Lesson 5

The purpose of this warm-up is to elicit students’ strategies for solving an equation for the value of \(x\). The negative integers and location of \(x\) in each equation were purposeful to spark a discussion about operations of integers and a negative coefficient of a variable. 

Tell students to close their books or devices and that they are going to see some equations they are to solve mentally. Display each problem one at a time for all to see. Give students 30 seconds of quiet think time for each equation. Ask students to share their strategies for finding the value of \(x\). Record and display their responses for all to see.

Some students may reason about the value of \(x\) using logic. For example, in \(\text- 3x = 9\), the \(x\) must be -3 since \(-3 \boldcdot \text- 3=9\). Other students may reason about the value of \(x\) by changing the value of each side of the equation equally by, for example, dividing each side of \(\text- 3x = 9\) by -3 to get the result \(x=\text- 3\). Both of these strategies should be highlighted during the discussion where possible.

To involve more students in the conversation, consider asking as the students share their ideas:

• “Can you explain why you chose your strategy?”
• “Can anyone restate ___’s reasoning in a different way?”
• “Did anyone reason about the problem the same way but would explain it differently?”
• “Did anyone reason about the problem in a different way?”
• “Does anyone want to add on to _____’s strategy?”
• “Do you agree or disagree? Why?”

The goal of this activity is for students to build fluency solving equations with variables on each side. Students describe each step in their solution process to a partner and justify how each of their changes maintains the equality of the two expressions (MP3).

Look for groups solving problems in different, but efficient, ways. For example, one group may distribute the \(\frac12\) on the left side in problem 2 while another may multiply each side of the equation by 2 in order to re-write the equation with less factors on each side.

Arrange students in groups of 2. Instruct the class that they will receive 4 cards with problems on them and that their goal is to create a solution to the problems.

For the first two cards they draw, students will alternate solving by stating to their partner the step they plan to do to each side of the equation and why before writing down the step and passing the card. For the final two problem cards, each partner picks one and writes out its solution individually before trading to check each others’ work.

To help students understand how they are expected to solve the first two problems, demonstrate the trading process with a student volunteer and a sample equation. Emphasize that the “why” justification should include how their step maintains the equality of the equation. Use MLR 3 (Clarify, Critique, Correct) by reminding students to push each other to explain how their step guarantees that the equation is still balanced as they are working. For example, a student might say they are combining two terms on one side of the equation, which maintains the equality as the value of that side does not change, only the appearance.

Distribute 4 slips from the blackline master to each group. Give time for groups to complete the problems, leaving at least 5 minutes for a whole-class discussion. If any groups finish early, make sure they have checked their solutions and then challenge them to try and find a new solution to one of the problems that uses less steps than their first solution. Conclude with a whole-class discussion.

If time is a concern, give each group 2 cards rather than all 4 and have them only doing the trading steps portion of the activity, but make sure that all 4 cards are distributed throughout the class. Give 6–7 minutes for groups to complete their problems. Make sure each problem is discussed in a final whole-group discussion. Alternatively, extend the activity by selecting more problems for students to solve with their partners.

Depending on your observations as students worked, you may wish to begin the discussion with a few common errors and ask students to explain why they are errors. For example, write out a solution to problem 2 where the second line has \(3.5x-6\) instead of \(3.5x-3\) and then ask students to find the error.

The goal of this discussion is for the class to see different, successful ways of solving the same equation. Record and display the student thinking that emerges during the discussion to help the class follow what is being said. To highlight some of the differences in solution paths, ask:

• “Did your partner ever make a move different than the one you expected them to? Describe it.”
• “For problem 4, could you start by halving the value of each side? Why might you want to do this?”
• “What’s an arithmetic error you made but then caught when you checked your work?”

In this activity, students investigate a number puzzle. After the puzzle is demonstrated, students are tasked with figuring out how it works and encouraged to create an algebraic representation of the puzzle. The goal of this activity is to build student fluency working with equations with complex structure. This activity also looks ahead to the future work on functions where students will revisit some of these ideas and learn the language of inputs and outputs. More immediately, this activity points to the study of equations that do not have a single answer, which students will learn about in more depth later in this unit.

While students work, identify those using expressions with different structures for their representation of the number puzzle to share during the whole-class discussion. For example, some students may use \(\frac{(3x-7)\boldcdot 2 -22}{6}\) while others write \(\frac16((3x-7)\boldcdot 2 -22)\) and connecting these together with the outcome of the number puzzle, namely \(x-6\), is the focus of the whole-class discussion.

Tell students to close their books or devices and choose a number (but not share the number with anyone else). Tell them they will perform a sequence of operations on their number and then tell you their final answer. Say each step of Tyler’s number puzzle, giving students time to calculate their new number after each step. Select 5–6 students to share their final number, and after each, tell them their original number as quickly as you can.

Pause here and ask students if they can tell how you are able to figure out their number so fast. If no students notice that each number you say is always 6 more than the number given at the end of the steps, you may wish to record and display the pairs of numbers for all students to see or call on more students so that everyone can hear more pairs of numbers.

Once the class agrees that you are able to figure out their original numbers by adding 6 to their final number, tell them that the number puzzle is really Tyler’s and that their task is to figure out how it works. Tell students to open their books or devices. 

Give 3–4 minutes quiet work time for students to write their explanations, followed by a whole-class discussion.

Select previously identified students to share their representations with the class. Record and display the different ways Tyler’s number puzzle can be written as an expression. To highlight the connection between the different expressions, ask:

• “What do all these expressions have in common that make the number puzzle work?” (All of them are equivalent to \(x-6\).)
• “How would you justify that, for example, \(\frac16 ( (3x - 7) \boldcdot 2 - 22) = x - 6\)?” (I would simplify the left side of the equation until it was \(x-6\).)
• “What does it mean if we have an equation that says \(x - 6 = x - 6\)?” (The two sides of this equation are always the same.)