Lesson 9.4Working Backward to a Goal
Start hereWords, worked example, and sentence starters
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.
| Word | What it means | Example |
|---|---|---|
| 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 exampleworking backward to find the input
These numbers are not on your problems. The steps are. Follow them with your own numbers.
A band earns $26 for each poster it sells. How many posters must it sell to reach a goal of $1,300?
- Define: n = posters sold, e = dollars earned. The rule is e = 26n.
- The goal takes the place of the output: 26n = 1,300.
- Undo the multiplication — divide both sides by the rate: 1,300 ÷ 26 → 50.
- Check in context: 26 × 50 → $1,300, so 50 posters reach the goal.
Answer: The band must sell 50 posters to reach $1,300.
Now you trySame steps, your turn
The steps are the same as the worked example. The numbers are yours. Do the work.
A club earns $18 for each car it washes. How many cars must it wash to reach a goal of $720?
- Define: c = , r = . Rule: r = c.
- The goal takes the place of the output: c = 720.
- Undo the multiplication — divide both sides by the rate: 720 ÷ → .
- Check in context: × → $720, so cars reach the goal.
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 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.
Lesson 9.4Working Backward to a Goal
Apply Day · Version ASupported practice
Learning target I can write an equation to represent the relationship between two variable quantities and use it to solve a real problem.
-
1Complete each step, then write the final answer.
(Lesson 9.4) A gym membership costs $4.5 per month, so the equation is c = 4.5m. Use it to find the cost c of 3 months.
- 1Substitute the months: c = 4.5 × ___.
- 2Multiply: 4.5 × 3 = ___.
Hint Re-read the question. What exactly is it asking you to find?
Final answer: -
2Complete each step, then write the final answer.
(Lesson 9.4) A gym membership costs $6 per month, so the equation is c = 6m. Use it to find the cost c of 4 months.
- 1Substitute the months: c = 6 × ___.
- 2Multiply: 6 × 4 = ___.
Hint Re-read the question. What exactly is it asking you to find?
Final answer: -
3Complete each step, then write the final answer.
(Lesson 9.4) A gym membership costs $7.5 per month, so the equation is c = 7.5m. Use it to find the cost c of 5 months.
- 1Substitute the months: c = 7.5 × ___.
- 2Multiply: 7.5 × 5 = ___.
Hint Re-read the question. What exactly is it asking you to find?
Final answer:
Lesson 9.4Working Backward to a Goal
Apply Day · Version BCore practice
Learning target I can write an equation to represent the relationship between two variable quantities and use it to solve a real problem.
-
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?
- A40 hours
- B400 hours
- C997.5 hours
- D2,500 hours
My work — one step per line What I did to both sides -
2Complete each step, then write the final answer.
A gym membership costs $7.5 per month, so the equation is c = 7.5m. Use it to find the cost c of 4 months.
- 1Substitute the months: c = 7.5 × ___.
- 2Multiply: 7.5 × 4 = ___.
Final answer: -
3Circle 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?
- A4 days
- BAbout 17 days
- C50 days
- D400 days
Show your work
Lesson 9.4Working Backward to a Goal
Apply Day · ChallengeExtension
Learning target I can write an equation to represent the relationship between two variable quantities and use it to solve a real problem.
-
1Circle 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?
- Ac = 2.5s
- Bc = 8s
- Cc = 10.5s
- Dc = 20s
Show your work -
2Complete each step, then write the final answer.
A gym membership costs $7.5 per month, so the equation is c = 7.5m. Use it to find the cost c of 4 months.
- 1Substitute the months: c = 7.5 × ___.
- 2Multiply: 7.5 × 4 = ___.
Final answer: -
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?
- A72 hours — no, that is less than 400 hours
- B168 hours — no, that is less than 400 hours
- C504 hours — yes, because 504 hours is more than the 400 hours needed
- D504 hours — no, because 504 is less than 1,000
Show your work