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Multiplication doesn't always mean the number gets bigger

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Multiplication doesn't always mean the number gets bigger

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People are taught that multiplication means a number gets bigger and division means a number becomes smaller but this is not always true. Specifically, when multiplying by a fraction less than 1.

Section 7.4 of Math for Elementary School Teachers Textbook

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It is important to understand fraction and decimal equivalents before adding or subtracting them.

Section 6.5 of Math for Elementary School Teachers Textbook

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If a whole is broken into equal parts, then the more parts there are, the smaller the pieces will be.

Section 6.5 of Math for Elementary School Teachers Textbook

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Generalizable rule for the numerator and denominator of a fraction

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Generalizable rule for the numerator and denominator of a fraction

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Teachers should create a rule concerning the numerator and denominator of fractions that applies to all fractions.

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Variety of models to represent fractions

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Variety of models to represent fractions

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It is important for students to fully understand fractions before learning how to add, subtract, divide and multiply fractions. Students should learn how to represent fractions in multiple different ways.

10 Practical Tips for Making Fractions Come Alive
and Make Sense (Clark, Roche, Mitchell; Mathematics Teaching in the Middle School 13(7), 2008).

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In the fraction a/b, b is the name or size of the part and a is the number of parts of that name or size.

Stated in 10 Practical Tips for Making Fractions Come Alive
and Make Sense (Clark, Roche, Mitchell; Mathematics Teaching in the Middle School 13(7), 2008).

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3/8   vs.   3/12

3/8 is larger because the pieces are larger

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One way to help students learn fraction equivalents is to use fraction blocks. These allow children to physically see what fractions equal the same thing.

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In this model, the end fraction of 6/35 is displayed in purple and is smaller than both the blue and green areas that represent the fractions being multiplied.

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Extend understanding of fraction equivalence and ordering.
CCSS.MATH.CONTENT.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.

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Develop understanding of fractions as numbers.
CCSS.MATH.CONTENT.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.

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Apply and extend previous understandings of multiplication and division.
CCSS.MATH.CONTENT.5.NF.B.4
Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

CCSS.MATH.CONTENT.5.NF.B.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 equivalence a/b = (n × a)/(n × b) to the effect of multiplying a/b by 1.

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Develop understanding of fractions as numbers.
CCSS.MATH.CONTENT.3.NF.A.3
Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.

CCSS.MATH.CONTENT.3.NF.A.3.A
Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.

CCSS.MATH.CONTENT.3.NF.A.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.

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Develop understanding of fractions as numbers.
CCSS.MATH.CONTENT.3.NF.A.3.D
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.