The Size of Things
Objective: To explore the relationship between the length of a side of a cube and its surface area and volume.

Prior Knowledge: surface area of a polyhedron: the sum of the areas of its faces volume of a solid: the number of cubic units contained in the interior of a solid. Students should be able to graph scatter plots on a graphing calculator.

Time Required: One 50 minute class period.

Materials needed: Ideally 100 wooden blocks per group, minimally 36 blocks per group.

Group size: 2 to 3 students per group.

Procedure:
1. Starting with 1 block, calculate the surface area and volume. Rather than using actual measurements, the students should use side length = 1 unit. Record your answers in a table like the one below.

2. Double the length of each side of the cube by adding enough blocks to create a 2 x 2 x 2 cube. Calculate the surface area and volume of this cube and record your answers.

3. Triple the length of the sides of your original cube by adding enough blocks to create a 3 x 3 x 3 cube. Calculate the surface area and volume of the new cube and record your answers.

4. Quadruple the lengths of the sides of your original cube by adding enough blocks to create a 4 x 4 x 4 cube. Calculate the surface area and volume of the new cube and record your answers.

5. Continue filling in the table by extrapolating the surface area and volume of cubes with side lengths of up to 10 units.
 Length of a Side Surface Area Volume 1 2 3 4 5 6 7 8 9 10
6. Make a scatter plot of the data: Length of a Side vs. Surface Area
Using a window of: xmin = – 11, xmax = 11, xscl = 1, ymin = – 1100, ymax = 1100, yscl = 100

7. Using this data, write the equation of the curve that would pass through these points.

8. Graph the equation on your calculator. Make any adjustments to your equation that are necessary so that the curve passes through all of your data points. What do you notice about the curve in relation to the data points?

9. Make a scatter plot of the data: Length of a Side vs. Volume using the same window as in #6.

10. Using this data, write the equation of the curve that would pass through these points.

11. Graph the equation on your calculator. Make any adjustments necessary so that the curve passes through all of your data points. What do you notice about the curve in relation to the data points?

12. The equation you wrote in # 7 should be the formula for finding the surface area of a cube and the equation in #10 should be the formula for finding the volume of a cube. Which domain and range make sense for your equations in a real-world situation?

13. Change your window settings to reflect the sensible domain and range.

14. The graphs of your data are discrete while the graphs of your equations are continuous. Which is more applicable to the real world?

15. Now graph both sets of data and equations together. Which point(s) do your equations have in common? Why?

16. Between these two points which graph grows more quickly? After the largest point of intersection which graph grows more quickly? Explain.

17. In a table like the one below, find the ratio of the surface area of each larger cube to the surface area of the cube with a side length of one unit. Also find the ratio of the volume of each larger cube to the volume of the cube with a side length of one unit.
 Length of a Side Ratio of new surface area to original (side length = 1 unit) surface area Ratio of new volume to original (side length = 1 unit) volume 1 1:1 1:1 2 24:6 = __:__ 8:1 3 4 5 6 7 8 9 10
18. When you double the dimensions of a cube, by how many times does the surface area increase? What about when you triple it?
19. When you double the dimensions of a cube by how many times does the volume increase? What about when you triple it?