Balance Equation Puzzles
One of the ways that I teach solving linear equations (things like 2x + 4 = 3x – 5) is by using balance puzzles:
squares are x’s, circles are constants. The puzzle above is the same as 2x + 2 = 8. The solution, at the bottom, is the same as x = 3. Credit: the puzzle above, as well as this whole idea, came from James Tanton’s book Math Without Words.
This ‘puzzle’ way of introducing equations is great for my role as an intervention teacher because it ‘tricks’ students into solving algebra problems without them realizing it.
But, when I went looking around the internet for more puzzles like these, I couldn’t find very many, which made me very sad.
So, I made a website that generates these puzzles for me. And, even better, I can use these generators in class with students as we solve puzzles together. Here, see for yourself:
How I Use These:
These puzzles have a very simple, concrete set of rules: equal terms on opposite sides weigh the same and can be ‘canceled’ out; equal terms can be added to both sides of the balance since they weigh the same; positive and negative terms ‘zero’ out when they’re on the same side of a balance; the puzzle is ‘solved’ when the circles and squares are on separate sides of the balance.
I find my students come to me with a very procedural understanding of algebra – it’s a series of arbitrary rules that don’t make sense and somehow get an answer that the teacher cares about but doesn’t have any personal meaning to me. I use these puzzles as a way to bypass this very negative mentality, and I use the puzzles to make the algebra concrete for the student again. X’s and numbers stop being arbitrary symbols and start being squares and circles (which explains why you can’t combine them). The equal sign is no longer this random symbol in an equation, but the divider between one side of a balance to the other side. This ‘negative’ perspective of algebra gradually gets overwritten with the positive memories of solving puzzles and explaining their reasoning.
I usually spend a day or two using these generators at the front of the classroom and doing problems with students. These days have been some of the most successful lessons I’ve ever done – students can verbalize how to solve the puzzle while I record their words in symbols on a whiteboard; soon their description isn’t in terms of circles and squares but in x’s and numbers; soon there’s no puzzle at all but an equation instead, but I can still go back to having students think of the puzzle if necessary (which is a big deal in terms of not stepping too far up the ladder of abstraction all at once).
Lastly, knowing the rules to the puzzle provides a self-checking mechanism for the rules of solving an algebra problem. If a student is unsure if they’re allowed to do something, I can relate it back to the puzzle and ask if they can make the same move in the puzzle. Students are usually more confident with how they would solve the puzzle rather than the equation, but this confidence slowly starts to transfer to the actual equation and soon they can speak with confidence about the rules of algebra that let them solve an equation.
My Lessons (.pdf) (multiple days)
Update: @Borschtwithanna shared this related and cool-looking resource with me: Mobile Puzzles. These, in turn, reminded me that another source of inspiration for this whole activity was Paul Salomon’s Imbalance Puzzles.
Disclaimer: I made these myself and they work for my Windows computer when I run Google Chrome. They also work on my Android phone. They also work on my SMART Board. I’m not a software designer who cares about checking these on every platform in every situation, so I sincerely hope they work for you too – but, if they don’t, I probably won’t spend a ton of time to fix it. You (yes you – reading this) are welcome to make your own edits if you’d like – I’d love to see these get better.