Using the Factoring Introduction I will try to make a connection to factoring concepts that students are already familiar with. Namely, students have factored whole numbers in the past. Typically, students in my district learn how to do this by using tree diagrams.
I let students do a think-pair-share around the two questions on this slide. Students may come up with either 10 and 1, 5 and 2 or 15 and 1, 5 and 3 respectively. I accept both answers from students, but ask them to think about why these are factors. Then, I show them the two key words (factor, multiply). I ask students to think with their partners about why factor means "to take apart".
On this slide, I provide some opportunities for students to deepen their understanding of these two key words. I ask students to relate each word to a metaphor. Metaphorical thinking is a good way to deepen understanding with vocabulary (e.g., Factoring is like a crowbar. Multiplying is like glue).
No I have students work with their partner to explain why 2x and 2x+6 are factors of the polynomial 4x^2 + 12x. I will stop students after about 2 minutes and do a non-verbal cue to identify who can explain why those expressions are factors of the given polynomial. I will then call on one or two pairs to explain their thinking (MP3). Next, I have students brainstorm other factors of the polynomial. I make a list on the board of possible factors to help students see the expression can be factored in different ways, just like a non-prime number.
Slide 5 gives students an opportunity to practice what was discovered on the previous slide. It does so in a guided way. Students are first asked to verify that the given expressions are factors. Then, I ask students to find new factors of their own. I let students work on this with their partners. I also leave the list from Slide 4 posted (along with the original expression) to give students some hints.
Instructional Note: Some students will undoubtedly factor out a 1 from the expression. This is a good teaching point because while it is a correct factorization, it does not accomplish the goal of writing the expression in a simplified way.
For this activity, I show students the three expressions in multiply_polynomials_direct. Then, I ask them to prove that they are equivalent to the given expression. I also ask students to come up with one other way to write the original expression.
I conclude this brief portion of the lesson by having students turn and talk with a partner. Each partner should verify (using the Distributive Property) that his/her partner has made an equivalent expression.
After I distribute the Factoring Common Factor Practice worksheet I let the students inspect the structure of the example to see why it is true (MP7). Students are familiar with this area model from earlier lessons. I will also use this area model for future lessons so I want students to be familiar with the structure.
Then, I ask the students to think of another way they could write the expression in the example. After collecting a couple of ideas, I remind students that there are several equivalent expressions that are each a correct answer to the questions on this worksheet.
I plan for students work on Questions 1-7 with their partners. After about 10 minutes, I will ask all of the students to look at Question #7. I will then give them the task of determining "Why 4x is the greatest common factor (GCF) of the expression?"
After we discuss their ideas about the GCF of the polynomial in Question 7, I will give the students to practice factoring by taking out the greatest common factor using Questions 8-10.
Teaching point: Since I initially let students factor out any factor up to this point, I will probably ask them to think about how to identify the GCF. Usually, students will see when there is "more that you can take out of an expression." For example, in the expression in parenthesis in 3(x^2+2x) each term has another x in common. Through a guided discussion, my class is usually able to come up with the fact that you want to take out the greatest common numerical factor of the coefficients, and, the greatest power that each term has on a variable.
Today's Ticket Out the Door serves two purposes. First, I will be able to see which students understand how to factor an expression in different ways. Second, I will see which students are able to identify the greatest common factor of an expression as a way to factor it completely.