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An Assortment of Fractions

Grade 4: Unit 3: Lesson 15

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Lesson Purpose

The purpose of this lesson is for students to use equivalence to reason about problems that involve

combining or removing fractional amounts.

Lesson Narrative

In a previous unit, students learned to recognize and generate equivalent fractions. Earlier in this

unit, they learned to add and subtract fractions with the same denominator, seeing these

operations as joining and separating parts of the same whole. In this lesson, students encounter

situations that involve combining and removing fractions with different denominators (limited to 2,

3, 4, 6, and 8), prompting them to rely on their understanding about equivalence to reason about

the problems. This work prepares students to use equivalent fractions to join tenths and

hundredths in upcoming lessons.

Students are not expected to reason symbolically, or to write fractional expressions with different

denominators and then rewrite them with a common denominator. Instead, they reason using their

intuitive understanding of equivalence, which they have begun to build since grade 3, and with the

support of visual representations as needed.

Learning Goals

Use equivalence to reason about addition and subtraction problems.

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Let’s find the heights of some stacked objects.

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Which One Doesn’t Belong: Halves, Fourths,Sixths, and Eighths

Which One Doesn’t Belong?

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Which one doesn’t belong?

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Let’s Talk About It!

Let’s find at least one reason why each one doesn’t belong.

What are some fractions that are equivalent to the numbers or expressions in A, B, and C?

What are some ways to decide if two fractions are equivalent?

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All the Way to the Top

Activity 1

In earlier lessons, students found sums and differences of fractions (including mixed numbers) with

the same denominator. In this activity, they reason about problems that involve combining or

removing fractional amounts with different denominators—2, 4, and 8—in the context of stacking

playing bricks. Because the denominators are familiar and are multiples or factors of one another,

students can rely on what they know about the relationships of halves, fourths, and eighths to

compare amounts (how much more or less one amount than another) or to combine them.

The last question asks students to reason about the height of a tower of bricks created by

combining three shorter stacks. Students may arrive at two different answers depending on their

familiarity with playing bricks and attention to precision. Some students may notice that each

playing brick has studs that disappear into the bottom of another brick when stacked, so the

combined height of two stacks will be less than the sum of the heights of individual stacks (MP6).

Both answers are acceptable as long as they are supported.

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For each question, show your reasoning.

How much taller is Lin’s tower compared to:

a. Priya’s tower?

b. Kiran’s tower?

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They are playing in a room that is 109 inches tall. Priya says that if they combine their towers to make a super tall tower, it would be too tall for the room and they’ll have to remove one brick.

Do you agree with Priya? Explain your reasoning.

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Let’s Talk About It!

How did you find the fractional differences between the measurements?

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Stacks of Blocks

Activity 2

MLR1: Stronger and Clearer Each Time

Previously, students used their knowledge of equivalence to reason about the sums and differences

of fractions with denominators 2, 4, or 8. In this activity, they do the same with fractions with

denominators 2, 3, and 6. As before, students are not expected to write addition expressions in

which the fractions are written with a common denominator (though some students may choose to

do so). Instead, they rely on what they know about the relationship between and , and

between and , to solve the problems. Students may choose to use visual representations to

support their reasoning. When students create and compare their own representations for the

context, they develop ways to model the mathematics of a situation and strategies for making

sense of and persevering to solve problems (MP1, MP4).

The measurements in the task— , , and —are given in feet. Because each of them has a whole-

number equivalent in inches, some students may choose to reason entirely in inches, which is a

valid strategy. Ask these students to think about how they’d approach the problems if the given unit

is an unfamiliar one, or one that doesn’t convert handily to whole numbers in another unit.

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What do you notice? What do you wonder?

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Andre is building a tower out of foam blocks. The blocks come in three different thicknesses:

½ foot, ⅓ foot, and ⅙ foot.

Andre stacks one block of each size. Will that stack be more than 1 foot tall? Explain or show how you know.
Can Andre use only the ⅙ foot and ⅓ foot blocks to make a stack that is 1 ½ feet tall? If you think so, show one or more ways. If not, explain why not.
Can Andre use only the ⅙ foot and ½ foot blocks to make a stack that is 1 ⅓ feet tall? If so, show one or more ways. If not, explain why not.

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Let’s Talk About It!

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Lesson Synthesis

How would you find the combined lengths of ½ inch and ⅜ inch? How would you find the difference of the two lengths?

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Which Stack is Taller?

Cool-Down

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Which stack of foam blocks is taller:

• Two ⅓ -foot blocks and one ⅙ -foot block, or

• One ½ -foot block and two ⅙ -foot blocks?

Cool-Down: Which Stack is Taller?

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