# Lesson 9

Equivalent Equations and Functions

These materials, when encountered before Algebra 1, Unit 7, Lesson 9 support success in that lesson.

## 9.1: More Equivalent Equations (5 minutes)

### Warm-up

In this warm-up, students have the opportunity to explain why several equations are equivalent. These skills will be useful in the associated Algebra 1 lesson when students manipulate quadratic equations to solve them in factored form. For the second question, monitor for students who:

1. Distribute the 5 from the original equation and distribute the 10 from the solution to obtain matching equations.
2. Factor 2 from the original equation and combine with the 5 to get the solution.

### Student Facing

Explain why each of these equations is equivalent to $$5(2x-20) + 4 = 8$$.

1. $$10x - 100 = 4$$
2. $$10(x-10) + 4 = 8$$
3. $$10x=104$$
4. $$x = \frac{52}{5}$$

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

The purpose of this discussion is to remind students of some standard moves for rearranging linear equations. Select students to share their responses. Select previously identified students to share their methods for approaching the second question. Connect the responses by showing that $$10x - 100$$ can factor into $$10(x-10)$$.

## 9.2: Finding Solutions and Functions (20 minutes)

### Activity

In this activity, students find solutions to equations from a list of values, then rearrange the equations into functions whose graphs have $$x$$-intercepts at the same place as the solutions to the equations. In the associated Algebra 1 lesson, students solve quadratic equations using the factored form. This activity gives an opportunity to preview that work with additional support.

### Student Facing

Here is a list of possible solutions to equations.

• -9
• -7
• -6
• -4
• 0
• 3
• 4
• 5
• 6
• 7
1. For each equation, find any values on the list that are solutions. (Some equations have two solutions, and some only have one.)
1. $$35 = x^2 - 1$$
2. $$(x - 5)(x + 7) = 0$$
3. $$0 = (7 - x) \boldcdot x$$
4. $$(x + 3)^2 = 36$$
5. $$x^2 + 8x + 16 = 0$$
2. For each function, explain how it is related to the associated equation from the previous question. Then, graph the function using technology. Where can you see the solution to each equation on its graph?
1. $$f(x) = x^2 - 36$$
2. $$g(x) = (x - 5)(x + 7)$$
3. $$h(x) = (7-x)\boldcdot x$$
4. $$k(x) = (x+3)^2 - 36$$
5. $$m(x) = x^2 + 8x + 16$$

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

The purpose of the discussion is to recognize how solutions to equations are related to the graphs of similar functions. Select students to share their solutions. Ask students,

• “For all of the solutions in the first question, what are the associated function values from the second question?” (Zero)
• “Why is it helpful to have the factored form equal to zero for the second and third equations of the first question?” (It can be solved using the zero product property.)

## 9.3: Card Sort: Matching Equations (15 minutes)

### Activity

In this partner activity, students take turns matching equivalent equations. 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. Distribute one set of cards to each group of students. Give students time to work with their partner, followed by a whole-class discussion.

### Student Facing

Your teacher will give you a set of cards.

Take turns with your partner to match two equivalent expressions.

1. For each match that you find, explain to your partner how you know it’s a match.
2. For each match that your partner finds, listen carefully to their explanation. If you disagree, discuss your thinking and work to reach an agreement.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

Select groups to share their matches and how they sorted their equations. Attend to the language that students use to describe their matches, giving them opportunities to describe their equivalent equations more precisely.