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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

• 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

• 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.

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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.

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11.  Estimate 4953 divided by 68

Student 1

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

Student 2

1. 4900/70 = 70
2. Because I rounded 68 to 70, that’s 2 more 70s or 140.
3. I added 140 to 53 to get 193.
4. 193/68 = 2 something
5. So, my answer is 72 something.

Student 3

1. I rounded 4953 to 4956 because I knew that 49 and 56 are both divisible by 7.
2. 4956/70 = 70.8
3. I think the answer is bit lower because I rounded up.

Student 4

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

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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