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**1. Are there more inches in a mile or seconds in a day?**

**Student 1**

- I thought there were 2000 feet in a mile, but I realized I was wrong when I talked with my neighbor.

**Student 2**

- 60 x 60 = 3600
- 3600 x 24
- 5280 x 12
- I know for steps 2 and 3 to be equal, 3600 would have to be half of 5280 because 24 is twice 12, and it’s not. And because 3600 is more than half of 5280, I know that 3600 times 24 is more than 5280 times 12. My answer, therefore, is there are more seconds in a day.

**Student 3**

- I changed 5280 to 5000 and 12 to 10. Multiply 5000 with 10 to get 50000. Then I multiplied 280 times 2 to get about 500, add a 0 to this to make 5000. So there are about 55,000 feet in a mile.
- To figure out the time, I changed 24 hours to 20; and 20 times 3600 is 72,000.

**2. Which is greater, 86 x 38 or 88 x 36?**

**Student 1**

- I think they’re the same because…
- 86 x 38 = (80 x 30) + (6 x 8)
- 88 x 36 = (80 x 30) + (8 x 6)

**Student 2**

- I ignored the 30 in both 38 and 36, therefore I just have…
- 86 x 8 and 88 x 6
- To do 86 times 8, I’d have 8 times 6 and 8 times 8
- To do 88 times 6, I’d have 6 times 8 and 6 times 8
- Well, the (8 x 6) and (6 x 8) are the same in steps 2 and 3, so I can see that step 2 gives a bigger answer because 8 x 8 is more than 6 x 8.

**Student 3**(in response to Student 1)

- I don’t think your way works because I tried it on this problem.
- 12 x 10 = 120
- You’d then do 10 x 10 = 100
- And then 2 x 0 = 0
- But steps 2 and 3 don’t add up to 120.

**Student 4**

- I did a simpler problem. I made up numbers that would add up to 8, then I also multiply them. (I do this because I notice that if the two original problems were addition problems, then their answers would be the same.)
- 4 x 4 = 16; and 4 + 4 = 8
- 3 x 5 = 15, and 3 + 5 = 8
- 2 x 6 = 12, and 2 + 6 = 8
- So I notice that the further apart the two numbers are, the smaller their product. 86 x 38 are closer together, therefore their product is greater.

**3. What is 0.52 x 789?**

**Student 1**

- 1/2 of 790 is 395
- .01 of 800 is 8
- because it’s .02, I did 8 x 2 = 16
- 395 + 16 = 411

**Student 2**

- 0.5 of 800 is 400. That’s all I have.

**Student 3**

- 1/2 of 700 = 350
- 1/2 of 80 = 40
- 1/2 of 9 = 4.5
- Add these together, I get 394.5
- I started doing .02 of 700, .02 of 80, … but time was up

**Student 4**

- 70 ÷ 2 = 35
- 9 ÷ 2 = 4.5
- 35 + 4.5 = 39.5
- I changed 39.5 to 395 because it was 700, not 70.

**4. Which is greater, 12/17 or 5/8?**

**Student 1**

- I think 12/17 is bigger. I imagined rectangles. The 12/17 rectangle has smaller pieces but more of them. But I’m not confident.

**Student 2**

- I went with 12/17. I did 1/2 of 17 is 8.5, which is 3.5 away from 12. Then I did 1/2 of 8 is 4, which is 1 away from 5. I’m only 30% confident.

**Student 3**

- I did the same as Student 2. But I went further and compared 1/8 with 3.5/17. I changed 3.5/17 to 3/16, reduce 3/16 to 1.5/8. Because 1.5/8 is greater than 1/8, I went with 12/17.

**Student 4**

- I found a common numerator instead because I knew that if they had common numerator, the one with the smaller denominator is the bigger fraction.

I asked if he could give the class an example, he said, “Okay, compare 5/3 with 5/2. If this is pizza, you have the same amount of slices, but the half slices are bigger than the thirds, so 5/2 gives you more."

I asked if he could give the class an example, he said, “Okay, compare 5/3 with 5/2. If this is pizza, you have the same amount of slices, but the half slices are bigger than the thirds, so 5/2 gives you more."

**Student 5**

- I changed the fraction 12/17 to 60/85 by multiplying by 5. I changed 5/8 to 60/96 by multiplying by 12. Denominator 85 is smaller, so that fraction is actually bigger.