Unit 4

Multiplication and Division of Fractions and Decimal Fractions

 

Grade 5

Math

Unit Length and Description:

 

30 days

 

In Unit 4 students begin to explore multi-digit decimal multiplication and division. Students use models to multiply a fraction by a fraction. Students explore what happens when they resize one of the factors and consider what happens to the size of the product. Students begin to understand what happens to the size of the product when they multiply a fraction by a whole number, a whole number times a fraction, or a fraction by a fraction. Throughout the unit, students solve real world problems involving multiplication of fractions and mixed numbers. Students apply their understanding of whole number division to dividing whole numbers by fractions and fractions by whole numbers.

Throughout the unit students apply their knowledge of order of operations and writing expressions as they work with finding equivalent fractions and the fraction operations.

 

Standards:

 

Major Clusters

NF Number and Operations - Fractions

Apply and extend previous understandings of multiplication and division to multiply and divide fractions.

5.NF.4

Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

Interpret the product of (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. For example, use a visual fraction model to show (2/3 4 = 8/3, and create a story context for this equation. Do the same with (2/3) (4/5) = 8/15. (In general, (a/b) (c/d) = ac/bd.)

5.NF.5

Interpret multiplication as scaling (resizing), by:

a. Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.

b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (na)/(nb) to the effect of multiplying a/b by 1.

5.NF.6

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

5.NF.7

Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. (Students able to multiple 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 level.)

a. Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (1/3) 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) 4 = 1/12 because (1/12) 4 = 1/3.

b. Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 (1/5) = 20 because 20 (1/5) = 4.

c. Solve real world problems involving division of unit fractions by nonzero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?

Supporting Clusters

Measurement and Data

Convert like measurement units within a given measurement system.

5.MD.2

Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots. For example, given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally.

Additional Clusters

Operations and Algebraic Thinking

Write and interpret numerical expressions.

5.OA.A.1

Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.

5.OA.A.2

Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation add 8 and 7, then multiply by 2 as 2 (8 +7). Recognize that 3 (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.

Foundational Standards

3.OA.A.1

Interpret products of whole numbers, e.g., interpret 5 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 7.

3.OA.A.2

Interpret whole-number quotients of whole numbers, e.g., interpret 56 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 8.

3.OA.B.6

Understand division as an unknownfactor problem. For example, find 32 8 by finding the number that makes 32 when multiplied by 8.

4.OA.A.1

Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 x 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.

4.OA.A.2

Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.

3.NF.A.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.

4.NF.A.1

Explain why a fraction a/b is equivalent to a fraction (n a)/(n 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.

4.NF.B.4

Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.

a.    Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 (1/4), recording the conclusion by the equation 5/4 = 5 (1/4).

b.    Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 (2/5) as 6 (1/5), recognizing this product as 6/5. (In general, n (a/b) = (n a)/b.)

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

4.OA.A.1

Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.

4.OA.A.2

Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.

4.MD.A.2

Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.

4.MD.B.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 information presented in line plots. For example, from a line plot find and interpret the difference in length between the longest and shortest specimens in an insect collection.

5.MD.B.2

Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots. For example, given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally.

5.NF.B.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.

Standards for Mathematical Practices

1.   Make sense of problems and persevere in solving them.

2.   Reason abstractly and quantitatively.

3.   Construct viable arguments and critique the reasoning of others.

4.   Model with mathematics.

5.   Use appropriate tools strategically.

6.   Attend to precision.

7.   Look for and make use of structure.

8.   Look for and express regularity in repeated reasoning.

Standards: Instructional Outcomes

 

         5.NF.B.4:

o   I can multiply fractions.

o   I can determine the sequence of operations when multiplying a fraction to a whole number.

o   I can determine the sequence of opertions when multiplying two fractions.

o   I can multiply fractional side lengths to find areas of a rectangle.

o   I can prove multiplying fractional side lengths to find the area is the same as tiling a rectangle with unit squares.

o   I can model the area of rectangles with fractional side lengths with unit squares.

         5.NF.B.5:

o   I can tell about the size of a product based on the factors (relative to 2).

o   I can explain the relationship between two multiplication problems that share a common factor ( and ).

o   I can compare the product of two factors without multiplying. Example: 2 x ? = < 1 Answer must be less than .

o   I can explain why multiplying a number by a fraction greater than one will result in a product greater than the given number.

o   I can explain why multiplying a fraction by one (which can be written as various fractions, ex. , etc.) results in an equivalent fraction.

o   I can explain why multiplying a fraction by a fraction will result in a product smaller than the given number.

         5.NF.B.6:

o   I can solve real world problems involving multiplication of fractions and mixed numbers using models or equations.

o   I can explain or illustrate my solution using fraction models or equations.

         5.NF.B.7:

o   I can represent division of a unit fraction by a non-zero whole number in a variety of ways.

o   I can represent division of a whole number by a unit fraction in a variety of ways.

o   I can represent division of a unit fraction by a non-zero whole number and a whole number by a unit fraction in a variety of ways to solve real world problems.

         5.MD.B.2

o   I can solve problems using line plots with halves, fourths, and eighths using any operation.

o   I can make a line plot for measurements of fourths, halves, and eighths.

         5.OA.A.1

o   I can evaluate expressions using the order of operations including parenthesis (), brackets [], or braces {}.

o   I can write expressions using the order of operations including parenthesis (), brackets [], or braces {}.

         5.OA.A.2

o   I can write simple expressions that record calculations with numbers.

o   I can interpret numerical expressions without evaluating them.

 

 

Enduring Understandings:

 

         Multiplication does not always make the product larger than the factors.

         Division does not always make the quotient smaller than the factors.

         A fraction is relative to the size of the whole or unit.

         Creating visual models aids in multiplying and dividing fractions.

 

Essential Questions:

 

         How do operations with fractions compare/relate to operations with whole numbers and decimals?

         How is multiplying or dividing whole numbers similar to multiplying or dividing fractions?

         How can multiplying and dividing fractions be modeled using area, a number line, or measurement models?