The purpose of this Warm-up is to invite students to interpret a comparison situation that can be represented with . Students generate mathematical questions based on the context and consider how they might represent the answer to comparison questions with both numerical and algebraic expressions.
This activity uses the Co-Craft Questions math language routine to advance reading and writing as students make sense of a context and practice generating mathematical questions.
Launch
Tell students to close their books or devices (or to keep them closed). Arrange students in groups of 2. Introduce the context of esports and elicit what students know about esports, including teams, players, and the games played. If needed, explain that the term “esport” is short for “electronic sport,” and it refers to video game competitions involving multiple players (individuals or teams). Then use Co-Craft Questions to further orient students to the context and elicit possible mathematical questions.
Give students 1–2 minutes to write a list of mathematical questions that could be asked about the situation before comparing questions with a partner.
Activity
None
Priya is on her school’s esport team. She scored 473 points in the last match. Noah, Elena, and Clare each scored fewer points than Priya did.
Student Response
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Building on Student Thinking
Activity Synthesis
Invite several partners to share one question with the class and record responses. Ask the class to make comparisons among the shared questions and their own. Ask, “What do these questions have in common? How are they different?” Listen for and amplify language related to the learning goal, such as “the points Priya scored,” “the points (others) scored,” and “how many more (or fewer).”
Tell students that they will take a closer look at the relationship between the students’ scores in the next activity.
7.2
Activity
Standards Alignment
Building On
Addressing
6.EE.2.a
Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation "Subtract y from 5" as 5 - y.
Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.
In this activity, students encounter two situations in which performing the same calculation with different values of one quantity gives the values of another quantity. Students interpret a situation with the support of tables. Through repeated reasoning, students see that they can summarize the calculations they performed with an algebraic expression (MP8) and use these expressions to answer questions about specific values. As students relate quantities in situations to numerical and variable expressions, they practice reasoning quantitatively and abstractly (MP2).
Monitor for students who use different strategies to answer the last question about the number of stickers Noah sold. Here are some likely strategies, listed from more concrete to more abstract:
Guess a number and multiply it by 0.5 to see if it gives 127.5.
Reason about a relationship seen in the table but is not explicit in the expression. (For example, the number of stickers is twice the number of dollars.)
Write and solve an equation.
Launch
Keep students in groups of 2. Tell students that they now have some information about how many fewer points Noah, Elena, and Clare scored compared to Priya and will calculate their scores. Then they will solve another problem about Noah’s sticker sale.
Give students 7–8 minutes of partner work time, followed by a whole-class discussion.
Representation: Internalize Comprehension. Differentiate the degree of difficulty or complexity by beginning with more accessible values. Extend the given table, and begin by exploring values for money collected based on 1, 2, 3, 5, and 10 stickers sold. Draw students’ attention to what changes and what stays the same each time they calculate the money collected. Supports accessibility for: Conceptual Processing, Memory
Activity
None
Priya scored 473 points in an esport match.
The table shows how Noah, Elena, and Clare scored compared to Priya.
Complete the table.
Noah
Elena
Clare
points fewer than Priya's score
101
53
18
score
Another esport player scored fewer points than Priya. Write an expression that can be used to find their score. Explain your reasoning.
Noah made stickers that show his team’s logo. He sells them for \$0.50 per sticker.
Complete the table to show how much money Noah would collect if he sold each number of stickers.
stickers sold
12
183
money collected (dollars)
How many stickers did Noah sell if he collected $127.50? Be prepared to explain your reasoning.
Student Response
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Building on Student Thinking
Activity Synthesis
The goal of the discussion is to make it explicit to students that we can write a mathematical expression to represent a calculation, even if we do not know what one of the numbers is in the calculation.
Invite students to share their expression for finding the score of an esport player who scored points fewer than Priya did, and to explain their reasoning. If no students mentioned that the same calculation—subtracting the difference in points from 473—was done to find the other students’ scores, emphasize this idea. (Consider displaying , , and for all to see.)
Next, ask previously selected groups to share their reasoning for the last question. Sequence the discussion of the strategies in the order listed in the Activity Narrative. If possible, record and display the students’ work for all to see.
Connect the different responses by asking questions, such as:
“How are these strategies alike? How are they different?”
“How does each strategy use the relationship between the number of stickers and the number of dollars?”
If no students wrote and solved the equation (or equivalent), display the equation for all to see. Ask students to explain how this equation represents the question and why and how it can be used to answer the question. Discuss questions such as:
“What does represent?” (the amount of money collected for selling stickers)
“What does represent?” (the amount of money collected for selling stickers)
”How can we find the value of ?” (Divide each side by 0.5.)
During the discussion, emphasize the use of terms, such as “coefficient,” “variable,” and “solution,” to reinforce students’ knowledge of mathematical language.
7.3
Activity
Standards Alignment
Building On
Addressing
6.EE.2.a
Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation "Subtract y from 5" as 5 - y.
Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.
This activity prompts students to write expressions to find unknown values in a situation and to describe the calculation process more generally. Then students use the expression with a variable to write and solve an equation involving the same situation.
The activity is structured in a way that allows students to progress gradually from concrete reasoning toward abstraction and to notice regularity. For the first three questions, students reason about numbers repeatedly before they write an expression that uses a variable (MP8).
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.
Launch
Arrange students in groups of 2. Give students 8–10 minutes of quiet work time and time to share with a partner, followed by a whole-class discussion.
Engagement: Develop Effort and Persistence. Connect a new concept to one with which students have experienced success. For example, invite students to draw a picture or tape diagram to help understand the situation and relationships as an intermediate step before writing an equation. Supports accessibility for: Social-Emotional Functioning, Conceptual Processing
Activity
None
Clare is 5 years older than her cousin.
Suppose Clare’s cousin is 10 years old. How can you find Clare’s age?
Write expressions to show Clare’s age when her cousin is at different ages. Then record the answer.
Cousin is . . .
8 years old
2 years old
years old
expression to show Clare's age
Clare is 12 years old. She said this equation, , can be used to find out how old her cousin is. Do you agree? Explain your reasoning.
Diego has 3 times as many comic books as Han does.
Suppose Han has 10 comic books. How can you find the number of comic books Diego has?
Write expressions that can show how many comic books Diego has for different numbers of comic books that Han has. Then record the answer.
Han has . . .
6 books
17 books
books
expression to show how many books Diego has
Diego has 27 comic books. How many comic books does Han have? Write an equation that represents the question and use it to find the answer.
Two-fifths of the vegetables in Priya’s garden are tomatoes.
Suppose Priya’s garden has 20 vegetables. How can you find the number of tomatoes?
How many tomatoes are in Priya’s garden if it has vegetables?
If Priya’s garden has 6 tomatoes, how many vegetables are there? Write an equation that represents the question and use it to find the answer.
A school paid \$31.25 for each calculator.
If the school bought calculators, how much did they pay?
The school spent \$500 on calculators. How many did the school buy?
Student Response
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Building on Student Thinking
There are multiple quantities in each problem. Students may write expressions that match, but make errors when solving the second part of each problem—with or without writing an equation. Ask students to explain what their expression represents. Then ask what they need to do to answer the last part of each problem. If needed, invite students to draw a tape diagram to represent the problem.
Activity Synthesis
The goal of this discussion is to make sure students see the connection between the expressions they write and the value given in the last part of each problem.
Invite students to share the variable expression they wrote for each situation and the equation in the last part of each question. Ask students how they know that each equation represents the question and can be asked to find the answer. If not mentioned by students, point out that the expression and the given value both represent the same quantity. (Consider displaying the pairs of quantities, as shown.) An equation can be written to describe that equality, and then solved to find the answer.
Clare’s age:
Clare’s age: 12
Diego’s books:
Diego’s books: 27
Tomatoes:
Tomatoes: 8
Cost:
Cost: 500
Then ask students to share how they solved each equation.
Continue to highlight the use of “expression,” “variable,” and “coefficient” as students discuss expressions and equations in context.
Lesson Synthesis
To highlight the key ideas of the lesson, invite students to refer back to the work they did in all activities. Ask them to notice any pattern in the situations and in how they answered questions about the situation. Discuss questions such as:
“What do all the situations in this activity have in common?” (They are about two quantities, how they are related, and how to find one quantity when we know the other.)
“In each situation, we know the relationship of the quantities. When we have the value of one quantity, we are able to calculate the other. What if we don’t have the value of that first quantity? Can we still describe how to find the other?” (Yes) “How?” (Use a variable and write an expression.)
“Here’s another situation: You have \$20 and spend \$4. How much money do you have left?” (\$16) “How much would you have left if you spent \$8 or \$13.50?” (\$12 or \$6.50) “Can you describe how much you would have left if you spent dollars?” (Yes, by writing .)
“Why might it be helpful to write an expression like ?” (It is an efficient way to describe the leftover money for any amount spent.)
“If we know the amount left over, say, \$4.75, how can we use the expression to find out how much was spent?” (Write and think about what value of makes the equation true.)
Student Lesson Summary
Suppose you were born on the same day as your neighbor, but she is 3 years older than you. When you were 1, she was 4. When you were 9, she was 12. When you are 42, she will be 45.
If we let represent your age at any time, your neighbor’s age can be expressed .
your age
1
9
42
neighbor's age
4
12
45
We often use a variable, such as or , as a placeholder for a number in expressions. Variables make it possible to write expressions that represent a calculation even when we don't know all the numbers in the calculation.
How old will you be when your neighbor is 32? We know your neighbor is 32. We also know your neighbor’s age is your age plus 3, or . We can write the equation to represent these relationships. When your neighbor is 32 you will be 29, because is true when is 29.
Standards Alignment
Building On
Addressing
6.EE.2
Write, read, and evaluate expressions in which letters stand for numbers.
