After my success (I hope) with linear equations, I started a unit doing the same thing with quadratics.
“A triangle’s height is three feet longer than its base. Create a table linking the height to the area. Graph.”
“A rectangle’s length is twice as long as its width. Create a table linking the width to its area.”
“A rectangle has sides of X+3 and X-2. Create a table linking X to the rectangle’s area.”
Just as with linear equations, the students really improved at generating values. They also, I think, quickly grasped that generating data for quadratics is considerably more complicated than linear equations. More than one pointed out to me that they couldn’t just “add three each time” as they could with linear equations.
I taught them how to break it down into parts and assign each part to a column. For example, the first triangle problem:
|Base||Height||Base * Height||Half BH (Area)|
The stronger students could see how this led to the equation; even the weaker students could see that each equation had steps, and they started to get suspicious if it got too easy. One struggling student called me over to tell me he must be doing something wrong because “look, it’s going up by the same amount”. He was linking length to width, rather than length to area.
Even if they don’t get much stronger in working with quadratics generally, this exercise clearly helped them gain competence at working through a problem. Next step: generating values quickly from an equation in standard form. Hello, synthetic substitution.