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**13. Which two are closer: 3^2, 2^3, sqrt(72)?**

**Student 1**

- I think 3^2 and 2^3 are closer because I know these are 9 and 8, and I don’t know what sqrt(72) is.

**Student 2**

- I know sqrt(72) is between 8 and 9 because 8^2 is 64 and 9^2 is 81. Well, 72 is 8 away from 64 and 9 away from 81. So sqrt(72) is closer to 8.

**Student 3**

- I know 8 times 9 is 72. So I think about what 8.5 times 8.5 is, and it’s 72.25. Therefore sqrt(72) is closer to 8, probably 8.4 something.

*This 6th grader, after others had shared, tried a few more numbers and made a conjecture that when you multiply two consecutive numbers, like 5 and 6, then the square of the number between them, 5.5, is the same, plus 0.25 more. This works for 2 or 3-digit numbers too. For example, 20 x 21 = 420, and 20.5^2 = 420.25.*

**14. Which is greater, 65 x 47 or 67 x 45?**

**Student 1**

- I think they are equal because 67 minus 65 is 2, and 47 minus 25 is also 2.

**Student 2**

- We had something like this before. They both have 65 x 45 in them, so there are still 2 x 65 in the first one and 2 x 45 in the second one. It’s obvious 2 x 65 is bigger, so I know 65 x 47 is bigger.

**Student 3**

- I rounded the first one to 65 x 50, and I can do that in my head to get 3250. I ignored the zero then put it back later. Then I rounded the second one to 70 x 45, which is 3150. So the first one is bigger. I’m 99.8% confident.

*When I asked why her confidence was not 100%, she replied, “Okay, it’s 100%.”*

**Student 4**

- I have this picture in my mind [she came up and drew a sketch that I lost when site went down]. You told us that a multiplication problem is like the area of a rectangle, so I imagined these two overlapping rectangles. Kinda like what Student 2 shared — the overlap is 45 times 65. So you can see the leftover areas: the 2 by 65 is bigger than the 2 by 45.

**15. What is 0.48 × 650?**

**Student 1**

- 650/2 = 325
- 0.01(650) = 6.5
- 6.5 + 6.5 = 13
- 325 – 13 = 312

**Student 2**

- I only did 650 divided by 2 to get 325. But I knew the real answer would be less than 325.

**Student 3**

- 600(.08) = 48
- 50(.08) = 4
- 48 + 4 = 52
- 600(0.4) = 240
- 240 + 52 = 292
- 50(0.4) = 20
- 292 + 20 = 312

**Student 4**

- 650/2 = 325
- .02(100) = 2
- 2(6) = 12 (there are 6 one-hundreds in 600)
- 50 [from the 650 given] is 1/2 of 100, therefore I added 1 to the 12 to get 13
- 325 – 13 = 312

**16. There are 2,790 miles of flight distance from Los Angeles to New York City. This distance is equal to how many inches?**

**Student 1**

- I knew the math was 2790 times 5280 because that’s how many feet are in a mile, then times 12 for number of inches, but I just blanked out, didn't know where to even begin. I’m tired. Sorry.

**Student 2**

- I rounded all the numbers.
- I multiplied 2800 by 5000. I did 28 times 5, which is 140, then I added 5 zeros, so that’s 14 million. Right?
- Anyway, 14 million times 12… I just did 14 million times 10, so that’s 140 million.

**Student 3**

- I wasn't sure if there were 5280 feet in a mile or 528 feet in a mile. I went with the wrong one, thinking there was about 500 feet in a miles, so I multiplied 3000 — I bumped 2790 to 3000 — by 500. That’s 15, then add 5 zeros. I got lost a little bit. I think I got 1,500,000. Then I ran out of time.

**Student 4**

- I rounded 2790 up to 3000, then I rounded 5280 down to 5000. I figured if I rounded one number up and one number down, then my answer would be good. It’d be pretty close.
- 3000(5000) = 15,000,000
- To get inches, I separated the 12 into 10 and 2.
- 15 million times 10 is 150 million
- 15 million times 2 is 30 million
- 150 + 30 = 180… 180 million