Struggles with estimating square roots

My #1 struggle in teaching my kids is failing to anticipate the topics that are going to give them extra difficulty. This week produced a shining example of that struggle.

I did two short projects with my younger son involving estimate square roots. He seemed comfortable with basic estimation questions – find the nearest integer to \sqrt{35} for example, but the slightly more advanced problems – say finding the nearest integer to 2*\sqrt{5} – gave him a tremendous amount of difficulty.

Here’s our first time through with 4*\sqrt{5}. What I didn’t appreciate is that he would want to focus on the value of \sqrt{5} as a starting point. While I was able to place \sqrt{5} in between two integers, the later multiplication by 4 caused some problems.

 

We discussed this type of problem a little more and I thought that we’d had some really productive discussions. However, reviewing a pretty similar problem led to difficulties that were quite similar to what we’d encountered the first time through:

 

Following this second struggle, we talked a little more about this type of approximating and I think this second round of discussions has helped him understand these approximations a little better. I wish that I would have understood ahead of time how difficult the transition to these more difficult problems was going to be.

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