Lesson 12

The purpose of this warm-up is for students to revisit ideas they learned in the previous lesson about balanced hangers:

• You can add or subtract the same thing on each side and the hanger stays in balance.
• You can divide each side by the same number and the hanger stays in balance.

During the activity synthesis when discussing the moves from Hanger C to Hanger D, ask students what would change if they knew the blue square weighed 3 grams. (We would subtract 3 from each side instead of \(x\), but it’s still taking away one blue square from each side, so nothing really changes other than how we say it.) Select students to share their responses. An important takeaway for students is that they can talk about the moves needed to go from Hanger A to B to C to D without ever knowing the actual weight of each of the shapes. The blue square could be 3 grams or 50 grams, but the moves would all still be the same because each blue square represents the same weight.

Give students 2 minutes of quiet work time followed by a whole-class discussion. 

Some students may think a variable stands for more than one object. Tell these students that a variable only stands for one object, as it also only represents one number. 

Ask students to explain how they decided on the matching equation. As students discuss the final questions, highlight responses that emphasize the same objects being removed from each side creates the next figure in line. The exception to this is the move from Hanger B to Hanger C, where the number of objects on each side is halved. Ask students to explain why this is an okay move, even though different objects are being removed from each side (1 square and 3 triangles on the left, 2 squares and 1 triangle on the right). Refer to MLR 2 (Collect and Display).

In this activity, students match a card with two equations to another card describing the move that turns the first equation into the second. The goal is to help students think about equations the same way they have been thinking about hangers: objects where equality is maintained so long as the same move is made on each side. Additionally, this is the first activity where students encounter equation moves involving negative numbers, which is not possible when using hangers.

Review with students what we know about equations based on reasoning about hangers:

• We can add the same quantity to each side, and the equation is still true (the hanger is still in balance).
• We can subtract the same quantity from each side, and the equation is still true.
• We can double or triple or halve or third the things that appear on each side, and the equation is still true. More generally, we can multiply the number of things on each side by the same number.

Tell students that hanger diagrams are really only useful for reasoning about positive numbers, but the processes above also work for negative numbers. Negative numbers are just numbers, and they have to follow the same rules as positive numbers. In fact, if we allow negative numbers into the mix, we can express any maneuver with one of two types of moves:

• Add the same thing to each side. (The “thing” could be negative.)
• Multiply each side by the same thing. (The “thing” could be a fraction less than 1.)

Arrange students in groups of 2. Give each group 12 pre-cut slips from the blackline master. Give 3–4 minutes for partners to match the numbered slips with the lettered slips then 1–2 minutes to trade places with another group and review each other’s work. Ask partners who finish early to write down on a separate sheet of paper what the next move would be for each of the numbered cards if the goal were to solve for \(x\). Follow with a whole-class discussion.

The goal of this discussion is to get students using the language of equations and describing the changes happening on each side when solving. Ask:

• “What is a move you could do to the equation \(7=2x\) on card 1 that would result in an equation of the form \(x=\underline{  }\)? What is another move that would also work?” (Multiply each side by \(\frac12\). Divide each side by 2.)
• “Which numbered card was the most challenging to match?” (Card 2, because it at first I only looked at the \(x\)-terms and thought the move involved a change of \(8x\).)
• “Does anyone have a value for \(x\) that would solve one of the numbered cards? How did you figure it out?” (\(x=2\) is a solution for card 5. I added 3 to each side and then multiplied each side by \(\frac14\).)

End the discussion by inviting groups to share the equations they wrote for card 6 and describe how they match the move "add \(3x\) to each side."

The purpose of this activity is to get students thinking about strategically solving equations by paying attention to their structure. Distribution first versus dividing first is a common point of divergence for students as they start solving.

Identify students who choose different solution paths to solve the last two problems. 

Arrange students in groups of 2. Give students 2 minutes quiet think time for problem 1, then 3–5 minutes partner time to discuss problem 1 and complete the other problems. Follow with a whole-class discussion.

Some students may not distribute or collect like terms before performing the same operation on each side.

Have previously identified groups share the different solution paths they chose for solving the last two questions.

To highlight the different strategies, ask:

• “What are the advantages of choosing to distribute first? To divide first?” (Answers vary. Distributing first eliminates confusion about which terms can be subtracted from each side. Dividing first makes the numbers smaller and easier to mentally calculate.)
• “What makes it easier to distribute versus divide first on the last question?” (Dividing by 4 before distributing will result in non-integer terms, which can be harder to add and subtract mentally.)
• “Is one path more ‘right’ than another?” (No. As long as we follow valid steps, like adding or multiplying the same thing to each side of an equation, the steps are right and will give a correct solution.)