When and Why Do We Write Quadratic Equations?

5 min

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 tt until they find one that yields the specified value of cc.
  • Reason backwards (subtract 2.5 from 62.50 and then divide the result by 12) without writing out the steps.
  • Solve 62.5=12t+2.562.5=12t+2.5 and 278.5=12t+2.5278.5=12t+2.5 by performing the same operation on each side to isolate tt.

Student Task

The expression 12t+2.5012t + 2.50 represents the cost to purchase tickets for a play, where tt is the number of tickets. Be prepared to explain your response to each question.

  1. A family paid $62.50 for tickets. How many tickets were bought?
  2. A teacher paid $278.50 for tickets for her students. How many tickets were bought?

Sample Response

  1. 5. Sample reasoning: Subtracting 2.50 from 62.50 gives 60. If 12t12t is 60, then tt is 5.
  2. 23. Sample reasoning: Subtracting 2.50 from 278.50 gives 276. If 12t12t is 276, then tt is 23.
Activity Synthesis (Teacher Notes)

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.

Standards
Building On
  • A-REI.3·Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
  • A-REI.3·Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
  • A-REI.3·Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
  • HSA-REI.B.3·Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

15 min

15 min