Elementary Mathematics

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1 Mobile County Public School System Division of Curriculum & Instruction Instructional Planning Guide Elementary Mathematics 5th Grade Unit 3: Adding, Subtracting, and Dividing Fractions This document is provided by MCPSS as a resource to assist teachers with understanding the CCRS and planning instruction. This document is available on the MCPSS website and will also be provided in the MCPSS LiveBinders. The order in which the units are organized is based on the progression and arrangement of standards on Jason Zimba s Wire Graph of the CCSSM. While they can be taught in a different order, they are most effective in the order provided. 1

2 Yearly Pacing Guide 5.OA.1 5.OA.2 5.NBT.4 5.NBT.5 5.NBT.6a 5.NBT.6b 5.NBT.7 5.NBT.8 5.NBT.9 5.NBT.10 Quarter 1 Quarter 2 Quarter 3 Quarter 4 5.OA.1 5.OA.1 5.OA.2 5.OA.2 5.NF.13 5.NF.11 5.NF.14a 5.NF.12 5.NF.14b 5.NF.17a 5.NF.15a 5.NF.17b 5.NF.15b 5.NF.17c 5.NF.16 5.MD.19 Standards for This Unit COS # 5.OA.1 5.OA.2 5.NF.11 5.NF.12 5.NF.17 5.OA.3 5.MD.18 5.MD.20a 5.MD.20b 5.MD.21 5.MD.22a 5.MD.22b 5.MD.22c 5.G.23 5.G.24 5.G.25 5.G.26 COS Standard Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. [5-OA1] Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. [5-OA2] Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. [5-NF1] Example: 2 / / 4 = 8 / / 12 = 23 / 12. (In general, a / b + c / d = (ad + bc) / bd.) [5.NF.1] 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. [5-NF2] Example: Recognize an incorrect result 2 / / 2 = 3 / 7 by observing that 3 / 7 < 1 / 2. [5.NF.2] Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. (Students able to multiply fractions in general can develop strategies to divide fractions in general by reasoning about the relationship between multiplication and division. However, division of a fraction by a fraction is not a requirement at this grade.) [5-NF7] a. Interpret division of a unit fraction by a nonzero whole number, and compute such quotients. Example: Create a story context for ( 1 / 3) 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that ( 1 / 3) 4 = 1 / 12 because ( 1 / 12) x 4 = 1 / 3. [5-NF7a] b. Interpret division of a whole number by a unit fraction, and compute such quotients. Example: Create a story context for 4 ( 1 / 5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ( 1 / 5) = 20 because 20 x ( 1 / 5) = 4. [5-NF7b] c. Solve real-world problems involving division of unit fractions by nonzero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. Examples: How much chocolate will each person get if 3 people share 1 / 2 lb of chocolate equally? How many 1 / 3 -cup servings are in 2 cups of raisins? [5-NF7c] 2

3 Make a line plot to display a data set of measurements in fractions of a unit ( 1 2, 1 4, 1 8 ). 5.MD.1 9 Use operations on fractions for this grade to solve problems involving information presented in line plots. Example: Given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally. [5-MD2] Student Outcomes from ALSDE INSIGHT Tool Students understand that: Students know: Students can: There are conventions in mathematics such as, order of operations, that are arbitrary but have been agreed to for communication purposes, Mathematical symbols in expressions communicate the order of operations. The operations of addition, subtraction, multiplication, and division all arise in multiple contexts (See Tables 1 and 2 for contexts), Mathematical symbols in equations communicate the order of operations. Two fractions are equivalent if they are the same size share of the same whole or are the same point on a number line, Addition and subtraction of fractions are applied to fractions referring to the same whole, The unit fraction (1/b) names the size of the unit with respect to the referenced whole, and that the numerator counts the parts referenced and the denominator tells the number of parts into which the whole was partitioned, The operations of addition and subtraction are performed on counts with like names/labels/denominators and that the sum or difference retains the same name/label/denominator. Addition and subtraction of fractions are applied to Strategies for rewriting numerical expressions that contain parentheses, brackets, and/or braces in equivalent forms that do not contain grouping symbols. Conventions for using parentheses, brackets and/or braces in writing numerical expressions. Meanings of operations (addition, subtraction, multiplication, and division) and conventions for grouping symbols. Strategies for generating equivalent fractions, Strategies for adding fractions with like denominators. Characteristics of addition and subtraction contexts for whole numbers and fractions (Table 1). Strategies for representing and solving addition Efficiently apply strategies for rewriting and evaluating expressions that contain parentheses, brackets, and/or braces. Write multi-step numerical expressions involving all four operations using parentheses, brackets, and/or braces to convey the desired order of operations, Evaluate multi-step numerical expressions involving all four operations and parentheses, brackets, and/or braces by applying conventions for order of operations. Use logical reasoning and mathematical vocabulary to interpret the meaning of mathematical expressions involving more than one operation. Given verbal mathematical contexts involving multiple operations, Write and interpret the corresponding numerical expressions. e.g., express the calculation "add 8 and 7, then multiply by 2" as 2 x (8 + 7) Given numerical expressions involving multiple operations, Explain the meaning of the expression without performing indicated calculations (e.g., recognize that 3 x ( ) is three times as large as , without having to calculate the indicated sum or product). Generate equivalent fractions using the generalization a/b = (n x a)/(n x b), Given a variety of addition and subtraction problems involving fractions and mixed numbers with unlike denominators, Find the sums or differences by finding an equivalent sum or difference of fractions with like denominators. Represent quantities (whole numbers and fractions) and operations (addition and subtraction) physically, pictorially, or symbolically, Strategically choose and apply a variety of representations to solve addition and subtraction word problems involving fractions, Use symbols to represent unknown quantities in addition and subtraction equations and solve such equations Accurately computes sums and differences of fractions, 3

4 fractions referring to the same whole; The operation of addition with whole numbers and/or fractions represents both putting together and adding to contexts; The operation of subtraction with whole numbers and/or fractions represents taking apart, taking from, and additive comparison contexts; The operations of addition and subtraction are performed on counts with like names/labels/denominators and that the sum or difference retains the same name/label/denominator; Connections between representations and symbols provide justifications for solutions and solution paths, Properties of operations allow manipulation of mathematical expressions for sense making and easier computation, The operation of division represents; contexts of partitioning into equal-sized shares, contexts of partitioning equally among a given number of groups, or contexts involving multiplicative comparisons. Questions concerning mathematical contexts (in particular, measurement contexts) can be generated and answered by collecting, organizing and analyzing data and data displays. and subtraction problems involving fractions, Strategies for generating equivalent fractions, Strategies for estimating sums and differences of fractions. Properties of operations, Contexts for division of fractions and whole numbers, Strategies for using visual models (e.g., manipulatives, diagrams, pictures) to solve division problems that involve fractions and whole numbers. Techniques for constructing line plots, Standard units and the related tools for measuring, Strategies for adding, subtracting, multiplying, and dividing fractions. Use logical reasoning and connections among representations to justify solutions, the reasonableness of solutions, and solution paths. Given word problems involving the addition or subtraction of fractions referring to the same whole, including cases of unlike denominators, Explain and justify solutions and the reasonableness of solutions using connections among unit fractions, bench mark fractions, visual representations, and understanding of operations (e.g., addition and subtraction). Strategically choose and apply visual models to represent and solve problems involving the division of unit fractions by nonzero whole numbers, or whole numbers by unit fractions, Accurately compute quotients of unit fractions and whole numbers using models, Use logical reasoning to communicate connections between visual models and computational procedures for problems involving division of unit fractions and whole numbers. Given a story context involving division of a unit fraction by a non-zero whole number, or division of a whole number by a unit fraction, Use visual models and properties of operations to explain and represent the division problem context, find the quotient, and explain the solution's relationship to the given multiplication problem. Given a problem involving division of a unit fraction by a nonzero whole number, or division of a whole number by a unit fraction, Create a corresponding story context, a model to represent the division context, and accurately solve the problem. Students are able to: Use standard units and the related tools to make measurements to the nearest eighth unit, Organize and represent measurement data on a line plot, Choose and apply appropriate strategies to solve problems generated by conjectures from examining data displays, Communicate justification for strategy choice and solutions to problems involving measurements, Apply strategies for solving problems involving all four operations with fractions. Make and use line plots (scale to match unit of measure) to represent data generated by making measurements (to the nearest eighth unit) of several objects or by making repeated measurements, Use information from the constructed line plots to generate questions and solve problems including problems that involve all four operations with fractions. 4

5 Coming From and Going To Standards for this unit are built on: 4.NF.12 Explain why a fraction a is equivalent to a fraction ( nxa ) by using b ( nxb) visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. [4-NF1] 4.NF.13 Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators or by comparing to a benchmark fraction such as 1. Recognize that comparisons 2 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. [4-NF2] 4.NF.14 Understand a fraction a with a > 1 as a sum of fractions 1. [4- b b NF3] a. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. [4-NF3a] 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. [4-NF3b] 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. [4-NF3c] 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. [4-NF3d] 4.NF.15 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. [4-NF4] a. Understand a fraction a b as a multiple of 1. [4-NF4a] b Standards in this unit are building to: 6.EE.18 Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q, and x are all nonnegative rational numbers. 6.SP.25 Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. [6-SP1] 6.SP.26 Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape. [6-SP2] 6.SP.27 Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number. [6-SP3] b. Understand a multiple of a as a multiple of 1, and use this b b understanding to multiply a fraction by a whole number. [4-NF4b] 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. [4-NF4c] 5

6 The Content Standards in this Unit CLUSTER #1 Write and interpret numerical expressions 5.OA.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. [5.OA1] This standard builds on the expectations of third grade where students are expected to start learning the conventional order. Students need experiences with multiple expressions that use grouping symbols throughout the year to develop understanding of when and how to use parentheses, brackets, and braces. First, students use these symbols with whole numbers. Then the symbols can be used as students add, subtract, multiply and divide decimals and fractions. Examples: ( ) 4 Answer: 11 {[2 x (3+5)] 9} + [5 x (23-18)] Answer: (0.4 x 2) Answer: 11.2 (2 + 3) x ( ) Answer: Answer: 5 1/6 2 3 {80 [2 x (3 ½ + 1 ½ ) ] }+ 100 Answer: 108 To further develop students understanding of grouping symbols and facility with operations, students place grouping symbols in equations to make the equations true or they compare expressions that are grouped differently. Examples: = (7 2) = 10 3 x = 22 [3 x (125 25)] + 7 = = 2 x ½ 24 [(12 6) 2] = (2 x 9) + (3 ½) Compare 3 x and 3 x (2 + 5) Compare and 15 (6 + 7) 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 x (8 + 7). Recognize that 3 x ( ) is three times as large as , without having to calculate the indicated sum or product. [5-OA1] This standard refers to expressions. Expressions are a series of numbers and symbols (+, -, x, ) without an equals sign. Equations result when two expressions are set equal to each other (2 + 3 = 4 + 1) Example: 4(5 + 3) is an expression. When we compute 4(5 + 3) we are evaluating the expression. The expression equals 32. 4(5 + 3) = 32 is an equation. This standard calls for students to verbally describe the relationship between expressions without actually calculating them. This standard calls for students to apply their reasoning of the four operations as well as place value while describing the relationship between numbers. The standard does not include the use of variables, only numbers and signs for operations. Examples: Write an expression for the steps double five and then add 26. Student: (2 x 5) + 26 Describe how the expression 5(10 x 10) relates to 10 x 10. Student: The expression 5(10 x 10) is 5 times larger than the expression 10 x 10 since I know that 5(10 x 10) means that I have 5 groups of (10 x 10). Students use their understanding of operations and grouping symbols to write expressions and interpret the meaning of a numerical expression. Examples: Students write an expression for calculations given in words such as divide 144 by 12, and then 6

7 subtract 7/8. They (144 12) 7/8. Students recognize that 0.5 x (300 15) is ½ of (300 15) without calculating the quotient. CLUSTER #2: Use equivalent fractions as a strategy to add and subtract fractions. 5.NF.11 Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. [5-NF1] Example: 2 /3 + 5 /4 = 8 / /12 = 23 /12. (In general, a /b + c /d = (ad + bc) /bd.) This standard calls for students to build on the work from fourth grade where students add fractions with like denominators. In fifth grade, the example provided in the standard has students find a common denominator by finding the product of both denominators. For 1/3 + 1/6, a common denominator is 18, which is the product of 3 and 6. This process should be introduced using visual fraction models (area models, number lines, etc.) to build understanding before moving into the standard algorithm. Students should apply their understanding of equivalent fractions developed in fourth grade and their ability to rewrite fractions in an equivalent form to find common denominators. They should know that multiplying the denominators will always give a common denominator but may not result in the smallest denominator. Examples: Example: Present students with the problem 1/3 + 1/6. Encourage students to use the clock face as a model for solving the problem. Have students share their approaches with the class and demonstrate their thinking using the clock model. 5.NF.12 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. [5-NF2] This standard refers to number sense, which means students understanding of fractions as numbers that lie between whole numbers on a number line. Number sense in fractions also includes moving between decimals and fractions to find equivalents, also being able to use reasoning such as 7/8 is greater than3/4 because 7/8 is missing only 1/8 and 3/4 is missing 1/4 so 7/8 is closer to a whole. Also, students should use benchmark fractions to estimate and examine the reasonableness of their answers. Example, 5/8 is greater than 6/10 because 5/8 is 1/8 larger than ½ (4/8) and 6/10 is only 1/10 larger than 1/2 (5/10). 1) Your teacher gave you 1/7 of the bag of candy. She also gave your friend 1/3 of the bag of candy. If you and your friend combined your candy, what fraction of the bag would you have? Estimate your answer and then calculate. How reasonable was your estimate? 7

8 Student 1: 1/7 is really close to 0. 1/3 is larger than 1/7, but still less than 1/2. If we put them together we might get close to 1/2. 1/7 + 1/3= 3/21 + 7/21 = 10/21. The fraction does not simplify. I know that 10 is half of 20, so 10/21 is a little less than ½. Student 2: 1/7 is close to 1/6 but less than 1/6, and 1/3 is equivalent to 2/6, so I have a little less than 3/6 or ½. 2) Jerry was making two different types of cookies. One recipe needed ¾ cup of sugar and the other needed 2 3 cup of sugar. How much sugar did he need to make both recipes? Mental Estimation: A student may say that Jerry needs more than 1 cup of sugar but less than 2 cups. An explanation may compare both fractions to ½ and state that both are larger than ½ so the total must be more than 1. In addition, both fractions are slightly less than 1 so the sum cannot be more than 2. Area Model Linear Model Example: Using a bar diagram Melisa had 2 1/3 candy bars. She promised her brother that she would give him ½ of a candy bar. How much will she have left after she gives her brother the amount she promised? If Mary ran 3 miles every week for 4 weeks, she would reach her goal for the month. The first day of the first week she ran 1 ¾ miles. How many miles does she still need to run the first week? o Using addition to find the answer: 1 ¾ + n = 3 o A student might add 1 ¼ to 1 ¾ to get to 3 miles. Then he or she would add 1/6 more. Thus 1 ¼ miles + 1/6 of a mile is what Mary needs to run during that week. 8

9 CLUSTER #3: Apply and extend previous understandings of multiplication and division to multiply and divide fractions. 5.NF.17 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. (Students able to multiply fractions in general can develop strategies to divide fractions in general by reasoning about the relationship between multiplication and division. However, division of a fraction by a fraction is not a requirement at this grade.) [5-NF7] 5.NF.17a Interpret division of a unit fraction by a nonzero whole number, and compute such quotients. Example: Create a story context for ( 1 / 3) 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that ( 1 / 3) 4 = 1 / 12 because ( 1 / 12) x 4 = 1 / 3. [5-NF7a] 5.NF.17b Interpret division of a whole number by a unit fraction, and compute such quotients. Example: Create a story context for 4 ( 1 / 5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ( 1 / 5) = 20 because 20 x ( 1 / 5) = 4. [5-NF7b] 5.NF.17c Solve real-world problems involving division of unit fractions by nonzero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. Examples: How much chocolate will each person get if 3 people share 1 / 2 lb of chocolate equally? How many 1 / 3 -cup servings are in 2 cups of raisins? [5-NF7c] 9

10 5.NF.17 is the first time that students are dividing with fractions. In fourth grade students divided whole numbers, and multiplied a whole number by a fraction. In fifth grade, students experience division problems with whole number divisors and unit fraction dividends (fractions with a numerator of 1) or with unit fraction divisors and whole number dividends. For example, the fraction 3/5 is 3 copies of the unit fraction 1/5. 1/5+1/5+1/5=3/5 = 1/5 x 3 or 3 x 1/5 Students extend their understanding of the meaning of fractions, how many unit fractions are in a whole, and their understanding of multiplication and division as involving equal groups or shares and the number of objects in each group/share. In sixth grade, they will use this foundational understanding to divide into and by more complex fractions and develop abstract methods of dividing by fractions. 5.NF.7a This standard asks students to work with story contexts where a unit fraction is divided by a non-zero whole number. Students should use various fraction models and reasoning about fractions. Examples: 1) Create a story context for 5 1/6. Find your answer and then draw a picture to prove your answer and use multiplication to reason about whether your answer makes sense. How many 1/6 are there in 5? 2) Knowing the number of groups/shares and finding how many/much in each group/share. Four students sitting at a table were given 1/3 of a pan of brownies to share. How much of a pan will each student get if they share the pan of brownies equally? The diagram shows the 1/3 pan divided into 4 equal shares with each share equaling 1/12 of the pan. 5.NF.7c Extends students work from other standards in 5.NF.7. Students should continue to use visual fraction models and reasoning to solve these real-world problems. Examples: 1) How many 1/3-cup servings are in 2 cups of raisins? Student: I know that there are three 1/3 cup servings in 1 cup of raisins. Therefore, there are 6 servings in 2 cups of raisins. I can also show this since 2 divided by 1/3 = 2 x 3 = servings of raisins. 2) Knowing how many in each group/share and finding how many groups/shares Angelo has 4 lbs of peanuts. He wants to give each of his friends 1/5 lb. How many friends can receive 1/5 lb of peanuts? A diagram for 4 1/5 is shown below. Students explain that since there are five fifths in one whole, there must be 20 fifths in 4 lbs. 10

11 3) How much rice will each person get if 3 people share 1/2 lb of rice equally? o A student may think or draw ½ and cut it into 3 equal groups then determine that each of those part is 1/6. o A student may think of ½ as equivalent to 3/6. 3/6 divided by 3 is 1/6. It s important that students represent the problems they are solving, have a visual image of the why behind the algorithm and can explain their reasoning. CLUSTER #4: Represent and interpret data 5.MD.19 Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots. Example: Given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally. [5.MD2] This standard provides a context for students to work with fractions by measuring objects to one-eighth of a unit. This includes length, mass, and liquid volume. Students are making a line plot of this data and then adding and subtracting fractions based on data in the line plot. Examples: 1) Students measured objects in their desk to the nearest ½, ¼, or 1/8 of an inch then displayed data collected on a line plot. How many object measured ¼? ½? If you put all the objects together end to end what would be the total length of all the objects? 2) Ten beakers, measured in liters, are filled with a liquid. The line plot above shows the amount of liquid in liters in 10 beakers. If the liquid is redistributed equally, how much liquid would each beaker have? (This amount is the mean.) Students apply their understanding of operations with fractions. They use either addition and/or multiplication to determine the total number of liters in the beakers. Then the sum of the liters is shared evenly among the ten beakers. 11

12 ASSESSMENT FOR THE UNIT Assessment Resources Suggested Pre-Assessments Grade 5: Chapter 3 Pre-Test Grade 5: Chapter 4 Pre-Test Suggested Formative Assessments MIF Chapter Quick Checks Putting on Your Thinking Cap Math Journal Grade 5, Chapter 3 Test Suggested Summative Assessments MIF Test Prep: Chapter 3 Test Prep: Chapter 4 System Unit 3 AMP 1: Addition/Subtraction of Fractions Unit 3 AMP 2: Dividing Fractions Unit 3 AMP 3: Line Plots with Fractions Assessment Samples Standard 5.OA.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. Standard 5.OA.2 Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. Standard Sample See previous samples. Sample See previous samples. Sample 1. What fraction has the same value as 2 + 5? Rename each fraction so that they a difference can be determined. 5.NF.11 Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. [5-NF1] Example: 2 / / 4 = 8 / / 12 = 23 / 12. (In general, a / b + c / d = (ad + bc) / bd.) =? =? 5. Look at the problem To solve this problem, Ynestra 5 3 wrote 9. Explain what Ynestra did to rename the fractions

13 Standard 5.NF.12 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. [5-NF2] Example: Recognize an incorrect result 2 / / 2 = 3 / 7 by observing that 3 / 7 < 1 / 2. Sample 1. Terry made a pan of lasagna. She sent 1 2 of it to her mother and she put 1 3 of it in a container for lunch the next day. The rest she put in the freezer for dinner another day. How much of the lasagna did Terry put in the freezer? 2. Jasmine made a juice mixture of 2 3 water and 3 2 apple juice. How many 4 3 cups of the juice mixture did she make? 3. Write an equation to represent the word problem: One-third of Mrs. Jones class is wearing white shirts. One-sixth of the class is wearing blue shirts. How much of the class is wearing white and blue shirts? 4. There were 3 partial pies left from the teacher luncheon. A model of the leftover pies is below. If they pies were combined, about how many whole pies would there be left? Standard 5.NF.17 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. (Students able to multiply fractions in general can develop strategies to divide fractions in general by reasoning about the relationship between multiplication and division. However, division of a fraction by a fraction is not a requirement at this grade.) [5-NF7] a. Interpret division of a unit fraction by a nonzero whole number, and compute such quotients. Example: Create a story context for ( 1 / 3) 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that ( 1 / 3) 4 = 1 / 12 because ( 1 / 12) x 4 = 1 / 3. [5- NF7a] 5. Terry wants to mix two colors of paint in a 1 gallon bucket. She has 5 6 gallon of white paint and 1 gallon of blue paint. Will Terry be able to mix the 2 paint in the gallon bucket? Why or why not? Sample 1. How many 1 are there in 5? =? 3. Write a division equation to represent the model. 4. Draw a model that represents the expression Katy is has a piece of ribbon that is 4 feet long. How many 1 foot pieces 3 of ribbon can she cut from the ribbon she has? Draw a model to show your answer. 6. Sarah is making chocolate chip cookies. To make the whole recipe, she needs 1 cup of chocolate chips. Sarah only has 1 cup. How much of the 4 whole recipe can Sarah make with the chocolate chips she has? 7. Mrs. Lewis made a cheese tray for a party. She sliced an 8 inch long block of cheddar cheese into 1 slices. How many pieces of cheddar chees did 2 Mrs. Lewis have for the cheese tray? 13

14 b. Interpret division of a whole number by a unit fraction, and compute such quotients. Example: Create a story context for 4 ( 1 / 5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ( 1 / 5) = 20 because 20 x ( 1 / 5) = 4. [5- NF7b] c. Solve real-world problems involving division of unit fractions by nonzero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. Examples: How much chocolate will each person get if 3 people share 1 / 2 lb of chocolate equally? How many 1 / 3 -cup servings are in 2 cups of raisins? [5-NF7c] 8. Four children shared 1 of a gallon of tea. Write an equation to show how 8 much tea each child received. 9. If 9 was divided by 1, would the quotient be greater than or less than 9? 3 Show with models how you know. 10. Complete the problem below. Use only the single digits from the box = = 11. Complete each number sentence with a symbol from the box. < > = ( 1 2 x 1 2 ) (2 1 2 ) 2 Standard 5.MD.19 Make a line plot to display a data set of measurements in fractions of a unit ( 1 2, 1 4, 1 8 ). Use operations on fractions for this grade to solve problems involving information presented in line plots. Example: Given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally. [5-MD2] Sample 1. A biologist measured the weight of different leaves. The information is in the table below. Draw a line plot that represents the information gathered in the table. Weight (grams) Leaf Weights Number of Leaves What is the sum of the weights of the leaves in the column with fewest leaves and the column with the most leaves? 14

15 PREPARING FOR THE UNIT Required Areas for Instruction The following topics are considered required for instruction. This means that in teaching the standards for this unit, these items are essential to the success and mastery of the content. Vocabulary and Symbols numerator denominator unit fraction benchmark fraction estimate reasonableness improper fraction mixed number equivalent x = a b ( ) [ ] < > numerical expression variables evaluate simplify equation solve order of operations exponents sum difference product quotient inverse operations For unknowns: Students should be able to understand and use symbols in the place of digits and numerals. This includes letters and other non-numerical symbols. Tools number lines fraction bars grid paper paper fractions Experiences 5.OA.1 This standard builds on the expectations of third grade where students are expected to start learning the conventional order. Students need experiences with multiple expressions that use grouping symbols throughout the year to develop understanding of when and how to use parentheses, brackets, and braces. First, students use these symbols with whole numbers. Then the symbols can be used as students add, subtract, multiply and divide decimals and fractions. To further develop students understanding of grouping symbols and facility with operations, students place grouping symbols in equations to make the equations true or they compare expressions that are grouped differently. 5.OA.2 Students should be given ample opportunities to explore numerical expressions with mixed operations. This is the foundation for evaluating numerical and algebraic expressions that will include whole-number exponents in Grade 6. There are conventions (rules) determined by mathematicians that must be learned with no conceptual basis. For example, multiplication and division are always done before addition and subtraction. Begin with expressions that have two operations without any grouping symbols (multiplication or division combined with addition or subtraction) before introducing expressions with multiple operations. Using the same digits, with 15

16 the operations in a different order, have students evaluate the expressions and discuss why the value of the expression is different. For example, have students evaluate and Discuss the rules that must be followed. Have students insert parentheses around the multiplication or division part in an expression. A discussion should focus on the similarities and differences in the problems and the results. This leads to students being able to solve problem situations which require that they know the order in which operations should take place. After students have evaluated expressions without grouping symbols, present problems with one grouping symbol, beginning with parentheses, then in combination with brackets and/or braces. Have students write numerical expressions in words without calculating the value. This is the foundation for writing algebraic expressions. Then, have students write numerical expressions from phrases without calculating them. Suggested Essential Questions How are equivalent fractions helpful when solving problems? How can a fraction be greater than 1? How can a model help us make sense of a problem? How can comparing factor size to 1 help us predict what will happen to the product? How can decomposing fractions or mixed numbers help us model fraction multiplication? How can decomposing fractions or mixed numbers help us multiply fractions? How can fractions be used to describe fair shares? How can fractions with different denominators be added together? How can looking at patterns help us find equivalent fractions? How can making equivalent fractions and using models help us solve problems? How can we describe how much someone gets in a fair-share situation if the fair share is less than 1? How can we describe how much someone gets in a fair-share situation if the fair share is between two whole numbers? How can we model an area with fractional pieces? How can we model dividing a unit fraction by a whole number with manipulatives and diagrams? How can we tell if a fraction is greater than, less than, or equal to one whole? How does the size of the whole determine the size of the fraction? What connections can we make between the models and equations with fractions? What do equivalent fractions have to do with adding and subtracting fractions? What does dividing a unit fraction by a whole number look like? What does dividing a whole number by a unit fraction look like? What does it mean to decompose fractions or mixed numbers? What models can we use to help us add and subtract fractions with different denominators? What strategies can we use for adding and subtracting fractions with different denominators? When should we use models to solve problems with fractions? How can I use a number line to compare relative sizes of fractions? How can I use a line plot to compare fractions? 16

17 Transition Activities INSTRUCTIONAL RESOURCES FOR UNIT Grade 4 Chapter 6; lessons 7-8 Calendar Math (Fills in gaps and continually supports the development of grade level primary standards) Math in Focus (Provided by MCPSS) Grade 5: Chapter 3: Lessons , Chapter 4: Lessons 4.6, 4.6a, 4.7, 4.7a Chapter 5: Lessons AMSTI (AMSTI Schools have these materials) [5.OA.1] Year One Units Number Puzzles/Multiple Towers: CC 2.4A Prisms & Pyramids: CC 1.5A, 2.4A Year Two Units Decimals on Grids & Number Lines: CC 3A.8, 3A.9 Other Units Growth Patterns: Sess [5.OA.2] Year One Units Number Puzzles/Multiple Towers: Inv. 1 - Sess , Inv. 2 - Sess. 2.4 CC 2.4A Year Two Units How Many People/Teams?: Inv. 1 - Sess. 1.2, 1.4, Inv. 3 - Sess. 3.2 Other Units Growth Patterns: Sess , [5.NF.11] Year One Units What's That Portion: Inv. 3 - Sess (Roll Around the Clock and Fraction Tracks) Learnzillion (Learnzillion.com) 5.OA.1 5.OA.2 5.NF.11 5.NF.12 5.NF.17a - c 5.MD.19 [5.NF.12] Year One Units What's That Portion: Inv. 1 - Sess. 1.1, 1.5, Inv. 2 - Sess. 2.2, 2.4, Inv. 3 - Sess Year Two Units Decimals on Grids & Number Lines: Inv. 2 - Sess. 2.2, [5.NF.17] Year One Units What's That Portion: CC 4A.1, 4A.2, 4A.3, 4A.7 a. What's That Portion: Inv. 1 - Sess. 1.5, Inv. 2 - Sess. 2.4, 2.6 CC 4A.8, 4A.10 b. What's That Portion: Inv. 1 - Sess. 1.5, Inv. 2 - Sess. 2.4, 2.6 CC 4A.9, 4A.10 c. What's That Portion: Inv. 1 - Sess. 1.5, Inv. 2 - Sess. 2.4, 2.6 CC 4A.8, 4A.9, 4A.10 Year Two Units Measuring Polygons: Inv. 2 - Sess. 2.4 c. Measuring Polygons: Inv. 2 - Sess. 2.4 Other Units How Long Can You Stand on One Foot?: CC 1.5A, 1.6A [5.MD.19] Year One Units How Long Can You Stand on One Foot?: Sess CC 1.5A, 1.6A ASLDE Using Parenthesis in Numerical Expressions or, "Does it really matter what order I do this in?" Operation to Exploration Fraction Action 17

18 Student Glossary of Terms benchmark A commonly known fraction that serves as a meaningful reference point. fraction common factor A whole number that divides two (or more) other numbers exactly. denominator The number named by the numeral below the fraction bar. difference A name of the result of subtraction. (Used for OA.2). dividend The number that is being divided. divisor The number that the dividend is being divided by. equation A mathematical sentence with an = sign. equivalent Has the same value as. estimate To find a number close to an exact amount, or a calculated guess that is close to the exact number. exponent Tells how many times the base number is used as a factor. factor fraction greater than (>) improper fraction inverse operations less than (<) mixed number numerator numeric expression order of operations product proper fraction quotient reciprocal simplify A whole number that divides exactly into another number. A number expressible in the form a/b where a is a whole number and b is a positive whole number. Representing more of a quantity. A fraction where the numerator is equal to or greater than the denominator. An operation that undoes another operation. Representing less of a quantity. A number comprising a whole-number part and a fractional part. The number named by the numeral above the fraction bar. An expression that contains only numbers and operation symbols (no equal sign). The order in which operations should be done. (See diagram below. The result of multiplying two or more quantities. A fraction where the numerator is less than the denominator. The result of a division. The inverse of a fraction A fraction is in simplest for if its numerator and denominator are relatively prime; that is, the only common factor they have is 1. solve Finding the solutions of equations or problems. sum The result of addition. unit fraction A fraction whose numerator is 1. variable A letter or symbol that represents a number or other mathematical thing. Teacher (Unit Terms) 18

19 Associative Property Commutative Property computation algorithm computation strategy Distributive Property evaluate an expression Identity Property interpret an expression (Also called the grouping property.) Only for addition and multiplication. Changing the grouping of the numbers does not change the result of an operation. (also called the order property) Only for addition and multiplication Changing the order of the number does not change the result of the operation. A set of predefined steps applicable to a class of problems that gives the correct result in every case when the steps are carried out correctly. See also: computation strategy. Purposeful manipulations that may be chosen for specific problems, may not have a fixed order, and may be aimed at converting one problem into another Multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products. The Distributive Property says that if a, b, and c are real numbers, then: a x (b + c) = (a x b) + (a x c) Substitute the values given for the variables and perform the operations according to the order of operations. (For multiplication) The identity for multiplication is the number 1. Because 1 multiplied by any number is equal to that number. (a 1 = a) Translate between words and symbols representations of algebraic expressions. 19