Lesson 10
Quadratic Zeros
These materials, when encountered before Algebra 1, Unit 7, Lesson 10 support success in that lesson.
10.1: Which One Doesn’t Belong: Factored Quadratics (5 minutes)
Warm-up
This warm-up prompts students to compare four expressions. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another.
Launch
Arrange students in groups of 2–4. Display the expressions for all to see. Give students 1 minute of quiet think time and then time to share their thinking with their small group. In their small groups, ask each student to share their reasoning why a particular item does not belong, and together find at least one reason each item doesn't belong.
Student Facing
Which one doesn’t belong?
A: \((x+3)^2\)
B: \((x+3)(x-3)\)
C: \((x-3)(x-3)\)
D: \(x^2+6x+12\)
Student Response
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Activity Synthesis
Ask each group to share one reason why a particular item does not belong. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which one does not belong, attend to students’ explanations and ensure the reasons given are correct. During the discussion, ask students to explain the meaning of any terminology they use, such as factored form or standard form. Also, press students on unsubstantiated claims.
10.2: Finding Solutions by Graphing (15 minutes)
Activity
In this lesson, students graph functions to find the zeros then write associated equations that would be solved with the values they find. In the associated Algebra 1 lesson, students do a similar process to determine whether a quadratic expression can be easily factored. Students reason abstractly and quantitatively (MP2) when they use equations to represent the zeros on a graph.
Student Facing
- Use technology to graph the functions, then find the zeros.
- \(f(x) = (x+2)(x-5)\)
- \(g(x) = (5x-4)(x-3)\)
- \(h(x) = x^2 + 5x + 4\)
- \(k(x) = x^2 + 5x + 3\)
- \(m(x) = 2x^2 - 13x - 15\)
- \(n(x) = 2x^2 - 13x - 10\)
- For each function, write an equation that would be solved by the zeros. Are the solutions exact or approximate?
Student Response
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Activity Synthesis
The purpose of the discussion is to show the relationship between zeros of a function and solutions to associated equations. Ask students,
- “A function has zeros at 2 and 3. Write a quadratic equation with solutions at those same values.” (The equation \((x-2)(x-3) = 0\) has solutions at 2 and 3.)
- “A function has zeros at \(\frac{1}{2}\) and -2. Write a quadratic equation with solutions at those same values.” (The equation \((2x-1)(x+2) = 0\) or the equation \((x-\frac{1}{2})(x+2) = 0\) each have solutions at \(\frac{1}{2}\) and -2.)
10.3: Matching More Factored Expressions (20 minutes)
Activity
In this partner activity, students take turns matching expressions in factored form with associated functions in standard form. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3).
Launch
Arrange students in groups of 2. Tell students that for each expression in column A, one partner finds an associated function in column B and explains why they think it is equivalent. The partner's job is to listen and make sure they agree. If they don't agree, the partners discuss until they come to an agreement. For the next expression in column A, the students swap roles. If necessary, demonstrate this protocol before students start working.
Student Facing
Take turns with your partner to match an expression in factored form with a function in standard form.
- For each match that you find, explain to your partner how you know it’s a match.
- For each match that your partner finds, listen carefully to their explanation. If you disagree, discuss your thinking and work to reach an agreement. Match each expression in factored form to its associated function in standard form.
Expressions in factored form
- \((2a+5)(a+4)\)
- \((3a-1)(a-10)\)
- \((a+7)(5a-2)\)
- \((4a-5)(4a-5)\)
- \((4a-5)(4a+5)\)
- \((2a+7)(9a+4)\)
Functions in standard form
- \(f(x) = 2a^2 + 13a + 20\)
- \(g(x) = 16a^2 -25\)
- \(h(x) = 5a^2 +33a -14\)
- \(j(x) = 16a^2 - 40a + 25\)
- \(k(x) = 18a^2 + 71a + 28\)
- \(m(x) = 3a^2 -31a + 10\)
Student Response
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Activity Synthesis
Once all groups have completed the matching, discuss the following:
- “Which matches were tricky? Explain why.” (The two expressions that had \(4a\) as a term were difficult because I had to look more closely at all of the terms and not just a few of them.)
- “Did you need to make adjustments in your matches? What might have caused an error? What adjustments were made?”