Unit 4 Part 2 (Volume)
~Geometry~ The following booklet belongs to____________ in Period __. I, ________________________, promise to keep this tiny notebook in a safe place and not lose it. If I do, I am responsible for getting the notes and risk losing points if anything is missing. Lesson 8a: Volume of Rectangular Prisms using Mini Cubes Opening Exercise : Which prism will hold more 1 in. by 1 in. by 1 in. unit cubes? How many more unit cubes will the prism hold? Prism #1 has ____ unit cubes. Prism #2 has ____ unit cubes. Prism #____ will hold ____ more unit cubes. Fractional “Mini Squares” f or Area of 2­Dimensional Shapes 1. How many 12 by 12 squares fit in a 1 by 1 square? 2. How many 13 by 13 squares fit in a 1 by 1 square? A 1 by 1 square can fit ____ 12 by 12 squares. A 1 by 1 square can fit ____ 13 by 13 squares. Fractional “Mini Cubes” f or Volume of 3­Dimensional Shapes 3. How many 12 by 12 by 12 mini cubes fit in a 1 by 1 by 1 cube? 4. How many 13 by 13 by 13 mini cube fit in a 1 by 1 by 1 cube? A unit cube can fit ____ 12 by 12 by 12 mini cubes. A unit cube can fit ____ 13 by 13 by 13 mini cubes. Lesson 8a Problem Set Answer #1­13 without drawing the figures and without looking at the previous page. Area (2­Dimensional shapes) 1. How many 12 by 12 squares fit in a 1 by 1 square? How do you know? 3. How many 14 by 14 squares fit in a 1 by 1 square? How do you know? 2. How many 13 by 13 squares fit in a 1 by 1 square? How do you know? 4. In general, how do we find the number of “mini squares” that fit in a 1 by 1 unit square? Volume (3­Dimensional shapes) using 12 by 12 by 12 mini cubes 5. How many 12 by 12 by 12 mini cubes fit in a unit 6. How many times bigger is the total number of “mini cubes” compared to a unit cube? cube? How do you know? The t otal number of mini cubes is ___ t imes bigger than t he number of unit cubes. 7. Find the volume of a 12 by 12 by 12 mini cube. 8. Compare your answer in #10 to #11. How are they related? Since t his mini cube is 1 out of t he t otal ___, it is 1 the v olume of t he unit cube. Volume (3­Dimensional shapes) using 13 by 13 by 13 mini cubes 9. How many 13 by 13 by 13 mini cube fit in a unit cube? How do you know? 10. How many times bigger is the total number of “mini cubes” compared to unit cube? The t otal number of mini cubes is ___ t imes bigger than t he number of unit cubes. 11. Find the volume of a 13 by 13 by 13 cube? 12. Compare your answer in #10 to #11. How are they related? Since t his mini cube is 1 out of t he t otal ___, it is 1 the v olume of t he unit cube. Lesson 8b: Finding Volume using Mini Cubes Example 1 : Recall from lesson 8a Prism #2. It will be used to ship miniature dice whose side lengths have been cut in half. The dice are 12 in. by 12 in. by 12 in. cubes. How many dice can fit in the box? Using Unit Cubes V = Bh V = (10 · 4) · 6 V = (40) · 6 V = 240 in 3 * NOTE * Since the volume for both prisms is the same, what relationship is there between the mini cubes and the unit cubes? Using Dice (Mini Cubes) Steps for Using M ini Cubes 1. Find the new dimensions in mini cubes 2. Find the total number of mini cubes 3. Find the volume of a single mini cube 4. Find the original volume using the mini cubes V = Volume of one mini cube • # of mini cubes (Step 2 multiplied by Step 3) Example 2 : A 14 in. cube is used to fill the prism. How is t he number of mini cubes related t o t he v olume? 1. “New Dimensions” 2. Total mini cubes 3. Volume of a Mini Cube 4. Original Volume V = Volume of one mini cube • # of mini cubes Exercise 1­5 (Use your notebook as necessary!) : 1. Use the prism to answer the following questions. (Recall the steps) 1. 2. 3. THINK: How is the number of mini cubes (step 2) related to the volume (step 4)? 4. 2. Calculate the volume of the following rectangular prisms using…. a. The “best fit” Mini Cube. b. Either formula V = Bh . 3. A toy company is packaging its toys to be shipped. Some of the very small toys are placed inside a cube shaped box with side lengths of 12 in. These smaller boxes are then packed into a shipping box with dimensions of 12 in. by 4 12 in. by 3 12 in. a. How many small toys can be packed into the larger b. Find the volume of the box using the number of toys box for shipping? (Steps 1­2) from part a. (Steps 3­4) 4. Calculate the volume of the rectangular prism. Use both methods to determine the volume. Method 1: Volume using Mini­Cubes Method 2: Volume using formula 5. Challenge : A rectangular prism with a volume of 8 cubic units is filled with cubes. First it is filled with cubes with side lengths of 12 unit. Then it is filled with cubes with side lengths of 13 unit. a. How many more of the cubes with 13 unit side lengths than cubes with 12 unit side lengths will be needed to fill the prism? b. Why does it take more cubes with 13 unit side lengths to fill the prism? Lesson 8c: Volume Using Mini Cubes Practice 1. For a through c below, each rectangular prism is built with 12 ­in. blocks. Find the length, width, height, and volume of the prism. a. b. c. 2. Answer the following questions about cubic blocks. a. What size cubic blocks would you use to make a rectangular prism that is 2 12 in. by 3 12 in. by 4 in.? Explain your choice. b. How many blocks would you need? c. Give the volume of the model in cubic inches. 3. A rectangular prism has edge lengths 1 13 in., 2 in., and 1 23 in. a. How many cubes with edge lengths 13 in. would be needed to fill the prism? b. What is the volume of the prism? 4. Megan uses 216 cubic blocks to make a rectangular prism 9 blocks long and 3 blocks tall. Each block measures 14 inch on each side. a. How many blocks wide is the prism? b. What are the prism’s dimensions in inches? c. What is the prism’s volume, in cubic inches? d. Megan uses all the bricks to make a new prism that is 6 bricks tall. Give 3 possible sets of dimensions of the base of the prism Megan could have made. 5. Use the right rectangular prism below to answer the questions. a. Use fractional unit cubes to find the volume of the right rectangular prism. Explain your steps. b. Verify that the volume found is the same as one found by multiplying the edge lengths of the prism. 6. Cube­shaped boxes will be loaded into the cargo hold of a truck. The cargo hold of the truck is in the shape of a rectangular prism. The edges of each box measures 2.5 feet and the dimensions of the cargo hold are 7.5 feet by 15 feet by 7.5 feet, as shown below. a. What is the volume, in cubic feet, of each box? b. Determine the number of boxes that will completely fill the cargo hold of the truck. Use words and/or numbers to show how you determined your answer. 7. Find the volume using the formula V = lwh . a. b. 8. Which set of dimensions describes the rectangular prism with the greatest volume? a. 3 12 in. by 2 in. by 5 in. b. 3 i n. by 3 in. by 3 12 in. c. 4 in. by 2 in. by 4 in. d. 2 14 in. by 2 in. by 7 in.