Open Middle: Factors of Polynomials

Open middle problem about rewriting polynomials as a factor of its products. This is not a trivial task. It is challenging. I intended students to work in pairs. My reflection after first use. "Students were very engaged. All were puzzled by screen 1 or 2 and didn’t want to stop. I did it in pairs. Some went to a vertical whiteboard space. One spot light on my instruction that was exposed was that students were surprised how many trinomials can not be written as a product of its factors. We encountered that reality with algebra tiles early on and then we remained in the fantasyland where almost everything is factorable until today. They found out quickly that they can’t just drag numbers into slots and hope they work out. One pairing later shared their strategy of knowing whether a set of numbers might work. I missed the connection to the discriminant of the quadratic formula in the moment. They said that you know that you have to split the middle term two ways (they drew an area model) and then multiply. That number has to be at least as big as the product of the other numbers (showing the a and c). I missed that they essential said that it wouldn't work unless `\left(\frac{b}{2}\right)^{2}\ge ac`. I need to revisit this thought when we encounter the quadratic formula." Note: the images and logic for the nice snapping animation were taken from an activity written by the Desmos staff. https://twitter.com/pejorgens/status/1116380884641304576

By Paul Jorgens