Finding Elegance in Equivalence

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Over the past week, we have been piloting diagnostic questions with students to get an idea of their understanding of equivalence. We gave students a series of questions, all having to do with the understanding of what the equal sign represents.

One question, though, really challenged my idea of what "higher order" strategies we were hoping to see in our students.

Here's the question:

If 4x + 8 = 52, what is 2x + 4?

Take a moment to figure it out.
What is your answer?
How did you come to that answer?

If you're like my colleagues and myself (and most of the students of whom we asked the question), your instinct might have been to solve for x using the first expression,

...and then substitute that value for x in the second expression:

So in this case, 2x + 4 = 26.

As students advance to higher order strategies when learning math, we often nudge them toward more efficient methods: generalized procedural strategies (as opposed to guess and check, or building concrete models) that can work with any case.

Solving for x in the above question is a very valid strategy, employing algebraic thinking in a method that would work for any similar question (even one that was more complex). But is it the most efficient strategy? I count four separate calculations: 52 subtract 8, 44 divided by 4, 2 times 11, and 22 added to 4.

Is there another way?

Look at the question again - can you determine 2x + 4 in only one calculation?

If 4x + 8 = 52, what is 2x + 4?

You can, if you recognize that 2x + 4 is half of 4x + 8. Because of the equivalence in the expression, the solution is therefore half of 52. Half of 52 is 26... and there's your answer. One step.

(Okay, perhaps you could argue that to make sure 2x + 4 was half of 4x + 8, you had to determine 4 divided by 2, and 8 divided by 2. But one could also argue that those are likely easy or fast calculations. There is also much less to keep track of while determining the solution, than when solving for x and substituting back. So elegant!)

Being able to look at a problem and recognize (mimic?) steps that could be taken to determine a solution is one thing. Being able to look at a problem in different ways and perhaps find a more elegant solution based on the context or the numbers given demonstrates flexibility, fluency, and comfort playing with the numbers.

When we are giving students problems to solve, what strategies are we hoping they'll employ? And are we giving them the chance to discover new strategies based on the numbers we choose?


  1. In the Ontario curriculum, the second overall expectation in Patterning and Algebra provides those strategies for flexibility with numbers and expressions. This is why it is so important to cluster Patterning and Algebra with Number Sense and Numeration. Thanks for reminding me of the continuum of learning from number to algebra. You have certainly shown how the expectations around equality make their way to the intermediate levels.

    1. That's still a big area of learning for me... how the basic ideas of equivalence transcend into (and can be recognized in) intermediate and senior mathematics. Looking forward to lots of great learning through next year!


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