Summary
In this lesson, students will use number sense to identify patterns that help them factor trinomials with a leading coefficient of one. Students will build on their understanding of multiplying binomials to recognize the relationship between expanding and factoring. Prerequisite knowledge for this lesson includes multiplying binomials.
Essential Question(s)
How can we rewrite a trinomial in a different form?
Learning Objectives
Factor quadratic trinomials into two binomials.
Recognize the relationship between factoring and expanding.
Snapshot
Engage
Students recall factor pairs of a given set of numbers, and then find the sums and differences of those pairs to create a table of values.
Explore
Students analyze their table of values to find two numbers with a given product and given sum or difference.
Explain
Students make the connection between finding the right pair of numbers and factoring a quadratic expression.
Extend
Students apply their understanding by solving a real-world problem after watching an interview of a civil engineer who discusses key elements of his job.
Evaluate
Students practice factoring quadratic expressions through a game of Quiz, Quiz, Trade.
Materials
Lesson Slides (attached)
Factor Pairs handout (attached; one per student; print two-sided)
Factor Pairs (Key) document (attached)
Guided Notes handout (attached; one per student; print two-sided)
Guided Notes (Model Notes) document (attached)
Civil Engineering handout (attached; one per student; print one-sided)
Factoring Quiz, Quiz, Trade cards (attached, one set; print one-sided)
Square tiles (30–50 per student); digital option available
Individual whiteboards (optional; one per student)
Dry erase markers and erasers (optional; one per student)
Pencils
Paper
Preparation
During the Engage phase of the lesson, students are using square tiles to model rectangles with a given area. To save time distributing materials, consider using zip-top bags to group 30–50 tiles.
Before the lesson, print the attached Factoring Quiz, Quiz, Trade cards. Make sure to print these two-sided, selecting the “flip on long edge” setting, which is typically the default printer setting. This printer setup ensures that each answer appears on the back of its corresponding question. There are 32 unique cards across the 16-page document, resulting in eight printed pages when printed two-sided. Consider printing on cardstock paper, especially if you plan to reuse these cards.
Once printed, cut out the cards. All of these cards are the same size for easy cutting.
Engage
20 Minute(s)
Introduce the lesson using the attached Lesson Slides. Display the essential question on slide 3 and move to slide 4 to identify the lesson's learning objectives. Review each of these with students to the extent you feel necessary.
Display slide 5 and give each student a copy of the attached Factor Pairs handout and approximately 30–50 square tiles. Explain to students that they are to use the square tiles to model the area of each rectangle (from the first column of their table). Specifically, using the prompt from the slide, ask students to model all possible rectangles with an area of 1. Students should quickly recognize that there is one possible option, and that the dimensions of this rectangle are 1 by 1. Have students record these side lengths in the second column of their table. Transition to slide 6 to model where to write this.
Similarly, display slide 7 and ask students to use their square tiles to model all possible rectangles with an area of 4. Here, students should notice that there are two options: 1 by 4 and 2 by 2. Move to slide 8 to indicate where students should record these dimensions. Then click on the slide to trigger the animation that shows the second pair.
Display slide 9 and direct students to continue this process to complete the second column of their table, using the square tiles to model the possible rectangles for each given area.
Use the Factor Pairs (Key) document as needed.
Display slide 10 and direct students’ attention to the third column of their table. Here students are to find the positive and negative sums of their factor pairs for 1. Help students understand the reasoning: (+1) · (+1) = 1 and (–1) · (–1) = 1, and finding the sums of those factor pairs would be (+1) + (+1) = +2 and (–1) + (–1) = –2. Transition to slide 11 to indicate how students should record their sums.
Similarly, show slide 12 and prompt students to find the sums of their factor pairs for 4. Then move to slide 13 to show students how to record these sums.
Display slide 14 and direct students to continue this process to complete the third column of their table.
Transition to slide 15 and direct students’ attention to the fourth column of their table. Here students are to find the positive and negative differences of their factor pairs for 1. Help students understand the reasoning: (+1) · (+1) = 1 and (–1) · (–1) = 1, and finding the differences of those factor pairs would be (+1) – (+1) = 0 and (–1) – (–1) = 0. Since zero is neither positive nor negative, they only need to record zero once. Transition to slide 16 to indicate how students should record this difference.
Similarly, display slide 17 and prompt students to find the differences of their factor pairs for 4. Then move to slide 18 to show students how to record these differences.
Display slide 19 and direct students to continue this process to complete the fourth (last) column of their table.
Explore
10 Minute(s)
Display slide 20 and introduce the “Which Pair?” game. Explain that students identify a pair of numbers that have a given product (first number on the slide) and have a sum or difference of the second number on the slide. Each question will be of the following format, “What two numbers multiply to _____ and add/subtract to _____?” Encourage students to have their Factor Pairs handout available for easy reference for this game.
Communicate with students how you would like them to share their responses for this activity, and then display slide 21. Read the prompt, “What two numbers multiply to 12 and add/subtract to 7?”
After giving students time to think and share their responses, display slide 22 to reveal that 3 and 4 are the two numbers.
Repeat this again using slides 23–24, “What two numbers multiply to –12 and add/subtract to 1?” If needed, give hints or ask guiding questions to help students. After this second example, draw their attention to the similarities and differences between this and the previous example.
Repeat this once again using slides 25–26, “What two numbers multiply to –20 and add/subtract to –8?”
Explain
15 Minute(s)
Display slide 27 and give each student a copy of the attached Guided Notes handout. Review the vocabulary of trinomials and binomials, and then share that today they are going to learn how to factor a trinomial into a product of binomials.
Move to slide 28 and let students know that in the same way there are many ways to expand binomials (FOIL, Box Method, etc.), there are many ways to factor. Let students know that the “What Pair?” game they played earlier should sound similar to the factoring tips on the slide for x2 + bx + c:
What you are multiplying to is the value of c.
What you are adding/subtracting to is the value of b.
Use the attached Guided Notes (Model Notes) document as needed.
Display slide 29 and walk through factoring (x2 + 7x + 12), using the following steps:
Step 1: Identify b and c.
Step 2: List the factor pairs that multiply to c.
Step 3: Find the factor pair that multiplies to c and adds/subtracts to b.
Step 4: Plug the factors into the binomials.
Then click on the slide, once per step, to trigger the animation to reveal the results for each step.
Display slide 30 and walk through factoring (x2 + x – 12).
Then click on the slide, once per step, to trigger the animation to reveal the results for each step.
Display slide 31. Have students independently try to factor (x2 + 9x + 20) and check their work by expanding their binomials. As students work, circulate the room to check progress and answer questions. Once students are finished checking their work, move to slide 32 to reveal the factored form. Give students time to make corrections and ask questions.
Take a moment to bring the class together for a quick discussion. Ask the class what they noticed when they checked their work.
Similarly, display slide 33. Have students factor (x2 – 8x – 20) and check their work. When students are done, show slide 34 so students can check their work. Give students time to make corrections and ask questions.
Extend
20 Minute(s)
Display slide 36 and give each student a copy of the attached Civil Engineering handout. Introduce the Civil Engineer and Rectangular Prisms video, which features an interview with Bobby Williams talking about his career as a civil engineer. Direct students’ attention to the “Career-Focused Video” section of their handout and have them review the questions they need to be able to answer from the video. Then play the video on the slide, stopping at 3:56.
After the video, display slide 37 and facilitate a whole-class discussion using the prompts from the handout and slide:
Describe one part of the civil engineer’s job.
What is the most enjoyable part of the civil engineer’s job?
What is at least one challenging aspect of being a civil engineer?
What are actions you could take or classes you should take if you are interested in becoming a civil engineer?
Have students form groups of three. Show slide 38 and direct students’ attention to the “Factoring and Area” portion of their handout.
Use the handout or slides 38–40 to read the real-world scenario. In this scenario, students examine a square foundation with a 2-meter concrete border and a total area of 81 square meters. They are shown the work of the quadratic equation set up, expanded, and manipulated to equal zero. Students are then asked to factor the resulting quadratic expression (on one side of the equation) and check their work by multiplying their resulting binomials. Here students should find that (x2 + 8x – 65) = (x – 5)(x + 13).
Evaluate
20 Minute(s)
Display slide 41 and introduce the Quiz, Quiz, Trade strategy. Let students know that they should remain seated for the first round to make sure everyone understands the procedure for this activity before getting up to find a new partner for the second round. Here students are going to practice factoring.
Give each student one of the Factoring Quiz, Quiz, Trade cards. Explain that they should hold the card such that the side with the question (and question mark) is facing away from them. The side with the question is what they will show their partner. Have students remain seated and introduce themselves to their neighbor (now partner). Have students hold their cards as directed.
Show slide 42. Explain the reasoning for the name of the activity, “Quiz, Quiz, Trade,” by sharing that Student A reads Student B’s card, answering the “quiz” question aloud. Student B checks Student A’s answer using the answer on the card (that is facing them). Then Student B does the same by answering Student A’s “quiz” question. Once partners have correctly answered the “quiz” and “quiz” questions, they now “trade” cards and find a new partner. Once students understand the process, have them get up and find a new partner around the room. If needed, practice one more round with students seated.
Use the 2-minute timer on the slide to pace the activity, giving students two minutes to find a new partner, each answer a question, and trade cards. When the timer expires, direct students to trade cards, if they have not done so already, and find a new partner. Repeat this process as many times as you deem necessary.
Resources
K20 Center. (n.d.). Polypad. Tech Tools. https://learn.k20center.ou.edu/tech-tool/4556
K20 Center. (n.d.). Quiz, quiz, trade. Strategies. https://learn.k20center.ou.edu/strategy/3086
K20 Center. (2021, September 21). 2-minute timer. [Video]. YouTube. https://www.youtube.com/watch?v=HcEEAnwOt2c
K20 Center. (2021, September 21). 3-minute timer. [Video]. YouTube. https://www.youtube.com/watch?v=iISP02KPau0
K20 Center. (2023, March 21). Civil engineer and rectangular prisms [Video]. YouTube. https://www.youtube.com/watch?v=qrQ2Mrg6Vg0
Khan Academy. (2015, July 14). Factoring polynomials as (x+a)(x+b) [Video]. YouTube. https://www.youtube.com/watch?v=1kfq0aR3ASs