I gave a packet to groups of four students, a large whiteboard, and no calculator. Here's my favorite question in the packet:

What is the best estimate for 12/13 + 8/7. Your choices are 1, 2, 19, and 21. The students must explain their answer.

All groups started with finding a common denominator and the exact answer. Once they saw their answer wasn't there, they started over assuming that they did something wrong.

I made the announcement to reread the directions and recognize that it's multiple choice.

Then I heard one boy say to the rest of his group that since both fractions were close to 1, the answer must be 2. He was met with blank stares. It took him a few minutes to convince two of his groups mates that he was correct. But one girl was still not sure. In fact, she sounded annoyed. I heard her say to the rest of the group that they were guessing and not even trying the problem. But the group remained patient and tried a different approach. A different boy in the group wrote the fraction 13/13 and asked her what it was equal to. Then he wrote 12/13 and asked if that was really close to 13/13. Perfect.

Here's another great discussion that happened in the same group:

Which show appropriate use of "canceling digits"? There were 6 choices, here are two of them.

(cd)/(ac)=d/a

(5xyz)/(7xy)=5z/7

They all agreed that the second one was okay. But only one thought the first one was okay and asked the rest of her group what the different was in the two problems. One student wrote the fraction 42/27 and said you can't cancel out the 2s. Don't worry, she didn't give up and finally won them over.

I love that I decided to take today to complete this packet. If you are interested, I took the problems from Uncovering Students Thinking in Mathematics.

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