I have students work individually to find the two binomial factors for each of the given trinomials on this slide of trinomial_factoring2_warmup. I instruct students to check their solution by multiplying the factors together to see if they obtain the original trinomial. Students can then verify their answer with a partner.
After going over the Warm Up I plan to lead students through a presentation (trinomial_factoring) that scaffolds their work with factoring polynomials.
I plan to ask students to reason about this first question individually. I ask students to think about the second factor and consider how they would multiply the two binomials together (MP2). At this point, I am trying not to teach students any "tricks" about how to factor trinomials. (I consider tricks to be strategies that focus on the signs or the numbers themselves (b and c values). I want students to develop the conceptual understanding of factoring polynomials by looking at the inverse operation of multiplication.
Teacher's Note: In elementary school, students learn multiplication of whole numbers before division. To answer a question like 21/7, students may ask 7 times what gives me 21. The same line of reasoning applies here.
I have found that if students are able to reason about the values and the structure of a polynomial expression, they will eventually see that the last two numbers multiply to the c value and the sum of the first term in one parenthesis and the last term in the other give the b value.
Once students think they have a solution to the problem, I have them turn-and-talk with a neighbor to justify their idea (MP3). I encourage students to explain their thinking and not simply tell the answer they obtained. If I hear a student who does a nice job explaining their thinking, I will ask them to repeat their idea out loud for the class as a model to reflect on.
Slides 2 and 3
Both of these problems approach the concept from the same perspective. I am trying to expose students to various combinations of positive and negative b and c values. After students discuss their solutions, I will have one or two students share their ideas with the class. I guide students towards making the connection between the factors and the values of b and c. I plan to show students where the b and c values are coming from in each example by showing the product and sum from multiplying the two factors.
I expect that some students with these two slides. Some struggle because of a lack of number sense. I have included an intervention resource that may help some students to practice with the type of reasoning required for factoring. This will not "cure" a lack of number sense, but it will help students practice the thinking required for determining how to factor trinomials. If students have a serious lack of number sense, I will make an appointment with them to do some remedial work.
Now I ask students to work with their partner to try to factor a trinomial without any factors being given. Students should be thinking first about the values that could multiply to -30. This trinomial was chosen specifically because both 10*3 and 15*2 will give a product of -30 and a sum of -13. However, only one combination will give both the correct sum and product at the same time. Guide students towards this understanding by showing them both sets of binomials (x-10)(x-3) and (x-15)(x+3). They can then discuss with their partner why only one set will multiply to the given trinomial.
I have included factoring practice at two different levels: Standard and Basic.
The standard practice assignment is: trinomial_factoring2_practice. This practice assignment has much more than 20 minutes worth of practice. It offers a wide range of questions from simple to more complex. Based on the flow of the lesson so far, I will decide which exercises to assign to students. Questions 19, 20 and 23 have students investigate the difference of perfect squares. If assigned these problems, students will examine the structure of these expressions and see that the sum of the two like terms would be zero.
The second set of exercises is given in the document: trinomial_factor_day2_basic. This is a scaled down version of the first set of problems. There are no trinomials with leading coefficients greater than 1 and no difference of squares polynomials.
I let students work with a partner of similar ability level for today's practice. And, I give the students the appropriate practice assignment. I encourage students to "think out loud" with their partner so that they can critique each other's reasoning and help each other to make sense of the concept of factoring (MP3).
To close today's lesson, students are going to choose one set of trinomials to work with (either 1, 2, & 3 or 4, 5 and 6). The first set of trinomials is more challenging than the second. Both sets of trinomials have the same common factor (x - 1). I designed this exit ticket to help me identify which students can currently factor trinomials fluently.