Algebra 1 Reasoning with Equations and Inequalities Seeing Structure in Expressions Simplifying expressions

A Class Set of Identities Work

Predict: What responses to this prompt would you expect from my Algebra 1 students? (Prior to this problem my kids had mostly worked with integer arithmetic, solving linear equations in one-variable and graphing scenarios and equations.)


Study: What do you notice in this (small) class set of responses? Note anything that surprises you.

Kid 1:


Kid 2


Kid 3


Kid 4


Kid 5




Kid 6


Kid 7


Kid 8


Kid 9


Wrap Up

How did your predictions hold up? What surprised you the most? What’s something you wish you knew more about?

6 replies on “A Class Set of Identities Work”

I thought the first student’s replies were interesting. I did not expect anything like that, though I suppose it is a reasonable extension of a “one variable -> one solution” hypothesis.

I liked the x+inf = inf that some students put for part (c). The formulation is imprecise, but it is conceptually correct, so I wonder how they got there.

Hmm. Good that at least one student knows that typical quadratic equations that have solutions have two. No where are we restricted to the real numbers, so the example works.

Good to see that some students recognize what mathematical contradictions look like, though what always comes to mind when I want to demonstrate a first take at equations with no solutions is x + 1 = x.

Also nice to see that some understand what an identity (tautology) looks like. You can’t really beat x = x for showing that such things exist.

Unless I went through these too quickly, however, doesn’t seem like any one student nailed them all. And some of the misconceptions are a little bit disturbing.

I was surprised that a couple thought something along the lines a + b = 7 had only two solutions, as even with positive integers that isn’t true.

I really love this prompt – wondering how something similar could be done for graphing families along the lines of: Which graphing families COULD produce: 1, 2, infinite, 0 intercepts”. IE 1: Linear and Linear (y=x and y=-x) 2: quadratic and linear/quadratic and linear etc

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