### What is 3.75 x 32?

Student 1

- 32 x 3 = 96
- I think of 0.75 as 3/4, and 1/4 of 32 is 8, so 3/8 is 24
- 96 + 24 = 120

Student 2

- I rounded 3.75 to 4, and did 32 x 4 = 128
- Since it’s 1/4 less, and 1/4 of 32 is 8, I subtracted 8 from 128 to get 120

Student 3

I changed the problem by multiplying 3.75 by 4 and divide 32 by 4.

- 3.75 x 4 = 3(4) + 0.75(2) + 0.75(2) = 12 + 1.5 + 1.5 = 15
- 32 ÷ 4 = 8
- 15 x 8 = 120

Student 4

- 3 x 32 = 96
- To multiply 0.75 and 32, I kept doubling 0.75 until it’s easy. So, 0.75 doubled is 1.5, and add another pair of 0.75 would give me 3
- Since it took 4 sets of 0.75 to make 3, then there are 8 of these sets in 32
- 3 x 8 = 24
- Add this 24 to 96, I get 120

Mrs. Nguyen

(I shared this strategy with them as no one had mentioned it.)

- 3/8 = 0.375
- So 3.75 is 3/8 of 10
- 32 x 10 = 320
- 3/8 of 320 = 120
Another strategy, similar to Student 3′s:

- 3.75 x 2 = 7.5 and 32 ÷ 2 = 16
- 7.5 x 2 = 15 and 16 ÷ 2 = 8
- 15 x 2 = 30 and 8 ÷ 2 = 4
- 30 x 4 = 120

### Pattern #136

Student 1

I see 2 squares and 1 rectangle: S = 2(n^2) + n(n+1).

Student 2

I see a full rectangle, then subtract the empty space: S = 2n(2n+1) – n(n+1).

Student 3

I see 2 rectangles: S = n(2n) + n(n+1).

I move the whole left portion down to bottom of the right portion to have 1 rectangle: S = n(3n+1).

### What is 37.5 x 16?

Student 1

- 37.5 x 10 = 375
- 37.5 + 37.5 = 75
- 75 x 3 = 225
- 375 + 225 = 600

Student 2

- 37.5 x 2 = 75
- 16 ÷ 2 = 8
- 75 x 8 = 600

Student 3

- 40 x 16 = 640 (I did 4 times 16, then add the 0)
- I went over by 2.5, so 2.5 x 16 = 2(16) + 0.5(16) = 32 + 8 = 40
- 640 – 40 = 600

Student 4

- 37.5 x 2 = 75
- 16 ÷ 2 = 8
- 75 x 2 = 150
- 8 ÷ 2 = 4
- 150 x 4 = 600

### Pattern #134

Student 1

I see the middle rectangle separately from the 2 sides. Then I move the left side pieces to join the right side pieces to form 1 rectangle: S = n + (n^2) + n(n+1).

I move the left pieces over to the right to form one large rectangle: S = 2n(n+1).

I see a Gauss equation. I add the top and bottom row, which is (n+3n). The number of pairs of with this 4n sum is the height of the pattern divided by 2: S = 4n[(n+1)/2].