Saturday, November 10, 2012

Visualizing Order of Operations

This year my Grade 7s came in to my class with some missing prerequisite skills. Half of them started the year not knowing integers or order of operations or how to calculate simple fractions and percents, which are all supposed to be prerequisite knowledge for Grade 7 at our school. So, (although I did go back and fill in those gaps more or less,) teaching them algebra skills on top of this shaky foundation has been a new challenge.

At some point, I realized that it was difficult for these kids to look at a formula (in simple algebraic evaluation, for example) and to visualize what operations need to happen first, second, third, etc. They know the rules for PEMDAS, but it is just hard for them to always do it consistently. So, I came up with a trick of teaching kids to circle the operations in the given exercises in a layered manner, so that they can train their eyes to look for the operations, rather than to look at the equation from left to right.

Something like:

My special ed helper agrees that this is really helpful for them to visualize the rules. My only frustration is that they don't do this consistently because they still think they can just see it all in their heads, and in doing so they end up missing an extra negative sign here and there and throwing off the entire answer.

Anyway, I think the same trick can be used with certain types of equations to help kids see the layering and the process of peeling away the onion.

So, this is what I'll focus on for the next couple of weeks, to see if it can help solidify their foundation with this type of layered equation. (They're pretty OK with the ones with x's on both sides, since we had started off with doing balance visualization and crossing out shapes.) Basically, anything that can help them sink their teeth into abstract representation is worth a try for me. Any other ideas on how I can help these kids?


  1. Truth be told, your onion peeling method is rather innovative and original. Peace.

  2. I like it! Kinda reminds me of this applet:

    Instead of drawing a circle around each 'layer', you cover it up, so the equation above becomes █ - 8 = 35 so █ = 43 or (5x^2 + 6)/2 = 43

    Then you go on to the next layer. I always liked the idea because you are focusing specifically on order of operations. I've never tried it though, because I thought it may cause problems when you have unknowns on both sides. Tbh, your way may well be better, since you can see how the whole equation is broken down in a single line, but I do like that you get instant feedback on the app if it's right or not.