Grade 4 Teacher Guide — Unit 2

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Unit 2: Fraction Equivalence and Comparison

Goals

• Students will build and reason about equivalent fractions (fractions that name the same amount even though they look different) and compare and order fractions with denominators of 2, 3, 4, 5, 6, 8, 10, 12, and 100.

Overview

This unit builds on what students learned in third grade about equivalent fractions and comparing fractions.

In grade 3, students split shapes into equal-sized parts and named each part with a unit fraction—a fraction with 1 as the numerator, like $\frac{1}{4}$. They learned that any unit fraction comes from splitting a whole (1) into that many equal parts. From there, students combined unit fractions to build other fractions, including fractions greater than 1, and displayed them using fraction strips and tape diagrams (rectangular models used to represent fractions). At that stage, they only worked with denominators of 2, 3, 4, 6, and 8. Students also placed fractions on a number line, learning that fractions are numbers in their own right and that equivalent fractions land on the exact same point.

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

Students learn a general rule: a fraction is equivalent to another fraction when each unit fraction has been split into a certain number of smaller equal parts—this makes each piece proportionally smaller, but it also multiplies the total number of pieces in the whole by that same amount. For example, $\frac{3}{5}$ is equivalent to $\frac{6}{10}$: if each fifth is split into 2 smaller parts, you now have 6 shaded parts instead of 3—twice as many—but each part is half as small.

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

Throughout the Unit

Students keep building mental math strategies for multiplication, using the fluency they developed in grade 3 along with the properties of multiplication. The Number Talks (short daily discussions where students solve problems mentally and share strategies) in this unit support this goal, focusing on the factors 2, 4, 5, 6, 8, 10, and 12. Students practice strategies like doubling and halving numbers, connecting these strategies to folding fraction strips and splitting tape diagrams into smaller unit fractions.

Below is a sample of the Number Talk warm-ups included in the unit:

• Lesson 5
• Lesson 9
• Lesson 16

These factors were chosen on purpose to help students build flexibility with the unit fractions used throughout the unit. By noticing the relationships between these factors and their products during Number Talks, students become quicker and more confident at finding 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): Activity 1; Fraction Cards with Denominators 5, 8, 10, 12, and 100 (1 copy per 2 students): Activity 1
Lesson 17	Markers, paper, paper clips, tape (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 using physical objects and diagrams.
• Reason about where fractions fall on the number line.

Overview

In this section, students return to ideas and models of fractions from grade 3, but now they work with denominators 5, 10, and 12 as well. They use physical fraction strips, drawings of fraction strips, tape diagrams, and number lines to understand fraction size and how fractions relate to one another.

Students think about the relationship between fractions where one denominator is a multiple of the other (for example, comparing fractions with denominators of 4 and 8, or 3 and 6). They explore different ways to picture and explain these relationships. Students also compare fractions to benchmarks such as $\frac{1}{2}$ and 1.

This work prepares students for the deeper study of equivalence and comparison of fractions in the lessons that follow.

Suggested Centers

Lesson 1

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

Lesson 2

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

Lesson 3

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

Lesson 4

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

Lesson 5

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

Lesson 6

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

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Section A Checkpoint

Student Task

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

a. [Number line with 8 equal parts between 0 and 1]

b. [Number line with 5 equal parts per whole]

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 parts out of 8.

b. $\frac{8}{5}$ — There are 5 equal parts in each whole, and the point sits on the 8th tick mark, showing 8 parts total.

Responding to Student Thinking

Some students may not yet correctly read the partitions on a number line as unit fractions, or understand 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 understand fractions less than 1 but not yet grasp that a numerator can be larger than the denominator once a fraction is greater than 1. For example, a student might label $\frac{8}{5}$ as $\frac{3}{5}$.

In the next section, as students represent equivalent fractions using number lines, watch for and invite students to share how they figure out what each tick mark represents. Ask how number lines with fractions are similar to and different from number lines with whole numbers. Emphasize what the numerator and denominator mean and how those meanings connect to reading partitions and points on a number line.

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Student Task

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 reasoning: $\frac{7}{12}$ is the 7th tick mark from 0 on a number line divided into 12ths, and it's more than halfway to 1. 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 reasoning using benchmarks. For example, they might think $\frac{7}{12}$ is greater simply because 7 and 12 are both larger numbers than 2 and 8.

Encourage these students to play Compare, Stage 6, found in Section C. Watch for the ways students reason about comparisons using ideas about $\frac{1}{2}$ and 1, or by finding common denominators (matching denominators to compare fractions more easily). Invite students to share their reasoning and illustrate it with a diagram.

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Student Task

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

Solution

Sample response: If I split each third into 4 equal pieces, each new piece is $\frac{1}{12}$, and $\frac{1}{3}$ lands on the 4th tick mark from 0—showing it's the same as $\frac{4}{12}$.

Responding to Student Thinking

Some students may not yet be reasoning about fraction size or about multiples of numerators and denomin