Lesson 9

The purpose of this Math Talk is to elicit understandings students have for the equals sign and strategies they have for manipulating equations while preserving equality. These understandings help students develop fluency and will be helpful in students’ Algebra 1 class when they will need to be able to solve complex equations and justify why certain moves preserve equality.

Math Talks build fluency by encouraging students to think about expressions or equations and rely on what they know about properties of operations and equality to mentally solve a problem.

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.

Ask previously identified students, using the strategies you monitored for, 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?”

Emphasize that the expressions on both sides of the equals sign should have the same value, and point out strategies that make explicit use of the equivalence. If the following strategies are not mentioned, ask students about these strategies, and demonstrate if necessary:

• Compensation strategies, such as noticing that 25 is 4 more than 21, so the number that goes in the blank should be 4 more than 3.
• “Doing the same thing to both sides” strategies, such as subtracting 21 from both sides to get the blank alone.

The purpose of this activity is to further remind students of the different forms that equations representing relationships can take, and to begin to focus students’ attention on equivalent forms of equations in two variables. This will be useful when students learn to solve equations for a variable in their Algebra 1 class.

It’s not critical that students justify why the equations are equivalent to one another. It is more important that they connect each equation back to how it relates to the context.

Monitor for students who notice that the equations are equivalent and the moves that could be used to relate one to another.

Identify students who struggle with connecting contexts to expressions.

The goal of the discussion is to identify multiple equivalent expressions that describe the same situation, and to think about what it means for equations to be equivalent. Focus on how the different equations match or represent the story. Don’t focus on how the different equations are equivalent to each other unless students bring it up.

Ask students:

• What does Han's equation calculate? When would this be useful? (Han's equation calculates the total amount of fizzy juice. This is useful in making sense of the amount of each ingredient used because we have a constraint of no more than 36 ounces of juice altogether.)
• What does Clare's equation calculate? When would this be useful? (Clare's equation calculates the amount of sparkling water used in the fizzy juice. This is useful for identifying the exact amount of each ingredient used because if know the total, 36 ounces, and the exact amount of grape juice used, then we know the amount left represents the amount of sparkling water in the fizzy juice.)
• Continue for other scenarios and equations. 

The purpose of this activity is for students to identify why equations are not equivalent. In an associated Algebra 1 lesson, students rewrite equations to express one variable in terms of the other. The work in this activity prepares students to be successful in the associated Algebra 1 lesson by allowing them to gain a deeper understanding of what makes two equations equivalent and to practice making appropriate moves to maintain the balance in an equation.

The purpose of this discussion is to reinforce the concept that different equations can represent the same relationship.

Display the school fundraiser situation from the practice activity for all to see.

Here are discussion questions:

• What are some of the true equations that describe the school fundraiser situation? (\(2w + 1.5p = 200, w = 100 - 0.75p, p = \frac{200 - 2w}{1.5}\))
• What can be seen in each equation students share, and why might these equations be useful? ( \(2w + 1.5p = 200\) shows how the water and pretzel revenue should equal $200. The other two equations help if you want to know how many waters you need to sell or how many pretzels you need to sell, if you know the other amount.)

Explain that because all these equations describe the same relationship, just in different ways, they are equivalent. Discuss how equivalent equations can be found by thinking about the situation in different ways, as well as by applying acceptable moves for solving equations, such as adding the same amount to each side or multiplying each side by the same amount.