the use of visual representations. Focus on connection to diagrams/visuals as students do this work.

5.NF.7 ) Connect previous understandings of division to divide by unit fractions by a whole number

Day 16 and Day 17

Investigations Unit 4 CCSS Lesson 4 A. 9 * Students will need additional paper to encourage the use of visual representations. Focus on connection to diagrams/visuals as students do this work.

5.NF.7 ) Connect previous understandings of division to divide by whole number by a unit fraction

) Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions

Day 18 Investigations Unit 4 CCSS Lesson 4 A.10


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) Multiply and divide with fractions Day 19 Practice pages with multiplication and division story problems (appendix)

5.MD.2 ) Create a line plot to display a data set of measurements in fractions of a unit

) Use operations when interpreting line plot with fractions

Days 20-24

Investigations Unit 9 CCSS 1.5A and 1.6A: Line Plot with Fractions (adding, subtracting, dividing unit fractions) Take out problems with mean, median, mode, which are not part of 5th grade standards *Range - Students determine the difference between the minimum and the maximum values in a set of data Day 20-21 CCSS Lessons 1.5A and 1.6A Day 23 Extension Questions for Lessons 1.5A and 1.6A (appendix) Day 24 Class creates data set or use the data set in appendix for Operating with Fractions



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Unit of Study 7: 2-D Shapes and Coordinate Geometry Primary Curricular Resources: Investigations Unit 7: Measuring Polygons Coordinate Grid Materials (appendix) Positive and Negative Integers (appendix) Estimated Instructional Time: 17 days (and an additional two days for the mathematics MCAS) April 28 - May 22

Overarching Questions: ) Using the attributes of side length, angle size

and parallel sides, how can I describe, compare, and classify two-dimensional shapes including triangles and quadrilaterals?

) How can I use a coordinate grid to solve real world problems?

Standards for Mathematical Practice Focus MP 5: Use appropriate tools strategically: Mathematically proficient fifth graders consider the available tools (including estimation) when solving a mathematical problem and decide when certain tools might be helpful. They use graph paper to accurately create graphs and solve problems or make predictions from real world data. MP 6: Attend to precision. Students will specify attributes and classify shapes with clear explanations and use of appropriate academic language.

Instructional Notes: The Geometry Progressions recommend and PARCC expects that students use the inclusive definition of trapezoids, i.e. as a “quadrilateral with at least one pair of parallel sides”. This means that a parallelogram would be considered a trapezoid. We have included the diagram from the Progressions document, which represents this definition, in the appendix.

Concepts developed in this unit:

! Understand attributes belonging to 2-D figures and classify triangles and quadrilaterals based on their properties

! Understand how to use a coordinate grid to locate ordered pairs

! Represent real world and geometric problems using the coordinate grid

Prior knowledge expected: 4.G.1 Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures. 4.G.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. 4.G.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.

Learning Outcomes 5.G.1. Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x-axis and x-coordinate, y-axis and y-coordinate). 5.G.2. Represent real-world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values


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of points in the context of the situation. 5.G.3. Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles. 5.G.4. Classify two-dimensional figures in a hierarchy based on properties.



5.G.3 5.G.4

Throughout unit, use academic language such as: scalene, isosceles, equilateral, obtuse triangle, acute triangle, polygon, parallel, perpendicular, line, ray, line segment, quadrilateral, parallelogram, trapezoid, square, rhombus, kite, rectangle, attributes, properties, classify, category, subcategory, coordinate grid, axis

) Classify triangles by the sizes of their angles

) Classify triangles by the lengths of their sides.

Day 1 Investigations Unit 5 Measuring Polygons Lesson 1.1

) Use transparency shape cards.

) Reinforce/review vocabulary for classifying: acute, obtuse, right, scalene, equilateral, isosceles.

) Create chart from teacher note, page 29 “ways to describe triangles”

) Wrap up lesson with activity 3- guess my rule

5.G.3 5.G.4

) Identify attributes belonging to a category of two dimensional figures also belonging to all subcategories of that category

) Identify attributes of quadrilaterals to describe and compare

Day 2 Investigations Unit 5 Measuring Polygons Lesson 1.2 * Create chart “types of quadrilaterals” (see teacher guide page 35) (omit logo paths)

5.G.3 5.G.4

) Identify attributes belonging to a category of two dimensional figures as also belonging to all subcategories of that category

) Identify attributes of quadrilaterals to describe and compare

Day 3 Investigations Unit 5 Measuring Polygons Lesson 1.3 Relationships among Quadrilaterals (2 days) ) Page 10 (student work) should be extended to develop

math practice 3 (construct viable arguments and critique the reasoning of others)

) Students should be asked to state and defend their claim based on the 5 statements found on page 10

) Students play “Guess my rule” to scaffold towards the hierarchy categories

* Optional use of D&I Guide, page R39 for extra practice

5.G.3 ) Classify two-dimensional figures in a hierarchy based on properties.

Day 4 Investigations Unit 5 Measuring Polygons


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5.G.4 ) Identify attributes belonging to a category of two-dimensional figures also belonging to all subcategories of that category (i.e., all rectangles have four right angles and squares are rectangles, so all squares have four right angles)

Lesson 1.3 - Finish lesson ) Based on work from previous day, students will work on

discovering a way to create a Venn diagram that could be used to categorize shapes based on attributes. Students can use the power polygons to help with creating categories.

) The goal is to work towards the hierarchy as seen in the Progressions Note (appendix)

) Discussion: Categories and creation of a classroom anchor chart with shape hierarchies

) Students continue playing “Guess My Rule” and applying new understandings of shape hierarchy

5.G.3 5.G.4

) Classify two-dimensional figures in a hierarchy based on properties.

) Identify attributes belonging to a category of two-dimensional figures also belonging to all subcategories of that category (i.e., all rectangles have four right angles and squares are rectangles, so all squares have four right angles)

Day 5 Investigations Unit 5 Measuring Polygons

) Lesson 1.4 - Use practice pages from remainder of lessons from unit 1

) Use the following pages for practice/independent work time: 12, 15, 20, 23 and 25

5.G.3 5.G.4



Day 6 Guess My Shape Lesson Plan (appendix) Use recording sheet to check for understanding * Use pages R37-40 from the D&I Guide for additional practice identifying polygons attributes and angle measurements; depending on your students’ needs, add more days to play and practice game * Optional use of D&I Guide, page R37

5.G.3 5.G.4



Day 7 Investigations Unit 5 Measuring Polygons Mini-assessment: Assessment page M17 #1 (unit 5) Students defend claims about the categorization of quadrilaterals: Illustrative Math Activity (trapezoids vs. parallelograms): http://www.illustrativemathematics.org/illustrations/1505

5.G.1 5G.2

) Graph points on coordinate plane to solve real world math problems

) Represent real world math problems on first quadrant of

Day 8 Coordinate Grid Introduction (appendix) Use pages in appendix - Students practice color coding and identifying parts of the coordinate grid.


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coordinate plan ) Interpret coordinate values of points in a story context

Focus on the following questions: Why is it important? Why do we use it in the real world?

5.G.2 ) Graph points on coordinate plane to solve real world math problems

) Interpret coordinate values of points in a story context

Day 9 - Day 10

Coordinate Grid Part 1 and Part 2 (appendix)

5.G.1 5.G.2

) Graph points on coordinate plane to solve real world math problems

) Represent real world math problems on first quadrant of coordinate plan

) Interpret coordinate values of points in a story context

Day 11 and Day 12

Complete the “Polygon Mystery” activity (appendix). This activity is a combination of both polygon knowledge and understanding of the coordinate grid. Coordinate Grid Battleship Game: http://www.illustrativemathematics.org/illustrations/48

MA 1 ) Students use negative and positive integers to describe quantities such as temperature above/below zero, elevation below/above sea level or credit/debit.

Day 13 and Day 14

Positive and Negative Integers (appendix)

5.OA.3 5.G.1 5.G.2

) Students recognize numerical patterns using two given rules

) Use data sets to graph points on coordinate grids

Day 15 and Day 16

Numerical Patterns and Coordinate Grids (appendix) * Optional use of D&I Guide, pages R66 and R69 for practice with numerical patterns



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Unit of Study 8: Strengthening Fluency Primary Curricular Resources: Interpreting and Evaluating Expressions (appendix) Strengthening Fluency with Whole Number Multiplication (appendix) Strengthening Fluency with Decimal Division (appendix) Estimated Instructional Time: 12 days (There are an additional four days for the end-of-year review and assessment) May 23 - June 16

Overarching Questions: ! How can the U.S. standard algorithm for multiplication of whole numbers be used for 3 and 4-digit numbers? ! What are the strategies for dividing decimals? ! How do I interpret and/or evaluate expressions with fractions and decimals, including those with parentheses, brackets, or braces?

Standards for Mathematical Practice Focus MP 5: Use appropriate tools strategically: Mathematically proficient fifth graders consider the available tools (including estimation) when solving a mathematical problem and decide when certain tools might be helpful. MP 6: Attend to precision. As students record and explain their equations and strategies, they use clear, concise notation and accurate representations

Instructional Notes: The work with expressions builds upon work students did with whole numbers in units of study #2 and #3. It is also an opportunity to continue to develop fluency with fraction and decimal operations. Students will extend their fluency with the standard algorithm for multiplication to 3-digit by 3-digit and 4-digit by 3-digit problems (PARCC expectation). Students will develop strategies for division with decimals with quotients in tenths and hundredths (PARCC expectation). The number of days per section is suggested below. Please use the materials in ways that meet your students’ needs.

Concepts developed in this unit:

! Evaluate, and interpret without evaluating, numerical expressions involving decimals and fractions, including those with parentheses, brackets, or braces

! Fluency with multiplication of whole numbers using the standard algorithm

! Dividing with decimals

Prior knowledge expected: Students multiplied up to 3-digit x 2-digit numbers at the beginning of the year. Students divided with decimals earlier in the year. Students worked with fraction and decimal operations in earlier units of study. Students worked with expressions with whole numbers earlier in the year.

Learning Outcomes 5.OA.1: Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. 5.OA.2. Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “Add 8 and 7, then multiply by 2” as 2 ! (8 + 7). Recognize that 3 ! (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.


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5.NBT.5. Fluently multiply multi-digit whole numbers using the standard algorithm. 5.NBT.7. Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.



5.OA.1 5.OA.2

) Use parentheses, brackets, or braces in numerical expressions, which include fractions and decimals, and evaluate expressions with these symbols.

) Write simple expressions that record calculations with numbers, including decimals and fractions, and interpret numerical expressions without evaluating them.

Days 1- 4

Interpreting and Evaluating Expressions (appendix)

5.NBT.5 ) Multiply up to 3-digit by 4-digit numbers, using the U.S. standard algorithm

Days 5-8

Strengthening Fluency with Whole Number Multiplication (appendix)

5.NBT.7 ) Solve decimal division problems with quotients in the tenths and hundredths

Days 9- 12

Strengthening Fluency with Decimal Division (appendix)

boston public schools, 2013-2014 grade 5...

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