Fluency Trajectory for Fractions

Developed by:Danielle Seabold Kalamazoo RESAMatthew Mayer Kalamazoo RESA

Developed using the following resources:http://ime.math.arizona.edu/progressions/http://www.ncpublicschools.org/acre/standards/common-core-tools/#unmathhttp://alex.state.al.us/ccrs/node/76https://www.turnonccmath.nethttp://www.corestandards.org/Mathhttp://katm.org/wp/?page_id=91

Equipartitioning is a strand of the standards that builds from students' experiences with fair sharing to the creation of equal-sized groups or parts from evenly divisible collections or wholes. There is a framework for it in the standards but because it can be a critical foundation on which to build division, multiplication, ratio and fractions and draws on standards from both geometry and number, it is presented as a separate strand here.

EquipartitioningFirst Grade

Standard Explanation Concrete Representational Abstract1.G.3: Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.

1.G.3 is the first time students begin partitioning regions into equal shares using a context such as cookies, pies, pizza, etc... This is a foundational building block of fractions. Students should have ample experiences using the words, halves, fourths, and quarters, and the phrases half of, fourth of, and quarter of. Building on equipartitioning collections of discrete wholes leads to the idea of equipartitioning single whole circles and rectangles. Students equipartition a whole into two equal shares and name the resulting shares as, for example, "halves" or "half of". This can also be considered a form of decomposition, i.e. decomposing the whole into two equal parts. A fraction that describes each person's share, 1/n, is called a "unit fraction," and a fair share (1/n per person) is a "unit ratio". Students justify that they have a fair-share by using the three criteria for equipartitioning:

1.      Having the correct number of parts

2.      Exhausting the whole, not leaving any parts unused

3.      Having equal-sized parts

Build-it Draw-it Write-it

Partition circles and rectangles into two and four equal shares.

Partition circles and rectangles into two and four equal shares by drawing in the lines to show the partition(s).

Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of or four of the shares.

Second GradeStandard Explanation Concrete Representational Abstract

2.G.3: Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

It is important for young children to have experiences actually marking, folding, and/or cutting the shapes into fair shares. Students find halves and fourths far easier than odd-numbered splits. Observations have shown that with the circle, students tend to be more successful with six splits than three splits because they can start by splitting the circle in half and then working with a half to split it into thirds. Presenting students the challenge of odd-numbered splits helps them develop their understanding of how to start from the center to the edge, making a "radial cut," as opposed to starting at one edge of the circle to make a diameter all the way across.

Build-it Draw-it Write-itPartition circles and rectangles into two, three, or four equal shares by using objects.

Partition circles and rectangles into two, three, or four equal shares by using pictures and drawing in the lines to show the partition.

Partition circles and rectangles into two, three, or four equal shares writing in the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths.

Third GradeStandard Explanation Concrete Representational Abstract

3.G.2: Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole.

This standard builds on students‘ work with fractions and area. Students are responsible for partitioning shapes into halves, thirds, fourths, sixths and eighths.

Build-it Draw-it Write-itPartition shapes into parts with equal areas.

Partition shapes into parts with equal areas by drawing in the partition(s)

Express the area of each part as a unit fraction of the whole.

Standard Explanation Concrete Representational Abstract3.NF.1: Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.

3.NF.1 refers to the sharing of a whole being partitioned. Fraction models in third grade include only area (parts of a whole) models (circles, rectangles, squares) and number lines. Set models (parts of a group) are not addressed in Third Grade.

Build-it Draw-it Write-itUnderstand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; by using concrete models to represent the problem.

Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; by using visual fraction models or illustrations to represent the problem.

Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; by using equations to represent the problem.

Fifth GradeStudents should also create story contexts to represent problems involving division of whole numbers.

Standard Explanation Concrete Representational Abstract5.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.

Fifth grade student should connect fractions with division, understanding that 5 ÷ 3 = 5/3 Students should explain this by working with their understanding of division as equal sharing.

Built-it Draw-it Write-itInterpret 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 fraction models to represent the problem.

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 to represent the problem.

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 equations to represent the problem.

Examples: If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? This can be solved in two

ways. First, they might partition each pound among the 9 people, so that each person gets 50 x 1/9 = 50/9 pounds. Second, they might use the equation 9 x 5= 45 to see that each person can be given 5 pounds, with 5 pounds remaining. Partitioning the remainder gives 5 5/9 pounds for each person.

Ten team members are sharing 3 boxes of cookies. How much of a box will each student get? When working this problem a student should recognize that the 3 boxes are being divided into 10 groups, so s/he is seeing the solution to the following equation, 10 x n = 3 (10 groups of some amount is 3 boxes) which can also be written as n = 3 ÷ 10. Using models or diagram, they divide each box into 10 groups, resulting in each team member getting 3/10 of a box.

Two afterschool clubs are having pizza parties. For the Math Club, the teacher will order 3 pizzas for every 5 students. For the student council, the teacher will order 5 pizzas for every 8 students. Since you are in both groups, you need to decide which party to attend. How much pizza would you get at each party? If you want to have the most pizza, which party should you attend?

Your teacher gives 7 packs of paper to your group of 4 students. If you share the paper equally, how much paper does each student get?

Each student receives 1 whole pack of paper and ¼ of the each of the 3 packs of paper. So each student gets 1 ¾ packs of paper.

Fractions have their roots in equipartitioning. A fraction is always defined relative to the whole to which it refers. Young students learn this as they equipartition collections and single wholes. They begin to understand “½” as a description of half of the whole collection or as a description of one-half of a single whole, shared fairly between 2 people. They know that the whole collection or single whole are each 2 times as large as that fair share. This Fluency Trajectory focuses on the transition from unit fractions to fractions in the form of a/b in Grade 3, then on equivalence and comparison of fractions, leading up to operations with fractions in Grades 4-5. In Grades 1 and 2, students use fraction language to describe partitions of shapes into equal shares. In Grade 3 they start to develop the idea of a fraction more formally, building on the idea of partitioning a whole into equal parts. Grade 4 students learn a fundamental property of equivalent fractions: multiplying the numerator and denominator of a fraction by the same non-zero whole number results in a fraction that represents the same number as the original fraction. This property forms the basis for much of their other work in Grade 4, including the comparison, addition, and subtraction of fractions and the introduction of finite decimals. In Grade 5, they connect fractions with division, understanding that a

b = a ÷ b for whole numbers a and b, with b not equal to zero.

Fractions – Third GradeThe Meaning of Fractions

Set models (parts of a group) are not explored in Third Grade.Standard Explanation Concrete Representational Abstract

3.NF.1: Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.

In 3.NF.1 students should focus on the concept that a fraction is made up (composed) of many pieces of a unit fraction, which has a numerator of 1. This standard refers to the sharing of a whole being partitioned or split. Fraction models in third grade include area (parts of a whole) models (circles, rectangles, squares) and number lines.

Build-it Draw-it Write-itUnderstand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts be using objects to directly model what is happening.

Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts by illustrating the situation.

Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts using derived facts.

Number Lines and Diagram - FractionsStandard Explanation Concrete Representational Abstract

3.NF.2.a: Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line.

In 3.NF.2a, when you examine the number line diagram below, the space between 0 and 1 is divided (partitioned) into 4 equal regions. The distance from 0 to the first segment is 1 of the 4 segments from 0 to 1 or ¼. Students need ample experiences folding linear models (e.g., string, sentence strips) to help them reason about and justify the location of fractions.

Build-it Draw-it Write-itRepresent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts using a physical number line or concrete objects as a number line.

Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts.

The number line diagram is the first time students work with a number line for numbers that are between whole numbers (e.g., that ½ is between 0 and 1). Students need ample experiences folding linear models (e.g., string, sentence strips) to help them reason about and justify the location of fractions, such that ½ lies exactly halfway between 0 and 1. In the number line diagram below, the space between 0 and 1 is divided (partitioned) into 4 equal regions. The distance from 0 to the first segment is 1 of the 4 segments from 0 to 1 or ¼ (3.NF.2a).

Standard Explanation Concrete Representational Abstract3.NF.2.b: Represent a fraction a/b on a number line diagram by marking off a lengths of 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/bon the number line.

The distance from 0 to the third segment is 3 segments that are each one-fourth long. Therefore, the distance of 3 segments from 0 is the fraction ¾.

Build-it Draw-it Write-itRepresent a fraction a/b on a number line diagram by marking off a lengths of 1/b from 0 using concrete objects to model iteration.

Represent a fraction a/b on a number line diagram by marking off a lengths of 1/b from 0 drawing the out each iteration of the fraction.

Represent a fraction a/b on a number line diagram by marking off a lengths of 1/b from 0 labeling the fraction iterations and the resulting fraction.

The number line diagram is the first time students work with a number line for numbers that are between whole numbers (e.g., that ½ is between 0 and 1). Students need ample experiences folding linear models (e.g., string, sentence strips) to help them reason about and justify the location of fractions, such that ½

lies exactly halfway between 0 and 1.In the number line diagram below, the distance from 0 to the third segment is 3 segments that are each one-fourth long. Therefore, the distance of 3 segments from 0 is the fraction ¾ (3.NF.2b).

Equivalent FractionsStandard Explanation Concrete Representational Abstract

3.NF.3.c: Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.

This standard includes writing whole numbers as fractions. The concept relates to fractions as division problems, where the fraction 3/1 is 3 wholes divided into one group. This standard is the building block for later work where students divide a set of objects into a specific number of groups. Students must understand

the meaning of a1 .

Build-it Draw-it Write-itExpress whole numbers as fractions using objects.

Express whole numbers as fractions by drawing pictures.

Express whole numbers as fractions using numbers and symbols.

This standard includes writing whole numbers as fractions. The concept relates to fractions as division problems, where the fraction 3/1 is 3 wholes divided into one group. This standard is the building block for later work where students divide a set of objects into a specific number of groups. Students must understand the meaning of a/1. Example: If 6 brownies are shared between 2 people, how many brownies would each person get?Visual Examples:

Standard Explanation Concrete Representational Abstract3.NF.3.b: Recognize and generate simple equivalent fractions, (e.g.,1/2 = 2/4, 4/6 =2/3). Explain why the fractions are equivalent (e.g., by using a visual fraction model).

3.NF.3.b, calls for students to use visual fraction models (area models) and number lines to explore the idea of equivalent fractions. Students should only explore equivalent fractions using models, rather than using algorithms or procedures.

Build-it Draw-it Write-itRecognize and generate simple equivalent fractions, using a concrete fraction model.

Recognize and generate simple equivalent fractions, using a representational fraction model.

This standard calls for students to use visual fraction models (area models) and number lines to explore the idea of equivalent fractions. Students should only explore equivalent fractions using models, rather than using algorithms or procedures.

Standard Explanation Concrete Representational Abstract3.NF.3.a: Understand two fractions as equivalent (equal) if they are the same size or the same point on a number line.

3.NF.3.a, calls for students to use number lines to explore the idea of equivalent fractions. Students should only explore equivalent fractions using number lines, rather than using algorithms or procedures.

Build-it Draw-it Write-itRecognize and generate simple equivalent fractions, using a concrete fraction model.

Recognize and generate simple equivalent fractions, using a representational fraction model.

This standard calls for students to use visual fraction models (area models) and number lines to explore the idea of equivalent fractions. Students should only explore equivalent fractions using models, rather than using algorithms or procedures.Example:In the number line diagram below, the space between 0 and 1 is divided (partitioned) into 4 equal regions. The distance from 0 to the first segment is 1 of the 4 segments from 0 to 1 or ¼.

Comparing FractionsStandard Explanation Concrete Representational Abstract

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

In this standard, students should also reason that comparisons are only valid if the wholes are identical. For example, ½ of a large pizza is a different amount than ½ of a small pizza. Students should be given opportunities to discuss and reason about which ½ is larger.

Build-it Draw-it Write-itCompare two fractions with the same numerator or the same denominator by reasoning about their size. By using a concrete fraction model.

Compare two fractions with the same numerator or the same denominator by reasoning about their size. By using a representational fraction model.

Compare two fractions with the same numerator or the same denominator by reasoning about their size. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions.

An important concept when comparing fractions is to look at the size of the parts and the number of the parts. For example, 18 is smaller than

12 because when 1

whole is cut into 8 pieces, the pieces are much smaller than when 1 whole is cut into 2 pieces.

Students recognize when examining fractions with common denominators, the wholes have been divided into the same number of equal parts. So the fraction with the larger numerator has the larger number of equal parts.

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To compare fractions that have the same numerator but different denominators, students understand that each fraction has the same number of equal parts but the size of the parts are different. They can infer that the same number of smaller pieces is less than the same number of bigger pieces.

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After understanding how to recognize and generate equivalent fractions, students begin to compare fractions numerically. They reason about the denominators when the numerators are equal, and about the numerators when the denominators are equal. For example, 3/8 is less than 3/4, because 3 items shared among 8 people results in a smaller fair share than 3 items shared among 4 people. Likewise 3/8 is greater than 2/8, because 3 items shared among 8 people results in a larger fair share than 2 items shared among 8 people, or because 3/8 is to the right of 2/8on the number line.

Note to teachers: Similar comparisons of fractions can be made using part-whole contexts, but this can be confusing to students or lead to misconceptions when dealing with improper fractions in which the numerator is larger than the denominator. Therefore, it is suggested to acknowledge such explanations by students, but to use caution in overgeneralizing and to avoid presenting that reasoning by itself. For example, students may express“3/4 of people have brown eyes” as “3out of 4 people have brown eyes”. However, the “part out of whole” language cannot be used with improper fractions. Students should also be asked whether their comparisons still make sense if the referent unit changes. For example, they should be reminded of questions like,

“Are all halves the same?” and challenged with questions such as, “Suppose John has 12 apples in a bag and that Louise has 18 apples in a bag. Would you rather have half of John’s bag of apples or half of Louise’s bag of apples if you wanted to eat the most apples?”

Interpreting Data - FractionsStandard Explanation Concrete Representational Abstract

3.MD.4: Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units — whole numbers, halves, or quarters

Students in second grade measured length in whole units using both metric and U.S. customary systems. It‘s important to review with students how to read and use a standard ruler including details about halves and quarter marks on the ruler. Students should connect their understanding of fractions to measuring to one-half and one-quarter inch. Third graders need many opportunities measuring the length of various objects in their environment. This standard provides a context for students to work with fractions by measuring objects to a quarter of an inch.

Build-it Draw-it Write-itGenerate measurement data by measuring lengths using rulers marked with halves and fourths of an inch.

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

Example: Measure objects in your desk to the nearest ½ or ¼ of an inch, display data collected on a line plot. How many objects measured ¼? ½? etc… Some important ideas related to measuring with a ruler are: • The starting point of where one places a ruler to begin measuring • Measuring is approximate. Items that students measure will not always measure exactly ¼, ½ or one whole inch. Students will need to decide on an appropriate estimate length. • Making paper rulers and folding to find the half and quarter marks will help students develop a stronger understanding of measuring length Students generate data by measuring and create a line plot to display their findings.

Example:Measure objects in your desk to the nearest ½ or ¼ of an inch, display data collected on a line plot. How many objects measured ¼? ½? etc…

Fractions – Fourth GradeEquivalent Fractions

Students can use visual models or applets to generate equivalent fractions.Standard Explanation Concrete Representational Abstract

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 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.

This standard refers to visual fraction models. This includes area models, linear models (number lines) or it could be a collection/set models. This standard extends the work in third grade by using additional denominators (5, 10, 12, and 100).

Build-it Draw-it Write-itExplain why a fraction a/b is equivalent to a fraction (n x a) / (n x b) by using concrete fraction models.

Explain why a fraction a/b is equivalent to a fraction (n x a) / (n x b) by using representation fraction models.

Students are challenged to use different representations for a/b and (n xa)/(n x b) to reason about and justify the equivalence for various values of n, and are asked to generate representations for other equivalent fractions.One way to initially approach this problem is to give students a number of equivalent fraction pairs and ask them how fractions with different numbers of parts are located at the same location on the number line.  

Students also recognize the same pattern from other pairs of equivalent fractions and conclude that two fractions are equivalent if and only if their numerators and denominators are both increased or decreased by the same factor. For example, students recognize that the first pair of fractions are equivalent, but that the second and third pairs of fractions are not equivalent:

1.       , both numerator and denominator increased multiplicatively by the same factor, 2

2.       , numerator and denominator decreased (multiplicatively) by different factors

3.       , numerator and denominator increased additively, and not by multiplicative factors

All the area models show 1/2. The second model shows 2/4 but also shows that 1/2 and 2/4 are equivalent fractions because their areas are equivalent. When a horizontal line is drawn through the center of the model, the number of equal parts doubles and size of the parts is halved.

Students will begin to notice connections between the models and fractions in the way both the parts and wholes are counted and begin to generate a rule for writing equivalent fractions. 1/2 x 2/2 = 2/4.

There is NO mathematical reason why fractions must be written in simplified form, although it may be convenient to do so in some cases.Standard Explanation Concrete Representational Abstract

4.NF.2: 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 as1/2. 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

This standard calls students to compare fractions by creating visual fraction models or finding common denominators or numerators. Students’ experiences should focus on visual fraction models rather than algorithms. When tested, models may or may not be included. Students should learn to draw fraction models to help them compare. Students must also recognize that they must consider the size of the whole when comparing fractions (ie, ½ and 1/8 of two medium pizzas is very different from ½ of one medium and 1/8 of one large).

Build-it Draw-it Write-itCompare two fractions with different numerators and different denominators, by creating common denominators or numerators, or by comparing to a benchmark fraction and justify the conclusions, e.g., by using a concrete fraction model.

Compare two fractions with different numerators and different denominators, by creating common denominators or numerators, or by comparing to a benchmark fraction and justify the conclusions, e.g., by using a representational fraction model.

Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction. Record the results of comparisons with symbols <, =, or >, and justify the conclusions.

using a visual fraction model.Students compare fractions with different numerators and denominators using four strategies that build on prior work by: 

1) Using visual models (e.g., circle diagrams and fraction bars) to compare two fractions.2) Comparing fractions to benchmark fractions.3) Comparing the denominators of two fractions whose numerators are equal, or comparing the numerators of two fractions whose denominators are equal.4) Relating the use of models to numeric strategies.

In employing each of these strategies students should be reminded to clearly state the referent unit, especially if that referent unit is not equal to 1.

This standard calls students to compare fractions by creating visual fraction models or finding common denominators or numerators. Students’ experiences should focus on visual fraction models rather than algorithms. When tested, models may or may not be included. Students should learn to draw fraction models to help them compare and use reasoning skills based on fraction benchmarks. Students must also recognize that they must consider the size of the whole when comparing fractions (ie,1/2 and 1/8 of two medium pizzas is very different from1/2 of one medium and 1/8 of one large). Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

Example: There are two cakes on the counter that are the same size. The first cake has ½ of it left. The second cake has 5/12 left. Which cake has more left?

Adding and Subtracting FractionsStandard Explanation Concrete Representational Abstract

4.NF.3.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

Students should justify their breaking apart (decomposing) of fractions using visual fraction models. The concept of turning mixed numbers into improper fractions needs to be emphasized using visual fraction models.

Build-it Draw-it Write-itDecompose 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

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

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.

using a visual fraction model.

using a concrete fraction model.

using a representational fraction model.

Students can decompose a fraction into additive parts, writing it multiple ways as a result of composing unit fractions. They learn to do this for all rational number fractions, a/b, where b does not equal 0. Students should be challenged to combine or join single unit fractions (e.g., 1/2 or 1/3 or 1/4) in multiple ways to make other fractions. They should justify their decompositions using visual models such as circle diagrams, fraction bars, and equipartitioning.

A fraction with a numerator of one is called a unit fraction. When students investigate fractions other than unit fractions, such as 2/3, they should be able to decompose the non-unit fraction into a combination of several unit fractions. Example: 2/3 = 1/3 + 1/3 Being able to visualize this decomposition into unit fractions helps students when adding or subtracting fractions. Students need multiple opportunities to work with mixed numbers and be able to decompose them in more than one way. Students may use visual models to help develop this understanding.

Example: 1 ¼ - ¾ = 4/4 + ¼ = 5/4 5/4 – ¾ = 2/4 or ½

Example:

Similarly, converting an improper fraction to a mixed number is a matter of decomposing the fraction into a sum of a whole number and a number less than 1. Students can draw on their knowledge from third grade of whole numbers as fractions. Example, knowing that 1 = 3/3, they see:

Standard Explanation Concrete Representational Abstract4.NF.3.a: Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.

A fraction with a numerator of one is called a unit fraction. When students investigate fractions other than unit fractions, such as 2/3, they should be able to join (compose) or separate (decompose) the fractions of the same whole.

Build-it Draw-it Write-itUnderstand addition and subtraction of fractions as joining and separating parts using concrete fraction models.

Understand addition and subtraction of fractions as joining and separating parts using representational fraction models.

Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.

Before students are formally introduced to the mechanics of fraction addition and subtraction, they need to be able to reason about the structure of a fraction a/b. One way for students to understand a/b is to decompose a/b into a parts of size 1/bth (see Standard 3.NF.2.b earlier in this LT). This helps them to further understand that fractions can be “joined” (added) and “separated” (subtracted), if (and only if) the fractions are taken as parts of the same whole. Proceeding in this way, students’ understanding of writing additive number facts with two or more addends can be applied to adding multiple 1/bths to create a/b. Decomposing a/b into a parts of size 1/bth can be used as a basis for transitioning to common denominators in addition. That is, students proceed from this fact to add a/b + c/b to get (a+c)/b. This helps students avoid the misconception that, for example, 1/4 + 1/4 = 2/8 (i.e., the misconception of adding fractions by adding the numerators and the denominators). Models such as rulers, with which children can see that 1/4 inch + 1/4 inch ≠ 2/8 inch can also help to establish this correct

reasoning. Students can add and subtract using fractions with like denominators for all problem types from addition and subtraction, including joining, separating, and comparison problems.Example: 2/3 = 1/3 + 1/3 Being able to visualize this decomposition into unit fractions helps students when adding or subtracting fractions. Students need multiple opportunities to work with mixed numbers and be able to decompose them in more than one way. Students may use visual models to help develop this understanding.

Example of word problem: Mary and Lacey decide to share a pizza. Mary ate 3/6 and Lacey ate 2/6 of the pizza. How much of the pizza did the girls eat together? Possible solution: The amount of pizza Mary ate can be thought of a 3/6 or 1/6 and 1/6 and 1/6. The amount of pizza Lacey ate can be thought of a 1/6 and 1/6. The total amount of pizza they ate is 1/6 + 1/6 + 1/6 + 1/6 + 1/6 or 5/6 of the whole pizza.

Standard Explanation Concrete Representational Abstract4.NF.3.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.

A separate algorithm for mixed numbers in addition and subtraction is not necessary. Students will tend to add or subtract the whole numbers first and then work with the fractions using the same strategies they have applied to problems that contained only fractions. Mixed numbers are introduced for the first time in Fourth Grade. Students should have ample experiences of adding and subtracting mixed numbers where they work with mixed numbers or convert mixed numbers so that the numerator is equal to or greater than the denominator.

Build-it Draw-it Write-itAdd and subtract mixed numbers with like denominators, e.g., by using concrete fraction models.

Add and subtract mixed numbers with like denominators, e.g., by using representational fraction models.

Add and subtract mixed numbers with like denominators.

Example: Susan and Maria need 8 3/8 feet of ribbon to package gift baskets. Susan has 3 1/8 feet of ribbon and Maria has 5 3/8 feet of ribbon. How much ribbon do they have altogether? Will it be enough to complete the project? Explain why or why not. The student thinks: I can add the ribbon Susan has to the ribbon Maria has to find out how much ribbon they have altogether. Susan has 3 1/8 feet of ribbon and Maria has 5 3/8 feet of ribbon. I can write this as 3 1/8 + 5 3/8. I know they have 8 feet of ribbon by adding the 3 and 5. They also have 1/8 and 3/8 which makes a total of 4/8 more. Altogether they have 8 4/8 feet of ribbon. 8 4/8 is larger than 8 3/8 so they will have enough ribbon to complete the project. They will even have a little extra ribbon left, 1/8 foot. Example: Trevor has 4 1/8 pizzas left over from his soccer party. After giving some pizza to his friend, he has 2 4/8 of a pizza left. How much pizza did Trevor give to his friend? Possible solution: Trevor had 4 1/8 pizzas to start. This is 33/8 of a pizza. The x’s show the pizza he has left which is 2 4/8 pizzas or 20/8 pizzas. The shaded rectangles without the x’s are the pizza he gave to his friend which is 13/8 or 1 5/8 pizzas.

Example: While solving the problem, 3 ¾ + 2 ¼ students could do the following:

Standard Explanation Concrete Representational Abstract4.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.

This standard continues the work of equivalent fractions by having students change fractions with a 10 in the denominator into equivalent fractions that have a 100 in the denominator. In order to prepare for work with decimals (4.NF.6 and 4.NF.7), experiences that allow students to shade decimal grids (10x10 grids) can support this work. Student

Build-it Draw-it Write-itExpress 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. Students can also use base

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. Students can also use

experiences should focus on working with grids rather than algorithms. Students can also use base ten blocks and other place value models to explore the relationship between fractions with denominators of 10 and denominators of 100.

ten blocks and other place value models to explore the relationship between fractions.

representational fraction models or illustrations to explore the relationship between fractions.

Students add and subtract decimals to the hundredths by using place value (e.g., ones to ones, tenths to tenths, hundredths to hundredths; see Standard 5.NBT.7 in the Place Value and Decimals LT), which may involve lining up the decimal points of the numbers. Attention to place value helps students avoid a common misconception of right-justifying decimals with different numbers of decimal places.

This work in fourth grade lays the foundation for performing operations with decimal numbers in fifth grade. Students use their knowledge of equivalent fractions and of adding fractions with like denominators to solve word problems and add fractions with denominators of 10 and 100 by writing a/10 as (a x 10) / 100.Students in fourth grade work with fractions having denominators 10 and 100. Because it involves partitioning into 10 equal parts and treating the parts as numbers called one tenth and one hundredth, work with these fractions can be used as preparation to extend the base-ten system to non-whole numbers.

Example: Example: Represent 3 tenths and 30 hundredths on the models below.

Students can use base ten blocks, graph paper, and other place value models to explore the relationship between fractions with denominators of 10 and denominators of 100. Base Ten Blocks: students may represent 3/10 with 3 longs and may also write the fraction as 30/100 with the whole in this case being the flat (the flat represents one hundred units with each unit equal to one hundredth). Students begin to make connections to the place value chart as shown in 4.NF.6. Students make connections between fractions with denominators of 10 and 100 and the place value chart. By reading fraction names, students say 32/100 as thirty-two hundredths and rewrite this as 0.32 or represent it on a place value model as shown below.

Students use the representations explored in 4.NF.5 to understand 32/100 can be expanded to 3/10 and 2/100.

Students represent values such as 0.32 or 32/100 on a number line. 32/100 is more than 30/100 (or 3/10) and less than 40/100 (or 4/10). It is closer to 30/100 so it would be placed on the number line near that value.

Standard Explanation Concrete Representational Abstract4.NF.3.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.

Students apply their knowledge of adding and subtracting fractions with like denominators by solving word problems.

Build-it Draw-it Write-itSolve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using concrete fraction models to represent the problem.

Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using representational fraction models to represent the problem.

Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using equations to represent the problem.

Example:A cake recipe calls for you to use ¾ cup of milk, ¼ cup of oil, and 2/4 cup of water. How much liquid was needed to make the cake?

Mixed numbers are introduced for the first time in Fourth Grade. Students should have ample experiences of adding and subtracting mixed numbers where they work with mixed numbers or convert mixed numbers into improper fractions. Keep in mind Concrete-Representation-Abstract (CRA) approach to teaching fractions. Students need to be able to ―show‖ their thinking using concrete and/or representations BEFORE they move to abstract thinking.

Interpreting Data - FractionsStandard Explanation Concrete Representational Abstract

4.MD.4: Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Solve problems involving addition and subtraction of fractions by using

This standard provides a context for students to work with fractions by measuring objects to an eighth of an inch. Students are making a line plot of this data and then adding and subtracting fractions based on data in the line plot.

Build-it Draw-it Write-itMake a line plot to display a data set of measurements in fractions of a unit. Solve problems involving addition and subtraction of fractions by using concrete objects. 

Make a line plot to display a data set of measurements in fractions of a. Solve problems involving addition and subtraction of fractions by using representational

Make a line plot to display a data set of measurements in fractions of a unit. Solve problems involving addition and subtraction of fractions by using symbols and/or equations. 

information presented in line plots. 

drawings/illustrations. 

This standard provides a context for students to work with fractions by measuring objects to an eighth of an inch. Students are making a line plot of this data and then adding and subtracting fractions based on data in the line plot. Example: Students measured objects in their desk to the nearest ½, ¼, or 1/8 inch. They displayed their data collected on a line plot. How many object measured ¼ inch? ½ inch? If you put all the objects together end to end what would be the total length of all the objects. Ten students in Room 31 measured their pencils at the end of the day. They recorded their results on the line plot below.

Data has been measured and represented on line plots in units of whole numbers, halves or quarters. Students have also represented fractions on number lines. Now students are using line plots to display measurement data in fraction units and using the data to solve problems involving addition or subtraction of fractions. Have students create line plots with fractions of a unit (1/2, 1/4, 1/8) and plot data showing multiple data points for each fraction.

Pose questions that students may answer, such as: • ―How many one-eighths are shown on the line plot? Expect ―two one-eighths as the answer. Then ask, ―What is the total of these two one-eighths? Encourage students to count the fractional numbers as they would with whole-number counting, but using the fraction name. • ―What is the total number of inches for insects measuring 3/8 inches? Students can use skip counting with fraction names to find the total, such as, ―three-eighths, six-eighths, nine-eighths. The last fraction names the total. Students should notice that the denominator did not change when they were saying the fraction name. have them make a statement about the result of adding fractions with the same denominator. • ―What is the total number of insects measuring 1/8 inch or 5/8 inches?‖ Have students write number sentences to represent the problem and solution such as, 1/8 + 1/8 + 5/8 = 7/8 inches.

Multiplication and Division Problems Involving Non-Whole Rational Number Operations (Fractions)Standard Explanation Concrete Representational Abstract

4.NF.4a: 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

This standard builds on students’ work of adding fractions and extending that work into multiplication.

Build-it Draw-it Write-itUnderstand a fraction a/b as a multiple of 1/b. Use a concrete fraction model to represent 5/4 as the product 5 × (1/4).

Understand a fraction a/b as a multiple of 1/b. Use a representational fraction model to represent 5/4 as the product 5 × (1/4).

Understand a fraction a/b as a multiple of 1/b. Recording the conclusion by using an equation.

equation 5/4 = 5 × (1/4).Example: 3/6 = 1/6 + 1/6 + 1/6 = 3 x (1/6) Number line: Area model:

Students should see a fraction as the numerator times the unit fraction with the same denominator. Example:

One interpretation of this standard is that it represents student understanding of: 1) how to multiply a unit fraction as input by a whole number as operator and 2) how to multiply a whole number as input by a unit fraction as an operator.Unit fraction as input quantity operated by a whole number operator Students explain the meaning of a fraction times a whole number by referring back to equipartitioning of a whole. Students share one cake (a whole) among three people and code the process as the division 1 (cake) ÷ 3 (people) = 1/3 (cake per person). Reversing this process (using reassembly of the whole) results in a meaning for: 1/3 × 3 = 1 or a unit fraction times a whole number. This is an example applying the referent preserving model by scaling.

Stating the same relationship using mathematical symbols, students recognize how the result of one whole is obtained from 1/3 × 3, that is, 1/3, 3 times, results in 3/3 which equals one. From this, it follows that to multiply,

 Applying the same reasoning to a referent transforming problem, such as, “Bob paints 1/10  square meter on a wall in one minute. How much surface can he paint in 12 minutes?” Students make a D/M box with headers of surface (in square meter) and minutes. They know that painting 1/10 square meter goes with 1 minute. To find out what corresponds with 12 minutes, they know they must multiply : 

1/10 square meter per minute × 12 minutes =  12/10 square meters

Applying the same reasoning to referent composing, students reason with area. Students visualize a rectangular area of 1/3 × 5 square units dynamically as the product of “sweeping” a line segment of width 1/3 unit over a length of 5 units. They interpret the distance of sweeping (5 units) as an operator on the width

quantity (1/3 unit). The area created by the sweep is thus 5/3 square units.

Whole number as input quantity operated by a unit fraction as an operatorStudents explain the meaning of a whole number times a fraction by referring back to equipartitioning of a collection. When sharing a collection of 24 coins among 6 pirates as shown in the D/M box below, 24 can be described as 6 times as large as 4 or 4 can be described as 6 times smaller than 24. Students also name each pirate’s share in relation to the collection as 4/24 or 1/6 of the collection. Therefore students reason that 24 ÷ 6 = 4 and 1/6

th of 24 = 4. They write 1/6 of 24 as a number sentence: 24 × 1/6. Students view this using an input-output machine with 1/6

th as the operator. In this case, the operator is referent preserving by scaling. 

Thus, for conceptual consistency, students interpret “multiplying by 1/b” to be the equivalent operation as “dividing by b”. As a result, students recognize that 24 ÷ 6 = 4 and 24 × 1/6= 4 are two ways to operate on 24 to produce 4.Stating the same relationship using mathematical symbols, students write: 

Just by exploring different ways of expressing the referent preserving relationships from equipartitioning, students can move flexibly vertically up or down the D/M box.  In the example below they can move from: (1) top to bottom using division by 6, (2) top to bottom using multiplication by 1/6, or (3) bottom to top using multiplication by 6. Although not yet thoroughly discussed, some students reason what the missing fourth arrow’s label is (4) bottom to top using division by 1/6.

 

Students can represent the equivalence of dividing by b and multiplying by 1/b using a number line as referent preserving, such as showing that 1/6 of 24 on the number line is equal to 4. 

Applying the same reasoning to a referent transforming problem, such as, “It takes a car one hour to travel 30 miles on dirt roads. How far does it travel in one-third of an hour?”, students make a D/M box with headers of distance (miles) and time (hours) and fills in the bottom row with 30 miles and 1 hour. To find out what corresponds to 1/3 of an hour, they must multiply:

30 miles/hour ×  1/3 hours = 10 miles Applying the same reasoning to a referent composing area problem, such as 5 × 1/3 they create a length of 1/3 from a unit square, and sweep it across a width of 5 to get 5/3 square units. 

Standard Explanation Concrete Representational Abstract4.NF.4b: 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 ×

This standard extended the idea of multiplication as repeated addition.

Build-it Draw-it Write-itUnderstand 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 concrete fraction model.

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 representational fraction model.

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 derived facts or work in the abstract to represent the situation and

(1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)

answers.

Students need many opportunities to work with problems in context to understand the connections between models and corresponding equations. Contexts involving a whole number times a fraction lend themselves to modeling and examining patterns. This standard builds on students’ work of adding fractions and extending that work into multiplication.Example:3 x (2/5) = 2/5 + 2/5 + 2/5 = 6/5 = 6 x (1/5). Students are expected to use and create visual fraction models to multiply a whole number by a fraction.

If each person at a party eats 3/8 of a pound of roast beef, and there are 5 people at the party, how many pounds of roast beef are needed?

The same thinking, based on the analogy between fractions and whole numbers, allows students to give meaning to the product of whole number and a fraction. Example:

Standard Explanation Concrete Representational Abstract4.NF.4c: Solve word problems involving multiplication of a fraction by a whole

When introducing this standard make sure student use visual fraction models to solve word problems related to multiplying a whole number by a fraction.

Build-it Draw-it Write-itSolve word problems involving multiplication of a fraction by a whole

Solve word problems involving multiplication of a fraction by a whole

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.

number, e.g., by using concrete fraction models to represent the problem.

number, e.g., by using representational fraction models to represent the problem.

number, e.g., by using equations to represent the problem.

Example: In a relay race, each runner runs ½ of a lap. If there are 4 team members how long is the race?

Students apply the earlier Standard 4.NF.4.b to word problems in order to multiply a fraction by a whole number. Recall that the three models of multiplication were: 

 Model 1: Referent Transforming: To demonstrate their understanding of how to multiply a generalized non-unit fraction by a whole number, students solve referent transforming problems related to associativity. For example, students solve problems such as: “Suppose each bottle of soda has 1/8 gallon and there are 6 bottles of soda in a case. How much soda is there in 5 cases? Show two different ways to solve the problem and justify their equivalence.” Students solve the problem 1/8 × 6 × 5 by either reasoning that there are 1/8 gallon per bottle × 6 bottles = 6/8 gallon of soda in a bottle and there are 5 cases, so there are 6/8 gallon per case × 5 cases = 30/8 (or 15/4) gallons. They can also reason that there are 6 bottles per case × 5 cases = 30 bottles and 1/8 gallon per bottle, so there are 1/8 gallon per bottle × 30 bottles. Therefore, students recognize the associativity of multiplication as showing (1/8 × 6) × 5 = 1/8 × (6 × 5). The total is 15/4 or 3 3/4 gallons. Model 2: Referent Preserving: To demonstrate their understanding of how to multiply a generalized non-unit fraction by a whole number, students also solve referent preserving models of multiplication by extending the use of the fair sharing table of values using additional rows. For example, if students know that one small ice cream cake feeds three people, they know that each person receives 1/3 of a cake. Suppose then a problem asks them how much cake to get for a family of 5? From the previous standard, they can multiple both the 1 person and the 1/3 cake by 5 to get 5 persons and 5/3 cakes (or 1 2/3 cakes).  Now suppose the problem is extended again to say that four families show up, then how much cake is needed? This requires them to multiply the 5 persons and the 5/3 cakes by 4, to produce 20/3cakes (or 6 2/3 cakes) for 20 people. They recognize that they cannot purchase 6 2/3 cakes, so the answer to the problem, how many cakes should they purchase for 20 people is 7 ice cream cakes.  This problem demonstrates how to multiply a generalized non-unit fraction by a whole number.Model 3: Referent Composing: As a model of referent composing multiplication of a fraction times a whole number, student can explain why (1/b × a) × c and  1/b × (a × c) both produce the same area. Consider the problem: A rectangular strip of paper measures 5/3 inches × 4 inches. What is the area of the paper? Starting by creating the width of 5/3 inches which is the same as 1/3 × 5 (either as sweeping a length 1/3 unit by 5 units), students interpret the number 4 in (1/3 × 5) × 4 as an operator on that width or the sweeping distance of 4 inches producing the area desired as 20/3. They also recognize that the result of extending the width of 5 units by 4 times is the same as increasing the area 1/3 × 5 by 4 times, and thus (1/3 × 5) × 4 = 1/3 × (5 × 4) = 20/3. Later students also use a volume

model to illustrate the associative property of the multiplication of three numbers, including fractions.

Fractions – Fifth GradeAdding and Subtracting Fractions

It is not necessary to find a least common denominator to calculate sums of fractions, and in fact the effort of finding a least common denominator is a distraction from understanding adding fractions.

Standard Explanation Concrete Representational Abstract5.NF.1: 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.NF.1 builds on the work in fourth grade where students add fractions with like denominators. In fifth grade, the example provided in the standard 2/3 + ¾ has students find a common denominator by finding the product of both denominators. This process should come after students have used visual fraction models (area models, number lines, etc.) to build understanding before moving into the standard algorithm describes in the standard The use of these visual fraction models allows students to use reasonableness to find a common denominator prior to using the algorithm.

Build-it Draw-it Write-itAdd and subtract fractions with unlike denominators by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators by using concrete fraction models.

Add and subtract fractions with unlike denominators by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators by using representational fraction models.

Add and subtract fractions with unlike denominators by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators by using equations.

Example: 1/3 + 1/6

I drew a rectangle and shaded 1/3. I knew that if I cut every third in half then I would have sixths. Based on my picture, 1/3 equals 2/6. Then I shaded in another 1/6 with stripes. I ended up with an answer of 3/6, which is equal to 1/2. On the contrary, based on the algorithm that is in the example of the Standard, when solving 1/3 + 1/6, multiplying 3 and 6 gives a common denominator of 18. Students would make equivalent fractions 6/18 + 3/18 = 9/18 which is also equal to one-half. Please note that while multiplying the denominators will always give a common denominator, this may not result in the smallest denominator. Students should apply their understanding of equivalent fractions 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:

Fifth grade students will need to express both fractions in terms of a new denominator with adding unlike denominators. For example, in calculating 2/3 + 5/4 they reason that if each third in 2/3 is subdivided into fourths and each fourth in 5/4 is subdivided into thirds, then each fraction will be a sum of unit fractions with denominator 3 x 4 = 4 x 3 + 12:

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.

Standard Explanation Concrete Representational Abstract5.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. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2.

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 than ¾ because 7/8 is missing only 1/8 and ¾ is missing ¼ so 7/8 is closer to a whole Also, students should use benchmark fractions to estimate and examine the reasonableness of their answers. Example here such as 5/8 is greater than 6/10 because5/8 is 1/8 larger than ½(4/8) and 6/10 is only 1/10 larger than ½ (5/10)

Build-it Draw-it Write-itSolve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using concrete fraction models to represent the problem.

Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using representational fraction models to represent the problem.

Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using equations to represent the problem.

Example: 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?

Example: Jerry was making two different types of cookies. One recipe needed 3/4 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

Sonia 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?

Using addition to find the answer:1 ¾ + n = 3 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.

Example: Using an area model to subtract This model shows 1 ¾ subtracted from 3 1/6 leaving 1 + ¼ = 1/6 which a student can then change to 1 + 3/12 + 2/12 = 1 5/12. 3 1/6 can be expressed

with a denominator of 12. Once this is done a student can complete the problem, 2 14/12 – 1 9/12 = 1 5/12.

This diagram models a way to show how 3 1/6 and 1 ¾ can be expressed with a denominator of 12. Once this is accomplished, a student can complete the problem, 2 14/12 – 1 9/12 = 1 5/12.

Estimation skills include identifying when estimation is appropriate, determining the level of accuracy needed, selecting the appropriate method of estimation, and verifying solutions or determining the reasonableness of situations using various estimation strategies. Estimation strategies for calculations with fractions extend from students’ work with whole number operations and can be supported through the use of physical models.

Multiplication and Division Problems Involving Non-Whole Rational Number Operators (Fractions)Standard Explanation Concrete Representational Abstract

5.NF.4.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.

Students need to develop a fundamental understanding that the multiplication of a fraction by a whole number could be represented as repeated addition of a unit fraction

Build-it Draw-it Write-itInterpret 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 using concrete models.

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 using representational models.

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 using derived facts.

This standard extends student’s work of multiplication from earlier grades. In fourth grade, students worked with recognizing that a fraction such as 3/5 actually could be represented as 3 pieces that are each one-fifth (3 x (1/5)). This standard references both the multiplication of a fraction by a whole number and the multiplication of two fractions. Visual fraction models (area models, tape diagrams, number lines) should be used and created by students during their work with this standard.

As they multiply fractions such as 3/5 x 6, they can think of the operation in more than one way. • 3 x (6 ÷ 5) or (3 x 6/5) • (3 x 6) ÷ 5 or 18 ÷ 5 (18/5) Students create a story problem for 3/5 x 6 such as, • Isabel had 6 feet of wrapping paper. She used 3/5 of the paper to wrap some presents. How much does she have left? • Every day Tim ran 3/5 of mile. How far did he run after 6 days? (Interpreting this as 6 x 3/5) Example: Three-fourths of the class is boys. Two-thirds of the boys are wearing tennis shoes. What fraction of the class are boys with tennis shoes? This question is asking what 2/3 of ¾ is, or what is 2/3 x ¾. What is 2/3 x ¾, in this case you have 2/3 groups of size ¾ (a way to think about it in terms of the language for whole numbers is 4 x 5 you have 4 groups of size 5). The array model is very transferable from whole number work and then to binomials.

Standard Explanation Concrete Representational Abstract5.NF.4.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.

This standard extends students’ work with area. In third grade students determine the area of rectangles and composite rectangles. In fourth grade students continue this work. The fifth grade standard calls students to continue the process of covering (with tiles). Grids (see picture) below can be used to support this work.

Build-it Draw-it Write-itFind 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 by using concrete objects to directly model the array or area model.

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 by illustrating the array or area model.

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 by using repeated addition or derived facts.

Students model the multiplication of two generalized non-unit fractions (e.g. 2/3 × 5/4) area models in this standard. To multiply 2/3 × 5/4 using the area model, students begin with a 1 × 1 unit square. They partition each of the sides to represent the denominators of the two fractions and create a rectangular region whose measure is the product of the two denominators, for example, a 1/12th unit with sides of 1/3rd and 1/4th. Students scale each of the sides to be the size the number of 1/bth units, i.e. scale by 2 in the 1/3rd direction to represent 2/3 , and 5 in the 1/4th direction to represent 5/4. The region bounded by the new area is comprised of 10 units of 1/12th. Students name the product, explain why it is the product of the numerators and the denominators, and identify the unit of 1 that the answer is defined in relation to. Students can also name the region by its equivalent fractional value in lowest common terms and explain their answer. In this example, that would be 5/6.

Students reason that 2/3 × 5/4 is the same as (1/3 × 2) × (1/4 × 5) as well as (2 × 5) × (1/3 × 1/4). Students can also write the multiplication as a daisy chain (2 ÷ 3) × (5 ÷ 4) and recognize that the numerators are the multipliers and the denominators are the divisors. It is critical that students are able to identify the referent whole to which any particular fraction is applied. A fraction is always understood relative to an explicit or an implied whole, and students identify the whole with respect to their solution. 

Based on their knowledge of the commutative property and associative property of multiplication involving unit fractions and whole numbers (see Standard 4.NF.4.a and 4.NF.4.b earlier), students are able to carry out all the types of multiplication shown in the grid. They complete the last three cells by learning to multiply unit fractions times generalized non-unit fractions and generalized non-unit fractions times generalized non-unit fractions in this standard which illustrates the model of referent composing. They can explain why they did not need common denominators for multiplication since in multiplication they compose new units.

Example: The home builder needs to cover a small storage room floor with carpet. The storage room is 4 meters long and half of a meter wide. How much carpet do you need to cover the floor of the storage room? Use a grid to show your work and explain your answer. In the grid below I shaded the top half of 4 boxes. When I added them together, I added ½ four times, which equals 2. I could also think about this with multiplication ½ x 4 is equal to 4/2 which is equal to 2.

Example:

In solving the problem 23 x

45 , students use an area model to visualize it as a 2 by 4 array of small rectangles each of which has side lengths 1/3 and 1/5. They

reason that 1/3 x 1/5 = 1/(3 x 5) by counting squares in the entire rectangle, so the area of the shaded area is (2 x 4) x 1/(3 x 5) = 2x 43 x5 . They can explain that the

product is less than 45 because they are finding

23 of

45 . They can further estimate that the answer must be between

25 and

45 because

23 of

45 is more than

12

of 45 and less than one group of

45 .

Standard Explanation Concrete Representational Abstract5.NF.5.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 equivalencea/b = (n × a)/(n × b) to the effect of multiplying a/b by 1.

This standard asks students to examine how numbers change when we multiply by fractions. Students should have ample opportunities to examine both cases in the standard: a) when multiplying by a fraction greater than 1, the number increases (and b) when multiplying by a fraction less the one, the number decreases. This standard should be explored and discussed while students are working with 5.NF.4, and should not be taught in isolation.

Build-it Draw-it Write-itExplaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number; 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 by using concrete fraction models.

Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number; 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 by using representational fraction models.

Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number; 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 by using equations.

Because students can translate multiplication by a fraction into a combination of multiplication and division, they can predict whether a fractional operator will increase or decrease the size of the quantity it operates on.  If the operator is greater than one, then the numerator is greater than the denominator. When switched into a “daisy chain”, the numerator is the multiplier and the denominator is the divisor. If a number is multiplied by a larger number than it is divided, then the result or product is larger than the quantity operated on. Students predict for example that in multiplying 16 by 5/3, the product will be greater than 16. Similarly, for the fraction as operator, if the denominator is greater than the numerator, then the fraction is less than one, and the quantity is divided by a larger number than it is multiplied, resulting in a smaller number. Students predict, for example, that in multiplying 2/3 by 3/8, the product will smaller than2/3rds. Students recognize that a/b = (n × a)/(n × b) by rewriting it as a daisy chain and reason that the combined operator, “n times  larger followed by n times smaller,” does not change the input quantity a/b. Example: Mrs. Bennett is planting two flower beds. The first flower bed is 5 meters long and 6/5 meters wide. The second flower bed is 5 meters long and 5/6 meters wide. How do the areas of these two flower beds compare? Is the value of the area larger or smaller than 5 square meters? Draw pictures to prove your answer.

223 x 8 must be more than 16 because 2 groups of 8 is 16 and 2

23 is almost 3 groups of 8. So, the answer must be close to but less than 24.

Standard Explanation Concrete Representational Abstract5.NF.5.a: Interpret multiplication as scaling by: 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.

This standard calls for students to examine the magnitude of products in terms of the relationship between two types of problems. This extends the work with 5.OA.1.

Build-it Draw-it Write-itInterpret multiplication as scaling by using concrete models.

Interpret multiplication as scaling by using drawing or illustrations.

Interpret multiplication as scaling by using derived facts.

Not only can students predict the outcome of using operators that are less than, equal to or greater than one on the input, but they can predict whether the overall effect of the combination of both factors in multiplication in reference to an original assumed referent unit of one. For instance, multiplying 2/3 by 3/4 is less than one because each factor is less than one. If both numbers are greater than one, it is also easy to predict. However, if one factor is greater than one and the other is less than one, prediction by students needs to take into consideration the full effects of the various scale factors. Multiplying mixed numbers, multiplying mixed numbers and fractions less than one, and multiplying fractions less than one in different contexts can provide students such challenges. Students are already familiar with mixed numbers. By reasoning about the scaling effect of multiplication, and whether the each quantities being multiplied is a proper fraction, an improper fraction, or a mixed numbers, students can predict the value of the product relative to the value of any of the quantities.

Example:

Standard Explanation Concrete Representational Abstract5.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.

This standard builds on all of the work done in this cluster. Students should be given ample opportunities to use various strategies to solve word problems involving the multiplication of a fraction by a mixed number. This standard could include fraction by a fraction, fraction by a mixed number or mixed number by a mixed number.

Build-it Draw-it Write-itSolve real world problems involving multiplication of fractions and mixed numbers, e.g., by using concrete fraction models to represent the problem.

Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using representational fraction models to represent the problem.

Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using equations to represent the problem.

Students solve equations and word problems by multiplying fractions and mixed numbers. This can either be a fraction by a mixed number (a/b × c d/e where c d/e  is a mixed number) or a mixed number by a mixed number (a b/c × d e/f where  a b/c  and  d e/f  are mixed numbers). Students are already familiar with mixed numbers and they also use them in addition and subtraction. There are three different types of real world problems involving multiplication of fractions and mixed numbers:

Model 1: Referent Transforming Model 2: Referent Preserving Model 3:Referent Composing

Fair Sharing (Partitive) Measurement (Quotative) ArraysRate Scaling Area

Equal Groups Cartesian Product

Model 1: Referent Transforminga.         A fair-sharing multiplication problem is, “There are 4 1/2 barrels of soda and each barrel holds 3 2/3 liters of soda. How much soda is there in total?” 

Students predict the final outcome is greater than one since both factors are greater than one. b.         An example of rate problem involving multiplication might be: “Jessica walked at 4 1/2 miles per hour for 1 2/3 hours. How far did she walk?” Students

predict the final outcome is greater than one since both factors are greater than one. c.         An example of equal groups problem involving multiplication with transformation of units might be: “There are 5 1/2 bags of candies. Each bag

holds2/3 pound candies. What is the total amount of candies in all the bags?” Students predict the final outcome is greater than one because the problem, once coded as multiplication can be viewed as × 11 ÷ 2 × 2 ÷ 3. Thus overall, the original one is scaled up by 22 and down by 6, so it is greater than one.

Model 2: Referent Preservinga.         The Measurement (Quotative) multiplication problem is“12 1/3 ribbons each measuring 3 1/2 inches long are taped together to create a single ribbon. How

long is the ribbon created?” Students predict the final outcome is greater than one, since both factors are greater than one. b.         An example of scale problem involving multiplication might be: “Given a square with sides 3 1/2 units long and you shrink the square by a scale factor

of 1/5, what will be the length of a side of the new square created?” Students predict the final outcome is less than one because the problem, once coded as multiplication can be viewed as × 7 ÷ 2 × 1 ÷ 5. Thus overall, the original one is scaled up by 7 and down by 10 so the final outcome becomes less than one.

 c.         An example of Equal Groups problem involving multiplication with preservation of units might be: “Lucy uses 1 1/5 cups of vinegar in her salad dressing

recipe. How much vinegar would Lucy use to make 3 1/2 recipes?” Students predict the final outcome is greater than one, since both factors are greater than one.

Model 3:   Arrays, Area, and Cartesian Products An example of an area problem involving multiplication is: “A closet floor is covered with tiles whose dimensions are one foot by one foot. The closet measures 5 3/4 feet by 7 1/3 feet. How many tiles cover the floor?” Students predict the final outcome is greater than one, since both factors are greater than one. Note to teachers: Before addressing the standards directly associated with division of fractions, it is important to note how extensively the topic of multiplication of fractions has been treated. It has been treated across a set of nine cases, based on the use of whole numbers, unit fractions and generalized non-unit fractions. It has been developed across all three models: referent transforming, referent preserving and referent composing. And, it has been developed by carefully linking multiplication and division as inverse operations and applying the properties of multiplication (commutativity, associativity, identity and even inverse). This treatment is in marked contrast to a teacher who tells his or her students that to multiply fractions, multiply the numerators and then multiply the denominators and simplify. Such instruction diminishes the foundation created by multiplication and division of fractions and weakens students’ ability to reason multiplicatively and to model problems with mathematics. It is hoped that this extensive treatment shows the importance of the topic and its full conceptualization.

Example: There are 2 ½ bus loads of students standing in the parking lot. The students are getting ready to go on a field trip. 2/5 of the students on each bus are girls. How many busses would it take to carry only the girls?

Example:

Evan bought 6 roses for his mother. 23 of them were red. How many red roses were there? Using a visual, a student divides the 6 roses into 3 groups and counts

how many are in 2 of the 3 groups.

Example:

Students able to multiply fractions in general can develop strategies to divide fractions in general, by reasoning about the relationship between multiplication and division. But division of a fraction by a fraction is not a requirement at this grade.

Standard Explanation Concrete Representational Abstract5.NF.7.a: Interpret division of a unit fraction by a non-zero whole number, and compute such quotients.

5.NF.7, is the first time that students are dividing with fractions. 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.

Build-it Draw-it Write-itInterpret division of a unit fraction by a non-zero whole number, and compute such quotients using concrete objects to model the problem.

Interpret division of a unit fraction by a non-zero whole number, and compute such quotients using drawings or number lines to model the problem.

Interpret division of a unit fraction by a non-zero whole number, and compute such quotients by using equations.

Whole number as divisors. For example, to define 1/9 ÷ 3, one can ask, what number times 3 equals 1/9. The answer is 1/27.  Since students know the answer, they must find ways to generalize the approach and justify their solutions.  In this case, students know that dividing by 3 is the same as multiplying by 1/3 and so 1/9 ÷ 3 = 1/9 × 1/3 = 1/27. Students generate contextual problems for each type of division and 1) explain their answers in terms of meanings of the operations in a variety of contexts and models and 2) predict the size of the output relative to the input based on the type and value of number used. To explore contexts, students work with the three most common representations, the D/M box, the number line representation and the area model. To create a problem where a unit fraction is divided by a whole number, students create a fair sharing context associating a unit fraction with a number greater than one.  For example, if a chocolate torte is so rich that 1/3 of this luscious three-layer cake serves 4 people, then how much would serve one person. To solve this problem using the D/M box, students know to co-split the top rows by 4 to produce the problem 1/3 ÷ 4. The answer is generated by recalling that dividing by 4 is the same as multiplying by 1/4, hence the answer is 1/12. Because it is a co-split, the example is referent preserving. 

A referent transforming division problem would ask if a slug travels 1/3 of a meter in 4 minutes, how far does it travel in one minute? The problem would be1/3 meters divided by 4 minutes = 1/12 meters per minute. Area Models illustrate length × width equal area. Therefore they also represent area divided by length (width) equals width (length). For the problem 1/3 ÷ 4, we are asking what width coupled with a length of 4 would result in an area of 1/3. The representation below shows the answer: 

Example: You have 1/8 of a bag of pens and you need to share them among 3 people. How much of the bag does each person get?

Standard Explanation Concrete Representational Abstract5.NF.7.b: Interpret division of a whole number by a unit fraction, and compute such quotients.

This standard calls for students to create story contexts and visual fraction models for division situations where a whole number is being divided by a unit fraction.

Build-it Draw-it Write-itInterpret division of a whole number by a unit fraction, and compute such quotients by using concrete objects.

Interpret division of a whole number by a unit fraction, and compute such quotients by using representational models.

Interpret division of a whole number by a unit fraction, and compute such quotients by using equations.

To first reason about these problems, students write, for example, 30 ÷ 1/2 = C. The answer has to solve the multiplication problem C × 1/2 = 30. This means also that C ÷ 2 = 30. The students know that C = 60.  They also reason this by writing the division problem itself as a fraction:

From their work with fractions and in Standard 5.NF.5.b, they know that they can multiply this by 2/2 and preserve the value of the fraction. This results in the problem 60/1. Hence 30 ÷ ½ = 60.Example: 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?

Standard Explanation Concrete Representational Abstract5.NF.7.c: Solve real world problems involving division of unit fractions using visual models and equations.

Extends students’ work from other standards in 5.NF.7. Student should continue to use visual fraction models and reasoning to solve these real-world problems.

Build-it Draw-it Write-itSolve real world problems involving division of unit fractions using concrete models.

Solve real world problems involving division of unit fractions using representational models.

Solve real world problems involving division of unit fractions using equations.

In the context of the D/M Box, this problem can represent the question, if there are 30 oranges in a half a case, how many oranges are in a full case?  To go from a half case to a whole case, they can divide the half case by a half, producing one. Likewise, they can divide 30 by ½ to get a result of 60 oranges. 

Students check their reasoning that 30 ÷ 1/2 = 60 by reasoning that 60 × 1/2 = 30. They learn to re-represent the division by a unit fraction as a multiplication by the denominator of the fraction, based on the previous idea of n 1/n  ths equals one (see Standard 3.NF.3.c of the Fractions LT). A referent transforming rate problem would be: “Teesha drove 30 miles for 1/2 hour. What speed (miles per hour) did she drive at?” To find the distance travelled for an hour, they solve the problem:distance/ time = rate, or 30 miles ÷ 1/2 hour. Students recognize that 2 1/2th-hours makes an hour, and the associated distance of 30 miles is multiplied by 2 correspondingly to produce 60 miles per hour A number line type measurement problem would ask, “How many 1/2-inch pieces of cake can be cut from a 30-inch wide Christmas log cake?” The quotative strategy of imagining how many pieces could be produced gives the answer of 60 pieces. This is referent preserving. An area problem for 6 ÷ 1/5 could be if the area of a piece of cloth is 6 square yards, and one side is 1/5 of a yard, how long is the piece of cloth. Each of the square yards would be cut into strips one fifth wide and one yard long. They would be lined up to produce a piece of cloth 30 feet long.  This is referent composing.Example: How many 1/3-cup servings are in 2 cups of raisins?

Examples: 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.

Sixth GradeMultiplication and Division Problems Involving Non-Whole Rational Number Operations (Fractions)

A common way to approach division by fractions is by using the “invert and multiply rule”. However, this procedure is often taught without context.Standard Explanation Concrete Representational Abstract

6.NS.1: Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem.

In 5th grade students divided whole numbers by unit fractions and divided unit fractions by whole numbers. Students continue to develop this concept by using visual models and equations to divide whole numbers by fractions and fractions by fractions to solve word problems. Students develop an understanding of the relationship between multiplication and division.

Build-it Draw-it Write-itInterpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using concrete fraction models to represent the problem.

Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using representational fraction models to represent the problem.

Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using equations to represent the problem.

Example 1:

Students understand that a division problem such as 3 ÷ 25 is asking, “how many

25 are in 3?” One possible visual model would begin with three whole and

divide each into fifths. There are 7 groups of two-fifths in the three wholes. However, one-fifth remains. Since one-fifth is half of a two-fifths group, there is a

remainder of 12 . Therefore, 3 ÷

25 = 7

12 , meaning there are 7

12 groups of two-fifths. Students interpret the solution, explaining

how division by fifths can result in an answer with halves.

Students also write contextual problems for fraction division problems. For example, the problem, 23 ÷

16 can be illustrated with the following word problem:

Example 2:

Susan has 23 of an hour left to make cards. It takes her about

16 of an hour to make each card. About how many can she make? This problem can be modeled

using a number line. a. Start with a number line divided into thirds.

b. The problem wants to know how many sixths are in two-thirds. Divide each third in half to create sixths.

c. Each circled part represents 16 . There are four sixths in two-thirds; therefore, Susan can make 4 cards.

Example 3:

Michael 12 has of a yard of fabric to make book covers. Each book cover is made from of

18 a yard of fabric. How many book covers can Michael make?

Solution: Michael can make 4 book covers.

Example 4:

Represent 12 ÷

23 in a problem context and draw a model to show your solution.

Context: A recipe requires 23 of a cup of yogurt. Rachel has

12 of a cup of yogurt from a snack pack. How much of the recipe can Rachel make?

Explanation of Model:

The first model shows 12 cup. The shaded squares in all three models show

12 the cup.

The second model shows12 cup and also shows

13 cups horizontally.

The third model shows 12 cup moved to fit in only the area shown by

23 of the model.

23 is the new referent unit (whole) .

3 out of the 4 squares in the 23portion are shaded. A

12 cup is only

34 of a

23cup portion, so only ¾ of the recipe can be made.

There are three types of dividing fractions by fraction: i) Dividing two "normal" fractions, ii) Dividing a mixed number by a fraction, and iii) Dividing two mixed numbers.

Model 1: Referent Transforming Model 2: Referent Preserving Model 3:Referent Composing

Fair Sharing (Partitive) Measurement (Quotative) ArraysRate Scaling Area

Equal Groups Cartesian Product

Note that Arrays and Cartesian products are not viable models for interpreting division of fractions or mixed numbers in this standard.

1. Referent Transforminga) A fair-sharing example might be: “Ally made 4/5 of a pound of trail mix to be shared among her friends. She places them into bags of each

having 2/25pounds for each person. How much trail mix will there be in each bag?” (Solution: 4/5 pounds ÷ 2/25 pounds per person = 4/5 × 25/2 people = 100/10people = 10 people)

b) An example of rate problems involving partitive division might be: “A long distance runner covers 2 3/4 miles in 2/5 hour. How many miles can the runner cover in 1 hour?” (Solution: 2 3/4 miles ÷ 2/5 = 9/4 ÷ 2/5 = 9/4 × 5/2 = 45/8 miles).

c) An example of Equal Groups problem involving division with transformation of units might be: “If a box of cereal holds 28 1/2 ounces, then how many servings will there be if each serving is 31/2 ounces?” (Solution: 28 1/2 ÷ 3 1/2 = (28 1/2 × 2) ÷ (3 1/2 × 2) = 8 1/7)

 2.   Referent Preserving  

a) A measurement example might be, “A whole ribbon measuring 3/4 foot in length is cut into 3/8 -foot long strips. How many strips can be made from the whole ribbon?” (Solution: 3/4 ÷ 3/8 = 3/4 × 8/3 = 24/12 = 2) The whole ribbon is twice as long as the strip (An analysis of the units in this problem anticipates scale factors at 8th grade: inches ÷ inches = a scale factor).

b) Scaling problems involve the division of a quantity with a scale factor. An example of a scaling problem involving division might be: “After shrinking a square by a scale factor of 1/4, the length of the square is 12/15 feet long, what will the length of a side of a square before enlarging?” (Solution: 12/15÷ 1/4 = 12/60 = 1/5 feet long)

c) An example of Equal Groups problem involving division with preservation of unit units might be: “A bottle of medicine contains 8 2/3 oz. You can have 12 1/2 doses from this bottle. How many ounces are in each dose?” (Solution: 8 2/3 ÷12 1/2 = 26/3 ÷ 25/2= 26/3 × 2/25 = 52/75 ounces in each dose)

3.   Referent Composing An example of an area problem involving division is: “Mr. Jones plans to build a rectangular garden outside of his house that will cover 2/3 of a square mile. One dimension of the garden area will be determined by a fence that is 3/4 of a mile long. What is the other dimension of the garden area?” (Solution: 2/3 ÷3/4 = 2/3 × 4/3 = 8/9)