2. Figure A in Figure 5.22 on p. 223 shows a 5-unit-high, 3-unit-wide, and 2-unit-long box made out of blocks. Figures B through G of Figure 5.25 show different ways of subdividing the box into natural groups of blocks. For each of these ways of subdividing the box, use the meaning of multiplication as we have described it to write an expression for the total number of blocks in the box. Each expression should involve the numbers 5, 3, and 2, the multiplication symbol, and parentheses. Explain why you write your expressions as you do.
In picture B the 5x3x2 box has been cut into 5 horizontal slices, each of which is 3x2.
So I would write this as 5 x (3 x 2). You could write 5 x (2 x 3) also.
In C the box is cut into vertical slices, each being 5 x 2. There are 3 slices so I would write this as 3 x (5 x 2).
In D there are 2 vertical slices, each being 5 x 3. I’d write 2 x (5 x 3).
In E the box has been cut into 6 columns, each of which is 5 blocks high. It winds up being 6 columns because the box has been cut with both front-to-back cuts (cutting into 3 pieces) and then with a right-to-left cut (resulting in 2 slabs). So I would write this as 2 groups of 3 groups, each with 5 blocks, or (2 x 3) x 5. By commutativity it doesn’t matter if you write (3 x
2) x 5.
In F a horizontal cut has been made, along with a vertical cut from front to back, resulting in 5 groups of 3 groups of 2 blocks each, or (5 x 3) x 2.
Finally in G a horizontal cut and a left-to-right vertical cut have been made, resulting in 5 groups of 2 groups of 3 blocks each, or (5 x 2) x 3.
You can write any of these in different orders by commutativity.
4. Suppose you have 60 pennies arranged into 12 stacks with 5 pennies in each stack. The
12 stacks are arranged into 3 rows with 4 stacks in each row, as shown in Figure 5.27. For each of the following expressions, explain how to see a grouping of the pennies so that the expression describes the total number of pennies according to that grouping. a.) 3 x ( 4 x 5): We could view this as this as 3 rows. Each row has (4 x 5) pennies. So, there are
3 x (4 x 5) pennies. b.) (3 x 4) x 5: We could view the array as a rectangle of stacks. There are (3 x 4) stacks. Since each stack has 5 pennies there are (3 x 4) x 5 pennies. c. 5 x (3 x 4): We could view the array as stacks of 5 pennies, and there are 3 rows with 4 columns (3 x 4) of these stacks. So, there are 5 x (3 x 4) pennies in all.
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d. 4 x (3 x 5): We could view the array as 4 columns. Each column has (3 x 5) pennies. So, there are 4 x (3 x 5) pennies in all.
7. Explain how to use the associative property of multiplication to make 16 x 25 easy to calculate mentally. Write equations that show why your method is valid, and show specifically where you would have used the associative property of multiplication.
There are going to be more than one way to do this. Here is one:
16 × 25 = (4 × 4) × 25
= 4 × (4 × 25) the associative law is explicitly used here.
= 4 × (100)
= 400
9. Explain how to make the following product easy to calculate mentally (there are five 2’s and five 5’s in the product): 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 5 ⋅ 5 ⋅ 5 ⋅ 5 ⋅ 5 .
We can use the commutative and associative laws to put the product in a different order:
2 × 2 × 2 × 2 × 2 × 5 × 5 × 5 × 5 × 5 =
(2 × 5) × (2 × 5) × (2 × 5) × (2 × 5) × (2 × 5) =
10 × 10 × 10 × 10 × 10 =
100,000
14. Figure 5.29 shows a grocery store display of cases of sodas. The display consists of a large box shape with a “staircase” on top. The display is 7 cases wide and 5 cases deep; it is
4 cases tall in the front and 8 cases tall at the back. How many cases of sodas are there in the display? Solve this problem in at least two different ways, and explain your method in each case.
If you cut the display into slabs using front-to-back cuts you will get a bunch of different sized rectangles. The rectangles will all be 7 cases long and they will have various heights, from 4 to 8.
This way, for the total number of cases you will have:
(4 × 7) + (5 × 7) + (6 × 7) + (7 × 7) + (8 × 7) = 28 + 35 + 42 + 49 + 56 = 210
Another way is to look at the end of the display that is closest to us. That end has
(4 + 5 + 6 + 7 + 8) = 30 cases in it. There are 7 slabs just like that end (running from front to back) so there are
7 × 30 = 210 cases in the display.
17. Investigate the following two questions and explain your conclusions. a.) If you make a rectangular garden that is twice as wide and twice as long as a rectangular garden you already have, how will the area of the larger garden compare with the area of the original garden?
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If the new garden is twice as long and twice as wide as the old garden then its area will be 4 times as large as the old area. The commutative and associative laws explain this. If the old length and width are L and W, then the new length is 2 × L and the new width is 2 × W so we get
new area = (2 × L) × (2 × W)
= [ (2 × L) × 2 ] × W associative law used here
= [ 2 × ( 2 × L) ] × W commutative law used here
= [ (2 × 2) × L ] × W associative law here
= [ 4 × L ] × W
= 4 × (L × W)
= 4 × the old area.
associative law here. b.) If you make a cardboard box that is twice as wide, twice as tall, and twice as long as a cardboard box that you already have, how will the volume of the larger box compare with the volume of the original box?
If the new box is twice as long, twice as wide, and twice as tall as the old box then its volume will be 8 times as large as the old volume. The commutative and associative laws explain this.
If the old length, width, and height are L, W, and H, respectively, then the new length is 2 × L, the new width is 2 × W, and the new height is 2 × H, so we get
new area = (2 × L) × (2 × W) × (2 × H)
= [ (2 × L) × 2 ] × W × (2 × H) associative law used here
= [ 2 × ( 2 × L) ] × W × (2 × H) commutative law used here
= [ (2 × 2) × L ] × W × (2 × H) associative law here
= [ 4 × L ] × W × (2 × H)
= [ 4 × L ] × [W × (2 × H)]
= [ 4 × L ] × [2 × (W × H)]
= [ 4 × L ] × [2 × (W × H)]
= 4 × ( L × 2) × (W × H)
= 4 × ( 2 × L) × (W × H)
= ( 4 × 2 ) × (L × W × H)
= 8 × (L × W × H)
= 4 × (L × W)
= 4 × the old area.
associative law here.
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