Students will learn why factoring monic quadratics amounts to finding two numbers whose sum is the linear term, and whose product is the constant term. Here's an excellent explanation. Then students will learn the box method for factoring. After that, students will learn the abstract method. Here's a couple examples involving the abstract method, and here's a bunch more.

One type of interesting question, is to ask where two parabolas intersect. Here's an example: Find \(x\) so that \(f(x) = g(x),\) given that \(f(x) = 2x^2 - 11,\) and \(g(x) = x^2 - 4x + 10.\) And here's the solution:

$$\begin{align} & f(x) = g(x) \\ & 2x^2 - 11 = x^2 - 4x + 10 \\ & x^2 - 11 = -4x + 10 \\ & x^2 + 4x - 11 = 10 \\ & x^2 + 4x - 21 = 0 \\ & (x + 7)(x - 3) = 0 \end{align}$$Thus, \(f(x) = g(x)\) when \(x = -7,\) and when \(x = 3.\)

Provide a graph to confirm this.

Next, give your students this challenge: Find \(x\) in the figure below:

Here's the solution. Next, give your students this puzzle: In the figure below, what's the area of the rectangle?

Finally, give your students this logic puzzle. The solution is tedious, so allow them to explain the solution without writing it down in full.

Conclude by giving your students these challenges:

Tangram Tangle by NRICH: Additionally, ask your students to classify each shape they're able to make.