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.
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.