I begin this lesson by asking students to do a think-pair-share: solving_quadratics_day1_launch.
How students understand this question is crucial to the remainder of the lesson, so I don't rush the conversation. After students have a minute or two to think by themselves, I ask them to share their ideas with their partner. Then, I let several students share out their ideas and encourage them to build on each other's understanding. I let students do as much of the talking as possible at this point. I just guide their thinking in subtle ways. Typically, my students will come up with the fact that at least one of the numbers (or both) need to be zero in order for the product to be zero.
At the right moment I ask students, "Are there any other combinations of numbers that will work?" Often students will say that, for example, -3 and 3 could be substituted for each other. If this arises, I will ask the students in the class to either prove or disprove this idea.
On the second slide, I have students go through another iteration of think-pair-share so that they can process this idea fully. When students are sharing with their partner, I listen for those that are making the connection between the first and second slide. Namely, I want students to see that the quantity (x-2) is being multiplied by the quantity (x). Because of slide 1, I eventually want students to assert that either (x-2) is to equal zero or (x) is equal to zero (or both). When we arrive at this point, I ask students to make suggestions about how we can make this happen.
At this point, I do not ask students set up an equation like x-2=0. Rather, I want them to reason that the value that will make the quantity (x-2) equal zero is x = 2.
For the final slide of the launch, I let students go right to discussing the prompt with their partner (MP3). Students should now be starting to see the pattern in the structure of these expressions (MP7). Before moving on with the lesson, I will have several students share out their thinking and justification. Again, we will revisit the idea that the two quantities (x-2) and (x+3) are being multiplied together.
To continue our progress from the launch, I explain to students that we are thinking about the equation on the first page of solving_quadratics_day1_direct in exactly the same way as the first three equations from the warm-up. The problem is, however, it is difficult to tell, by inspection, what values make x^2-3x-10 equal zero. Then, I ask students, "Is there a way to rewrite this expression so that it looks more like the first three equations from the launch?" I pause, then I let students turn-and-talk with a partner. I am patient in this case, because I want the students to come up with the idea of factoring the equation into two binomials.
Once the equation is written in factored form (x-5)(x+2), students will be able to see which values will make the product of zero. To connect this equation to the graphical representation, I then ask students to use technology to graph the function f(x)=x^2-3x+10 and visually see that the roots are 5 and -2.
As we move to the second page, I want students to recognize that in order to use the zero product property, the equation must be equal to zero. This idea was implicit in the launch conversation. I let students work with their partner to determine how this equation could be rewritten so that it is equal to zero. Next, I let students solve the equation by factoring. Finally, I ask them to verify their solution by graphing the associated function.
Teaching Note: I find it is often necessary to explain to students that in order to graph the function all of the terms must be on the same side of the equal sign.
On Slide 3, I have students graph the polynomial function first so that they can see that there are three roots for this function. Then, we will discuss how to solve this equation by factoring. Since students have just completed a factoring unit, they are good about identifying the greatest common factor of x. So, x(x+4)(x+1) is just a generalization of the zero product property where abc=0.
Slide 4 presents an equation that can be contrasted with Slide 3. I will help students factor this equation as 2(x+2)(x+1)=0. Typically, students will be able to come up with x=-2 and x=-1 as roots. I expect many will be unsure how to handle the integer factor, 2. The question I like to ask is, "Can 2 equal 0?" Since this is impossible, I ask students to predict how many zeros this equation. Then, I let students graphically verify their predictions.
For today's Ticket out the Door, I ask students to factor a trinomial expression and solve a quadratic equation (see solving_quadratics_day1_close). I do not give students any instruction on how to complete these tasks. My goal is to see if students understand the similarities and differences between these two types of questions. Students will work on this question individually and submit their response before leaving class for the day.