Wednesday, July 28, 2010

My Take on Using Puzzles to Teach Substitution

Note: This is a break from the triangle area stuff I was talking about yesterday. I have full intention of returning to that sometime very soon.

I was reading through some math stuff yesterday and I came across a discussion of a grade-four question in Singapore math that requires a solution of only pseudo-algebra, thereby promoting an intuitive understanding of algebraic substitution.

It reminded me of one of my own favorite activities when I used to teach middle-schoolers, in introducing simultaneous equations and the idea of substitution. To preface this, I should state that my proudest accomplishment in creating these materials is that basically 80% or 90% of my (regular Bronx) 8th-graders would, within a few days, begin to realize that they can partially solve a simple system like this in their heads:

3x + 7y = 65
3x + 5y = -15

Essentially, they got so good with visualizing linear systems using shapes (triangles, circles, stars) that they can literally "see" that because the first equation has an overall right-hand-side value that is 80 greater than the second equation, and the only contributing factor is that the first equation has 2 extra y's, then that must mean that each y is "responsible" for 40, or y = 40. --To you guys, this may seem trivial, but it is not at all trivial to kids who, in many cases, have a natural fear of symbolic representation!

I presented this topic using a series of increasingly-difficult visual puzzles that you can find on the internet, and I let the kids work in groups of 3 or 4. I didn't introduce the idea of substitution, linear combination, or simultaneous systems, but I let them discover it for themselves through some scaffolded questions. Once they were able to solve the puzzles (WITHOUT guessing/checking), I then gave them algebra sheets and asked them to individually draw / "solve" the equations using shapes only. We didn't even begin to use algebra symbols until a couple of days down the road, at which point the transition from shapes to algebra was seamless for most kids. (I did this for two years in a row, and with other teachers at my school. The results are very duplicate-able.)

Anyway, here are the worksheets I have used in the past (that worked well in these first couple of days of group/individual exploration). I also modified and used them with my 10th-grade Algebra 2 Honors class during our intro to systems this year, and even those kids loved it!

Let me know what you think. I think it's similar to the Singapore method, but presented in a way that might be a bit more intuitive and generalized, supporting the learning of algebra. --And yes, sometimes I do miss teaching middle school (ie. having more time to focus on building introductory concepts), but whenever I feel nostalgic, I also hope that understanding middle-school approaches/foundation would help me be a better high-school teacher, in the long run. :)


  1. Eureka! I'd seen this on a visit to your blog several months ago, and it's been haunting me (in a good way) ever since-- finally re-found it today. I really love this lesson and particularly love the beginning of the classwork; if students had any doubts about how the puzzles related to math, or any insecurity about solving systems of equations (even if they didn't know it yet), #1 ties it all together with a pretty bow and makes it all so easy. I also think the last 3 problems are a low-effort, high-impact way to up the rigor and challenge students to think more critically about how numbers and variables play together in systems. Thanks for sharing!

  2. Nice! I'm glad you like it, Grace. :) Thanks for the thorough analysis on the lesson.

  3. This is really good stuff. Thanks for sharing it.

  4. Hi, Mimi. Any way you can post link to this? I'd love to have access to the actual file. It's really, really great!

  5. Here you go:

    (I don't like because when I've used it in the past, they messed up my formatting for some reason.)

  6. could you repost the worksheets? The link seems to be broken

  7. The link still works for me, but you can also find all those resources here in my Google Drive:

  8. Hi! Please can I have the answers for question 4. and 6? I've been struggling to work them out all evening!

    1. #4 and #6 in the puzzle? #4, if you look at the first row and compare it to the first column, you'd notice that when you have 3 squares and one triangle, the sum is 72; when you have 2 squares and one triangle, the sum is 56. That means each triangle must be worth 72 - 56 = 16. From there, you can figure out the rest. For #6, you can either use that same strategy by comparing the last column and the second row; OR, what I do, is encourage kids to look at the middle column. If two circles and two squares together give you 80, then what's one circle and one square worth? Where does that help us elsewhere in the diagram?