Your Name:

Date:

Class:

Date:

Class:

Constructing Square and Cubic Units

In this activity you will learn area and volume scale with changes in length. For example, would a person twice as tall weigh twice as much? Mass (and therefore weight) depends on volume so consider that as you ponder the answer. In this activity you will construct a two cubes using the patterns provided one is 1 cm on a side and the other is 5 cm on a side. Answer the following questions based on the two cubes that you have constructed. Keep in mind that each line segment on the cube is meant to be one centimeter.

- What is the surface area of the two cubes?

(be sure to include units)

Small Large - What is the volume of the two cubes?

(be sure to include units)

Small Large - How are the lengths, areas and volumes of the two cubes related mathematically?
- How many times greater is the side-length (length, width, or height) of the large cube?
- How many times greater is the surface area of the large cube?
- How many times greater is the volume of the large cube?

**Imagine you have enough small cubes to put them together to build a bigger cube. You would need eight of them. **

- If you were to form a single new cube using 8 of the 1-cm-sided cubes what would you have for
**dimensions**(l, w, h),**surface area**, and**volume**? - If you were to form a single new cube using 8 of the 5-cm-sided cubes, what would you have for
**dimensions**(l, w, h),**surface area**, and**volume**? - How much longer is the side of the combined cube compared to the original length? That is, what would you multiply by the length of the original cube to get the length of a side of the combined cube made of eight smaller ones? Consider both the smaller and the larger combined cubes that you made.
- How many times bigger is a combined cube’s
**surface area**compared to the eight individual cubes that were put together to build it? - How many times bigger is a combined cube’s
**volume**compared to the eight individual cubes that were put together to build it?

Consider all possible comparisons between different cubes as you answer the following questions. Remember, you have cubes 1 cm, 5 cm, 2 cm, and 10 cm on a side (when you include the ones you could build out of smaller cubes). We are trying to understand how areas and volumes change compared to a given change in the length of a side.

- In words, what is the general rule that tells you how many times larger the
**surface area**of a cube will be when you increase the length of the side by some factor?

- In words, what is the general rule that tells you how many times larger the
**volume**of a cube will be when you increase the length of the side by some factor? - Area units are ‘square’. Explain why by referring to the way the difference in surface area is related to the difference in side length of the cubes.
- Volume units are ‘cubic’. Explain why by referring to the way the difference in volume is related to the difference in side length of the cubes.

Use what you have learned in this activity to answer the following questions.

- A cube 1 cm on a side has a volume of 1 cm
^{3}. This is by definition equal to 1 mL. A volume of 1 mL (or 1 cm^{3}) of water has a mass of exactly 1 gram (g). How many mL of water would your new large combined cube (side length 10 cm) contain? How many liters (L) is this? What is the mass of this amount of water in g? In kilograms (kg)? - The density of lead is 11.3 g per cubic centimeter (g/cm
^{3}). How much would your small cube weigh (side length 1 cm), in grams, if it were made of lead? How much would the large combined cube (side length 10 cm) weigh? Describe your reasoning with a short sentence and show your work with a calculation. - Which would weigh more, the small cube (1 cm on a side) if it were made of lead or the large, combined cube if it were filled with water? Draw a picture of each cube labeled with the cube’s mass. Answer the question and describe your reasoning with a short sentence and show your work with a calculation.
- If you have the small cube (1 cm on a side) made of lead and another cube made of lead that is 2 cm on a side then how much heavier is the larger cube? Draw pictures of the cubes and label them with their masses. Answer the question and describe your reasoning with a short sentence and show your work with a calculation.
- You have built a scale model of a bookshelf. The volume of the scale model is 500 cubic inches (in
^{3}). What is the actual volume of the full scale bookshelf if the actual shelf is twice as big as measured by doubling the length, width, and height?

What is the actual volume of the full scale bookshelf if you increase L, W, and H by a factor of four?