Unit 2: Fraction Equivalence and Comparison

Goals

Students will learn to create and think about equivalent fractions (fractions that are equal in value even though they look different) and compare and order fractions that have these denominators: 2, 3, 4, 5, 6, 8, 10, 12, and 100.

Overview

This unit builds on what students already know about fractions. In third grade, students learned to divide shapes into equal parts and describe each part using a unit fraction — a fraction like $\frac{1}{4}$ that represents just one equal part of a whole. They discovered that any unit fraction comes from splitting 1 whole into equal pieces. Using unit fractions, students built other fractions, including fractions greater than 1, and showed them using fraction strips and tape diagrams (rectangle drawings split into equal sections). In third grade, they worked only with denominators of 2, 3, 4, 6, and 8. Students also placed fractions on number lines, learning that fractions are numbers just like whole numbers, and that equivalent fractions land on the very same point on the number line.

In this unit, students follow a similar path but go further. They use fraction strips, tape diagrams, and number lines to understand fraction size, create equivalent fractions, and compare and order fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.

Students learn an important pattern: a fraction is equivalent to another fraction when each unit fraction gets split into the same number of new equal parts. This makes each new part a certain number of times smaller, and it makes the whole broken into that same number of times as many parts. For example, $\frac{3}{5}$ is equivalent to $\frac{6}{10}$. If you split each fifth into 2 smaller parts, you now have 6 shaded parts instead of 3 — twice as many — and each part is half as small as before.

As the unit continues, students use equivalent fractions and benchmarks (easy reference points like $\frac{1}{2}$ and 1) to figure out where fractions belong on a number line, and to compare and order them.

Skills Built Throughout the Unit

Alongside fraction work, students keep practicing mental multiplication strategies they learned in third grade, using the properties of multiplication. Short daily warm-ups called Number Talks support this goal by focusing on the factors 2, 4, 5, 6, 8, 10, and 12. Students practice strategies like doubling and halving numbers, and they connect these strategies to folding fraction strips and splitting tape diagrams into smaller unit fractions.

Below are a few examples of the Number Talk warm-ups used in this unit, found in Lessons 5, 9, and 16.

These factors were chosen on purpose to help students become comfortable with the unit fractions used throughout the unit. By noticing the relationships between these factors and their products during Number Talks, students become quicker at figuring out equivalent fractions and comparing fractions with these denominators.

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Materials Needed

Lesson	Materials to Gather	Materials to Copy
Lesson 1	Straightedges (Activity 1, Activity 2)	Fraction Strips Template (1 copy per 2 students) — Activity 1
Lesson 2	Straightedges (Activity 1, Activity 2); materials from a previous lesson (Activity 2)
Lesson 3	—
Lesson 4	Straightedges (Activity 1)
Lesson 5	Straightedges (Activity 1)
Lesson 6	—	Card Sort: Where Do They Belong? Cards (1 copy per 2 students) — Activity 2
Lesson 7	Tools for creating a display (Activity 2)
Lesson 8	Tape — painter's or masking (Activity 1)
Lesson 9	Rulers or straightedges (Activity 1); sticky notes (Activity 2)	How Do You Know Cards (1 copy per 15 students) — Activity 2
Lesson 10	—
Lesson 11	—	Card Sort: Fractions Galore Cards (1 copy per 3 students) — Activity 3
Lesson 12	Colored pencils (Activity 2)
Lesson 13	—
Lesson 14	Tools for creating a display
Lesson 15	—
Lesson 16	—	Fraction Cards with Denominators 2, 3, 4, and 6 (1 copy per 2 students); Fraction Cards with Denominators 5, 8, 10, 12, and 100 (1 copy per 2 students) — Activity 1
Lesson 17	Markers, paper, paper clips, tape — painter's or masking (Activity 1, Activity 2)

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Section A: Size and Location of Fractions

Standards

Building On: 3.NF.A.1, 3.NF.A.2, 3.NF.A.2.a, 3.NF.A.3, 3.NF.A.3.b, 3.NF.A.3.d, 3.OA.B.5

Addressing: 4.NF.A.1, 4.NF.A.2

Building Towards: 4.NBT.B.4, 4.NF.A, 4.NF.A.1, 4.NF.A.2

Goals

• Make sense of fractions with denominators 2, 3, 4, 5, 6, 8, 10, and 12 by using physical objects and drawings.
• Reason about where fractions belong on the number line.

Overview

In this section, students revisit fraction ideas from third grade, but now they also work with denominators of 5, 10, and 12. Students use physical fraction strips, drawings of fraction strips, tape diagrams, and number lines to understand fraction size and the relationships between fractions.

Students think about how fractions relate when one denominator is a multiple of the other — for example, $\frac{1}{3}$ and $\frac{1}{6}$, or $\frac{1}{4}$ and $\frac{1}{12}$. They explore different ways to show these relationships. Students also compare fractions to benchmarks like $\frac{1}{2}$ and 1.

This section prepares students to reason about equivalence and comparison of fractions in the lessons that follow.

Suggested Centers

Lesson 1

• Get Your Numbers in Order (1–5), Stage 3: Denominators 2, 3, 4, 6 (Addressing)
• Mystery Number (1–5), Stage 3: Fractions with Denominators 2, 3, 4, 6 (Supporting)

Lesson 2

• Get Your Numbers in Order (1–5), Stage 3: Denominators 2, 3, 4, 6 (Addressing)
• Mystery Number (1–5), Stage 3: Fractions with Denominators 2, 3, 4, 6 (Supporting)

Lesson 3

• Get Your Numbers in Order (1–5), Stage 3: Denominators 2, 3, 4, 6 (Addressing)
• Mystery Number (1–5), Stage 3: Fractions with Denominators 2, 3, 4, 6 (Supporting)

Lesson 4

• Get Your Numbers in Order (1–5), Stage 3: Denominators 2, 3, 4, 6 (Addressing)
• Number Line Scoot (2–4), Stage 3: Halves, Thirds, Fourths, Sixths, and Eighths (Supporting)

Lesson 5

• Get Your Numbers in Order (1–5), Stage 3: Denominators 2, 3, 4, 6 (Addressing)
• Number Line Scoot (2–4), Stage 3: Halves, Thirds, Fourths, Sixths, and Eighths (Supporting)

Lesson 6

• Get Your Numbers in Order (1–5), Stage 3: Denominators 2, 3, 4, 6 (Addressing)
• Number Line Scoot (2–4), Stage 3: Halves, Thirds, Fourths, Sixths, and Eighths (Supporting)

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Section A Checkpoint

Problem 1

Label the point on each number line with a fraction it represents.

Solution:
a. $\frac{7}{8}$ — There are 8 equal parts between 0 and 1, and the point sits on the 7th tick mark, showing 7 out of 8 parts.

b. $\frac{8}{5}$ — Each whole is split into 5 equal parts, and the point sits on the 8th tick mark from 0, showing 8 parts total.

Responding to Student Thinking:
Some students may not yet understand that the partitions on a number line represent unit fractions, or that tick marks show distance from 0. For example, on the first number line, a student might count the tick marks between 0 and 1 and mistakenly label the point $\frac{1}{7}$.

Students may also correctly show fractions less than 1 but not yet realize that a fraction's numerator (top number) can be greater than its denominator (bottom number) once the fraction is greater than 1. For example, they might label $\frac{8}{5}$ as $\frac{3}{5}$ instead.

Later in the unit, when students represent equivalent fractions using number lines, watch for students who can explain how they figure out what the tick marks represent, and invite them to share with the class. Ask students how number lines with fractions are similar to and different from number lines with whole numbers. Stress the meaning of the numerator and denominator, and how understanding these helps in reading partitions and points on a number line.

Problem 2

Is $\frac{7}{12}$ greater than or less than $\frac{2}{8}$? Explain your reasoning. Use a number line if it helps.

Solution:
$\frac{7}{12}$ is greater than $\frac{2}{8}$. Sample explanation: $\frac{7}{12}$ is on the 7th tick mark from 0, and it is more than halfway to 1 (since half of 12 is 6, and $\frac{7}{12}$ is past $\frac{6}{12}$). So $\frac{7}{12}$ is greater.

Responding to Student Thinking:
Some students correctly identify the greater fraction but aren't yet using benchmark reasoning. For instance, they may think $\frac{7}{12}$ is greater simply because 7 and 12 are both bigger numbers than 2 and 8.

Encourage these students to play Compare, Stage 6, found in Section C. Watch for how students use what they know about $\frac{1}{2}$ and 1, or how they use common denominators, to make comparisons. Invite students to explain their thinking to classmates and to draw a diagram to support their reasoning.

Problem 3

Explain why $\frac{1}{3}$ is equivalent to $\frac{4}{12}$. Use a number line if it helps.

Solution:
Sample response: If each third is split into 4 equal pieces, each new small piece is $\frac{1}{12}$. Since $\frac{1}{3}$ lands on the 4th tick mark from 0, this shows it is equal to $\frac{4}{12}$.

Responding to Student Thinking:
Some students may not yet be reasoning about fraction size or about multiples of numerators and denominators.

Encourage these students to play Compare, Stage 6, in Section C. Watch for how students reason about equivalent fractions when one denominator is a multiple of the other. Invite students to share their thinking and illustrate it with a diagram. Connect strategies that involve drawing extra partitions on number lines to the idea of finding common denominators.

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Practice Problems

12 Problems — Pre-unit

Standards Practiced: 3.NF.A.1

Problem 1

What fraction of each figure is shaded?

Solution: $\frac{3}{4}$ of the circle and $\frac{5}{8}$ of the square

Problem 2

Explain why the sha