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

Lesson 9.4Working Backward to a Goal

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. h = 8 makes 2.5h = 20 true, so 8 is the solution.
substituteSpanish: sustituir To replace a variable with a number so you can compute. c = 15h with h = 5 becomes c = 15 × 5 = 75.
at leastSpanish: al menos That amount or more — the smallest amount that still works. at least 42 cars
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 = 24.95m means the total cost equals $24.95 times the number of months.

How it worksWorked example

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

Tickets cost $6. A club needs at least $100. How many tickets?

  1. Put in the goal: 100 = 6n.
  2. Divide: 100 ÷ 6 ≈ 16.7.
  3. 16 × 6 = $96 is too low; 17 × 6 = $102 is enough.

Answer: 17 tickets

Say it and write itSentence starters

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

  • My equation is and my goal is .
  • I put in place of , then undid the .
  • The group needs at least to reach the goal.
  • Another way to reach the same goal is to .

Word bank work backwardinverseundodividegoalat leastestimateequationsubstituteratesolutioncheck

Watch out!A common mistake

Adding the rate instead of multiplying

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

Lesson 9.4Working Backward to a Goal

Apply Day · 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. 1Complete each step, then write the final answer.
    Hours1?Toys896× ?× ?
    The machine makes 8 toys each hour.

    A machine makes 8 toys every hour. Let h = hours and t = toys. How many hours does it take to make 96 toys?

    Show your work

    Hint Solve 8h = 96. Divide both sides by 8.

    Final answer:
  2. 2Complete each step, then write the final answer.
    Bags1?Pounds2.550× ?× ?
    Each bag holds 2.5 pounds of apples.

    A farm puts 2.5 pounds of apples in each bag. Let b = bags and p = pounds. How many bags hold 50 pounds of apples?

    Show your work

    Hint Write p = 2.5b. Put in 50 for p. Then divide by 2.5.

    Final answer:
  3. 3Complete each step, then write the final answer.
    Cars1?Money earned$5$120× ?× ?
    Each car earns $5.

    A club earns $5 for each car it washes. The goal is $120. Let n = cars washed. How many cars must the club wash?

    Show your work

    Hint Solve 5n = 120. Divide both sides by 5.

    Final answer:
3-Point Exit Checklist: ☐ 1. Units & Labels Verified ☐ 2. Mathematical Reasoning Stated ☐ 3. Calculations Double-Checked
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 9.4Working Backward to a Goal

Apply Day · 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.
    Notebooks167Cost$3??× ?× ?× ?× ?
    Each notebook costs $3. Kim has $20.

    Kim has $20. Notebooks cost $3 each. She solves 3n = 20 and gets n ≈ 6.7. She says she can buy 7 notebooks. Is she right?

    1. ANo. 7 notebooks cost $21, so she can buy 6.
    2. BYes. 6.7 rounds up to 7.
    3. CNo. She can buy 6.7 notebooks.
    Show your work
  2. 2Complete each step, then write the final answer.

    A tank has 20 liters of water. A hose adds 6 liters each minute. Let m = minutes. How many minutes until the tank has 110 liters?

    Show your work
    Final answer:
  3. 3Complete each step, then write the final answer.
    Minutes1?Gallons4.590× ?× ?
    The pump moves 4.5 gallons each minute.

    A pump moves 4.5 gallons of water every minute. Let m = minutes and g = gallons. How many minutes does it take to move 90 gallons?

    Show your work
    Final answer:
3-Point Exit Checklist: ☐ 1. Units & Labels Verified ☐ 2. Mathematical Reasoning Stated ☐ 3. Calculations Double-Checked
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 9.4Working Backward to a Goal

Apply Day · 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.

  1. 1Complete each step, then write the final answer.
    Vans1?Students752× ?× ?
    Each van carries 7 students.

    A van carries 7 students. 52 students are going on a trip. Let v = vans. Solve 7v = 52. How many vans does the school need?

    Show your work
    Final answer:
  2. 2Complete each step, then write the final answer.

    A pool has 300 gallons of water. A pump adds 25 gallons each minute. Let m = minutes. How many minutes until the pool has 1,000 gallons?

    Show your work
    Final answer:
  3. 3Complete each step, then write the final answer.

    A bus company charges a $50 fee plus $3 for each mile. A school has $400 to spend. What is the greatest number of whole miles the bus can go?

    Show your work
    Final answer:
3-Point Exit Checklist: ☐ 1. Units & Labels Verified ☐ 2. Mathematical Reasoning Stated ☐ 3. Calculations Double-Checked
How did it go? ☐ 4 · I can teach it☐ 3 · I've got it☐ 2 · I need a hint☐ 1 · I need help