Learn on PengiIllustrative Mathematics, Grade 5Chapter 2: Fractions as Quotients and Fraction Multiplication

Lesson 8: Decompose Area and Apply Properties

In this Grade 5 Illustrative Mathematics lesson from Chapter 2, students learn how to decompose rectangles with mixed number side lengths to find area by applying the distributive property and partial products. Students practice multiplying whole numbers by mixed numbers such as 5 × 2½ and use diagrams to represent and calculate areas like 5 yards by 6¼ yards. The lesson builds on prior work with fractional side lengths and addresses standard 5.NF.B.4.b.

Section 1

Decomposing a Fraction as a Product

Property

Any fraction can be decomposed, or broken down, into a product of a whole number and a unit fraction.

ab=a×1b\frac{a}{b} = a \times \frac{1}{b}

Section 2

Multiplying a Whole Number and a Unit Fraction

Property

To multiply a whole number by a unit fraction, the whole number becomes the numerator of the resulting fraction, and the denominator stays the same.

n×1d=ndn \times \frac{1}{d} = \frac{n}{d}

Section 3

Calculating Products Using Decomposition and Division

Property

To multiply a fraction by a whole number, you can decompose the fraction, convert the unit fraction multiplication to division, and then multiply by the numerator.

ab×c=a×(1b×c)=a×(c÷b)\frac{a}{b} \times c = a \times \left(\frac{1}{b} \times c\right) = a \times (c \div b)

Examples

Section 4

Applying the Associative Property to Multiply a Whole Number and a Fraction

Property

To multiply a whole number by a non-unit fraction, we can decompose the fraction and use the associative property to regroup the whole numbers.

n×ab=n×(a×1b)=(n×a)×1bn \times \frac{a}{b} = n \times (a \times \frac{1}{b}) = (n \times a) \times \frac{1}{b}

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Chapter 2: Fractions as Quotients and Fraction Multiplication

  1. Lesson 1

    Lesson 1: Sharing Situations and Division as Fractions

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  2. Lesson 2

    Lesson 2: Interpret and Solve Division Situations

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  3. Lesson 3

    Lesson 3: Relate Division and Fractions

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  4. Lesson 4

    Lesson 4: Relate Division and Multiplication

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  5. Lesson 5

    Lesson 5: Divide to Multiply Fractions

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  6. Lesson 6

    Lesson 6: Area with Fractional Side Lengths (< 1)

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  7. Lesson 7

    Lesson 7: Fractional Side Lengths Greater Than 1

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  8. Lesson 8Current

    Lesson 8: Decompose Area and Apply Properties

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  9. Lesson 9

    Lesson 9: Area Situations

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  10. Lesson 10

    Lesson 10: Multiply More Fractions

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Lesson overview

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Section 1

Decomposing a Fraction as a Product

Property

Any fraction can be decomposed, or broken down, into a product of a whole number and a unit fraction.

ab=a×1b\frac{a}{b} = a \times \frac{1}{b}

Section 2

Multiplying a Whole Number and a Unit Fraction

Property

To multiply a whole number by a unit fraction, the whole number becomes the numerator of the resulting fraction, and the denominator stays the same.

n×1d=ndn \times \frac{1}{d} = \frac{n}{d}

Section 3

Calculating Products Using Decomposition and Division

Property

To multiply a fraction by a whole number, you can decompose the fraction, convert the unit fraction multiplication to division, and then multiply by the numerator.

ab×c=a×(1b×c)=a×(c÷b)\frac{a}{b} \times c = a \times \left(\frac{1}{b} \times c\right) = a \times (c \div b)

Examples

Section 4

Applying the Associative Property to Multiply a Whole Number and a Fraction

Property

To multiply a whole number by a non-unit fraction, we can decompose the fraction and use the associative property to regroup the whole numbers.

n×ab=n×(a×1b)=(n×a)×1bn \times \frac{a}{b} = n \times (a \times \frac{1}{b}) = (n \times a) \times \frac{1}{b}

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Chapter 2: Fractions as Quotients and Fraction Multiplication

  1. Lesson 1

    Lesson 1: Sharing Situations and Division as Fractions

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  2. Lesson 2

    Lesson 2: Interpret and Solve Division Situations

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  3. Lesson 3

    Lesson 3: Relate Division and Fractions

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  4. Lesson 4

    Lesson 4: Relate Division and Multiplication

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  5. Lesson 5

    Lesson 5: Divide to Multiply Fractions

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  6. Lesson 6

    Lesson 6: Area with Fractional Side Lengths (< 1)

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  7. Lesson 7

    Lesson 7: Fractional Side Lengths Greater Than 1

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  8. Lesson 8Current

    Lesson 8: Decompose Area and Apply Properties

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  9. Lesson 9

    Lesson 9: Area Situations

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  10. Lesson 10

    Lesson 10: Multiply More Fractions

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