Lesson 4

Practice Solving Equations and Representing Situations with Equations

4.1: Number Talk: Subtracting From Five (5 minutes)

Warm-up

The purpose of this number talk is to have students recall subtraction where regrouping needs to happen in preparation for the problems they will solve in the lesson.

Launch

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

Find the value of each expression mentally.

\(5-2\)

\(5-2.1\)

\(5-2.17\)

\(5-2\frac78\)

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)

4.2: Row Game: Solving Equations Practice (15 minutes)

Activity

The purpose of this activity is for students to practice solving equations. Some students may use the “do the same to each side” strategy they developed in their work with balanced hangers. Others may use strategies like substituting values until they find a value that makes the equation true, or asking themselves questions like “2 times what is 18?” As students progress through the activity, the equations become more difficult to solve by strategies other than “do the same thing to each side.”

Launch

Display an equation like \(2x=12\) or similar. Ask students to think about the balanced hangers of the last lesson and to recall how that helped us solve equations by doing the same to each side. Tell students that, after obtaining a solution via algebraic means, we end up with a variable on one side of the equal sign and a number on the other, e.g. \(x=6\). We can easily read the solution—in this case 6—from an equation with a letter on one side and a number on the other, and we often write solutions in this way. Tell students that the act of finding an equation's solution is sometimes called solving the equation.

Arrange students in groups of 2, and ensure that everyone understands how the row game works before students start working. Allow students 10 minutes of partner work followed by a whole-class discussion.

Action and Expression: Internalize Executive Functions. Chunk this task into more manageable parts. For example, after students have completed the first four rows of the table, check-in with either select groups of students or the whole class. Invite students to share the strategies they have used so far as well as any questions they have before continuing.
Supports accessibility for: Organization; Attention
Conversing, Representing, Writing: MLR2 Collect and Display. While pairs are working, circulate, collect and make a visual display of vocabulary, phrases and representations students use as they solve each situation. Make connections between how similar ideas might be communicated and represented in different ways. Look for and amplify phrases such as “I did the same thing to each side” or “I subtracted the same amount from both sides.” This helps students use mathematical language during paired and whole-group discussions.
Design Principle(s): Support sense-making

Student Facing

Solve the equations in one column. Your partner will work on the other column.

Check in with your partner after you finish each row. Your answers in each row should be the same. If your answers aren’t the same, work together to find the error and correct it.

column A column B
\(18=2x\) \(36=4x\)
\(17=x+9\) \(13=x+5\)
\(8x=56\) \(3x=21\)
\(21=\frac14 x\) \(28=\frac13 x\)
\(6x=45\) \(8x=60\)
\(x+4\frac56=9\) \(x+3\frac56=8\)
\(\frac57x=55\) \(\frac37x=33\)
\(\frac15=6x\) \(\frac13=10x\)
\(2.17+x=5\) \(6.17+x=9\)
\(\frac{20}{3}=\frac{10}{9}x\) \(\frac{14}{5}=\frac{7}{15}x\)
\(14.88+x=17.05\) \(3.91+x=6.08\)
\(3\frac34x=1\frac14\) \(\frac75x=\frac{7}{15}\)

 

Student Response

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

Activity Synthesis

Draw students' attention to \(21=\frac14 x\), and ask selected students to explain how they thought about solving this equation. Some may share strategies like “one-fourth of what number is 21?” Ideally, one student will say “divide each side by \(\frac14\)” and another will say “multiply each side by 4.” From their studies in earlier units, students should understand that multiplying by 4 has the same result as dividing by \(\frac14\). Next, turn students attention to \(\frac57x=55\) and ask them to describe the two ways to think about solving it. “Divide each side by \(\frac57\)” gets the same result as “Multiply each side by \(\frac75\).”

4.3: Choosing Equations to Match Situations (15 minutes)

Activity

The purpose of this activity is for students to practice matching equations to situations and then solving those equations using their new strategies. Monitor for students who draw diagrams (tape, hanger, or their own creations) that describe the relationships and those who solve the equations by doing the same to each side of one or more equations. 

Launch

Allow students 10 minutes of quiet work time, followed by a whole-class discussion.

Reading: MLR6 Three Reads. Demonstrate this routine with the first situation to support reading comprehension. Use the first read to help students understand context. Ask, "What is this situation about?" (e.g., Clare and Mai each have a different number of books). After the second read, ask students “What are the quantities in the situation?” (e.g., the number of books Mai has, the number of books Clare has). After the third read, ask students to brainstorm possible strategies to connect the situation with the appropriate equation(s). Encourage students to repeat this routine themselves for each situation. This helps students connect the language in the word problem with the equation(s) while keeping the intended level of cognitive demand in the task.
Design Principle(s): Support sense-making​

Student Facing

Circle all of the equations that describe each situation. If you get stuck, consider drawing a diagram. Then find the solution for each situation.

  1. Clare has 8 fewer books than Mai. If Mai has 26 books, how many books does Clare have?

    • \(26-x=8\)
    • \(x=26+8\)
    • \(x+8=26\)
    • \(26-8=x\)
      \(x = \text{______}\)
  2. A coach formed teams of 8 from all the players in a soccer league. There are 14 teams. How many players are in the league?

    • \(y=14\div8\)
    • \(\frac{y}{8}=14\)
    • \(\frac18y=14\)
    • \(y=14\boldcdot 8\)
      \(y = \text{______}\)
  3. Kiran scored 223 more points in a computer game than Tyler. If Kiran scored 409 points, how many points did Tyler score?

    • \(223=409-z\)
    • \(409-223=z\)
    • \(409+223=z\)
    • \(409=223+z\)
      \(z = \text{______}\)
  4. Mai ran 27 miles last week, which was three times as far as Jada ran. How far did Jada run?

    • \(3w=27\)
    • \(w=\frac13\boldcdot 27\)
    • \(w=27\div3\)
    • \(w=3\boldcdot 27\)
      \(w = \text{______}\)

Student Response

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

Student Facing

Are you ready for more?

Mai’s mother was 28 when Mai was born. Mai is now 12 years old. In how many years will Mai’s mother be twice Mai’s age? How old will they be then?

Student Response

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

Anticipated Misconceptions

Students who continue to focus on key words misidentify the relationship in each situation. Encourage students to express the relationships in their own words and draw diagrams comparing the given quantities. For example, in the situation with Clare and Mai, they can draw a long rectangle representing Mai's books subdivided into two pieces. Filling in the information given in the story will help clear up the relationships; Clare's rectangle is labeled \(x\), and she has 8 fewer books than Mai, so Mai's rectangle is labeled \(x+8\) and also 26. Alternatively, they can show that the piece labeled \(x\) must equal \(26-8.\) 

Activity Synthesis

Invite students to share their strategies for matching equations to the stories and for solving those equations. Include students who drew tape, hanger, or other types of diagrams to help them understand and reason about the relationships. Record the diagrams and strategies and have students compare them. Ask where they see information from the story in the parts of the diagrams and equations.

If no students bring it up, ask if any of the situations have a similar structure.

  • The Clare/Mai and Kiran/Tyler situations share a similar structure where both the larger quantity and the difference between the smaller and larger quantities are known while the smaller quantity is unknown. Note that the first relationship is expressed with “fewer” and the second with “more.” This provides an opportunity for students to reason about the quantities, decontextualizing to see the similar structure and then contextualizing to understand the situations and answer questions.
  • The soccer teams and Mai/Jada situations share similar structures in that equal parts add to a whole. The two problems differ in which quantities are known and unknown. In the soccer situation, the size of each group (8 players per team) and number of groups (14 teams) are known while the total is unknown. In the Mai/Jada multiplicative comparison situation, a total is known (27 miles) and the number of groups is known (3 times as many) but the size of each group is unknown.

Focusing on structure in this way helps students reason about the relationships between quantities in a situation, rather than focus on the words in the problem as hints to the operations needed in the equations. 

For students who solved for the unknown by using the equations, ask which of the chosen equations they decided to solve and why. 

Lesson Synthesis

Lesson Synthesis

The end of this lesson is a good place for students to take a moment and reflect on the learning of the past four lessons. Some questions to guide the discussion:

  • “Describe some ways to understand how a situation can be represented mathematically.”
  • “What have you learned about equations that surprised you?”
  • “Share your thoughts about using diagrams to help understand relationships. Where have you seen diagrams used earlier this year? Where were they most helpful to you? Least helpful?”
  • “Describe any connections you see between the types of diagrams used in the last four lessons.”

4.4: Cool-down - More Storytime (5 minutes)

Cool-Down

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

Student Lesson Summary

Student Facing

Writing and solving equations can help us answer questions about situations.

Suppose a scientist has 13.68 liters of acid and needs 16.05 liters for an experiment. How many more liters of acid does she need for the experiment?

  • We can represent this situation with the equation:

\(\displaystyle 13.68 + x=16.05\)

  • When working with hangers, we saw that the solution can be found by subtracting 13.68 from each side. This gives us some new equations that also represent the situation:

\(\displaystyle x=16.05 - 13.68\)

\(\displaystyle x=2.37\)

  • Finding a solution in this way leads to a variable on one side of the equal sign and a number on the other. We can easily read the solution—in this case, 2.37—from an equation with a letter on one side and a number on the other. We often write solutions in this way.

Let’s say a food pantry takes a 54-pound bag of rice and splits it into portions that each weigh \(\frac34\) of a pound. How many portions can they make from this bag?

  • We can represent this situation with the equation:

\(\displaystyle \frac34 x = 54\)

  • We can find the value of \(x\) by dividing each side by \(\frac34\). This gives us some new equations that represent the same situation:

\(\displaystyle x=54\div \frac34\)

\(\displaystyle x=72\)

  • The solution is 72 portions.