Lesson 9-1: Explore Relationships Between Two Variables

Warm-Up 1

Marcus earns money by mowing lawns. The more hours he works, the more money he earns. Which is the independent variable?

Vocabulary: The independent variable is the variable you control or that changes on its own. The dependent variable is the variable that responds or depends on the other one. Think: input causes the output.

Sentence frame: "The independent variable is ___ because it is the one Marcus ___."

✓Correct! Hours worked is the independent variable because Marcus controls how many hours he works. The money he earns depends on that.

✗Not quite. The independent variable is the one you control. Marcus decides how many hours to work (B), and his earnings depend on that choice.

Warm-Up 2

True or false: "The dependent variable is the one you control."

Hint: The word dependent means it depends on something else. The variable you control is the independent variable. The one that changes as a result is the dependent variable.

Sentence frame: "This statement is ___ because the dependent variable is the one that ___, not the one you ___."

✓Correct! The dependent variable is NOT the one you control. It is the one that depends on the independent variable. The independent variable is the one you control.

✗The statement is false. The dependent variable depends on the other variable. The independent variable is the one you control.

Practice 1

Sort each variable into the correct category: Independent or Dependent.

Remember: The independent variable is the input — the cause. The dependent variable is the output — the effect. Ask yourself: "Which one changes because of the other?"

Hours studied

Test score

Amount of rain

Height of a plant

Number of workers

Time to finish a job

Independent Variable (input)

Dependent Variable (output)

Sentence frame: "___ is independent because it is the input. ___ depends on it, so it is the dependent variable."

✓All correct! Independent variables are the inputs (hours studied, amount of rain, number of workers). Dependent variables are the outputs that change because of them.

✗Some items are in the wrong group. Ask: "Which one causes the other to change?" The cause is independent; the effect is dependent. Try again!

Practice 2

You earn $8 per hour at a part-time job. Identify the variables below.

Think about it: What do you choose or control? That is the independent variable. What changes because of your choice? That is the dependent variable.
Key words: input = what you put in, output = what you get out.

Independent variable:

Dependent variable:

Explain your reasoning:

Sentence frame: "The independent variable is ___ because you control how many ___ you work. The dependent variable is ___ because your total pay depends on ___."

✓Correct! Hours is the independent variable (you choose how many to work), and earnings is the dependent variable (your pay depends on hours).

✗Check your answers. You choose the number of hours (independent), and your earnings depend on that (dependent).

Practice 3

Which table shows a relationship where y depends on x?

Strategy: In a relationship between two variables, as x changes, y changes in a predictable pattern. Look for a table where y follows a rule based on x.

Explain your reasoning:

Sentence frame: "In table ___, as x increases by ___, y increases by ___. The rule is y = ___."

✓Correct! In table A, y = 5x. As x increases by 1, y increases by 5. This shows a clear dependent relationship where y depends on x.

✗Look for a consistent pattern. In table A, y = 5x (each y-value is 5 times the x-value). That is a clear dependent relationship.

Practice 4

In the sentence: "The faster you drive, the sooner you arrive." Which is the independent variable?

Think: Which variable is the input that causes the change? Which is the output that changes as a result? The speed you choose affects how quickly you arrive.

Explain your reasoning:

Sentence frame: "The independent variable is ___ because you control how fast you ___. The time to arrive ___ on your speed."

✓Correct! Driving speed is the independent variable. You control how fast you drive, and the time to arrive depends on your speed.

✗The independent variable is driving speed (C). You choose the speed, and the arrival time changes because of it.

Practice 5

True or false: In the equation y = 3x, x is the dependent variable.

Hint: In the equation y = 3x, you choose a value for x (the input) and then calculate y (the output). The dependent variable is the one that gets calculated — the output.

Explain your reasoning:

Sentence frame: "This is ___ because in y = 3x, you choose ___ and calculate ___. The dependent variable is ___, not ___."

✓Correct! The statement is false. In y = 3x, x is the independent variable (input). You pick x, and y depends on it, making y the dependent variable.

✗The statement is false. In y = 3x, x is the independent variable (input) and y is the dependent variable (output). You choose x; y is calculated from it.

Challenge

Describe a real-world situation that involves two variables. Identify the independent and dependent variables, write an equation, and explain which variable depends on the other.

Think about it: Choose a situation where one thing causes another to change. Examples: price per item and total cost, minutes of exercise and calories burned, or gallons of gas and miles driven.
Steps: (1) Describe your situation. (2) Name both variables. (3) Tell which is independent and which is dependent. (4) Write an equation like y = ___x.

Your real-world situation:

Independent variable and dependent variable:

Your equation:

Explain which variable depends on the other and why:

Sentence frame: "In my situation, the independent variable is ___ because I control ___. The dependent variable is ___ because it changes when ___ changes. My equation is ___ = ___ times ___."

🟠 Level 0 — Extra Support

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