Lesson 6

Write Expressions Where Letters Stand for Numbers

6.1: Algebra Talk: When $x$ is 6 (5 minutes)

Warm-up

The purpose of this algebra talk is to elicit strategies and understandings students have for evaluating an expression for a given value of its variable. These understandings help students develop fluency and will be helpful later in this lesson when students will need to be able to evaluate expressions.

Launch

Give students a minute to see if they recall that \(x^2\) means \(x \boldcdot x\) (which they learned about in an earlier unit in this course), but if necessary, remind them what this notation means.

Display one problem at a time. Give students 30 seconds of quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

Representation: Internalize Comprehension. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory; Organization

Student Facing

If \(x\) is 6, what is:

\(x + 4\)

\(7 - x\)

\(x^2\)

\(\frac13 x\)

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

  • “Who can restate ___’s reasoning in a different way?”
  • “Did anyone have 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?”
Speaking: 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.
Design Principle(s): Optimize output (for explanation)

6.2: Lemonade Sales and Heights (15 minutes)

Activity

Throughout this unit students have been matching equations to tape diagrams, matching equations to situations, and solving equations. This lesson shifts the focus to writing the expressions that describe situations with an unknown quantity. Students use operations to calculate quantities and notice repeated patterns in those calculations. They replace a part of the calculation with a letter to represent any possible value and create an expression that represents the situation (MP8). Students learn that they can use these expressions to answer questions about specific values. Monitor for students who use different strategies to answer the second part of each question. For each, select at least one student who uses a less-efficient method like trial-and-error and one student who writes and solves an equation. Note if any student represents a situation with a tape diagram—this can be presented to support reasoning about an unknown quantity using a variable.

Launch

Allow students 10 minutes quiet 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 cups of lemonade sold. Draw students’ attention to what changes and what stays the same each time they calculate the money collected.
Supports accessibility for: Conceptual processing
Conversing, Representing, Writing: MLR2 Collect and Display. During small-group discussion, listen for and collect the vocabulary and phrases students use to describe how to find the values of the table and how the expression represents the situation (e.g., “the number of cups is twice the number of dollars”). Make connections between how similar ideas are communicated and represented in different ways (e.g., “How do you see ‘twice’ in the tape diagrams and expressions?”). Remind students to borrow language from the display as needed. This will help students to use academic mathematical language during paired and group discussions when writing expressions representing situations with an unknown quantity.
Design Principle(s): Maximize meta-awareness

Student Facing

  1. Lin set up a lemonade stand. She sells the lemonade for \$0.50 per cup.

    1. Complete the table to show how much money she would collect if she sold each number of cups.

      lemonade sold (number of cups) 12 183 \(c\)
      money collected (dollars)
    2. How many cups did she sell if she collected \$127.50? Be prepared to explain your reasoning.
  2. Elena is 59 inches tall. Some other people are taller than Elena. 

    1. Complete the table to show the height of each person.

      person Andre Lin Noah
      how much taller than Elena (inches) 4 \(6 \frac12\) \(d\)
      person's height (inches)
    2. If Noah is \(64 \frac34\) inches tall, how much taller is he than Elena?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

The goal of the discussion is to ensure students see that they can write a mathematical expression to represent a calculation, even if they do not know what one of the numbers is in the calculation. Select students to present who solved the second part of each problem with different strategies. If no student took an approach with equations, demonstrate that \(0.5c=127.50\) can represent the situation and remind students they can find \(c\) with \(\frac{127.5}{0.5}\). Similarly for the second problem, demonstrate that \(59 + d = 64\frac34\) can represent the situation and \(d\) can be found with \(64\frac34 - 59\). Ask students to explain how the equations make use of the expressions they wrote in the tables and why they can write the equations in these ways.

6.3: Building Expressions (15 minutes)

Activity

The purpose of this activity is to help students write expressions given a situation, then to solve equations involving the same situation. For the first three questions, the work is still scaffolded by providing numbers to use to calculate before prompting students to write an expression that uses a variable (MP8).

Launch

Arrange students in groups of 2. Give students 5–10 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 as an intermediate step before writing an equation.
Supports accessibility for: Social-emotional skills; Conceptual processing

Student Facing

  1. Clare is 5 years older than her cousin.

    1. How old would Clare be if her cousin is:

      10 years old?

      2 years old?

      \(x\) years old?

    2. Clare is 12 years old. How old is Clare’s cousin?
  2. Diego has 3 times as many comic books as Han.

    1. How many comic books does Diego have if Han has:

      6 comic books?

      \(n\) books?

    2. Diego has 27 comic books. How many comic books does Han have?
  3. Two fifths of the vegetables in Priya’s garden are tomatoes.

    1. How many tomatoes are there if Priya’s garden has:

      20 vegetables?

      \(x\) vegetables?

    2. Priya’s garden has 6 tomatoes. How many total vegetables are there?
  4. A school paid \$31.25 for each calculator.

    1. If the school bought \(x\) calculators, how much did they pay?
    2. The school spent \$500 on calculators. How many did the school buy?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

Kiran, Mai, Jada, and Tyler went to their school carnival. They all won chips that they could exchange for prizes. Kiran won \(\frac23\) as many chips as Jada. Mai won 4 times as many chips as Kiran. Tyler won half as many chips as Mai.

  1. Write an expression for the number of chips Tyler won. You should only use one variable: \(J\), which stands for the number of chips Jada won.
  2. If Jada won 42 chips, how many chips did Tyler, Kiran, and Mai each win?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

There are multiple quantities in each problem. Students may lose track of what the variable represents.

Students may not see the value in setting up an equation if they can solve it mentally. Later problems are less likely to be solved mentally, so encourage students to write and solve an equation each time.

Activity Synthesis

We can often express one quantity in terms of another unknown quantity because we know a relationship between them. Sometimes we also know a value of this expression, and we can use that understanding to write an equation and solve for the unknown quantity. Consider asking some of the following questions to guide the discussion:

  • “What facts describing each situation helped you to write the expression?”
  • “How did you use the expression to write an equation? Why were you able to set the quantities on each side of the equation equal to each other? Can you give an example of this?” (They represent the same quantity in the story so they have to be equal; for example, the number of tomatoes in Priya’s garden is both 6 and \(\frac25\) of the number of vegetables, \(x\), in her garden.)
  • “What strategies did you use to solve each equation?”
  • “How did you check that your solution was correct?” (The best way to check when there is a context is to go back to the original situation and see if the solution makes the statements true. Checking in the equation has the problem that you won’t catch if the equation you wrote does not correctly represent the situation.)
Listening, Conversing: MLR8 Discussion Supports. Support whole-class discussion by displaying and inviting students to use these sentence frames: “To solve each equation I _____ because _____. ” or “To check my work is correct, I can _____ because _____.” As students share, encourage other students revoice or press for more explanation by asking, “So what I heard you say is ____” or “Can you tell me more about _____?"
Design Principle(s): Cultivate conversation; Optimize output

Lesson Synthesis

Lesson Synthesis

Keep students in the same groups. One partner makes up a story, similar to the situations they saw in the lesson, where they describe a relationship between two quantities. The second student assigns a value to one quantity and the other is unknown. As an example, tell students that you have half as many books as your friend, and you have 130 books. Your friend's number of books is the unknown quantity, let’s call it \(b\). You can then write the expression \(\frac12 b\) to represent your number of books, and the equation \(\frac12 b=130\) to describe the situation. \(b\) can then be found with \(\frac{130}{\frac12}\). Writing their own stories helps students reason about the meaning and structure of expressions and equations and the situations they represent.

6.4: Cool-down - Crazy Eights (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

Suppose you share a birthday with a 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 \(a\) represent your age at any time, your neighbor’s age can be expressed \(a+3\)

your age  1  9 42 \(a\)
neighbor's age 4 12 45 \(a+3\)

We often use a letter such as \(x\) or \(a\) as a placeholder for a number in expressions. These are called variables (just like the letters we used in equations, previously). 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? Since your neighbor's age is calculated with the expression \(a+3\), we can write the equation \(a+3=32\). When your neighbor is 32 you will be 29, because \(a+3=32\) is true when \(a\) is 29.