Using Variables for Two Related Quantities

6.EE.C.9 — Expressions and Equations

In this lesson:

• Identify independent and dependent variables
• Write equations connecting two quantities
• Build tables and graph the relationship

Learning Objectives for This Lesson

By the end of this lesson, you will be able to:

1. Identify the independent variable (input) and the dependent variable (output)
2. Write an equation expressing dependent in terms of independent
3. Construct a table of values from an equation
4. Graph ordered pairs and interpret the relationship

Skills You Bring to This Lesson

You already know:

• Variables in expressions: , — letters for changing quantities
• Unit rates: 65 miles per hour — a constant ratio
• Ordered pairs: plotted on the coordinate plane

Today these three skills come together.

Two Quantities That Change Together

When one quantity depends on another:

• Independent variable — input; the quantity you choose or control
• Dependent variable — output; its value depends on the input

Axis convention: Independent → -axis | Dependent → -axis

Independent and Dependent: The Car Example

(time, hours) — independent: we choose

(distance, miles) — dependent: determined by

Check-In: Identify Independent and Dependent

Identify the independent and dependent variable in each:

1. Maria earns $12/hour: hours worked, earnings.
2. Plant grows 2 cm/week: weeks, height in cm.
3. Notebooks cost $3 each: notebooks, total cost.

Which quantity do you control?

Writing the Equation: Car at 65 mph

Situation: A car travels at 65 miles per hour.

• Independent: (hours) → -axis
• Dependent: (miles) → -axis

Equation:

The coefficient 65 is the unit rate — it links to .

Building the Distance-Time Table from Scratch

Always start at . The car hasn't moved yet — .

Non-Proportional: Starting Value Plus Rate

Garden: starts with 10 plants; 3 added per week.

• Independent: (weeks) | Dependent: (plants)
• Equation:
• : rate of change | : starting value

Garden Table: Reading Rate and Starting Value

: starting value | each step : rate

Guided Practice with a Cafeteria Situation

Situation: A cafeteria serves 25 students per minute.

1. Which quantity is independent? Which is dependent?
2. Define variables with units.
3. Write the equation.
4. Build the table for .

Complete all four steps, then advance for the answer.

Cafeteria Situation: Answers and Full Table

• Independent: (minutes) | Dependent: (students served)
• Equation:

Practice Writing Equations for Three Situations

Write the equation. Identify both variables.

1. Earn $12/hr: = hours, = earnings.
2. Plant starts 8 cm; grows 2 cm/week: = weeks, = height.
3. Notebooks $3 each: = notebooks, = cost.

Write all three equations before advancing.

Equation Answers for Three Practice Situations

1. — proportional; starts at zero
2. — non-proportional; starting height 8 cm
3. — proportional; starts at zero

Pattern: (proportional) vs. (non-proportional)

From Table to Graph: Four Steps

To graph a two-variable relationship:

1. Draw axes; label with variable names and units
2. Independent → -axis; Dependent → -axis
3. Plot each ordered pair from the table
4. Describe the pattern formed

For : plot , , , ,

Graph of : Proportional Relationship

Five points form a straight line through the origin.

Interpreting Points on the Graph

From the graph of :

• : after 2 hours, the car traveled 130 miles
• : after 3 hours, the car traveled 195 miles
• Steeper line → larger rate of change
• -intercept : starting distance is 0

Graph of : Non-Proportional

• -intercept at — line does not pass through origin
• Starting value 10 lifts the entire line upward

Equation, Table, Graph: Three Views

All three describe the same relationship.

Quick Check: Graph to Equation

A line passes through and .

1. What is the -intercept?
2. How much does increase per unit of ?
3. Write the equation.

Answer all three before advancing.

Key Ideas from This Lesson

• Independent → -axis (input) | Dependent → -axis (output)
• Proportional: | Non-proportional:
• Always start tables at input = 0

In : = rate of change, = starting value

This Lesson Connects to Future Topics

This lesson builds toward:

• 7.RP.A.2 — Proportional relationships, constant of proportionality
• 8.EE.B — Slope and -intercept as formal tools
• Algebra 1 — Linear functions

The equation, table, and graph you used today are the foundation.