# Lesson 13

Making New, True Equations

These materials, when encountered before Algebra 1, Unit 2, Lesson 13 support success in that lesson.

## 13.1: Math Talk: Evaluating Expressions (5 minutes)

### Warm-up

The purpose of this Math Talk is to elicit strategies and understandings students have for finding the value of an expression after substituting a value for \(x \). These understandings help students develop fluency and will be helpful later in this lesson when students will need to be able to substitute expressions for a variable.

### Launch

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

### Student Facing

Find the value of \(y\) when \(x = 5\).

\(y=3x-4\)

\(y=\frac{2}{5}x+4\)

\(y=2x+3 + (3x - 1)\)

\(y=4x - (x+1)\)

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

- “Who can restate \(\underline{\hspace{.5in}}\)’s reasoning in a different way?”
- “Did anyone have the same strategy but would explain it differently?”
- “Did anyone solve the problem in a different way?”
- “Does anyone want to add on to \(\underline{\hspace{.5in}}\)’s strategy?”
- “Do you agree or disagree? Why?”

## 13.2: Solving for a Variable (15 minutes)

### Activity

The mathematical purpose for this activity is to allow students to practice solving for a variable in an equation with two variables. In the associated Algebra lesson, expressing one variable in terms of another is necessary before substituting an expression for a variable in another equation.

### Launch

Demonstrate how to solve for a variable by showing students the equation \(a + 2b = 5\). Solving for \(a\) results in \(a = 5 - 2b\) or \(a = \text{-}2b + 5\). Solving for \(b\) results in \(b = \frac{5-a}{2}\) or \(b = \frac{5}{2} - \frac{a}{2}\) or another equivalent expression.

### Student Facing

Solve for the indicated variable.

- Solve for \(k\). \(2t+k=6\)
- Solve for \(n\). \(10n=2p\)
- Solve for \(c\). \(12-6d=3c\)
- Solve for \(g\). \(h=8g+4\)
- Solve for \(x\). \(4x+3y=12\)
- Solve for \(y\). \(4x+3y=12\)

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

The purpose of the discussion is to share strategies for solving for one variable in an equation including more than one variable. Select students to share their answers and strategies. As students share, consider asking another student why each step makes sense to do.

Some questions for discussion:

- "How do you know when you are done solving for a variable?" (When the variable I want to solve for is the only thing on one side of the equation.)
- "How is solving for \(x\) in an equation like \(4x+3y=12\) similar or different from solving for \(x\) in an equation like \(4x + 6 = 12\)?" (The steps are very similar, but when there is only one variable, we can know the value of \(x\)
- "If you know the value for \(y\) is 2 and need to know the value for \(x\), would you rather begin with the equation \(4x + 3y = 12\) or the equivalent equation \(x = 3 - \frac{3}{4}y\)? Explain your reasoning." (I would rather use the second equation since I only need to substitute in the value for \(y\) and evaluate the expression rather than solve an equation.)

## 13.3: Solving Some Equations (20 minutes)

### Activity

In this activity, students get a chance to practice solving equations with a single variable. The equations resemble the types of equations students see in the associated Algebra 1 lesson after they substitute for a variable. Students will work in pairs and each partner is responsible for answering the questions in either column A or column B. Although each row has two different problems, they share the same answer. Ensure that students work their problems out independently and collaborate with one another when they do not arrive to the same answers. Students construct viable arguments and critique the reasoning of others (MP3) when they resolve errors by critiquing their partner's work or explaining their reasoning.

### Launch

Arrange students in groups of 2.

### Student Facing

Solve each equation.

row | column A | column B |
---|---|---|

1 |
\(4(2x+8)-10 = 14\) |
\(4 + 2(\text-3x+5)=20\) |

2 |
\(3(x-4)+6 = 60\) |
\(3(\frac12x + 9)-5=55\) |

3 |
\(4(\frac{x+3}{2})-5 = 10\) |
\(7-2(6x+1)=\text-49\) |

4 |
\(2x+(5-3x) = 14\) |
\(1=5x+10-4x\) |

5 |
\(4x + 2(3-x)=16\) |
\(x+2(x-4)+5=12\) |

6 |
\(2x - 2(3x-1) = 8\) |
\(\text-6x+2(4x+5)=7\) |

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

The purpose of the discussion is to recall strategies for solving equations with one variable. Select students to share their solutions including their reasoning for the steps they take.

Some questions for discussion:

- "When there is a variable inside parentheses, how did you approach the problem?" (I distributed the value multiplying the parentheses to remove the parentheses then solved the equation as usual.)
- "How could you check that your answer works using the original equation?" (I could substitute the value for \(x\) into the original equation and determine if the equation is true.)