Apply Two-Variable Relationships to Solve Problems
I can write an equation to represent the relationship between two variable quantities and use it to solve a real problem.
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🎯 Content Objective / Objetivo de contenido
I can write an equation to represent the relationship between two variable quantities and use it to solve a real problem.
Today's Flow
Total pacing: ~45 min · Progress bar at top tracks your place
LAUNCH
⏱ ~10 min
⏱️ 3 MIN · THINK-PAIR-SHARE
The plant produces 2.5 cars per hour, and the order needs 1,000 cars. Solving 2.5h = 1,000 gives h = 400 hours. Why do you divide instead of multiply to find h?
Check for Understanding #1
Teacher: If >30% thumbs down, re-teach with a fresh example before moving on.
Unit 9 · Lesson 4
An automobile plant's assembly line runs in 8-hour shifts and produces 2.5 cars per hour. An order arrives: 1,000 cars in 3 weeks. Should the manufacturer accept? Later, the student council plans a fundraiser car wash — last year they washed 34 cars and raised $408, and this year the goal is $500.
Concept Launch
💡 Use the equation to answer a real question
Writing the equation is only half the job. Applying it means substituting what you know, solving for what you don't, and checking whether the answer makes sense in the situation.
Analyzing & Applying Two-Variable Relationships. 1. Connect the four representations: Scenario, Table, Graph, and Equation. 2. Substitute given values into the equation to make predictions and solve problems. 3. Interpret coordinates and rate of change in the real-world context
Check for Understanding #2
Teacher: If >30% thumbs down, re-teach with a fresh example before moving on.
Now you try
VOCABULARY
⏱ ~8 min
| Term / Término | Meaning / Significado | Example / Ejemplo | Visual |
|---|---|---|---|
| solution solución |
A value of the variable that makes an equation true. Un valor de la variable que hace que una ecuación sea verdadera. |
h = 400 is the solution of 2.5h = 1,000. | h = 8 makes 2.5h = 20 true, so 8 is the solution. |
| substitute sustituir |
To replace a variable with a number so you can compute. Reemplazar una variable con un número para poder calcular. |
Substitute 8 for h in 2.5h to get 20 cars per shift. | c = 15h with h = 5 becomes c = 15 × 5 = 75. |
| at least al menos |
That amount or more — the smallest amount that still works. Esa cantidad o más — la cantidad mínima que todavía funciona. |
The student council must wash at least 42 cars to raise $500. | “At least $500” means $500 counts, and so does anything more. |
| predict predecir |
To make a reasonable guess about a future amount using information you already have. Hacer una estimación razonable sobre una cantidad futura usando la información que ya tienes. |
The council predicts about 35 cars this year because they washed 34 last year. | The line says 5 minutes → 6,000 feet, so 6 minutes should be about 7,200. |
| justify justificar |
To explain WHY an answer or recommendation makes sense, using the mathematics. Explicar POR QUÉ una respuesta o recomendación tiene sentido, usando las matemáticas. |
Justify accepting the order by showing 1,000 cars take about 2.5 weeks. | “35 cars at $14.30 raises $500.50, which clears the $500 goal.” |
Which Word Fits?
A value of the variable that makes an equation true is a ___.
Use It In a Sentence
Check for Understanding #3
Teacher: If >30% thumbs down, re-teach with a fresh example before moving on.
Turn & Talk — Launch
The plant produces 2.5 cars per hour, and the order needs 1,000 cars. Solving 2.5h = 1,000 gives h = 400 hours. Why do you divide instead of multiply to find h?
👂 Listen For
A strong answer names division as the inverse of multiplication and checks the solution by substituting it back in. A weak answer just says 'because that's what you do' without the inverse-operation reasoning.
Extend: 400 hours running 24 hours a day is about 17 days, which fits inside the 3-week deadline only if the plant runs every day with no stops. Justify whether the manufacturer should accept the order, using the math.
EXPLORE & PRACTICE
⏱ ~18 min
Visual Modeling Workspace
Use the drawing tray below to annotate the visual model. Teacher: say "Click to reveal" on key steps.
Explore Activity
Sort all eight cards. Check each number against its rate: cars = 2.5 × hours, or dollars = 12 × cars washed.
✍️ Explore Discourse
One 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?
Whiteboard Moment
Show your work clearly. Be ready to explain your thinking to a partner.
Turn & Talk — Explore
You sorted facts between the assembly line (2.5 cars/hour) and the car wash ($12/car). How did you check that 'washing 42 cars raises $504' belonged with the car wash?
👂 Listen For
Listen for actual multiplication (42 × 12 = 504) rather than matching by which numbers 'look like' cars versus hours. A strong answer verifies the arithmetic, not just the units.
Extend: The card 'washing 34 cars raised $408 last year' does NOT immediately show the $12 rate. Explain how you would work backward from that card to find the price per car.
Practice Check A
The order requires 1,000 cars. Using 2.5h = 1,000, how many hours will the plant need?
✍️ Show Your Work
Explain why your answer is correct using today's vocabulary.
Practice Check B
The plant needs 400 hours to fill the order. Running 24 hours a day, about how many days is that?
✍️ Show Your Work
Explain why your answer is correct using today's vocabulary.
Equivalent Ratio Sort
Complete the interactive activity using today's strategy.
✍️ Justify Your Thinking
The assembly line follows c = 2.5h, where h is hours and c is cars produced. Sort each statement: true or false?
A classmate turned in the work below. One step has a mistake. Read every step, find it, name it, and fix it.
No worked steps provided for this lesson.
Choose ONE option to show what you know — then do it in the workspace below.
Use evidence from today's lesson to complete each frame.
Today's key idea is: "Analyzing & Applying Two-Variable Relationships. 1. Connect the four representations: Scenario, Table, Graph, and Equation. 2. Substitute given values into the equation to make predictions and solve problems. 3. Interpret coordinates and rate of change in the real-world context" — and it works because ___.
Because solution means ___, but a tricky part is ___, so I have to ___.
I can justify my method by showing ___, which proves my answer makes sense.
I can prove my answer is correct by ___, using substitute to check my work.
✍️ TWR · WRITE 3 SENTENCES · 7 MIN
Analyzing & Applying Two-Variable Relationships. 1. Connect the four representations: Scenario, Table, Graph, and Equation. 2. Substitute given values into the equation to make predictions and solve problems. 3. Interpret coordinates and rate of change in the real-world context because ___
Analyzing & Applying Two-Variable Relationships. 1. Connect the four representations: Scenario, Table, Graph, and Equation. 2. Substitute given values into the equation to make predictions and solve problems. 3. Interpret coordinates and rate of change in the real-world context but ___
Analyzing & Applying Two-Variable Relationships. 1. Connect the four representations: Scenario, Table, Graph, and Equation. 2. Substitute given values into the equation to make predictions and solve problems. 3. Interpret coordinates and rate of change in the real-world context so ___
🌱 TWR · GROW THE KERNEL · 6 MIN
Answer these to add detail
Sentence starters (tap to use)
Student Workspace
Sort all eight cards. Check each number against its rate: cars = 2.5 × hours, or dollars = 12 × cars washed.
| Column A | Column B |
|---|---|
✏️ Sketch Your Strategy
Differentiation Paths
Step-by-step with a worked model and sentence frames.
The assembly line produces 2.5 cars per hour. How many cars does it produce in one 8-hour shift?
The order requires 1,000 cars. Using 2.5h = 1,000, how many hours will the plant need?
Suppose the variable s counts 8-hour SHIFTS instead of hours. Which equation gives the cars produced?
Partner Activity
Work with your partner on the practice problems at your differentiation path level. Explain each step using math vocabulary.
Check for Understanding #4
Teacher: If >30% thumbs down, re-teach with a fresh example before moving on.
Real-World Connection
🌍 Math in the Wild
Yelina is comparing the costs of having cupcakes delivered from two different bakeries. Each bakery has its own relationship between the number of cupcakes ordered and the total cost. Yelina has a budget of $110 to spend.
✍️ Connection Reasoning
From which bakery would you recommend Yelina purchase cupcakes? Explain.
Write an ___ for each bakery, substitute Yelina's $___ budget into each one, ___ the two results, and recommend the bakery that gives more ___ for the money.
Turn & Talk — Connect
Yelina has $110 to spend on cupcakes. Bakery A charges $2 per cupcake plus a $10 delivery fee (c = 2n + 10); Bakery B charges $2.50 per cupcake with no fee (c = 2.5n). Solving each equation for $110 gives 50 cupcakes at Bakery A and 44 at Bakery B. Which should Yelina choose, and why doesn't the higher per-cupcake rate automatically mean fewer cupcakes?
👂 Listen For
A strong answer names comparing the solved values (50 vs. 44), not just the coefficients (2 vs. 2.5), and explains that the fee changes the comparison. A weak answer assumes the smaller rate always wins without solving.
Extend: Show how you would CHECK that 50 cupcakes really fits Yelina's $110 budget at Bakery A, using the original equation.
Summarize: Apply Two-Variable Relationships to Solve Problems
Analyzing & Applying Two-Variable Relationships. 1. Connect the four representations: Scenario, Table, Graph, and Equation. 2. Substitute given values into the equation to make predictions and solve problems. 3. Interpret coordinates and rate of change in the real-world context
In your own words:
CLOSURE & REFLECT
⏱ ~8 min
Today I learned that ___ because ___.
One thing I am still not sure about is ___.
The plant produces 2.5 cars per hour. Using 2.5h = 1,000, how many hours does the order of 1,000 cars take?
Bonus Exit Check
The goal is $500 at $12 per car. Using 12n = 500, at least how many cars must be washed?
✍️ Show Your Work
Explain why your answer is correct using today's vocabulary.
Reflection & Self-Assessment
Continue Learning
Launch the Full Interactive Activity
Students continue practice in the HTML lesson engine with auto-check, hints, and differentiation.
Family Connection
Share tonight's family homework and discuss one vocabulary word at home.
Open Family Homework ↗Teacher Notes
⏱️ Pacing Guide
- Launch & vocab: 12 min
- I Do / We Do / You Do: 15 min
- Explore & practice: 15 min
- Connect & closure: 8 min
Total: ~45 min
🎯 Listen For · Common Errors
• A strong answer names division as the inverse of multiplication and checks the solution by substituting it back in. A weak answer just says 'because that's what you do' without the inverse-operation reasoning.
• Listen for actual multiplication (42 × 12 = 504) rather than matching by which numbers 'look like' cars versus hours. A strong answer verifies the arithmetic, not just the units.
• A strong answer names comparing the solved values (50 vs. 44), not just the coefficients (2 vs. 2.5), and explains that the fee changes the comparison. A weak answer assumes the smaller rate always wins without solving.
• Listen for students explaining WHY they round up instead of down — because rounding down leaves the goal unmet — not just stating '42 is the answer.'
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.
Answer Key (Teacher Appendix)
Hide this slide during presentation or move to the end of your copy.
✓ Practice 1: 400 hours — Divide both sides by 2.5: h = 1,000 ÷ 2.5 = 400 hours.
✓ Practice 2: About 17 days — Divide the hours by the hours in a day: 400 ÷ 24 ≈ 16.7, which is about 17 days — about 2.5 weeks.
✓ Practice 3: 42 cars — 500 ÷ 12 is about 41.7, and you cannot wash a fraction of a car, so round UP: 42 cars raise 42 × 12 = $504, which meets the goal.
✓ Practice 4: No — 41 cars raise $492, which is less than $500 — 41 × 12 = 492, and $492 < $500, so 41 cars fall $8 short.
✓ Exit ticket: 400 hours — Divide both sides by 2.5: h = 1,000 ÷ 2.5 = 400 hours. Check: 2.5 × 400 = 1,000 cars.