# Lesson 10

Edge Lengths, Volumes, and Cube Roots

## 10.1: Ordering Squares and Cubes (10 minutes)

### Warm-up

The purpose of this warm-up is to introduce students to cube roots during the discussion. This activity provides an opportunity to use cube root language and notation during the discussion. Students will explore the possibility of negative cube roots in the next lesson.

At first, students should be able to order the values of $$a$$, $$b$$, $$c$$, and $$f$$ without a calculator. However, $$d$$ and $$e$$ will be easier with a calculator. Encourage students to use estimated values for $$d$$ and $$e$$ to order the values before using a calculator. As students work, identify students who use different strategies for ordering.

Teacher Notes for IM 6–8 Math Accelerated

After introducing students to cube root language in the activity synthesis, display this text, image, and table for all to see:

Fill in the missing values using the information provided:

sides volume volume equation
$$27\,\text{in}^3$$

$$\sqrt[3]{5}$$

$$(\sqrt[3]{16})^3=16$$

Tell students that the table shows the values for three different cubes where all values are exact. Give students 1–2 minutes of work time with their partner to identify as many missing values as they can. Select groups to share their reasoning about which values go into the blanks on the table.

Previously, students learned that knowing the area of a square is sufficient information for stating the exact length of the side of the square using square roots. The purpose of this discussion is for students to make that same connection with the volume of cubes and cube roots. The cube root of the volume of a cube represents the exact value of the edge length of a cube in ways that measuring or approximating by cubing values do not (except in special cases such as perfect cubes).

If time allows, discuss the following questions:

• “What integers is $$\sqrt[3]{5}$$ between?” ($$\sqrt[3]{5}$$ is between 1 and 2 because $$1^3 = 1$$ and $$2^3 = 8$$.)
• “What integers is $$\sqrt[3]{16}$$ between?” ($$\sqrt[3]{16}$$ is between 2 and 3 because $$2^3 = 8$$ and $$3^3 = 27$$.)
• “Name another volume for a cube with edge lengths between 2 and 3.” (A cube with volume 26 has edge lengths between 2 and 3 since since $$3^3=27$$, meaning $$\sqrt[3]{26}$$ has a value slightly less than 3.)
Engagement: Develop Effort and Persistence. Connect a new concept to one with which students have experienced success. For example, review activities from lesson 2 in which students explored square root by completing a table based on the side length and the area of a square.
Supports accessibility for: Social-emotional skills; Conceptual processing

### Launch

Students in groups of 2. Give students 2–3 minutes to order the options to the best of their ability without a calculator, and to share their reasoning with a partner. Pause to discuss which are easy to order (likely $$f$$, $$b$$, $$c$$, and $$a$$) and which ones students are not sure about (likely $$d$$ and $$e$$, which are between $$b$$ and $$a$$). Then give students 1 minute with a calculator to finish ordering the options. Follow with a whole-class discussion.

### Student Facing

Let $$a$$, $$b$$, $$c$$, $$d$$, $$e$$, and $$f$$ be positive numbers.

Given these equations, arrange $$a$$, $$b$$, $$c$$, $$d$$, $$e$$, and $$f$$ from least to greatest. Explain your reasoning.

• $$a^2 = 9$$

• $$b^3 = 8$$

• $$c^2 = 10$$

• $$d^3 = 9$$

• $$e^2 = 8$$

• $$f^3 = 7$$

### Activity Synthesis

Ask a student to share their order of $$a$$, $$b$$, $$c$$, $$d$$, $$e$$, and $$f$$ from least to greatest. Record and display their responses for all to see. Ask the class if they agree or disagree. If the class agrees, select previously identified students to share their strategies for ordering the values. If the class is in disagreement, ask students to share their reasoning until an agreement is reached.

As students share, ask students which were the easiest to find and which were the hardest to find. Introduce students to cube root language and notation. Remind students that they previously learned that the equation $$c^2=10$$ has solution $$c=\sqrt{10}$$. Similarly, we can say that the equation $$d^3=9$$ has solution $$d=\sqrt[3]{9}$$. Ask students to write the solution to
$$f^3=7$$ ($$f=\sqrt[3]{7}$$).

Finally, tell students that while square roots are a way to write the exact value of the side length of a square with a known area, cube roots are a way to write the exact value of the edge length of a cube with a known volume, which students will do in a following activity.

## 10.2: Card Sort: Rooted in the Number Line (15 minutes)

### Activity

The purpose of this activity is for students to use rational approximations of irrational numbers and to match irrational numbers to equations they are solutions to. To do this, students sort cards into sets of three consisting of:

• A square or cube root value
• An equation of the form $$x^2=p$$ or $$x^3=p$$ that the value is a solution to
• A number line showing the location of the value

Identify groups using clear explanations for how they chose to arrange their sets of 3, particularly when matching the cards with a value plotted on a number line.

### Launch

Arrange students in groups of 2–4. Students should not use a calculator for this activity. Tell students that they will be given cards to sort into sets of three. Distribute 27 pre-cut slips from the blackline master to each group. Group work time followed by a whole-class discussion.

Representation: Internalize Comprehension. Chunk this task into more manageable parts to differentiate the degree of difficulty or complexity by beginning with fewer cards. For example, give students a subset of the cards (using multiples of 3) to start with and introduce the remaining cards once students have completed their initial set of matches.
Supports accessibility for: Conceptual processing; Organization

### Student Facing

Your teacher will give your group a set of cards. For each card with a letter and value, find the two other cards that match. One shows the location on a number line where the value exists, and the other shows an equation that the value satisfies. Be prepared to explain your reasoning.

### Anticipated Misconceptions

Some students may mix up cube and square roots. Encourage these students to pay close attention to the notation. They may want to fill in the “2” for the square roots similar to how the “3” is used for cube roots.

### Activity Synthesis

Select 3–4 previously identified groups to share one of their sets of three cards.

Also discuss:

• “What was a match that was hard to make?” (Two of the number lines have values between 4 and 5, and it took some extra reasoning to figure out that the one with the value closer to 4 was $$\sqrt{18}$$ while the one with the point closer to 5 was $$\sqrt[3]{100}$$.
• “If another class was going to sort these cards, what is something you would recommend they have or do that you found helpful?” (I would recommend they have a list of perfect squares and perfect cubes to help think about where the different roots are plotted on the number line.)
Speaking, Writing: MLR8 Discussion Supports. Display sentence frames to support students as they justify their reasoning for each match. For example, “The solution to _________ ($$x^3 = 25$$) is _________ ($$\sqrt[3]{25}$$) and it is between ______ (2 and 3) on the number line, because . . .” The helps students place extra attention on language used to engage in mathematical communication, without reducing the cognitive demand of the task. It also emphasizes the importance of justification.
Design Principle: Maximize meta-awareness

## 10.3: Cube Root Values (10 minutes)

### Activity

The purpose of this activity is for students to think about cube roots in relation to the two whole number values they are closest to. Students are encouraged to use the fact that $$\sqrt[3]{a}$$ is a solution to the equation $$x^3=a$$. Students can draw a number line if that helps them reason about the magnitude of the given cube roots, but this is not required. However students reason, they need to explain their thinking (MP3).

Monitor students multiplying non-integers by hand to try and approximate. While this isn’t what the problem is asking for, their work could be used to think about which integer the square root is closest to and should be brought up during the whole-class discussion.

Teacher Notes for IM 6–8 Math Accelerated
This activity is optional. Use this activity if students need additional practice interpreting the value of cube roots in relation to whole numbers.

### Launch

Do not give students access to calculators. Students in groups of 2. Two minutes of quiet work time, followed by partner then whole-class discussion.

Representation: Internalize Comprehension. Activate or supply background knowledge. Provide students with access to a number line that includes rational numbers to support information processing.
Supports accessibility for: Visual-spatial processing; Organization

### Student Facing

What two whole numbers does each cube root lie between? Be prepared to explain your reasoning.

1. $$\sqrt[3]{5}$$
2. $$\sqrt[3]{23}$$
3. $$\sqrt[3]{81}$$
4. $$\sqrt[3]{999}$$

### Activity Synthesis

Discuss:

• “What strategy did you use to figure out the two whole numbers?”
(I made a list of perfect cubes and then found which two the number was between.)
• “How can we write one of these answers using inequality symbols?” (For example, $$2<\sqrt[3]{23}<3$$.)
Speaking: MLR8 Discussion Supports. Use this routine to support whole-class discussion. Call on students to use mathematical language to restate and/or revoice the strategy (or strategies) presented. Consider providing students time to restate what they hear to a partner, before selecting one or two students to share with the class. Ask the original speaker if their peer was accurately able to restate their thinking. Call students' attention to any words or phrases that helped to clarify the original statement. This will provide more students with an opportunity to produce language that describes strategies to figure out which two whole numbers each cube root lies between.
Design Principle(s): Support sense-making; Maximize meta-awareness

## 10.4: Solutions on a Number Line (10 minutes)

### Activity

The purpose of this activity is for students to use rational approximations of irrational numbers to place both rational and irrational numbers on a number line, and to reinforce the definition of a cube root as a solution to the equation of the form $$x^3=a$$. This is also the first time that students have thought about negative cube roots.

### Launch

No access to calculators. Students in groups of 2. Two minutes of quiet work time, followed by partner then whole-class discussion.

Action and Expression: Develop Expression and Communication. Maintain a display of important terms and vocabulary. During the launch take time to review the following term with a number line and sample calculations from previous activities that students will need to access for this activity: cube root.
Supports accessibility for: Memory; Language

### Student Facing

The numbers $$x$$, $$y$$, and $$z$$ are positive, and:

$$\displaystyle x^3= 5$$

$$\displaystyle y^3= 27$$

$$\displaystyle z^3= 700$$

1. Plot $$x$$, $$y$$, and $$z$$ on the number line. Be prepared to share your reasoning with the class.
2. Plot $$\text- \sqrt[3]{2}$$ on the number line.

### Student Facing

#### Are you ready for more?

Diego knows that $$8^2=64$$ and that $$4^3=64$$. He says that this means the following are all true:

• $$\sqrt{64}=8$$
• $$\sqrt[3]{64}=4$$
• $$\sqrt{\text -64}=\text-8$$
• $$\sqrt[3]{\text -64}=\text -4$$

Is he correct? Explain how you know.

### Activity Synthesis

Display the number line from the activity for all to see. Select groups to share how they chose to place values onto the number line. Place the values on the displayed number line as groups share, and after each placement poll the class to ask if students used the same reasoning or different reasoning. If any students used different reasoning, invite them to share with the class.

Conclude the discussion by asking students to share how they placed -$$\sqrt[3]{2}$$ on the number line.

Representing, Conversing, Listening: MLR7 Compare and Connect. As students prepare their work before the discussion, look for approaches that favor the algebraic definition, and approaches that use the geometric interpretation. Call students’ attention to the different ways students describe how they operated on numbers cubed to determine the approximate cube roots, and to the different ways these operations are made visible in each representation (e.g., using the fact that $$\sqrt[3]{a}$$ is a solution to the equation $$x^3 = a$$, and finding rational approximations of the irrational numbers to plot the cube roots on the number line). Emphasize language used to make sense of strategies reinforcing the definition of a cube root as a solution to the equation of the form $$x^3 = a$$.
Design Principle(s): Maximize meta-awareness; Support sense-making

## Lesson Synthesis

### Lesson Synthesis

In this lesson, students learned about cube roots. Similar to square roots, cube roots can be thought of in the context of shapes. A cube with volume 64 cubic units has an edge length of 4 units, which is $$\sqrt[3]{64}$$, because $$4^3=64$$.

• “If a cube has a volume of 27 cubic inches, what is its side length?” (The cube with volume 27 cubic inches has an edge length of 3 inches, since $$\sqrt[3]{27}=3$$.)
• “What is the solution to $$x^3=150$$, and what two integers would it fall between on a number line?” (The solution to $$x^3=150$$ is $$\sqrt[3]{150}$$ and it is between 5 and 6 on a number line, because $$5^3 = 125$$ and $$6^3 =216$$.)
• “What is the solution to the equation $$a^3 = 64$$?” ($$\sqrt[3]{64}=4$$)
• “What is the solution to the equation $$a^3 = \text-64$$?” ($$\sqrt[3]{\text-64}=\text-4$$)
• “How can we plot cube roots on the number line?” (Find the two whole numbers they lie between, and determine the approximate location between them.)

## Student Lesson Summary

### Student Facing

To review, the side length of the square is the square root of its area. In this diagram, the square has an area of 16 units and a side length of 4 units.

These equations are both true: $$\displaystyle 4^2=16$$ $$\displaystyle \sqrt{16}=4$$

Now think about a solid cube. The cube has a volume, and the edge length of the cube is called the cube root of its volume. In this diagram, the cube has a volume of 64 units and an edge length of 4 units:

These equations are both true:

$$\displaystyle 4^3=64$$

$$\displaystyle \sqrt[3]{64}=4$$

$$\sqrt[3]{64}$$ is pronounced “The cube root of 64.”

We can approximate the values of cube roots by observing the whole numbers around it and remembering the relationship between cube roots and cubes. For example, $$\sqrt[3]{20}$$ is between 2 and 3 since $$2^3=8$$ and $$3^3=27$$, and 20 is between 8 and 27. Similarly, since 100 is between $$4^3$$ and $$5^3$$, we know $$\sqrt[3]{100}$$ is between 4 and 5. Many calculators have a cube root function which can be used to approximate the value of a cube root more precisely. Using our numbers from before, a calculator will show that $$\sqrt[3]{20} \approx 2.7144$$ and that $$\sqrt[3]{100} \approx 4.6416$$.

Also like square roots, most cube roots of whole numbers are irrational. The only time the cube root of a number is a whole number is when the original number is a perfect cube.