Lesson 8

In this algebra talk, students recall how to solve equations by considering what number can be substituted for the variable to make the equation true. (Note: \(x^2=49\) of course has another solution if we allow solutions to be negative, but students haven't studied negative numbers yet, and don't study operations with negative numbers until grade 7, so it is unlikely to come up.)

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?”

Students use diagrams to show that expressions can be equivalent or expressions can be equal for only one value of their variable. Working through these tape diagrams with small whole numbers, where students can count grids and use lengths to check their results, allows students to begin to generalize about equal and equivalent expressions. 

Provide access to graph paper. Ask students to draw diagrams that show:

\(2+3=3+2\)

\(2+3\) does not equal \(2 \boldcdot 3\)

We can tell that \(2+3\) and \(3+2\) are equal because the length of the diagrams represent the value of each expression, and the diagrams are the same length. We can tell that \( 2\boldcdot 3\) is not equal to these because this value is represented by the length of its diagram, and it's not the same length as the others. \(2+3\) and \(3+2\) are examples of expressions that are not identical, but are equal. Another example students have seen of this phenomenon are fractions like \(\frac12\) and \(\frac36\), which are not identical but equal.

When we start talking about expressions that have letters in them, the language gets more complicated, because expressions can be equal or not equal depending on the value the letter represents. 

Arrange students in groups of 2. Ask students to work independently on each question and then check in with their partner, discussing and resolving any disagreements. Allow 15 minutes to work and share responses with a partner, followed by a whole-class discussion.

For the first sets of diagrams, if we consider \(x+2=3x\), we can see that this is true when \(x\) is 1, but not for the other values of \(x\) that we tried. For the second set of diagrams, if we consider \(x+3 = 3+x\) we can see that this equation is always going to be true no matter what the value of \(x\) is. We call \(x+3\) and \(3+x\) equivalent expressions, because their values are equal no matter what the value of \(x\) is.

In this activity, students apply what they know about the meaning of operations and their properties to understand what is meant by “equivalent expressions.” The focus is more on building that understanding than it is about doing all the types they eventually need to be able to do.

It is expected that students will reason using what they know about operations on numbers and potentially use diagrams. For example they learned earlier this year that something \(\div \frac13\) is equivalent to that same thing \(\boldcdot 3\). They can also reason that they know for example that \(4+4+4=3\boldcdot 4\), so \(a+a+a=3a\).

Allow students 5 minutes of quiet work time, followed by a whole-class discussion.

Invite students to share their pairs and reasoning. Include students who used diagrams.