The variable YOU choose or control. It goes on the x-axis.
Example: hours worked.
Dependent Variable
Variable dependiente
The variable that CHANGES based on the independent variable. It
goes on the y-axis. Example: total pay.
Input-Output Table
Tabla de entrada-salida
A table showing x-values (input) and y-values (output). It helps
you see the pattern.
Coordinate Graph
Gráfico de coordenadas
A picture that shows the relationship between two variables.
Points are plotted as (x, y).
Equation (Rule)
Ecuación (Regla)
A math sentence showing the relationship. You plug in x to find
y.
Pattern / Rate
Patrón / Tasa
The amount that y changes each time x goes up by 1. It stays the
same in a linear relationship.
Sentence Frames
The independent variable is
___ because I can choose its value.
The dependent variable is
___ because it changes based on the
other variable.
The equation y = ___ shows the
relationship between x and y.
When x increases by 1, y increases by
___. This is the
rate of change.
To fill in the table, I substitute the
x-value into the equation to find y.
On the graph, the point (___,
___) means that when x is
___, y is
___.
The graph goes up / down from left to
right, which means as x increases, y
increases / decreases.
Step-by-Step Visual Guides
Example 1: Identify Independent & Dependent Variables
Scenario: A baker earns $8 per hour. How does the number of hours
affect total pay?
Ask: Which variable do YOU choose?
Hours worked (h) ← You
decide how many hours to work.
Ask: Which variable CHANGES because of the other?
Total pay (p) ← It depends
on the hours you work.
Write the equation:
p = 8 ×
h
Example 2: Complete a Table and Graph the Relationship
Equation: y = 2x + 1
Make a table. Choose x-values and plug them in.
x (input)
y = 2x + 1
y (output)
(x, y)
0
2(0) + 1
1
(0, 1)
1
2(1) + 1
3
(1, 3)
2
2(2) + 1
5
(2, 5)
3
2(3) + 1
7
(3, 7)
4
2(4) + 1
9
(4, 9)
Plot each (x, y) point on the graph.
Connect the points. They form a straight line!
The rate of change is +2 (y goes
up by 2 each time x goes up by 1).
Simplified Practice
1. A taxi charges $3 per mile. Which is the independent
variable: miles or total cost?
Which variable do you choose? You decide how far to ride.
Miles is the independent variable. Total cost is the dependent
variable.
2. Write an equation: You earn $6 for each lawn you mow.
Let x = number of lawns. Total money = $6 times the number of
lawns.
y = 6x (where x = lawns mowed, y = total money earned)
3. For y = 4x, find y when x = 3.
Replace x with 3 in the equation: y = 4(3).
y = 12
4. Complete the table for y = x + 5: x: 0, 1, 2, 3 y:
?, ?, ?, ?
Add 5 to each x-value to get y.
y: 5, 6, 7, 8
5. What is the ordered pair when x = 2 and y = 10?
An ordered pair is written as (x, y).
(2, 10)
6. In y = 3x + 2, what is the rate of change?
The rate of change is the number multiplied by x. It tells you
how much y goes up when x goes up by 1.
The rate of change is 3. Each time x goes up by 1, y goes up by
3.
7. A plant grows 2 inches each week. After 0 weeks it is 3
inches tall. Write the equation.
The starting height is 3. It grows 2 more inches per week (w).
h = 2w + 3 (where w = weeks, h = height in inches)
8. Plot the points (1, 4), (2, 6), (3, 8). What equation fits
these points?
Look at the pattern: when x goes up by 1, y goes up by 2. When x
= 1, y = 4. Try y = 2x + ?.
y = 2x + 2. Check: 2(1)+2=4, 2(2)+2=6, 2(3)+2=8. All correct!
9. For y = 5x, find x when y = 35.
Replace y with 35: 35 = 5x. Divide both sides by 5.
x = 7
10. Does the point (3, 11) fit the equation y = 3x + 2?
Plug in x = 3. Does y equal 11?
y = 3(3) + 2 = 11. Yes! The point (3, 11) fits the equation.
Real-World Connections
🚘 Road Trip
A car drives 60 miles per hour. Distance = 60 × hours. The
hours you drive is independent. The distance traveled is
dependent. After 3 hours: d = 60(3) = 180 miles.
🍭 Lemonade Stand
You sell cups of lemonade for $2 each. Total money = 2 ×
cups sold. If you sell 15 cups: y = 2(15) = $30. You choose how
many cups to sell (independent). Money earned depends on that
(dependent).
📷 Phone Battery
Your phone loses 5% battery per hour. Battery left = 100 −
5h. After 4 hours: 100 − 5(4) = 80%. Hours is independent,
battery percentage is dependent.
🌱 Growing Plants
A sunflower is 10 cm tall and grows 3 cm per week. Height = 3w +
10. After 6 weeks: 3(6) + 10 = 28 cm. Weeks is independent,
height is dependent.
Challenge Problems
Medium
1. A gym membership costs $25 per month plus a one-time $50 signup
fee. Write an equation for total cost (c) based on months (m). How
much do you pay after 8 months?
Total cost = monthly fee × months + signup fee.
c = 25m + 50. After 8 months: c = 25(8) + 50 = $250.
Medium
2. A table shows: x = {1, 2, 3, 4} and y = {7, 12, 17, 22}. Find
the equation relating x and y.
Find the rate of change (difference in y-values). Then figure out
the starting value.
Rate of change: +5. When x=1, y=7. Testing: 5(1) + 2 = 7.
Equation: y = 5x + 2.
Medium
3. Two friends save money. Ali starts with $20 and saves $5/week.
Bri starts with $50 and saves $3/week. Write equations for both.
After how many weeks will they have the same amount?
Ali: a = 5w + 20. Bri: b = 3w + 50. Set them equal.
5w + 20 = 3w + 50 → 2w = 30 → w = 15. After 15 weeks,
both have $95.
Hard
4. A candle is 12 inches tall and burns 0.5 inches per hour. Write
an equation. When will the candle be completely gone? Graph the
relationship.
Height = starting height − burn rate × hours. Set
height = 0 to find when it runs out.
h = 12 − 0.5t. When h = 0: 12 − 0.5t = 0 → t = 24
hours. The graph starts at (0, 12) and goes down to (24, 0).
Hard
5. A pool is being filled at a rate of 50 gallons per minute. It
already has 200 gallons. The pool holds 3,200 gallons. Write an
equation and determine how long until it is full.
g = 50m + 200 where g = gallons and m = minutes. Set g = 3200.
6. The graph of a relationship passes through (0, 4) and (3, 13).
Find the equation of the line.
Rate of change = (change in y) / (change in x) = (13 − 4) /
(3 − 0). The y-value when x = 0 is the starting value.
Rate = 9/3 = 3. Starting value = 4. Equation: y = 3x + 4.
Expert
7. A rental car costs $40/day plus $0.15 per mile. Another company
charges $60/day with unlimited miles. For what number of miles per
day is the first option cheaper?
Company 1: c = 40 + 0.15m. Company 2: c = 60. When is Company 1
less?
40 + 0.15m < 60 → 0.15m < 20 → m < 133.3. The
first option is cheaper when driving fewer than 134 miles per day.
Expert
8. Create a table and equation for a relationship where: the
dependent variable starts at 100 and DECREASES by 8 each time the
independent variable increases by 1. When does the dependent
variable reach 0?
Starts at 100, goes down by 8. Equation: y = 100 − 8x. Set y
= 0.
y = 100 − 8x. Table: (0,100), (1,92), (2,84)... When y = 0:
100 − 8x = 0 → x = 12.5. It reaches 0 between x = 12
and x = 13.
Expert
9. A scientist records bacteria growth: Hour 0 = 10 bacteria, Hour
1 = 20, Hour 2 = 40, Hour 3 = 80. Is this a linear relationship?
Why or why not? Predict Hour 5.
Check if the DIFFERENCE between y-values is constant (linear) or
if the RATIO is constant (exponential).
NOT linear! Differences: 10, 20, 40 (not constant). The bacteria
DOUBLE each hour (ratio = 2). Hour 4 = 160, Hour 5 = 320. This is
exponential growth.
Expert
10. Three relationships: y = 2x, y = 2x + 5, y = 2x − 3.
Without graphing, explain what these three lines have in common
and how they differ.
Look at the rate of change (coefficient of x) and the starting
value (constant added).
All three have the same rate of change (2), so they are PARALLEL
lines (same steepness). They differ in where they cross the
y-axis: at 0, 5, and −3 respectively. Same slope, different
y-intercepts.
Real-World Investigations
Investigation 1: School Fundraiser Comparison
Your school is choosing between two fundraiser options. Option A:
Sell candy bars for $2 each (no upfront cost). Option B: Sell
custom t-shirts for $10 each, but you must pay $120 for the
printing setup. Write equations for profit from each option.
Create tables for 0–30 items sold. Graph both on the same
coordinate plane. At what point does Option B become more
profitable? Present your recommendation with evidence.
Research the average monthly temperature in your city (or use:
Jan=35, Feb=38, Mar=48, Apr=58, May=68, Jun=78, Jul=83, Aug=81,
Sep=74, Oct=62, Nov=50, Dec=39). Create a table with month number
(1–12) as the independent variable and temperature as the
dependent variable. Graph it. Is this relationship linear? Why
does this matter for understanding climate vs. weather? What
patterns do you notice?
Materials: graph paper, colored pencils, internet for local data
(optional)
Investigation 3: Design Your Own Business
Create a small business concept. Determine: (a) your fixed costs
(startup expenses), (b) your variable cost per item, and (c) your
selling price per item. Write an equation for total cost, total
revenue, and profit. How many items must you sell to break even?
Create a complete table (0–50 items) and graph showing where
profit turns positive.
A function machine takes a number in and gives a number out. Input
3 → Output 11. Input 5 → Output 17. Input 10 →
Output 32. What is the rule? What output does input 100 give?
Rule: y = 3x + 2. Check: 3(3)+2=11, 3(5)+2=17, 3(10)+2=32. For
input 100: 3(100)+2 = 302.
Teaser 2: The Staircase Problem
You are building staircases with blocks. A 1-step staircase uses 1
block. A 2-step uses 3 blocks. A 3-step uses 6 blocks. How many
blocks for a 10-step staircase? Can you find the equation?
Pattern: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55. These are triangular
numbers! Equation: blocks = n(n+1)/2. For 10 steps: 10(11)/2 = 55
blocks. Note: This is NOT linear!
Teaser 3: The Mystery Relationship
Two variables are related. When x = 0, y = 50. Each time x
increases by 1, y gets closer to 0 but NEVER reaches it. After
many steps, y is still positive but very tiny. Is this a linear
relationship? What type of relationship could this be?
NOT linear! A linear relationship would eventually reach 0 and go
negative. This describes exponential DECAY (like y = 50 ×
0.5^x). Real examples: radioactive decay, cooling temperatures,
bouncing balls losing height.
Teaser 4: The Table Detective
Someone spilled coffee on a table of values! You can only see: x =
{1, ?, 3, 4, ?} and y = {5, ?, 13, 17, ?}. The relationship is
linear. Fill in ALL missing values.
Rate of change: (13−5)/(3−1) = 8/2 = 4. So y goes up
by 4 for each x increase of 1. Missing x values: 2 and 5. y
values: 9 and 21. Equation: y = 4x + 1.
Where This Math Leads — 7th Grade Preview
Proportional Relationships & Slope
In 7th grade, you will study proportional relationships in depth.
The "rate of change" you learned becomes the formal
concept of slope (rise over run). You will learn
that y = mx means a proportional relationship always passes
through the origin (0, 0).
Slope-Intercept Form: y = mx + b
The equations you wrote (like y = 3x + 2) are already in
slope-intercept form! In 7th–8th grade, you will formally
learn that m is the slope and
b is the y-intercept, and you will graph lines
quickly using these two values.
Systems of Equations
Remember finding when two friends have the same savings? That is a
"system of equations." In 8th grade, you will solve
these systematically using graphing, substitution, and elimination
methods.
Self-Assessment Checklist
Click each box to track your understanding:
I can identify the independent and dependent variable in a
situation
I can write an equation showing the relationship between two
variables
I can create an input-output table from an equation
I can plot ordered pairs on a coordinate graph
I can find the rate of change from a table or graph
I can determine the equation from a table of values
I can connect equations, tables, and graphs to real-world
situations
I can compare two linear relationships and find where they
intersect
I can distinguish between linear and non-linear relationships
I can create my own real-world problem involving two-variable
relationships