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# 2.3: Activities and Answer Keys

Difficulty Level: At Grade Created by: CK-12

## Activity 1-1: Why Are Cells Small?

### PLAN

Summary Students explore a model cell having the shape of a cube (cuboidal), such as some of the cells in the human throat (trachea) and kidney (the nephrons). They investigate the change in cell efficiency with increasing size, focusing on the relationship between surface area and volume. Students use two-dimensional drawings, a three-dimensional model, and mathematical calculations to make these comparisons. Finally, they relate their explorations of the surface area/volume ratio in the model cell to the importance of the microscopic size of human cells in the body.

Objectives

Students:

\begin{align*}\checkmark\end{align*} construct paper models in the shape of cubes to represent cells.

\begin{align*}\checkmark\end{align*} investigate the surface area-to-volume ratios of model, cube-shaped cells.

\begin{align*}\checkmark\end{align*} explain the relationship between surface area and volume of a cube as its size changes.

\begin{align*}\checkmark\end{align*} explain the importance of the microscopic size of cells in the body.

Student Materials

• Resource
• Activity Report
• Plain paper for constructing a model of a cube
• Scissors
• Metric ruler
• Clear tape

Teacher Materials

• Demonstration cube models, 1 cm, 2 cm, 4 cm, and 8 cm on an edge

Gather student materials.

Prepare demonstration cube models, 1 cm, 2 cm, 4 cm, and 8 cm on an edge. Models can be constructed from paper, as in the student activity procedure. More durable models can be cut from wood or made from modeling clay.

Consider planning this activity with the math teacher.

Estimated Time One to two periods

Interdisciplinary Connections

Math Complete math calculations and graphing on changes in surface area and volume.

Art Use cube shapes of varying sizes to create a painting, drawing, or collage showing relationships and perspectives of cube comparisons.

Prerequisites and Background Information

Students need basic math skills of addition, subtraction, multiplication, and division. They also need to be familiar with metric units for measuring area and volume.

### IMPLEMENT

Introduce Activity 1-1 by reviewing the formula and calculations for surface area and volume of a cube, and how to calculate the surface area to volume ratio.

• Calculator use is optional.
• Use metric graph paper.
• Students can build additional models of different sizes for comparison.

Steps 1-3 Show students sample models.

Have students verify the formula for surface area and volume of a cube by counting the squares in the two-dimensional (2-D) cube patterns. Have students graph the relationship between surface area and volume for each cube. These graphs can be done separately or superimposed.

### ASSESS

Use the completion of the activity and the written answers on the Activity Report to assess if students can

\begin{align*}\checkmark\end{align*} construct paper models in the shape of cubes to represent cells.

\begin{align*}\checkmark\end{align*} explain the changes in the surface area-to-volume ratios as the cube size increases.

\begin{align*}\checkmark\end{align*} explain the importance of small size to the efficiency of the cell.

## Activity 1-1: Why Are Cells Small? – Activity Report Answer Key

• Sample answers to these questions will be provided upon request. Please send an email to teachers-requests@ck12.org to request sample answers.

Part A

What are the surface area and the volume for a cube-shaped cell?

Look at the illustration of a cube 1 cm on an edge. Use this information to answer the questions below.

1. What is the area of one side?
2. Do all sides have the same area?
3. How many sides does the cube have?
1. Calculate the surface area of a cube 1 cm on a side using the formula for Area of a Cube: \begin{align*}\text{Area} = \text{Length} \times \text{Width} \times 6 \ \text{sides}\end{align*}. Remember to include the correct units of measurement. Check your answer on the table below.
2. Now calculate the volume of this cube using the formula for Volume of a Cube: \begin{align*}\text{Volume} = \text{Length} \times \text{Width} \times \text{Height}\end{align*}. Again, remember to include the correct units of measurement. Check your answer on the table below.
3. Which is the greater number, the surface area or the volume? Show this relationship in a surface/volume ratio. Check your answers with those in the table below.
Length of side (cm) Surface Area \begin{align*}(A = L \times W \times 6)\end{align*} Volume \begin{align*}(V= L \times W \times H)\end{align*} Surface/Volume
1 cm \begin{align*}6 \ cm^2\end{align*} \begin{align*}1 \ cm^3\end{align*} \begin{align*}\frac{6}{1}\end{align*} or 6:1

Part B

Build a model to show what happens to a cube-shaped cell when it doubles in size.

Look at the illustration for a cube 2 cm on an edge. Use the Resource to cut out this pattern. Cut out this pattern. Fold your pattern correctly to make a three-dimensional (3-D) model of a cube.

Use your model of a cube 2 cm on an edge to answer the following questions.

1. What is the area of one side?
2. Do all sides have the same area?
3. How many sides does the cube have?
1. Calculate the surface area of a cube 2 cm on a side using the formula for Area of a Cube: \begin{align*}\text{Area} = \text{Length} \times \text{Width} \times 6 \ \text{sides}\end{align*}. Remember to include the correct units of measurement.
2. Now calculate the volume of this 2-cm cube using the formula for Volume of a Cube: \begin{align*}\text{Volume} = \text{Length} \times \text{Width} \times \text{Height}\end{align*}. Again, remember to include the correct units of measurement.
3. Show the relationship of surface area to volume in a surface/volume ratio.
4. Record your calculations and complete the following table for cubes 4 cm and 8 cm on a side.
Length of side (cm) Surface Area \begin{align*}(A = L \times W \times 6)\end{align*} Volume \begin{align*}(V = L \times W \times H)\end{align*} Surface/Volume
1 cm \begin{align*}6 \ cm^2\end{align*} \begin{align*}1 \ cm^3\end{align*} \begin{align*}\frac{6}{1}\end{align*} or 6:1
2 cm
4 cm
8 cm

Part C

What happens to the relationship between surface area and volume when a cube-shaped cell doubles in size?

1. Look carefully at the surface/volume changes in your table from part B.
1. When the size of a cell doubles from 1 cm to 2 cm on a side, how many times greater is the surface area?
2. How many times greater is the volume?
3. Which increased more, surface area or volume? Show your work.
2. For further exploration, make a graph to show these relationships. (Hint: plot length of one cube side on the horizontal axis, and surface area and/or volume on the vertical axis.)
3. Imagine that the cubes represent cells of different sizes and the surface of the cubes represents the cell membrane. Which cube size would be most efficient in carrying out essential cell activities? Give reasons for your answer.
4. What could a cell do to increase the surface/volume ratio? Explain, using your calculations from this activity to support your answer.

A suggested response will be provided upon request. Please send an email to teachers-requests@ck12.org.

When animals use the energy they get from food, they lose a lot of that energy as heat radiating from the surface of their bodies. Explain why small animals, such as a mouse or a hummingbird, have to spend most of their time eating.

How Changes in Surface Area and Volume Affect Cells with Different Shapes Students use a spherical-shaped cell to explore the surface area-to-volume ratio as cells increase in size.

• Sample answers to these questions will be provided upon request. Please send an email to teachers-requests@ck12.org to request sample answers.
1. Describe four characteristics or functions of cells.
2. Describe the cell theory in your own words.
3. What is the relationship between a cell, tissue, organ, and system? Include examples.
4. Why are cells so small?
5. Why is homeostasis important to cells?

## Activity 1-1 Report: Why Are Cells Small? (Student Reproducible)

Part A

What are the surface area and the volume for a cube-shaped cell?

Look at the illustration of a cube 1 cm on an edge. Use this information to answer the questions below.

1. a. What is the area of one side of the cube? _______________________

b. Do all sides have the same area? _______________________

c. How many sides does the cube have? _______________________

2. Calculate the surface area of a cube 1 cm on a side using the formula for Area of a Cube: \begin{align*}\text{Area} = \text{Length} \times \text{Width} \times 6 \ \text{sides}\end{align*}. Remember to include the correct units of measurement. Check your answer on the table below.

3. Now calculate the volume of this cube using the formula for Volume of a Cube: \begin{align*}\text{Volume} = \text{Length} \times \text{Width} \times \text{Height}\end{align*}. Again, remember to include the correct units of measurement. Check your answer on the table below.

4. Which is the greater number, the surface area or the volume? Show this relationship in a surface area/volume ratio. Check your answers with those in the table below.

Length of side (cm) Surface Area \begin{align*}(A = L \times W \times 6)\end{align*} Volume \begin{align*}(V = L \times W \times H)\end{align*} Surface/Volume
1 cm \begin{align*}6 \ cm^2\end{align*} \begin{align*}1 \ cm^3\end{align*} \begin{align*}\frac{6}{1}\end{align*} or 6:1

Part B

Build a model to show what happens to a cube-shaped cell when it doubles in size.

Look at the illustration for a cube 2 cm on an edge. Use the Resource to cut out this pattern. Fold your pattern correctly to make a three-dimensional (3-D) model of a cube.

Use your model of a cube 2 cm on an edge to answer the following questions.

1. a. What is the area of one side?

b. Do all sides have the same area?

c. How many sides does the cube have?

2. Calculate the surface area of a cube 2 cm on a side using the formula for Area of a Cube: \begin{align*}\text{Area} = \text{Length} \times \text{Width} \times 6 \ \text{sides}\end{align*}. Remember to include the correct units of measurement.

3. Now calculate the volume of this 2-cm cube using the formula for Volume of a Cube: \begin{align*}\text{Volume} = \text{Length} \times \text{Width} \times \text{Height}\end{align*}. Again, remember to include the correct units of measurement.

4. Show the relationship of surface area to volume in a surface area/volume ratio.

5. Record your calculations and complete the following table for cubes 4 cm and 8 cm on a side.

Length of side (cm) Surface Area \begin{align*}(A = L \times W \times 6)\end{align*} Volume \begin{align*}(V= L \times W \times H)\end{align*} Surface Volume
1 cm \begin{align*}6 \ cm^2\end{align*} \begin{align*}1 \ cm^3\end{align*} \begin{align*}\frac{6}{1}\end{align*} or 6:1
2 cm
4 cm
8 cm

Part C

What happens to the relationship between surface area and volume when a cube-shaped cell doubles in size?

1. Look carefully at the surface/volume changes in your table from Part B.

a. When the size of a cell doubles from 1 cm to 2 cm on a side, how many times greater is the surface area?

b. How many times greater is the volume?

c. Which increased more, surface area or volume? Show your work.

2. For further exploration, make a graph to show these relationships. (Hint: plot length of one cube side on the horizontal axis and surface area and/or volume on the vertical axis.)

3. Imagine that the cubes represent cells of different sizes and the surface of the cubes represents the cell membrane. Which cube size would be most efficient in carrying out essential cell activities? Give reasons for your answer.

4. What could a cell do to increase its surface area/volume ratio? Explain, using your calculations from this activity to support your answer.

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