In John Hartsburg’s class, students are up at boards working equations on number lines. Working on the number line emphasizes the meaning of equality and “undoing” in a way that just writing steps of algebra work into papers sometimes misses. In addition, John uncovers interesting number sense issues as students explain their thinking.

The problem: 2x +5 = 21.

John: (addressing a student who has been hanging back) So tell me where you started.

Student 1: (points to end where 2x +5 is placed over 21 on the parallel number lines.)

John: And why did you put 21 under 2x + 5.

Student 1: Because they are in the problem.

John: Ok, but why over each other?

Student 2: Because they are equal.

Student 3: So they are the same distance on the number line.

John:. Ok.(addressing student 4) so what did you do next?

Student 4: Subtracted 5.

John: On which line?

Student 2: On both lines.

John: Why both?

Students are unsure. Two students in this group are especially quiet.

John: (directed to those two) Why did you take 5 from both lines? (crickets) What do you know about these two (points to 2x+5 and 21)?

Student 1: They are equal.

John: And if you subtract 5 from only one of them will they still be equal?

Student 2 tries to answer but John waits for student 1.

Student 1: No

John: Ok so when you subtracted 5 from each you got . . .?

Student 1: 2x and 16.

John: And what did you do next?

Student 3: divided by 2 . . .on BOTH lines.

John: and got?

Students: 8

John: And why did you put the 8 here, instead of more to the left or right?

Student 2: Because we were cutting the 16 in half.

John: So what’s the solution to this problem?

Students: 8. x equals 8.

John: Great. Write it in your journals and then I’ll give you the next one.

After three rounds of problems on number lines, students sit with a partner for a Round Robin 2 step equation practice activity. Most students just showed the algebraic steps, but a few preferred drawing out the number lines to work on. It *made sense* to them.

One of these students, as they were cleaning up to leave said to his partner, “I feel so smart in summer school! Look at all those problems we just did!”

Smiles all around.