Unit 8, Lesson 2
When and Why Do We Write Quadratic Equations?
Algebra 1
2.1
Warm-up
Standards Alignment
Building On
HSA-REI.B.3
Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
Addressing
Building Toward
Instructional Routines
NoneMaterials
NoneActivity Narrative
This Warm-up activates what students know about interpreting equations in context and about solving for a variable. The given equation is linear and relatively straightforward. The work prepares students to reason about quadratic equations in the lesson.
To find the unknown input in each question, students might:
- Try different values of until they find one that yields the specified value of .
- Reason backwards (subtract 2.5 from 62.50 and then divide the result by 12) without writing out the steps.
- Solve and by performing the same operation on each side to isolate .
Launch
NoneActivity
NoneThe expression represents the cost to purchase tickets for a play, where is the number of tickets. Be prepared to explain your response to each question.
- A family paid \$62.50 for tickets. How many tickets were bought?
- A teacher paid \$278.50 for tickets for her students. How many tickets were bought?
Activity Synthesis
Ask a student to share their response and reasoning. After they share, ask for one more different strategy.
If it does not come up, point out that we can think about cost as being a function of the number of tickets. In answering the questions, we were looking for the inputs that produce different outputs. This can be done by solving equations. In the case of linear equations, we can “do the same thing to each side” to isolate the variable.
2.2
Activity
Materials
NoneActivity Narrative
This task prompts students to try different ways to solve a quadratic equation. They are familiar with solving equations by performing the same operation on each side of an equation, but here they see that this is not really a workable strategy. Students are also discouraged from using a graph to solve the equation. Because they do not yet know an efficient way to use algebra to solve , they need to try different strategies and persevere in problem solving (MP1).
As students work, notice those who use strategies listed in the Activity Synthesis. Ask them to share their approach during discussion.
2.3
Activity
Instructional Routines
MLR5: Co-Craft QuestionsMaterials
NoneActivity Narrative
This activity aims to show that it is relatively easy to solve a quadratic equation when one side of the equation is zero and the other side is a quadratic expression in factored form, and that it may be a little tricky to solve the equation otherwise.
Although students are not yet formally introduced to the zero product property, they do have experience finding the zeros of a quadratic function when given an expression in factored form. This prior knowledge enables them to reason about the solutions to the equations.
As students make sense of the equations and ways to use them to solve a contextual problem, they practice reasoning quantitatively and abstractly (MP2).
This activity uses the Co-Craft Questions math language routine to advance reading and writing as students make sense of a context and practice generating mathematical questions.
Launch
Arrange students in groups of 2. Introduce the context of raffle tickets. Use Co-Craft Questions to orient students to the context and elicit possible mathematical questions.
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Display only the first sentence of the Task Statement, without revealing the questions. Give students 1–2 minutes to write a list of mathematical questions that could be asked about the situation before comparing questions with a partner.
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Invite several partners to share one question with the class, and record responses. Ask the class to make comparisons among the shared questions and their own. Ask, “What do these questions have in common? How are they different?” Listen for and amplify language related to the learning goal, such as “maximum” and “zero.”
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Reveal the first question, and give students 1–2 minutes to compare it to their own question and those of their classmates. Invite students to identify similarities and differences by asking, “Which of your questions is most similar to or different from the ones provided? Why?”
Ask students to recall the form in which each quadratic expression is written. Then, give them a minute to talk to a partner and recall at least two things about each form and what the form might tell us about the graph of the function that the expression defines.
Record their responses for all to see. If no students mention the connection between either of the forms and the horizontal intercepts of the graph or the zeros of the function, ask them about it.
Tell students that they do not need to find the ticket prices in the second question, only think about how to do so.
Lesson Synthesis
Help students to reflect on the key ideas of the lesson by discussing, as a class or with a partner, questions such as:
- “What are some limitations of solving by guessing and checking? What about by graphing?”
- “Which equation do you think is easier to solve: or ? Why?”
- “Which is easier to solve: or ? Why?”
Student Lesson Summary
The height of a potato that is launched from a mechanical device can be modeled by a function, , with representing time in seconds. Here are two expressions that are equivalent and both define function .
Notice that one expression is in standard form and the other is in factored form.
Suppose we wish to know, without graphing the function, the time when the potato will hit the ground. We know that the value of the function at that time is 0, so we can write:
Let's try solving , using some familiar moves. For example:
- Subtract 96 from each side:
- Apply the distributive property to rewrite the expression on the left:
- Divide both sides by -16:
- Apply the distributive property to rewrite the expression on the left:
These steps don’t seem to get us any closer to a solution. We need some new moves!
What if we use the other equation? Can we find the solutions to ?
Earlier, we learned that the zeros of a quadratic function can be identified when the expression defining the function is in factored form. The solutions to are the zeros to function , so this form may be more helpful! We can reason that:
- If is 6, then the value of is 0, so the entire expression has a value of 0.
- If is -1, then the value of is 0, so the entire expression also has a value of 0.
This tells us that 6 and -1 are solutions to the equation, and that the potato hits the ground after 6 seconds. (A negative value of time is not meaningful, so we can disregard the -1.)
Both equations we see here are quadratic equations. In general, a quadratic equation is an equation that can be expressed as , where , , and are constants and .
In upcoming lessons, we will learn how to rewrite quadratic equations into forms that make the solutions easy to see.