Lesson 7
The Root of the Problem
7.1: The Number That Cubes (5 minutes)
Warm-up
The purpose of this warm-up is to remind students that when a unit cube is dilated by a scale factor of \(k\), the volume is multiplied by \(k^3\). Asking for the scale factor required to achieve a certain volume allows students to consider the idea of a cube root in a geometric context.
Student Facing
A cube whose side lengths measure 1 unit has been dilated by several scale factors to make new cubes.
- For what scale factor will the volume of the dilated cube be 27 cubic units?
- For what scale factor will the volume of the dilated cube be 1,000 cubic units?
- Estimate the scale factor that would be needed to make a cube with volume of 1,001 cubic units.
- Estimate the scale factor that would be needed to make a cube with volume of 7 cubic units.
Student Response
For access, consult one of our IM Certified Partners.
Activity Synthesis
Ask students to share their thinking for the first 2 questions and then ask for estimates for the other 2 questions. Ask students, “How is this concept similar to square roots?” Explain that for a number \(x\), the cube root of \(x\) is the number that cubes to make \(x\). For example, \(\sqrt[3]{8}=2\) because \(2^3=8\). The cube root of \(x\) is written \(\sqrt[3]{x}\).
Compare students’ estimates for \(\sqrt[3]{7}\) and \(\sqrt[3]{1001}\) to decimal approximations 1.9129 and 10.0033 respectively. Demonstrate how to input cube roots into a scientific calculator.
7.2: Thinking Inside the Box (20 minutes)
Activity
In this task, students create a graph that represents the equation \(y=\sqrt[3]{x}\). They use the graph to analyze the relationship between volume and scale factor.
If individual devices are not available for students to use in the digital version of this activity, displaying the applet for all to see would be helpful during the synthesis.
Launch
Provide each student access to graph paper and a device to run the applet.
Supports accessibility for: Organization; Conceptual processing; Attention
Student Facing
A shipping company makes cube-shaped boxes. Their basic box measures 1 foot per side. They want to know how to scale the basic box to build new boxes of various volumes.
- If the company wants a box with a volume of 8 cubic feet, by what scale factor do they need to dilate the box?
- If they want a box with a volume of 10 cubic feet, by approximately what scale factor do they need to dilate the box?
- The company decides to create a graph to help analyze the relationship between volume (\(x\)) and scale factor (\(y\)). Complete the table, rounding values to the nearest hundredth if needed. Use the applet at the end of the activity to help, if you choose.
volume in cubic feet scale factor 0 1 5 8 10 15 20 27 - On graph paper, plot the points and connect them with a smooth curve.
- The graph shows the relationship between the volume of the dilated box and the scale factor. Write an equation that describes this relationship.
- Suppose the company builds a box with volume 21 cubic feet, then decides to build another box with volume 25 cubic feet. Use your graph to estimate how much the scale factor changes between these two dilated boxes.
- Use your graph to estimate how the scale factor changes between a box with volume of 1 cubic foot and one with volume of 5 cubic feet.
Use the applet to help, if you choose.
Student Response
For access, consult one of our IM Certified Partners.
Launch
Provide each student access to graph paper and a scientific calculator.
Supports accessibility for: Organization; Conceptual processing; Attention
Student Facing
A shipping company makes cube-shaped boxes. Their basic box measures 1 foot per side. They want to know how to scale the basic box to build new boxes of various volumes.
- If the company wants a box with a volume of 8 cubic feet, by what scale factor do they need to dilate the box?
- If they want a box with a volume of 10 cubic feet, approximately what scale factor do they need?
-
The company decides to create a graph to help analyze the relationship between volume (\(x\)) and scale factor (\(y\)). Complete the table, rounding values to the nearest hundredth if needed.
Then, on graph paper, plot the points and connect them with a smooth curve.
volume in cubic feet scale factor 0 1 5 8 10 15 20 27 - The graph shows the relationship between the volume of the dilated box and the scale factor. Write an equation that describes this relationship.
- Suppose the company builds a box with volume 21 cubic feet, then decides to build another with volume 25 cubic feet. Use your graph to estimate how much the scale factor changes between these 2 dilated boxes.
- Use your graph to estimate how the scale factor changes between a box with volume of 1 cubic foot and one with volume of 5 cubic feet.
Student Response
For access, consult one of our IM Certified Partners.
Anticipated Misconceptions
When finding the scale factor needed to achieve a volume of 10 cubic feet, some students may divide 10 by 3, resulting in a scale factor of approximately 3.3. Ask them if they divided 27 by 3 in the prior problem.
Some students may struggle to set an appropriate scale for the \(y\)-axis. Ask them to identify the largest \(y\)-value they'll need to graph.
Activity Synthesis
The goal of the discussion is to draw conclusions from the graph about the relationship between volume and scale factor.
- “If the company wanted to double the volume from 10 to 20 cubic feet, what happens to the scale factor? Does it double?” (It goes from 2.15 to 2.71, which is an increase by a factor of about 1.26.)
- You calculated the rate of change for an increase in volume of 4 cubic feet in different parts of the graph. Were these the same?” (No. For the increase from 1 to 5 cubic feet, the scale factor needed to increase more than when we went from 21 to 25 cubic feet.)
- “How does this relate to the real-life box situation?” (An increase in volume for lower values requires a big increase in scale factor. For larger volumes, the volume can be increased the same amount by only changing the scale factor a little bit.)
- "Compare and contrast this graph with the graph representing \(y=\sqrt{x}\) in an earlier activity." (The graphs look very similar. They both pass through the origin. The graph representing \(y=\sqrt[3]{x}\) might be a little steeper on the left and flatter on the right than that representing \(y=\sqrt{x}\).)
Design Principle(s): Support sense-making
7.3: Satellite Scale Factors (10 minutes)
Activity
Students apply concepts of scaling, surface area, and volume to solve a design problem.
Launch
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness
Supports accessibility for: Memory; Conceptual processing
Student Facing
A government agency is redesigning a satellite, or an object that goes in orbit around Earth. The surface of the satellite is covered with solar panels that supply the satellite with energy. The interior of the satellite is filled with scientific instruments. In the current design, the satellite has a surface area of 5.4 square feet and a volume of 1.2 cubic feet.
- If the agency wants to increase the surface area to 21.6 square feet so the satellite can generate more energy, by what scale factor do they need to dilate the satellite?
- If the agency instead wants to increase the volume to 4.05 cubic feet to fit in more scientific instruments, by what scale factor do they need to dilate the satellite?
Student Response
For access, consult one of our IM Certified Partners.
Student Facing
Are you ready for more?
Is it possible to dilate the satellite in the activity so that the number of square feet of surface area is equal to the number of cubic feet of volume? Explain or show your reasoning.
Student Response
For access, consult one of our IM Certified Partners.
Anticipated Misconceptions
Students may be unsure how to calculate the cube root of 3.375. Remind them that they can use their calculators.
Activity Synthesis
Ask students to describe what steps they took to solve these problems. What was the easiest part, and what was most difficult?
Lesson Synthesis
Lesson Synthesis
The goal is to discuss strategies for working backwards from a dilation to find the scale factor involved. Here are some questions for discussion:
- “What is the relationship between the cube of a number and the cube root of a number?“ (Cubing means multiplying together 3 copies of a number. For example, “5 cubed“ is \(5\boldcdot 5\boldcdot 5=125\). To determine the cube root of a number, find what number cubes to make that value. The cube root of 8 is 2 because \(2\boldcdot 2\boldcdot 2=8\).)
- “Suppose we have a solid with volume 10 cubic units. It’s dilated, and the volume of the image is 40 cubic units. How can you find the scale factor that was used?” (Divide the dilated volume by the original volume, then take the cube root of the result.)
- “How would this process be the same and different if you knew the surface area of the original and dilated solids?” (You would still divide the dilated solid’s surface area by the original solid’s surface area, but then you’d take the square root of the result instead of the cube root.)
7.4: Cool-down - Finding a Scale Factor (5 minutes)
Cool-Down
For access, consult one of our IM Certified Partners.
Student Lesson Summary
Student Facing
Suppose a prism has a volume of 5 cubic units. The prism is dilated, and the resulting solid has volume 320 cubic units. If we want to find the scale factor that was used in the dilation, we start by dividing the new volume, 320 cubic units, by the original volume, 5 cubic units, to find that the prism’s volume increased by a factor of 64. We know that when a solid is dilated by a scale factor \(k\), the volume is multiplied by \(k^3\). So, we need to find the number whose cube is 64. This number is called the cube root of 64 and is written \(\sqrt[3]{64}\). We know \(\sqrt[3]{64} = 4\) because \(4 \boldcdot 4 \boldcdot 4=64\). That is, the prism was dilated by a scale factor of \(k=4\).
We can create a graph that shows the relationship between the volume, \(V\), of a dilated solid and the scale factor, \(k\), needed to achieve it. Let’s use the prism with volume 5 cubic units as an example. Create a table of values, then plot the points and connect them with a smooth curve.
| dilated volume in cubic units (\(V\)) | scale factor (\(k\)) |
|---|---|
| 0 | 0 |
| 5 | 1 |
| 40 | 2 |
| 135 | 3 |
| 320 | 4 |
This graph represents the equation \(k=\sqrt[3]{\frac{V}{5}}\). The graph rises relatively steeply from \((0,0)\) but quickly flattens out.
We can also find the scale factor of dilation if we know the surface areas of the original and dilated solids. Suppose a cylinder has surface area 35 square units, and is dilated resulting in a surface area of 218.75 square units. Divide the numbers to find that the surface area increased by a factor of 6.25. Take the square root of 6.25 to conclude that the solid was dilated by a factor of 2.5.