This Math Talk focuses on solving equations with negative numbers. It encourages students to think about valid moves and to rely on what they know about creating equivalent equations to mentally solve problems. The understanding elicited here will be helpful later in the lesson when students solve equations in this lesson.
In describing their strategies, students need to be precise in their word choice and use of language (MP6).
Launch
Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:
Give students quiet think time, and ask them to give a signal when they have an answer and a strategy.
Invite students to share their strategies, and record and display their responses for all to see.
Use the questions in the activity synthesis to involve more students in the conversation before moving to the next problem.
Keep all previous problems and work displayed throughout the talk.
Action and Expression: Internalize Executive Functions. To support working memory, provide students with sticky notes or mini whiteboards. Supports accessibility for: Memory, Organization
Activity
None
Solve each equation mentally.
Student Response
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Building on Student Thinking
Activity Synthesis
To involve more students in the conversation, consider asking as the students share their ideas:
“Who can restate ’s reasoning in a different way?”
“Did anyone use 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?”
“What connections to previous problems do you see?”
Some students may reason about the value of using logic. For example, in , the must be -3 since . Other students may reason about the value of by changing the value of each side of the equation equally by, for example, dividing each side of by -3 to get the result . Both of these strategies should be highlighted during the discussion where possible.
MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, “First, I because . . . .” or “I noticed so I . . . .” Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class. Advances: Speaking, Representing
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 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 fewer factors on each side.
Launch
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 the equation by stating to their partner the step that 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. Remind 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 that they are combining two terms on one side of the equation, which maintains the equality because the value of that side does not change, only the appearance changes.
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 that they have checked their solutions. Then challenge them to find a new solution to one of the problems that uses fewer steps than their first solution used. Conclude with a whole-class discussion.
If time is a concern, give each group 2 cards rather than all 4, and have them do only the trading steps portion of the activity. In that case, make sure that all 4 cards are distributed throughout the class. Give 6–7 minutes for groups to complete their problems. Make sure that 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.
Activity
None
Here are 4 problems. Select 2 to solve with your partner by taking turns describing a move, then writing an equivalent equation. For the other 2 problems, you and your partner should each solve 1 of the problems on your own, and then trade to check your answers.
Activity Synthesis
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 from the one you expected them to? Describe it.”
“For the problem with on the left side, could you start by halving the value of each side? Why might you want to do this?” (Yes. It keeps the numbers smaller and easier to work with.)
“How can you check that your solutions are correct?” (Substitute the final value for the variable into the original equation, and check that the value of each side of the equation is the same.)
In this activity, students select a number and follow a series of instructions to change their number into a surprising result. Students then follow the same series of instructions for a variable to reveal how the result was not surprising at all, but a natural consequence of the instructions. This helps illustrate valid moves that can be done to an equation.
Monitor for students who combine like terms as they go as well as those who do not.
This activity uses the Collect and Display math language routine to advance conversing and reading as students clarify, build on, or make connections to mathematical language.
This activity is optional in this course. It introduces the idea of using expressions to make sense of number puzzles, which can help prepare students for the following activity.
Launch
Begin the activity by telling students not to communicate with one another until instructed to do so. This includes talking or showing each other what is on their papers.
Tell students to select a number and write it on the first line. Then follow the instructions to write the correct number on the next line. Continue filling in the blanks until all have been filled.
After all students have had a chance to complete all of the instructions, tell students to compare the number on the last blank with one another.
Ask students, “What do you notice about the final answers?” (Everyone got 3, regardless of their starting number.)
Demonstrate how to begin the next part of the activity. Write "" on the first line, then draw an arrow on the right side of the equation and label it “multiply by 2,” and then write "" on the next line.
To see why this works, tell students to write "" on the right side of their paper next to their original number. They should continue writing equivalent equations using all of the instructions and blanks for the equation. For the last move, they should subtract (their original number) on the right side to show what happens for whatever number might be chosen.
Select students who combine like terms as they follow the instructions, as well as those who do not. Ask them to share later.
Use Collect and Display to direct attention to words collected and displayed from an earlier activity. Ask students to suggest ways to update the display: “Are there any new words or phrases that you would like to add?” “Is there any language you would like to revise or remove?” Encourage students to use the display as a reference.
Activity
None
Student Response
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Building on Student Thinking
If students divide only 1 term by 2 instead of the whole expression, consider asking:
“How can you check that each of your equations are equivalent?”
“How could you use substituting a value or to help?”
Activity Synthesis
Invite selected students to share their equations involving . If all students combine like terms, ask them what each equation would look like if they did not. For example, on the third line, write as well as . Ask students, “What are the benefits and drawbacks of writing it each way?” (The first set of expressions show all of the instructions and how they affect , but there are more terms, so an instruction like “divide by 2” requires more work. The second set of expressions hides the instructions, but is easier to understand how to get the number in the blank.)
Tell students that all of these equations are equivalent because these are valid moves done correctly. Ask students what moves they have seen that create equivalent equations. Direct students’ attention to the reference created using Collect and Display. Invite students to borrow language from the display as needed, and update the reference to include additional phrases as they respond. Update the display to create a semi-permanent display of these valid moves to remain available in the classroom until the end of the unit.
Use the distributive property. ( is equivalent to )
Combine like terms. ( is equivalent to )
Add the same value on each side.
Subtract the same value on each side.
Multiply each side by the same non-zero value.
Divide each side by the same non-zero value.
8.4
Activity
Standards Alignment
Building On
Addressing
8.EE.C
Analyze and solve linear equations and pairs of simultaneous linear equations.
In this activity, students investigate a number puzzle. After the puzzle is demonstrated, students are asked to figure 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.
Monitor for student who use these strategies to write their expressions:
Rewrite the expression in a simpler way after each instruction.
Wait until the end to rewrite the expression.
This activity uses the Compare and Connect math language routine to advance representing and conversing as students use mathematically precise language in discussion.
Consider using the previous, optional, activity to help students see how algebraic expressions can help make sense of number puzzles like the one in this activity.
Launch
Tell students to close their books or devices. Ask them to choose a positive number, but not to share the number with anyone else. Tell them that 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 know how you are able to figure out their number so quickly. 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.
When the class agrees that you are able to figure out their original numbers by adding 6 to their final numbers, 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 of quiet work time for students to write their explanations, and follow that with a whole-class discussion. Select work from students with different strategies, such as those described in the Activity Narrative, to share later.
Representation: Internalize Comprehension. Provide a blank two-column table for students to keep track of the moves and their numbers and variables. Supports accessibility for: Organization, Attention
Activity
None
Tyler says he invented a number puzzle. He asks Clare to pick a number, and then asks her to:
Triple the number.
Subtract 7.
Double the result.
Subtract 22.
Divide by 6.
Clare says she now has -3. Tyler says her original number must have been a 3. How did Tyler know that?
Follow the same instructions starting with instead of a number. Explain or show your reasoning for why the last expression means that the person started with a number 6 greater than they ended with.
Activity Synthesis
The goal of this discussion is for students to discuss the different approaches to thinking through the number puzzle.
Display 2–3 approaches/representations from previously selected students for all to see. Use Compare and Connect to help students compare, contrast, and connect the different approaches. Here are some questions for discussion:
“What do the approaches have in common? How are they different?” (They are equivalent expressions, but one shows all of the instructions while the other is simpler to understand and some of the operations are done.)
“Why do the different approaches lead to the same expression?” (They are equivalent because you can use the distributive property and combine like terms as you go or all at once at the end.)
Lesson Synthesis
Give students 2–3 minutes to think about all the equations they solved in today’s lesson and to describe to a partner things that they need to be careful about in the future.
Explain to students that some mathematicians describe their moves by writing them next to the new equivalent equation. This makes it easier to both identify wrong moves and to make moves visible to everyone. For example, this pair of equivalent expressions could be written either of these ways.
Student Lesson Summary
When we have an equation in one variable, there are many different ways to solve it. We generally want to make moves that get us closer to an equation that clearly shows the value that makes the equation true.
For example, or show that 5 and are solutions. Because there are many ways to do this, it helps to choose moves that leave fewer terms or factors.
If we have an equation like , adding -5 to each side will leave us with fewer terms. The equation then becomes .
Dividing each side of this equation by 3 results in the equivalent equation , which is the solution.
Or, if we have an equation like , dividing each side by 4 will leave us with fewer factors on the left. The equation then becomes .
Here is a list of valid moves that can help create equivalent equations that move toward a solution:
Use the distributive property so that all the expressions no longer have parentheses.
Collect like terms on each side of the equation.
Add or subtract an expression on each side so that there is a variable on just one side.
Add or subtract an expression on each side so that there is just a number on the side without the variable.
Multiply or divide by a number on each side so that the variable on one side of the equation has a coefficient of 1.
For example, suppose we want to solve .
From lots of experience, we learn when to use different valid moves that help solve an equation.
Standards Alignment
Building On
Addressing
8.EE.C
Analyze and solve linear equations and pairs of simultaneous linear equations.
