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**9. estimate 8235 divided by 71**

**Student 1**

- 71 x 100 = 7100
- 8235 – 7100 = 1135
- 71 x 10 = 710
- I didn't have time to subtract 710 from 1135, but I know the answer is at least 110. I think it’s about 115.

**Student 2**

- I was trying to do 71 times what equals 8235. But I got stuck, couldn’t think.

**Student 3**

- How many 70s went into 1000, or how many 7s went into 100. This is about 14.
- I rounded 14 to 15.
- 15 times 8 (there are 8 groups of 1000 in 8000) = 120.

Student 4

Student 4

- I changed the problem to 8200 divided by 70.
- But I ignored the zeros and just asked how many times does 7 go into 82.
- I tried 7 times 10 which is 70, too small.
- I tried 7 times 11 which is 77, getting closer.
- I tried 7 times 12 which is 84.
- So 82 divided 7 is about 12, then I added a 0 to make it 120.

Student 5

Student 5

- I changed 8235 to 8435 because I knew 7 is a multiple of 84.
- 84/7 = 12
- 35/7 = 5
- 8435/70 = 1205
- Divided 1205 by 10 to get 120.5.
- I knew the answer should be less than 120.5 though because I rounded the 8235 up and the 71 down.

**10. Which two are closer: 1/4, 1/2, 3/5?**

**Student 1**

- I changed them all to percents, 1/2 is 50%, 3/5 is 60%, so these two are closer because 1/4 is only 25%.

**Student 2**

- I thought of the chocolate candy bar that my sister gave me.

**Student 3**

- I imagined pizza, slices of pizza. I knew the half slice is twice the quarter slice, and the 3/5 slice is about like that.

**Student 4**

- I looked at 1/4 and 1/2. I need to add another 1/4 to reach 1/2. When I compare 3/5 to 1/2, however, I need to subtract only to subtract a little bit. Well, in my head 3/5 is a little bit more than 1/2 because 2.5/5 is 1/2. So, I’d need to subtract .5/5 to get to 1/2, and that’s less than 1/4.

**11. Estimate 4953 divided by 68**

**Student 1**

- I rounded the numbers to 5050 and 70.
- That’s 505 divided by 7. I got 72.

Student 2

Student 2

- 4900/70 = 70
- Because I rounded 68 to 70, that’s 2 more 70s or 140.
- I added 140 to 53 to get 193.
- 193/68 = 2 something
- So, my answer is 72 something.

Student 3

Student 3

- I rounded 4953 to 4956 because I knew that 49 and 56 are both divisible by 7.
- 4956/70 = 70.8
- I think the answer is bit lower because I rounded up.

Student 4

Student 4

- I first did 4900/70 = 70.
- But there was time left, so I wanted to see if I could get closer, so I did 4970/70 = 71.

**12. Which two are closer: 66.6%, 2/3, 0.67?**

**Student 1**

- 2/3 and 0.67 are closer because I divided 2 by 3, and I got .66. And 66.6 is way more than a whole. Oh… wait! I never saw the percent sign! I thought it was just 66.6.

**Student 2**

- I think 66.6% and .67 are closer. 2/3 is near 50% because 1.5/3 is 50%.

**Student 3**

- 66.6% and .67 are closer because I know 1/3 is 30, and 2/3 is 60.

**Student 4**

- 66.6% and 2/3 are closer because I changed them all to decimals with 3 decimal places.
- 66.6% = .666
- 2/3 = .66666
- 0.67 = .670