I begin class with a Factoring Activity warm up that will review what students have learned about factoring so far. This will be the first time students see quadratics that have positive and negative signs in terms of factoring. During the warm up, I circulate and check for understanding, working with any students that might need one-on-one help or small group instruction. Ib ring the class together when most students have finished all five questions and ask for volunteers to share out their work.
During this discussion, I try to elicit from students if they have any tricks when working with different signs. I try to get students to talk about why if the quadratic has all positive signs the two binomials must also have two positive signs. Then we talk about what happens when the "b" term is negative but the "c" term is positive. I find it's more powerful to have students come up with these ideas rather than me presenting them as the teacher.
Next, I want to show students why factored form is useful and how it can be a quick way to see the x-intercepts for a quadratic. I show students a quadratic that is written in factored form. For example: y = (x – 2) (x + 5). I remind them that the x-intercepts are the values of x that make y = 0. Ask them what values of x will make the product (x – 2)( x + 5) equal to 0? I review the Zero Product Property here by telling students that if ab = 0, if and only if a =0 or b = 0. Students should be able to see that x could equal 2 or -5 and therefore, the x-intercepts of the quadratic are (2, 0) and (-5, 0). I then use an area model to rewrite the quadratic in standard form which will look more familiar to students.
I will make sure to go over the two Questions that are not factorable. I will try to elicit from my students how the two equations are different. One, Question 2 is not factorable, but has two x-intercepts. Question 5 is not factorable and does not cross the x-axis. Be sure to elicit from students how to determine this information. They may need to review Vertex Form in order to get a sense of the how the parabola looks.
For Question 6, I will make sure bring out that although the original equation and the equation divided by 2 have different graphs, they have the same x-intercepts. I will probably use graphing calculators to give students the opportunity to explore and confirm this relationship.
Like yesterday, I want to make sure to give some students time to process what they have learned today. I will give them an exit ticket question like the one below:
How would you decide what method to use for solving a quadratic equation? Explain your thinking.