Grade 6 Mathematics · Unit 9: Relationships Between Two VariablesStandard 6.AT.11

Lesson 9.4Apply Two-Variable Relationships to Solve Problems

Start hereWords, worked example, and sentence starters

Name Date Period

Learning target I can write an equation to represent the relationship between two variable quantities and use it to solve a real problem.

Start hereWords for this lesson

Read each word before you begin. Say it out loud.

WordWhat it meansExample
solutionSpanish: solución A value of the variable that makes an equation true. the number you solve for
substituteSpanish: sustituir To replace a variable with a number so you can compute. put 8 in for x
at leastSpanish: al menos That amount or more — the smallest amount that still works. “At least $500” means $500 counts, and so does anything more.
predictSpanish: predecir To make a reasonable guess about a future amount using information you already have. We measured up to 5 minutes → 6,000 feet. Reach PAST what was measured: 6 minutes should be about 7,200.
justifySpanish: justificar To explain WHY an answer or recommendation makes sense, using the mathematics. “35 cars at $14.30 raises $500.50, which clears the $500 goal.”
EquationSpanish: Ecuación A math sentence with an equals sign showing two amounts are the same. c = 4n

How it worksWorked examplesubstituting to find the output

These numbers are not on your problems. The steps are. Follow them with your own numbers.

A press makes 22 posters each hour. Write an equation, then find the posters made in 15 hours.

  1. Define: h = hours running, p = posters made. Posters depend on hours.
  2. The rate is 22 posters for each hour, so the rule is p = 22h.
  3. Substitute 15 for h: 22 × 15 → 330 posters.
  4. Check the context: many more hours gave a much larger count, so this fits.

Answer: p = 22h, so a 15-hour run makes 330 posters.

Now you trySame steps, your turn

The steps are the same as the worked example. The numbers are yours. Do the work.

A juicer fills 13 bottles each minute. Write an equation, then find the bottles filled in 40 minutes.

  1. Define: m = , b = .
  2. The rate is bottles for each minute, so the rule is b = m.
  3. Substitute 40 for m: × 40 → bottles.
  4. Check the context: does your answer fit that many minutes?
Answer:equation; bottles in 40 minutes

Say it and write itSentence starters

Finish each sentence out loud with a partner. Then use them in your writing.

  • I let stand for and stand for .
  • My equation is , so I substitute for .
  • The solution is , which means in this situation.
  • This answer is reasonable because .

Word bank equationsubstitutesolutionratevariablemultiplypertotalreasonablecheckinputoutput

Watch outA common mistake

Adding the rate instead of multiplying — writing 2.5 + 8 = 10.5 cars for an 8-hour shift instead of 2.5 × 8 = 20. The rate applies to EVERY hour, so the total is always rate × amount.

Grade 6 Mathematics · Unit 9: Relationships Between Two VariablesStandard 6.AT.11

Lesson 9.4Apply Two-Variable Relationships to Solve Problems

Version ASupported practice

Mastery check☐ Exceeds☐ Meets target☐ Needs practice
Name Date Period

Learning target I can write an equation to represent the relationship between two variable quantities and use it to solve a real problem.

  1. 1Circle the letter of the best answer. Show how you know.

    The assembly line produces 2.5 cars per hour. How many cars does it produce in one 8-hour shift?

    1. A10.5 cars
    2. B16 cars
    3. C20 cars
    4. D25 cars

    Hint How many cars come off the line in ONE hour?
    per 1
    1. 1LabelWrite what each column is.
    2. 2Find per 1Divide to get one.
    3. 3ScaleMultiply up to what is asked.
    4. 4CheckDoes the size make sense?
  2. 2Circle the letter of the best answer. Show how you know.

    Let h = hours of operation and c = cars produced. Which equation represents the assembly line?

    1. Ac = 2.5h
    2. Bc = 2.5 + h
    3. Ch = 2.5c
    4. Dc = h ÷ 2.5

    Hint Say it in words: the cars produced equal 2.5 ___ the hours.

    Show your work
  3. 3Circle the letter of the best answer. Show how you know.

    Using the table (1 hour, 2.5 cars; 8 hours, 20 cars; 16 hours, 40 cars), how many cars are produced in 24 hours?

    1. A26.5 cars
    2. B40 cars
    3. C50 cars
    4. D60 cars

    Hint How many 8-hour shifts fit in 24 hours?

    Show your work
  4. 4Circle the letter of the best answer. Show how you know.

    Last year the student council washed 34 cars and raised $408. How much did they charge per car?

    1. A$10
    2. B$12
    3. C$34
    4. D$374

    Hint The total is the price per car times the number of cars.
    per 1
    1. 1LabelWrite what each column is.
    2. 2Find per 1Divide to get one.
    3. 3ScaleMultiply up to what is asked.
    4. 4CheckDoes the size make sense?
  5. 5Circle the letter of the best answer. Show how you know.

    At $12 per car, how much does the student council raise by washing 20 cars?

    1. A$32
    2. B$120
    3. C$200
    4. D$240

    Hint Each car adds $12.
    per 1
    1. 1LabelWrite what each column is.
    2. 2Find per 1Divide to get one.
    3. 3ScaleMultiply up to what is asked.
    4. 4CheckDoes the size make sense?
  6. 6Circle the letter of the best answer. Show how you know.

    The price is set at $12 per car. Which is the DEPENDENT variable at the car wash?

    1. AThe number of cars washed
    2. BThe total amount of money raised
    3. CThe price of $12 per car
    4. DThe number of students on the council

    Hint Which quantity responds to the other?

    Show your work

Explain your thinkingOne rate is cars per hour and the other is dollars per car. Ask your partner to predict one value from each, then check by substituting — which prediction could you make without a calculator, and why?

Sentence starter Substituting ___ gives ___, so I predict ___.

How did it go? ☐ 4 · I can teach it☐ 3 · I've got it☐ 2 · I need a hint☐ 1 · I need help
Grade 6 Mathematics · Unit 9: Relationships Between Two VariablesStandard 6.AT.11

Lesson 9.4Apply Two-Variable Relationships to Solve Problems

Version BCore practice

Mastery check☐ Exceeds☐ Meets target☐ Needs practice
Name Date Period

Learning target I can write an equation to represent the relationship between two variable quantities and use it to solve a real problem.

  1. 1Circle the letter of the best answer. Show how you know.

    The order requires 1,000 cars. Using 2.5h = 1,000, how many hours will the plant need?

    1. A40 hours
    2. B400 hours
    3. C997.5 hours
    4. D2,500 hours
    My work — one step per lineWhat I did to both sides
  2. 2Circle the letter of the best answer. Show how you know.

    The plant needs 400 hours to fill the order. Running 24 hours a day, about how many days is that?

    1. A4 days
    2. BAbout 17 days
    3. C50 days
    4. D400 days
    Show your work
  3. 3Circle the letter of the best answer. Show how you know.

    The goal is $500 at $12 per car. Using 12n = 500, at least how many cars must be washed?

    1. A40 cars
    2. B41 cars
    3. C42 cars
    4. D50 cars
    per 1
  4. 4Circle the letter of the best answer. Show how you know.

    Can the student council wash 41 cars at $12 each and meet the $500 goal?

    1. ANo — 41 cars raise $492, which is less than $500
    2. BYes — 41 cars raise $502
    3. CYes — 41 rounds up to 42, so it counts
    4. DNo — 41 cars raise $412
    Show your work
  5. 5Circle the letter of the best answer. Show how you know.

    The council expects 35 cars and wants to raise at least $500. What is the minimum price per car, in whole cents, that reaches the goal? Use 35p ≥ 500.

    1. A$12.00
    2. B$14.00
    3. C$14.29
    4. D$465.00
    per 1
  6. 6Circle the letter of the best answer. Show how you know.

    How could the council CHECK that charging $14.30 for 35 cars meets the goal?

    1. ADivide 35 by 14.30 and confirm it is at least $500
    2. BMultiply 35 × 14.30 = 500.50 and confirm it is at least $500
    3. CAdd 35 + 14.30 and confirm it is at least $500
    4. DNo check is possible after solving
    Show your work

Explain your thinkingOne rate is cars per hour and the other is dollars per car. Ask your partner to predict one value from each, then check by substituting — which prediction could you make without a calculator, and why?

How did it go? ☐ 4 · I can teach it☐ 3 · I've got it☐ 2 · I need a hint☐ 1 · I need help
Grade 6 Mathematics · Unit 9: Relationships Between Two VariablesStandard 6.AT.11

Lesson 9.4Apply Two-Variable Relationships to Solve Problems

ChallengeExtension

Mastery check☐ Exceeds☐ Meets target☐ Needs practice
Name Date Period

Learning target I can write an equation to represent the relationship between two variable quantities and use it to solve a real problem.

Part 1Understand the ideamodels and meaning
  1. 1Write the letter of the matching item on each line.

    Connect the four representations of the assembly line: scenario, equation, table, and graph.

    1. Scenario: 2.5 cars every hour
    2. Table row: h = 8, c = 20
    3. Graph point (400, 1000)
    4. Equation c = 20s
    • AEquation: c = 2.5h
    • BGraph point (8, 20)
    • CThe 1,000-car order takes 400 hours
    • DCars counted by 8-hour shifts
Part 2Apply itreal-world problems
  1. 2Circle the letter of the best answer. Show how you know.

    Suppose the variable s counts 8-hour SHIFTS instead of hours. Which equation gives the cars produced?

    1. Ac = 2.5s
    2. Bc = 8s
    3. Cc = 10.5s
    4. Dc = 20s
    Show your work
  2. 3Circle the letter of the best answer. Show how you know.

    If the plant runs 24 hours a day, 7 days a week for the full 3 weeks, how many hours is that — and can it fill the 1,000-car order?

    1. A72 hours — no, that is less than 400 hours
    2. B168 hours — no, that is less than 400 hours
    3. C504 hours — yes, because 504 hours is more than the 400 hours needed
    4. D504 hours — no, because 504 is less than 1,000
    Show your work
  3. 4Circle the letter of the best answer. Show how you know.

    The solution to one problem was 41.7 cars. Would producing or washing a PARTIAL car make sense?

    1. ANo — cars come in whole numbers, so round up to the next whole car
    2. BYes — 0.7 of a car is a real amount you can wash
    3. CNo — so round down and accept a smaller total
    4. DYes — the equation is wrong if it gives a decimal
    Show your work
  4. 5Circle the letter of the best answer. Show how you know.

    Would you recommend the student council charge $12 or $14.30 per car? Which reasoning is strongest?

    1. AEither can work: $12 needs at least 42 cars washed, while $14.30 needs only about 35 — so the choice depends on how many customers they expect
    2. B$12 is always better because a smaller number is always better
    3. C$14.30 is always better because more money per car is always better
    4. DNeither price can reach the $500 goal
    Show your work

Explain your thinkingOne rate is cars per hour and the other is dollars per car. Ask your partner to predict one value from each, then check by substituting — which prediction could you make without a calculator, and why?

How did it go? ☐ 4 · I can teach it☐ 3 · I've got it☐ 2 · I need a hint☐ 1 · I need help