Explore Relationships Between Two Variables
I can recognize when two quantities change together, and identify which one is the independent variable and which is the dependent variable.
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🎯 Content Objective / Objetivo de contenido
I can recognize when two quantities change together, and identify which one is the independent variable and which is the dependent variable.
Today's Flow
Total pacing: ~45 min · Progress bar at top tracks your place
LAUNCH
⏱ ~10 min
⏱️ 3 MIN · THINK-PAIR-SHARE
Akela put her three bean trays in three different spots, so each got a different amount of sunlight. After three weeks she measured the height of the plants. Which of those two quantities did Akela choose, and which one responded?
Check for Understanding #1
Teacher: If >30% thumbs down, re-teach with a fresh example before moving on.
Unit 9 · Lesson 1
Akela planted three trays of bean seeds for a science experiment. Each tray sat in a different spot, so each got a different amount of sunlight per day. After three weeks she measured the average plant height in every tray. Two quantities in that experiment are related: change the sunlight and the height changes with it. One of them Akela chose, and the other one answered back. Today you will name which quantity is the independent one and which depends on it, and you will find pairs like this in a jogging tracker and a pizza shop.
Concept Launch
💡 Which quantity is doing the changing — and which is responding?
When two quantities are related, one of them depends on the other. Naming which is which is the whole job of this lesson.
Independent & Dependent Variables. 1. Independent Variable (x): The input / cause that is chosen or changes freely. 2. Dependent Variable (y): The output / effect that depends on the independent variable. 3. Sentence check: 'The (dependent) depends on the (independent)'
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 |
|---|---|---|---|
| independent variable variable independiente |
The quantity that is subject to choice. It does not depend on the other quantity. La cantidad que se elige. No depende de la otra cantidad. |
The hours of sunlight a tray receives — Akela chooses that. | Miguel picks the number of MONTHS — nothing else decides it for him. |
| dependent variable variable dependiente |
The quantity that changes in response to the independent variable. La cantidad que cambia en respuesta a la variable independiente. |
The height the plants reach — it responds to the sunlight. | Total cost = $24.95 × months — the cost RESPONDS to the months. |
| depends on depende de |
One quantity depends on another when changing the first causes the second to change. Una cantidad depende de otra cuando cambiar la primera hace cambiar la segunda. |
Plant height depends on sunlight. | “Total cost depends on the number of months.” Responder first, chooser last. |
| quantity cantidad |
Something that can be counted or measured. Algo que se puede contar o medir. |
Hours of sunlight, centimeters of height, minutes of jogging. | 24.95 dollars per month, 12 months — each is a number with a meaning. |
| relationship relación |
How two quantities change together. Cómo dos cantidades cambian juntas. |
More sunlight goes with taller plants. | 1 month → $24.95, 2 months → $49.90: the two quantities move together. |
Which Word Fits?
The quantity that is subject to choice is the ___ ___.
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
Akela put her three bean trays in three different spots, so each got a different amount of sunlight. After three weeks she measured the height of the plants. Which of those two quantities did Akela choose, and which one responded?
👂 Listen For
A strong answer names sunlight as chosen (independent) and height as responding (dependent), and can say the sentence 'height depends on sunlight.' A weak answer just says 'height is bigger' without naming which quantity was chosen.
Extend: Claire chose to jog for 30 minutes and burned 123 Calories. Later she chose to run exactly 3 miles and recorded how long it took. Explain why time is independent in one problem but dependent in the other.
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, then read one pair aloud as a sentence: ___ depends on ___ .
✍️ Explore Discourse
Take a card you and your partner put in different columns. Say the sentence 'The ___ depends on the ___' out loud both ways — which version is true, and how does that settle which variable is independent?
Whiteboard Moment
Show your work clearly. Be ready to explain your thinking to a partner.
Turn & Talk — Explore
When you sorted the eight cards into independent and dependent, how did you decide where 'number of Calories Claire burns' belonged?
👂 Listen For
Listen for the test 'who gets to choose this' rather than which number is bigger or which was mentioned first in the problem. A weak answer just repeats the card's category without justifying it.
Extend: The card 'number of months you keep a gym membership' and 'total amount you have paid the gym' were also sorted. Justify why 'total amount paid' can never be the independent variable in this pair.
Practice Check A
A gym charges $24.95 per month. Which pair correctly names the variables?
✍️ Show Your Work
Explain why your answer is correct using today's vocabulary.
Practice Check B
Claire jogs 30 minutes and burns 123 Calories. At the same rate, how many Calories would a 90-minute jog burn?
✍️ Show Your Work
Explain why your answer is correct using today's vocabulary.
Ratio Table Builder
Fill the ratio table. Each row must be equivalent.
| Factor | A | B |
|---|---|---|
| ×1 | ||
| ×2 | ||
| ×3 |
✍️ Justify Your Thinking
Sort each quantity: is it the independent variable (the input you choose) or the dependent variable (the output that responds)?
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: "Independent & Dependent Variables. 1. Independent Variable (x): The input / cause that is chosen or changes freely. 2. Dependent Variable (y): The output / effect that depends on the independent variable. 3. Sentence check: 'The (dependent) depends on the (independent)'" — and it works because ___.
Because independent variable 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 dependent variable to check my work.
✍️ TWR · WRITE 3 SENTENCES · 7 MIN
Independent & Dependent Variables. 1. Independent Variable (x): The input / cause that is chosen or changes freely. 2. Dependent Variable (y): The output / effect that depends on the independent variable. 3. Sentence check: 'The (dependent) depends on the (independent)' because ___
Independent & Dependent Variables. 1. Independent Variable (x): The input / cause that is chosen or changes freely. 2. Dependent Variable (y): The output / effect that depends on the independent variable. 3. Sentence check: 'The (dependent) depends on the (independent)' but ___
Independent & Dependent Variables. 1. Independent Variable (x): The input / cause that is chosen or changes freely. 2. Dependent Variable (y): The output / effect that depends on the independent variable. 3. Sentence check: 'The (dependent) depends on the (independent)' so ___
🌱 TWR · GROW THE KERNEL · 6 MIN
Answer these to add detail
Sentence starters (tap to use)
Student Workspace
Sort all eight cards, then read one pair aloud as a sentence: ___ depends on ___ .
| Column A | Column B |
|---|---|
✏️ Sketch Your Strategy
Differentiation Paths
Step-by-step with a worked model and sentence frames.
Akela chooses how many hours of sunlight each tray gets. Which is the INDEPENDENT variable?
A gym charges $24.95 per month. Which pair correctly names the variables?
Two quantities are related. Which statement is ALWAYS true?
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
At a local pizza shop a customer can order up to three toppings on a large pizza. The shop has 6 employees. The shop spends $1.15 on cheese and $1.95 on sauce for each pizza it makes.
✍️ Connection Reasoning
Which two quantities in this shop are related, so that one depends on the other — and which quantity does NOT depend on the number of pizzas at all?
In this scenario, the ___ is the independent variable. The ___ and the ___ both depend on it. The number of ___ stays the same no matter how many pizzas are made.
Turn & Talk — Connect
At the pizza shop, the number of employees (6) and the topping limit (3) stay fixed no matter how many pizzas are made, but the cost of cheese changes with every pizza. Why do employees and toppings NOT count as part of a dependent relationship here?
👂 Listen For
A strong answer explains that dependence requires movement — a constant like 6 employees can't be 'dependent' on anything because it never changes. A weak answer just says 'employees isn't a variable' without explaining why.
Extend: If the shop made 20 pizzas instead of a smaller batch, would the number of employees ever become 7? Use that answer to explain the difference between a variable and a fixed quantity.
Summarize: Explore Relationships Between Two Variables
Independent & Dependent Variables. 1. Independent Variable (x): The input / cause that is chosen or changes freely. 2. Dependent Variable (y): The output / effect that depends on the independent variable. 3. Sentence check: 'The (dependent) depends on the (independent)'
In your own words:
CLOSURE & REFLECT
⏱ ~8 min
Today I learned that ___ because ___.
One thing I am still not sure about is ___.
A dog walker charges by the hour. Which sentence is correct?
Bonus Exit Check
Which of these pairs is NOT a dependent relationship?
✍️ 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 sunlight as chosen (independent) and height as responding (dependent), and can say the sentence 'height depends on sunlight.' A weak answer just says 'height is bigger' without naming which quantity was chosen.
• Listen for the test 'who gets to choose this' rather than which number is bigger or which was mentioned first in the problem. A weak answer just repeats the card's category without justifying it.
• A strong answer explains that dependence requires movement — a constant like 6 employees can't be 'dependent' on anything because it never changes. A weak answer just says 'employees isn't a variable' without explaining why.
• Listen for a repeatable strategy ("ask what's chosen") rather than a guess based on which number looked bigger or came first in the sentence.
Common mistake: Naming the variables backward — calling the responding quantity 'independent' because it is the one the question asks about.
Answer Key (Teacher Appendix)
Hide this slide during presentation or move to the end of your copy.
✓ Practice 1: Independent: months; Dependent: total cost — You choose how many months to stay a member. The total cost responds to that.
✓ Practice 2: 369 — 30 minutes burned 123 Calories, and 90 is 3 times 30, so the Calories are 3 times 123, which is 369.
✓ Practice 3: Number of pizzas sold and number of employees the shop has — The shop has 6 employees whether it sells 5 pizzas or 500. That count does not respond to pizza sales.
✓ Practice 4: 4 hours to 8 cm — At 2 cm per hour, 4 hours of sunlight gives 4 times 2, which is 8 cm.
✓ Exit ticket: The total charge depends on the number of hours, so hours is the independent variable. — The walker chooses how many hours to work; the money that results responds to that choice. Hours is subject to choice, so hours is independent and the total charge is dependent.