In Algebra 1, I was the king of solving for x. Algebraic manipulation was fun and satisfying, and I was good at it. But my confidence was shaken when I encountered a test question of the variety 4x + 5 = 4x – 3. After subtracting 4x from both sides, I was left with 5 = –3. Surely, I thought, this must be wrong. Every equation I had seen until then was solvable for a unique value of x. I puzzled over the result for a while and scribbled some nonsensical answer on my paper. Later, when we reviewed the test, I kicked myself for not applying common sense to the problem.
I was reminded of this incident while developing a Web Sketchpad activity for McGraw Hill’s Reveal curriculum. The lesson presented students with equations that had either 0, 1, or an infinite number of solutions. Thinking back to my own difficulties, I wondered whether technology could help students gain some intuition into these outcomes. Rather than head straight to symbolic manipulation, could students sharpen their conceptual understanding through a sensorimotor approach?
I landed on the triple number line model below (and here), inspired by the dynagraph models we’ve described in previous posts. Suppose you’d like to solve 2(x + 1) – 6 = 3(x – 1). The variable x sits on the top number line, and you can vary the variable by dragging the point. As you do, the second number line displays the location of 2(x + 1) – 6 and the third number line displays 3(x – 1). Notice that the points on the second and third number lines move at different speeds. To observe this behavior over a larger domain, press Show unit point and drag the point at 1 closer to 0.
In this number-line context, what does it mean to solve for x? Students discuss this question and determine that a solution can be spotted when the points on the second and third number line sit at the same location, with the triangular pointers aligning vertically. Here, this occurs when x = –1, and at this value, both sides of the equation equal –6.
On page 2 of the websketch, there is a new equation to solve, 6(x – 3) + 10 = 2(3x – 4). Dragging x reveals that the points on the second and third number lines move in lockstep, one always directly below the other, with identical values for every location of x. This behavior suggests that the two sides of the equation are identical, and the equation has an infinite number of solutions.
On page 3, the equation is 3(4 – 2x) = 4(3-5x) + 14x – 1. As with the prior equation, the points move at the same speed, but now the point on the second number line is always 1 to the right of its companion below. This behavior suggests that there is no value of x for which the two sides of the equation are equal.
On page 4 of the websketch, students can create their own equations with either 0, 1, or an infinite number of solutions. To change the default expressions of 2x and 3x, double tap them to open the calculator and then edit the expressions. The short video below demonstrates these steps.