Variables and Two-Quantity Relationships

6.EE.C.9 — Grade 6 Expressions and Equations

In this lesson:

• Identify independent and dependent variables
• Write equations and build tables of ordered pairs
• Graph and interpret two-variable relationships

Four Goals for Today's Lesson

By the end, you should be able to:

1. Identify independent (controlled) and dependent (determined) variables — and explain which axis each belongs on
2. Write the equation with the dependent variable on the left
3. Generate a table starting at
4. Plot and interpret a graph

Which Quantity Do You Control?

A car travels at 65 miles per hour.

• You decide how many hours the trip lasts
• Once you decide that — what else do you know automatically?

In 6.EE.B.6, you wrote — a ranging expression. Today we name what and the result actually are.

Controlled vs. Determined: Two Variable Types

The independent variable — what you choose or control → x-axis

The dependent variable — what responds to your choice → y-axis

For the car:

• You choose (hours) — independent → x-axis
• (miles) is determined — dependent → y-axis

Building the Table:

Always include — the starting condition tells you where the graph begins.

Quick Check: Applying the Earnings Equation

Maria earns $12 per hour:

is independent (hours worked); is dependent (earnings)

What is when ?

Substitute, then check: does the unit rate match the coefficient?

A Non-Proportional Example:

At : 10 plants already existed. The 3 = weekly rate; the 10 = starting value.

Your Turn: Identify, Write, Build

A bathtub holds 60 gallons and drains at 4 gallons per minute.

• Which quantity do you control? Which responds?
• Write the equation (dependent on left)
• Build the table for

What is the water volume at ?

Check-In: What Does the Zero Row Mean?

For , what does at mean?

• A) The garden grew 10 plants in week 1
• B) There were 10 plants before week 1
• C) The growth rate is 10 plants per week

Choose, then advance.

From Table to Graph: Same Pairs, Drawn

You already have the ordered pairs. Graphing means placing them as dots on a coordinate plane.

• Horizontal axis: independent variable
• Vertical axis: dependent variable
• Connect the dots — the pattern emerges

The graph is the table, drawn. Nothing new is invented.

Graph of : A Proportional Line

The line passes through the origin — at , . Steepness = 65 miles per hour.

Garden Graph: Line Avoids the Origin

The line does NOT start at the origin — it starts at . That's the 10 plants already there.

One Test Tells You Which Type It Is

Quick check: Does the output equal 0 when the input is 0?

• Yes → proportional → line through the origin
• No → non-proportional → line does not pass through the origin

Equation, Table, Graph: One Relationship

Three windows onto the same relationship — given any one, build the others:

• Equation → table: substitute values
• Table → graph: plot each pair as a dot
• Graph → equation: read starting value and rate

One relationship, three views.

Reading a Graph to Write an Equation

A line through , , ,

Step 1: Starting value at → 6

Step 2: Rate — increases by 3 per unit of

Equation:

Your Turn: Write Equations from Graphs

Graph A: , , ,

Graph B: , , ,

For each:

1. State the starting value
2. State the rate per unit of
3. Write the equation

One line passes through the origin; one does not.

What You Now Know and Watch Out For

✓ Independent → x-axis; dependent → y-axis
✓ Table starts at 0 — that row shows starting value
✓ Equation, table, graph: three views of a relationship

Axis follows meaning, not equation order
3 = rate; 10 = starting value
Non-origin line is correct, not an error

Where These Ideas Lead Next

• Grade 7: coefficient in → constant of proportionality
• Grade 8: steepness + starting value → slope () + y-intercept () in

The intuitions you're building now make those formalizations feel obvious.