Distinguishing between Two Types of Situations
6.1: Which One Doesn’t Belong: Seeing Structure (5 minutes)
This warm-up prompts students to compare four equations. It encourages students to explain their reasoning, hold mathematical conversations, and gives you the opportunity to hear how they use terminology and talk about characteristics of the equations in comparison to one another. To allow all students to access the activity, each equation has one obvious reason it does not belong. Encourage students to find reasons based on the structure of the equation (MP7). During the discussion, listen for important ideas and terminology.
Arrange students in groups of 2–4. Display the equations for all to see. Ask students to indicate when they have noticed one that does not belong and can explain why. Give students 1 minute of quiet think time and then time to share their thinking with their group. In their groups, tell each student to share their reasoning about why a particular question does not belong, and together, find at least one reason each question doesn’t belong.
Which equation doesn’t belong?
\(4(x + 3) = 9\)
\(4 \boldcdot x + 12 = 9\)
\(4 + 3x = 9\)
\(9 = 12 + 4x\)
Ask each group to share one reason why a particular equation does not belong. Record and display the responses for all to see. After each response, ask the class whether they agree or disagree. Since there is no single correct answer to the question asking which one does not belong, attend to students’ explanations and ensure the reasons given are correct. During the discussion, ask students to explain the meaning of any terminology they use, such as coefficient or solution. Also, press students on unsubstantiated claims.
6.2: Card Sort: Categories of Equations (15 minutes)
The goal of this activity is for students to notice the structure of equations. Any way of sorting is fine, but the discussion should land on explaining how equations involving an expression like \(p(x+q)\) are different from ones that have an expression like \(px+q\). Monitor for different ways groups choose to categorize the equations, but especially for categories that distinguish between these two types of expressions. As students work, encourage them to refine their descriptions of equations using more precise language and mathematical terms (MP6).
Arrange students in groups of 2. Tell them that in this activity, they will sort some cards into categories of their choosing. When they sort the equations, they should work with their partner to come up with categories, and then take turns sorting each equation into one of their categories, explaining why they are doing so. If necessary, demonstrate this protocol before students start working.
Distribute one set of cards to each group of students. Give students 5 minutes to work with their partner, followed by a whole-class discussion.
Select groups of students to share their categories and how they sorted their equations. You can choose as many different types of categories as time allows, but ensure that one set of categories distinguishes between equations that involved expressions of the form \(px+q\) vs \(p(x+q)\). Attend to the language that students use to describe their categories and equations, giving them opportunities to describe their equations more precisely. Highlight the use of terms like coefficient, sum, product, variable, and solution.
Design Principle(s): Support sense-making; Maximize meta-awareness
6.3: Even More Situations, Diagrams, and Equations (15 minutes)
This activity is an opportunity to put together the learning of the past several lessons: correspondences between tape diagrams, equations, and stories, and using representations to reason about a solution. The focus of this activity is still contrasting the two main types of equations that students encounter in this unit.
Keep students in the same groups. 5 minutes of quiet work time followed by sharing with a partner and a whole-class discussion.
Supports accessibility for: Language; Conceptual processing
Design Principle(s): Support sense-making; Maximize meta-awareness
Story 1: Lin had 90 flyers to hang up around the school. She gave 12 flyers to each of three volunteers. Then she took the remaining flyers and divided them up equally between the three volunteers.
Story 2: Lin had 90 flyers to hang up around the school. After giving the same number of flyers to each of three volunteers, she had 12 left to hang up by herself.
- Which diagram goes with which story? Be prepared to explain your reasoning.
- In each diagram, what part of the story does the variable represent?
- Write an equation corresponding to each story. If you get stuck, use the diagram.
- Find the value of the variable in the story.
Are you ready for more?
A tutor is starting a business. In the first year, they start with 5 clients and charge $10 per week for an hour of tutoring with each client. For each year following, they double the number of clients and the number of hours each week. Each new client will be charged 150% of the charges of the clients from the previous year.
- Organize the weekly earnings for each year in a table.
- Assuming a full-time week is 40 hours per week, how many years will it take to reach full time and how many new clients will be taken on that year?
- After reaching full time, what is the tutor’s annual salary if they take 2 weeks of vacation?
- Is there another business model you’d recommend for the tutor? Explain your reasoning.
For each story, select 1 or more groups to present the matching diagram, their equation, and their solution method. Possible questions to ask:
- “How were the diagrams alike? How were they different?” (They have the same numbers and a letter. One has 3 equal groups and an extra bit, the other just has 3 equal groups, but each group is a sum.)
- “How were the stories alike? How were they different?” (They were both about distributing 90 flyers. In one story, Lin makes a series of moves to each volunteer. In the other story, she gives each volunteer the same amount, but then there are some left over.)
- “What parts of the story made you think that one diagram represented it?”
- “Explain how you reasoned about the story, diagram, or equation to find the value of the variable.”
Display the two equations from the last activity for all to see:
Tell students, “These equations have lots of things in common. They each have a 3, a letter, a 12, a 90, an equal sign, multiplication, and addition. Explain how these equations are different.” Ask students to think about this question quietly for a moment and share with a partner, then ask a few students to share with the whole class.
Highlight any responses that speak in general terms about the structure of the equations. For example, one equation is the sum of a product and a number and the other is the product of a number and a sum. Alternatively, if we evaluated one expression for a value of the variable, we would multiply it by 3 first and then add 12. For the other, we would add 12 first and then multiply by 3. One has three equal groups and an extra bit, and the other just has 3 equal groups, but the groups are each the result of adding 12 to an unknown.
6.4: Cool-down - After School Tutoring (5 minutes)
Student Lesson Summary
In this unit, we encounter two main types of situations that can be represented with an equation. Here is an example of each type:
After adding 8 students to each of 6 same-sized teams, there were 72 students altogether.
After adding an 8-pound box of tennis rackets to a crate with 6 identical boxes of ping pong paddles, the crate weighed 72 pounds.
The first situation has all equal parts, since additions are made to each team. An equation that represents this situation is \(6(x+8)=72\), where \(x\) represents the original number of students on each team. Eight students were added to each group, there are 6 groups, and there are a total of 72 students.
In the second situation, there are 6 equal parts added to one other part. An equation that represents this situation is \(6x+8=72\), where \(x\) represents the weight of a box of ping pong paddles, there are 6 boxes of ping pong paddles, there is an additional box that weighs 8 pounds, and the crate weighs 72 pounds altogether.
In the first situation, there were 6 equal groups, and 8 students added to each group. \(6(x+8)=72\).
In the second situation, there were 6 equal groups, but 8 more pounds in addition to that. \(6x+8=72\).