Five Tips for Teaching and Learning about Fractions by Paige Kopke

Generalizable rule for the numerator and denominator of a fraction

kopke6269 — 2019-10-22 17:16:50 UTC

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

More Pieces = Smaller Pieces

kopke6269 — 2019-10-22 17:16:54 UTC

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

Focus on equivalences

kopke6269 — 2019-10-22 17:16:57 UTC

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

Section 6.5 of Math for Elementary School Teachers Textbook

Multiplication doesn't always mean the number gets bigger

kopke6269 — 2019-10-22 17:17:01 UTC

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

Variety of models to represent fractions

kopke6269 — 2019-10-24 02:17:32 UTC

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).

The Rule

kopke6269 — 2019-10-24 02:23:10 UTC

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).

Example:

kopke6269 — 2019-10-24 02:40:55 UTC

3/8 vs. 3/12

3/8 is larger because the pieces are larger

kopke6269 — 2019-10-24 02:45:07 UTC

Example:

kopke6269 — 2019-10-24 02:52:31 UTC

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.

kopke6269 — 2019-10-24 02:56:36 UTC

Example:

kopke6269 — 2019-10-24 03:02:09 UTC

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.

Common Core: 3rd Grade

kopke6269 — 2019-10-24 16:14:57 UTC

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.

Common Core: 3rd Grade

kopke6269 — 2019-10-24 16:15:13 UTC

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.

Common Core

kopke6269 — 2019-10-24 16:15:29 UTC

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.

Common Core: 4th Grade

kopke6269 — 2019-10-24 16:15:39 UTC

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.

Common Core

kopke6269 — 2019-10-24 16:16:02 UTC

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.