Write Equations to Represent Relationships Between Two Variables
I can write an equation from a table or a graph to model the relationship between the dependent and independent variables.
How to Use This Deck
Click Present or press F11 for fullscreen. Use arrow keys to advance.
Blue boxes show exactly what to say, ask, and how long to spend.
Text boxes, polls, and drag-sort save automatically in the browser.
Press N or click 📝 in the toolbar for pacing tips and answers.
Launch the full HTML activity for independent practice.
File → Print or the print button for handout copies.
🎯 Content Objective / Objetivo de contenido
I can write an equation from a table or a graph to model the relationship between the dependent and independent variables.
Today's Flow
Total pacing: ~45 min · Progress bar at top tracks your place
LAUNCH
⏱ ~10 min
⏱️ 3 MIN · THINK-PAIR-SHARE
Recreation World charges $24.95 per month, and the table shows the cost growing by exactly $24.95 every row. Why is the equation c = 24.95m correct, but c = 24.95 + m would be wrong?
Check for Understanding #1
Teacher: If >30% thumbs down, re-teach with a fresh example before moving on.
The Cost of Fitness
Miguel is interested in joining a recreation center. Recreation World charges the monthly fee shown on the flyer. How can Miguel determine the cost of being a member of Recreation World for any number of months?
Concept Launch
💡 From a pattern in the table to an equation
A table can only list some values. An equation captures the whole pattern at once, so it works for any number of months or hours.
Writing Two-Variable Equations. Formula: y = kx | Dependent = (Unit Rate) × Independent. 1. Identify the independent variable (x) and dependent variable (y). 2. Find the constant unit rate (k = y ÷ x). 3. Write the equation relating the two variables: y = kx (or y = x + c)
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 |
|---|---|---|---|
| equation ecuación |
A mathematical sentence with an equals sign that shows two amounts are the same. Una oración matemática con un signo igual que muestra que dos cantidades son iguales. |
c = 24.95m says the total cost equals $24.95 times the months. | c = 24.95m means the total cost equals $24.95 times the number of months |
| variable variable |
A letter that represents a quantity that can change, or vary. Una letra que representa una cantidad que puede cambiar o variar. |
In c = 24.95m, both m (months) and c (cost) are variables. | In c = 24.95m, the m can be 1, 2, or 12 — it varies. |
| rate tasa |
A fixed amount of one quantity for every one unit of the other quantity. Una cantidad fija de una cantidad por cada unidad de la otra cantidad. |
$24.95 per month and $15 per hour are rates. | $24.95 for every 1 month — the amount attached to one unit. |
| represent representar |
To show a relationship using a tool such as a table, a graph, or an equation. Mostrar una relación usando una herramienta como una tabla, una gráfica o una ecuación. |
The table, the graph, and c = 15h all represent the canoe rental. | The same membership shown as a table, a graph, AND c = 24.95m. |
| define the variables definir las variables |
To state exactly what quantity each letter in an equation stands for. Decir exactamente qué cantidad representa cada letra de una ecuación. |
Let m = number of months and c = total cost in dollars. | “Let m = months and c = total cost” — written before you solve. |
Which Word Fits?
A mathematical sentence with an equals sign is an ___.
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
Recreation World charges $24.95 per month, and the table shows the cost growing by exactly $24.95 every row. Why is the equation c = 24.95m correct, but c = 24.95 + m would be wrong?
👂 Listen For
A strong answer tests the equation against a table row (2 months should give $49.90) rather than just stating a rule. A weak answer picks multiplication without checking it against the actual numbers.
Extend: The canoe rental costs $15 per hour. Justify why c = 15h will always work for ANY number of hours, even ones not shown in a table, while a table only shows the rows someone wrote down.
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 total by multiplying the rate by the number of months or hours.
✍️ Explore Discourse
Both equations charge by a rate. What do m and h each stand for in their own situation, and why would c = 24.95m be wrong for the canoe?
Whiteboard Moment
Show your work clearly. Be ready to explain your thinking to a partner.
Turn & Talk — Explore
You sorted cards between the gym equation (c = 24.95m) and the canoe equation (c = 15h). How did you decide where '2 hours costs $30' belonged?
👂 Listen For
Listen for actual substitution and multiplication, not matching by 'it sounds like hours so it's the canoe one.' A strong answer shows the check: 15 × 2 = 30.
Extend: The card 'the rate is $24.95 per month' and the card 'the rate is $15 per hour' were sorted instantly, without computing anything. Explain why those two cards needed a different strategy than the total-cost cards.
Practice Check A
Using c = 24.95m, what is the total cost of 10 months of membership?
✍️ Show Your Work
Explain why your answer is correct using today's vocabulary.
Practice Check B
Using c = 15h, what is the total cost of a 5-hour canoe rental?
✍️ 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
Sort each equation: does it say 'y is 4 times x', or something different?
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: "Writing Two-Variable Equations. Formula: y = kx | Dependent = (Unit Rate) × Independent. 1. Identify the independent variable (x) and dependent variable (y). 2. Find the constant unit rate (k = y ÷ x). 3. Write the equation relating the two variables: y = kx (or y = x + c)" — and it works because ___.
Because equation 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 variable to check my work.
✍️ TWR · WRITE 3 SENTENCES · 7 MIN
Writing Two-Variable Equations. Formula: y = kx | Dependent = (Unit Rate) × Independent. 1. Identify the independent variable (x) and dependent variable (y). 2. Find the constant unit rate (k = y ÷ x). 3. Write the equation relating the two variables: y = kx (or y = x + c) because ___
Writing Two-Variable Equations. Formula: y = kx | Dependent = (Unit Rate) × Independent. 1. Identify the independent variable (x) and dependent variable (y). 2. Find the constant unit rate (k = y ÷ x). 3. Write the equation relating the two variables: y = kx (or y = x + c) but ___
Writing Two-Variable Equations. Formula: y = kx | Dependent = (Unit Rate) × Independent. 1. Identify the independent variable (x) and dependent variable (y). 2. Find the constant unit rate (k = y ÷ x). 3. Write the equation relating the two variables: y = kx (or y = x + c) 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 total by multiplying the rate by the number of months or hours.
| Column A | Column B |
|---|---|
✏️ Sketch Your Strategy
Differentiation Paths
Step-by-step with a worked model and sentence frames.
Recreation World charges $24.95 per month. What is the total cost of 2 months of membership?
Using c = 24.95m, what is the total cost of 10 months of membership?
Miguel wants the cost of a full year (12 months) at Recreation World. Using c = 24.95m, what is the total?
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
Diamond is considering using an online music service. She can either purchase songs one at a time or pay to stream songs. Each choice has its own relationship between the number of songs and the total cost.
✍️ Connection Reasoning
Would you recommend that Diamond purchase or stream songs from the online music service?
Diamond should write an ___ for each option so she can ___ the costs. Each equation uses the ___ as the input and gives the ___ as the output.
Turn & Talk — Connect
Diamond is deciding whether to purchase songs one at a time or pay to stream them. Why does she need TWO separate equations instead of one equation that covers both options?
👂 Listen For
A strong answer explains WHY one equation can't cover both (different rates/structures) and describes substituting the same number of songs into each to compare fairly. A weak answer just says 'because they're different services' without naming the math reason.
Extend: Suppose purchasing costs $1.29 per song with no fee, and streaming costs a flat $9.99 no matter how many songs. Which representation — an equation for each — would let Diamond find the exact number of songs where the two options cost the same? Explain your reasoning without solving it fully.
Summarize: Write Equations to Represent Relationships Between Two Variables
Writing Two-Variable Equations. Formula: y = kx | Dependent = (Unit Rate) × Independent. 1. Identify the independent variable (x) and dependent variable (y). 2. Find the constant unit rate (k = y ÷ x). 3. Write the equation relating the two variables: y = kx (or y = x + c)
In your own words:
CLOSURE & REFLECT
⏱ ~8 min
Today I learned that ___ because ___.
One thing I am still not sure about is ___.
Recreation World charges $24.95 per month. Let m = number of months and c = total cost. Which equation represents the relationship?
Bonus Exit Check
A table pairs months with total cost: (1, 24.95), (2, 49.90), (4, 99.80). Which equation matches the table, with m = months and c = cost?
✍️ 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 tests the equation against a table row (2 months should give $49.90) rather than just stating a rule. A weak answer picks multiplication without checking it against the actual numbers.
• Listen for actual substitution and multiplication, not matching by 'it sounds like hours so it's the canoe one.' A strong answer shows the check: 15 × 2 = 30.
• A strong answer explains WHY one equation can't cover both (different rates/structures) and describes substituting the same number of songs into each to compare fairly. A weak answer just says 'because they're different services' without naming the math reason.
• Listen for the full sequence — pattern in words, then variables defined, then equation — not just a final equation pulled from memory.
Common mistake: Writing the equation with addition — c = 24.95 + m — instead of multiplication. The fee is charged EVERY month, so the total is the rate times the number of months; always test the equation against a row of the table.
Answer Key (Teacher Appendix)
Hide this slide during presentation or move to the end of your copy.
✓ Practice 1: $249.50 — Substitute m = 10: c = 24.95 × 10 = 249.50.
✓ Practice 2: $75 — Substitute h = 5: c = 15 × 5 = 75.
✓ Practice 3: c = 24.95m — Test each row: 24.95 × 1 = 24.95, 24.95 × 2 = 49.90, and 24.95 × 4 = 99.80 — every row fits c = 24.95m.
✓ Practice 4: (4, 60) — 4 hours cost 4 × 15 = 60 dollars, and hours come first, so the point is (4, 60).
✓ Exit ticket: c = 24.95m — Each month costs $24.95, so the total cost is $24.95 multiplied by the number of months: c = 24.95m. Check: 2 months gives 24.95 × 2 = 49.90, which matches the table.