Topic Eleven: Multiplying and Dividing Fractions and Mixed Numbers

Desired Results

Transfer:

1. Makes 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.
Established Goals:
  • 5.NF.3: Interpret a fraction as division of the numerator by the denominator (a/b = a ¸ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret ¾ as the result of dividing 3 by 4, noting that ¾ multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size ¾. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?
  • 5.NF.4: Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
    • a. Interpret the product (a/b) ´ q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a ´ q ¸ b. 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 .)
    • b. Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.
  • 5.NF.5. Interpret multiplication as scaling (resizing), by:
    • a. Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.
    • b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n ´ a)/(n ´ b) to the effect of multiplying a/b by 1.
  • 5.NF.6. Solve real-world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.
  • 5.NF.7. Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.
    • 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 non-zero 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 ½ lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?

Prerequisite Standards:
  • 4.NF.3d: Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem.
  • 4.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 equation 5/4 = 5 ´ (1/4).
  • 4.NF.4c: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?
  • 5.NF.2: Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem.Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2 .

"I Can" Statements:
  • I can understand that fractions are really the division of a numerator by the denominator.
  • I can solve word problems where I divide whole numbers to create an answer that is a mixed number.
  • I can multiply a fraction or whole number by a fraction.
  • I can think of multiplication as the scaling of a number (similar to a scale on a map.)
  • I can solve real world problems by multiplying fractions and mixed numbers.
  • I can divide fractions by whole numbers and whole numbers by fractions.
Big Ideas:
  • Equivalence: Any number, measure, numerical expression, algebraic expression, or equation can be represented in an infinite number of ways that have the same value.
  • Operation Meanings and Relationships: There are multiple interpretations of addition, subtraction, multiplication, and division of rational numbers, and each operation is related to other operations.
  • Estimation: Numbers can be approximated by numbers that are close. Numerical calculations can be approximated by replacing numbers with other numbers that are close and easy to compute mentally. Some measurements can be approximated using known referents as the unit in the measurement process.
  • Basic Facts and Algorithms: There is more than one algorithm for each of the operations with rational numbers. Some strategies for basic facts and most algorithm for operations with rational numbers, both mental math and paper and pencil, use equivalence to transform calculations into simpler ones.
  • Practices, Processes, and Proficiencies: Mathematics content and practices can be applied to solve problems.
Essential Questions:

  • What are standard procedures for estimating and finding products and quotients of fractions and mixed numbers?
Students will know...
  • A fraction describes the division of a whole into equal parts, and it can be interpreted in more than one way depending on the whole to be divided.
  • The product of a whole number and a fraction can be interpreted in different ways. One interpretation is repeated addition. Multiplying a whole number by a fraction involves division as well as multiplication. The products is a fraction of the whole number.
  • Rounding and compatible numbers can be used to estimate the products of fractions or mixed numbers.
  • A unit square can be used to show the area meaning of fraction multiplication. When you multiply two fractions that are both less than 1, the product is smaller than either fraction. To multiply fractions, write the product of the numerators over the product of the denominators.
  • One way to find the product of mixed numbers is to change the calculation to an equivalent one involving improper fractions.
  • The relative size of the factors can be used to determine the relative size of the product.
  • Some problems can be solved by first finding and solving a sub-problem and then using that answer to solve the original problem.
  • One way to find the quotient of a whole number divided by a fraction is to multiply the whole number by the reciprocal of the fraction.
  • The inverse relationship between multiplication and division can be used to divide fractions.
  • Information in a problem can often be shown with a diagram and used to solve the problem. Some problems can be solved by writing and completing a number sentence or equation.

Vocabulary: resizing, scaling, reciprocal
Students will be skilled at...
  • Using fractions to represent division, and locating and placing fractions on a number line.
  • Multiplying a fraction by a whole number.
  • Using compatible numbers and rounding to estimate with fractions.
  • Giving the product of two fractions.
  • Finding the area of rectangles.
  • Multiplying mixed numbers.
  • Comparing the size of the product to the size of one factor without multiplying as they begin to consider multiplication as scaling.
  • Solving multi-step word problems.
  • Dividing whole numbers by fractions.
  • Discovering the inverse relationship between multiplication and division that will help them to divide unit fractions by whole numbers.
  • Using diagrams and writing equations to solve problems.

Assessment Evidence

Performance Assessment:

MARS Task-Cindy's Cats
MARS Task-Fractions
Other Evidence:

Learning Plan

Learning Activities:

11-1:A fraction describes the division of a whole into equal parts, and it can be interpreted in more than one way depending on the whole to be divided.
11-2:The product of a whole number and a fraction can be interpreted in different ways. One interpretation is repeated addition. Multiplying a whole number by a fraction involves division as well as multiplication. The products is a fraction of the whole number.
11-3:Rounding and compatible numbers can be used to estimate the products of fractions or mixed numbers.
11-4: A unit square can be used to show the area meaning of fraction multiplication. When you multiply two fractions that are both less than 1, the product is smaller than either fraction. To multiply fractions, write the product of the numerators over the product of the denominators.
11-5:A unit square can be used to show the area meaning of fraction multiplication. When you multiply two fractions that are both less than 1, the product is smaller than either fraction. To multiply fractions, write the product of the numerators over the product of the denominators.
11-6: One way to find the product of mixed numbers is to change the calculation to an equivalent one involving improper fractions.
11-7: The relative size of the factors can be used to determine the relative size of the product.
11-8: Some problems can be solved by first finding and solving a sub-problem and then using that answer to solve the original problem.
11-9: One way to find the quotient of a whole number divided by a fraction is to multiply the whole number by the reciprocal of the fraction.
11-10: The inverse relationship between multiplication and division can be used to divide fractions.
11-11:Information in a problem can often be shown with a diagram and used to solve the problem. Some problems can be solved by writing and completing a number sentence or equation.
Resources:

Problem of the Month:
Fractured Numbers
Diminishing Return
Got Your Number
Party Time
Measuring Up

Centers:
Relating Fractions to Division
Multiplying Fractions by Dividing Rectangles
Fraction x fraction word problems
Area Word Problems with Fractional Side Lengths
Multipilcation and Scale Problems
Fraction X Mixed Number Word Problems
Whole Number X Mixed Number Word Problems
Mixed Numbers x Fraction Models
Divide a Unit Fraction by a Whole Number
Dividing a Whole Number by a Unit Fraction
Divide a Whole Number by a Unit Fraction
Division of Fractions Word Problems

Additional Units:
Multiplication of Fractions
Time for Recess