Last year, I thought I did a great job with standard 8.NS.2 — approximating irrational numbers with rational numbers. I interpreted it as questions like this:

My students got really good at these. A question similar to this was the only question my students got 100% right on a benchmark assessment last year.

Then, during review for the state test, I dropped a question similar to this one on them, from one of our new Common Core books:

I was instantly humbled. Not that everyone got it wrong, just that what seems like a pretty simple question caused a remarkable amount of confusion — and I think it’s because, while my students got really good at one skill, they saw that skill in the same form every time, and it cut out any understanding or analysis of square roots.

Enter this year. I’m doing a lot less teaching and a lot more listening. Class today felt good. Lots of students were able to find accurate estimates for without any guidance — lots of 9s, 9.5s, and 10s. After some discussion, all of the classes were happy with settling on “between 9 and 10” as a satisfactory answer. We stretched a bit with some analysis questions and bigger numbers. Tomorrow we’ll get more formal and precise in our estimates (quick — which is bigger, ?) but today we’re just starting out with whole numbers.

Here are three samples of exit tickets three different students produced:

All three students got the first two questions right. So did 90% of my students. Great. Last year I would’ve been happy and stop there. But I want to dig into that third question, and their answers are fascinating.

**What is going on in the first student’s mind?** What does he mean when he says “If you multiply 17 to a tenth it won’t equal 300”? Is he confused about what it means to estimate, and he’s just going through the motions for the first two questions, and is stumped when he gets to the third? He showed some high-quality work for the first two questions, demonstrating solid understanding of how he knows where each radical falls. Why doesn’t that extend to more specific reasoning on the third?

The second student got the second question wrong, then checked their work and fixed it. Looking strong. She thinks the estimate in #3 is a close one, and she shows some work to back it up — 17×17 is 289. Great. **But her explanation says “Yes because it’s in between”.** That doesn’t make any sense. She seems to be leaning hard on the language we used on simpler problems, rather than drawing from her knowledge of equality, size, and estimation. Does she even have a misunderstanding, or is it just a lack of language to express her thoughts?

The third student’s work shows better reasoning for why the estimate is a good one. I also like that she drew the square and labeled a side length — drawing pictures is something I’m pushing this year. However, I’ve struggled a bit with this topic. **Pictures are great, but does it help to solve the problem any more than just doing the multiplication?** I’m really lacking in authentic applications of these skills at the moment — they all boil down to the same few variations on needing to find the side, or perimeter, or something else, given the area of a square. Boring. Am I doing my students a disservice teaching this concept largely out of context? Should I have waited for the Pythagorean Theorem?

That’s my second day of class. I think my students learned some things, and I’m excited to dive into tomorrow. Hopefully I’ve learned some things as well.

mrdardyDylan – I think you are right on the mark in your analysis of the second student. You’ve been mentioning things being in between so that becomes code language to use. I’d love to hear how it goes when/if you try the number line question with them.

dkane47Post authorThanks! I hope I can push beyond that, they’ll tackle the number line question tomorrow and some other variations tomorrow. I’ll make sure to post about it!

katenerdypoogreat post, interesting problems.

i think the second student does understand, but that she isn’t communicating well. i interpreted it as “yes, because 300 is in between 17^2 (which is 289), and whatever 18^2 is (which i’m too lazy to compute but i’m sure is more than 300 because i can already tell that 17*18 is more than 300 so obviously 18*18 is more than 300).” maybe ask her a clarifying question like “in between what?”

regarding the first student, i think that he does understand how to estimate the square root, but not that the side length of a square is the square root of the area. i think he understands concepts but not applications. i’ve had students who have no trouble telling me that the side is 5 if the area is 25, but if i say the area is 20, they really get stuck and start inventing rectangles! i’m not sure what he means by the tenth, though.

i like the number line problem. another way to ask this is something like sqrtx is a bit more than 7. if x is a whole number, what could x be? justify your answer. this is also nicely open-ended, because what’s “a bit more than 7”?

there’s a few more nice problems on open middle, for example: http://www.openmiddle.com/decimal-approximations-of-roots/

good luck!

Brooke PowersMy favorite thing you say in this post is “I decided to do less teaching and more listening”. That is what good teachers do! Love you pushing your kids to a deep understanding of this standard and not just skill and drill. Awesome!