This Warm-up prompts students to compare four equations. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another.
During the discussion, listen for strategies for evaluating expressions with rational numbers that will be helpful in the work of this lesson.
Launch
Arrange students in groups of 2–4. Display the equations for all to see. Give students 1 minute of quiet think time, and ask them to indicate when they have noticed three equations that go together and can explain why. Next, tell students to share their response with their group and then together find as many sets of three as they can.
Activity
None
Which three go together? Why do they go together?
A
B
C
D
Student Response
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Building on Student Thinking
Activity Synthesis
Invite each group to share one reason why a particular set of three go together. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which three go together, attend to students’ explanations, and ensure the reasons given are correct.
During the discussion, prompt students to explain the meaning of any terminology they use, especially related to strategies for adding, subtracting, or multiplying signed numbers.
8.2
Activity
Standards Alignment
Building On
Addressing
7.EE.B.4.a
Solve word problems leading to equations of the form and , where , , and are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is cm. Its length is cm. What is its width?
In this activity, students solve four equations. Each solution can be found by asking “What value would make the equation true?” However, these equations also present an opportunity to demonstrate that “doing the same thing to each side” still works when there are negative numbers involved. Monitor for students who reason about what value would make the equation true and those who reason by doing the same thing to each side.
Since students have not been shown how to solve equations involving negative numbers, they have to do a notable amount of sense-making to complete the task (MP1).
Launch
Give 5–10 minutes of quiet work time followed by a whole-class discussion.
Activity
None
Solve each equation. Be prepared to explain your reasoning.
Student Response
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Building on Student Thinking
Some students may need some additional support remembering and applying strategies for performing operations on signed numbers. Draw their attention to any anchor charts or notes that are available from the previous unit.
Activity Synthesis
The goal of this discussion is to connect equation-solving moves with operations on signed numbers. Invite students to share their reasoning for each equation. Consider asking:
“Did anyone solve the equation the same way but would explain it differently?”
“Did anyone solve the problem in a different way?”
“Do you agree or disagree? Why?”
Draw students’ attention to the connection between the approaches of “finding the value that makes the equation true” and “doing the same to each side.”
MLR7 Compare and Connect. Lead a discussion comparing, contrasting, and connecting the different representations or strategies. Ask, “What do the approaches have in common? How are they different?” or “Why do the different approaches lead to the same outcome?” Advances: Representing, Conversing
8.3
Activity
Standards Alignment
Building On
Addressing
Building Toward
7.EE.B.4.a
Solve word problems leading to equations of the form and , where , , and are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is cm. Its length is cm. What is its width?
In this activity, students create an equation for their partner by applying a sequence of moves that create an equivalent equation. Their partner tries to guess which moves were made.
To describe moves that could create an equation with the same solutions, students need to attend to precision in the language they use (MP6).
The Building Student Thinking section mentions an anchor chart from a previous unit about performing operations on signed numbers. In this course, the referenced anchor chart was not created.
Launch
Display this sequence of moves for all to see. Ask students to think about how they know each equation has the same solution as the previous equation. As students share their reasons, write a valid reason next to each new equation. Here’s an example of what the reasons might look like:
Subtract 3 from each side.
Swap the two sides of the equation.
Multiply each side by -100.
Swap the factors -100 and ___. (Apply the commutative property of multiplication.)
Apply the distributive property.
Arrange students in groups of 2. Explain that they will start with the equation and use different combinations of things on this list to create new equations with the same solution. Give them 5–10 minutes to complete the task with their partner.
MLR8 Discussion Supports. Display sentence frames to support small-group discussion. Examples: “It looks like . . . .” “Another way to look at it is . . . .” “I know ___ because . . . .” Advances: Speaking, Representing
Engagement: Develop Effort and Persistence. Chunk this task into more manageable parts. Consider chunking problem 1 by each step. Check in with students to provide feedback and encouragement after each chunk. Be sure that the explanation of each step is mathematically sound. Supports accessibility for: Attention, Social-Emotional Functioning
Activity
None
Keep your work secret from your partner. Start with the equation . Do the same thing to each side at least three times to create an equation that has the same solution as the starting equation. Write the equation you ended up with on a slip of paper, and trade equations with your partner.
See if you can figure out what steps your partner used to transform into the equation they gave you. When you think you know, check with your partner to see if you are right.
Student Response
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Building on Student Thinking
Some students may need some additional support remembering and applying strategies for performing operations on signed numbers. Draw their attention to any anchor charts or notes that are available from the previous unit.
Activity Synthesis
Much of the discussion will take place in small groups. Questions for discussion:
“Did you have any disagreements, and how did you resolve them?”
“Did anything surprise you? Explain.”
“What are some important things to keep in mind when working with negative numbers?”
“Why can you do an operation on both sides of an equation without changing the solution?”
Lesson Synthesis
Ask students to think of one or two important things they learned in this lesson, and share them with a partner. Points to highlight include:
Doing the same thing to each side of an equation still keeps the equation balanced, even when there are negative numbers.
Doing the same thing to each side of an equation still keeps the equation balanced, even when the moves don’t get you closer to a solution.
Student Lesson Summary
To find a solution to some equations, we can just think about what value in place of the variable would make the equation true. Sometimes we also draw diagrams to reason about the solution. Using balanced hanger diagrams helped us understand that doing the same thing to each side of an equation keeps the equation true. So, another way to solve an equation is to perform the same operation on each side in order to get the variable alone on one side.
Doing the same thing to each side of an equation also works when an equation involves negative numbers. Here are some examples of equations that have negative numbers and steps we could take to solve them.
Example:
Example:
Doing the same thing to each side maintains equality even if it is not helpful for finding the solution. For example, we could take the equation and add -2 to each side:
If is true then is also true, but we are no closer to a solution than we were before adding -2. We can use moves that maintain equality to make new equations that all have the same solution. Helpful combinations of moves will eventually lead to an equation like , which gives the solution to the original equation (and every equation we wrote in the process of solving).
Standards Alignment
Building On
7.NS.A
Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
