Lesson 6

The purpose of this algebra talk is to elicit strategies and understandings students have for evaluating an expression for a given value of its variable. These understandings help students develop fluency and will be helpful later in this lesson when students will need to be able to evaluate expressions.

Give students a minute to see if they recall that \(x^2\) means \(x \boldcdot x\) (which they learned about in an earlier unit in this course), but if necessary, remind them what this notation means.

Display one problem at a time. Give students 30 seconds of 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 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 ___’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 _____’s strategy?”
• “Do you agree or disagree? Why?”

Throughout this unit students have been matching equations to tape diagrams, matching equations to situations, and solving equations. This lesson shifts the focus to writing the expressions that describe situations with an unknown quantity. Students use operations to calculate quantities and notice repeated patterns in those calculations. They replace a part of the calculation with a letter to represent any possible value and create an expression that represents the situation (MP8). Students learn that they can use these expressions to answer questions about specific values. Monitor for students who use different strategies to answer the second part of each question. For each, select at least one student who uses a less-efficient method like trial-and-error and one student who writes and solves an equation. Note if any student represents a situation with a tape diagram—this can be presented to support reasoning about an unknown quantity using a variable.

Allow students 10 minutes quiet work time followed by a whole-class discussion.

The goal of the discussion is to ensure students see that they can write a mathematical expression to represent a calculation, even if they do not know what one of the numbers is in the calculation. Select students to present who solved the second part of each problem with different strategies. If no student took an approach with equations, demonstrate that \(0.5c=127.50\) can represent the situation and remind students they can find \(c\) with \(\frac{127.5}{0.5}\). Similarly for the second problem, demonstrate that \(59 + d = 64\frac34\) can represent the situation and \(d\) can be found with \(64\frac34 - 59\). Ask students to explain how the equations make use of the expressions they wrote in the tables and why they can write the equations in these ways.

The purpose of this activity is to help students write expressions given a situation, then to solve equations involving the same situation. For the first three questions, the work is still scaffolded by providing numbers to use to calculate before prompting students to write an expression that uses a variable (MP8).

Arrange students in groups of 2. Give students 5–10 minutes of quiet work time and time to share with a partner, followed by a whole-class discussion.

There are multiple quantities in each problem. Students may lose track of what the variable represents.

Students may not see the value in setting up an equation if they can solve it mentally. Later problems are less likely to be solved mentally, so encourage students to write and solve an equation each time.

We can often express one quantity in terms of another unknown quantity because we know a relationship between them. Sometimes we also know a value of this expression, and we can use that understanding to write an equation and solve for the unknown quantity. Consider asking some of the following questions to guide the discussion:

• “What facts describing each situation helped you to write the expression?”
• “How did you use the expression to write an equation? Why were you able to set the quantities on each side of the equation equal to each other? Can you give an example of this?” (They represent the same quantity in the story so they have to be equal; for example, the number of tomatoes in Priya’s garden is both 6 and \(\frac25\) of the number of vegetables, \(x\), in her garden.)
• “What strategies did you use to solve each equation?”
• “How did you check that your solution was correct?” (The best way to check when there is a context is to go back to the original situation and see if the solution makes the statements true. Checking in the equation has the problem that you won’t catch if the equation you wrote does not correctly represent the situation.)