One of the things that I notice about my students is that they often lack number sense, by which I mean an intuition for numbers and arithmetic operations.

For example, an Algebra 1 student came by after school today to ask for some help. (That in itself was great and made me happy.) At one point, this student was struggling with how to find the solution to “-8+y=2”. We tried lots of different approaches: using the blank-out method where you replace y with a box, using the number line, etc. The student was throwing out guesses, but she wasn’t guessing wildly even though her guesses were way off–she really was trying to guess the correct answer. Here’s what I mean.

The picture above shows all of the guesses that she made. She would give a guess for “y” then I would write out what happens when we add her guess to -8 so we could see if it equaled 2 or not. I purposely wrote all of the guesses in numerical order, not in the order that she guessed. That allowed her to clearly see all the guesses she had made and I was hoping the visual pattern of the numbers would help her see the answer. Two things surprised me: (1) that it took her so many guesses before she got the right answer, and (2) the guesses that she produced were seemingly random: she would guess “y=5” then “y=-8” then “y=-2″ then y=0”. However she didn’t seem to be guessing wildly because she was trying quite hard to get it right and it was clear to her that I wasn’t going to just give her the answer (which is what students are trying to coax out of you when they are guessing wildly).

Certainly we could have just jumped to the algebraic method of adding “+8” to both sides of the equation to solve for y, but it seemed a much more valuable thing to help her develop more number sense, in particular, the intuition that negative numbers are less than positive numbers and that adding a lesser number means you’ll get a lesser answer.

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Today was a good day overall. Went for a walk outside with Geometry class (because they were starting to get ornery) and I happened to see some nice shadows cast on the ground by some trees so I snapped some photos and when we got back to class I pointed out the nice similar triangles in the photo (foreshadowing of what we will do in class next week).

Most of my Algebra 1 students are beginning to show mastery of some key skills and concepts. I feel like we are making slow, steady progress.

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## Published by Darryl Yong

Professor of Mathematics at Harvey Mudd College
View all posts by Darryl Yong

hooraaaaay for progress!

Good for you! In looking at the task, and thinking about the bridge between developing number sense and linking it to algebraic tasks, it made me think of using a number line:

Adding = moving to the right, subtracting, moving to the left…

how far, what direction do you need to travel to get to 2?