📖 6.EE.C.9 — Writing & Analyzing Equations

Use variables to represent two quantities in a real-world problem. Write an equation to show how they relate!

⚖️

The Equation

An equation shows the relationship between the independent and dependent variable.

General form: y = kx
where k is the rate (unit rate / constant of proportionality).

⚡

The Rate (k)

The rate tells you how much the dependent variable changes for every 1 unit of the independent variable.

Find it: k = y ÷ x
It stays the same for every row in the table!

🎮

Independent Variable (x)

The input — what you control.
Goes in the first column of a table.
Plotted on the horizontal axis.

🔑 Easy trick: It's the thing that CAUSES the other thing to change!

🎯

Dependent Variable (y)

The output — the result.
Goes in the second column of a table.
Plotted on the vertical axis.

🔑 Easy trick: It DEPENDS on the independent variable!

🔑 Steps to Write an Equation from a Table:
1️⃣ Identify which column is independent (x) and which is dependent (y)
2️⃣ Find the rate: divide any y-value by its matching x-value → k = y ÷ x
3️⃣ Write the equation: y = kx

📊 Worked Examples

🚗 Example 1 — Road Trip
A car travels 60 miles for every hour driven.
Hours (x) Miles (y)
1 60
2 120
3 180
Rate: k = 60 ÷ 1 = 60  →  Check: 120 ÷ 2 = 60 ✅
y = 60x    or    miles = 60 × hours
💵 Example 2 — Babysitting
A babysitter earns $8 per hour.
Hours (x) Earnings (y)
1 8
2 16
4 32
Rate: k = 8 ÷ 1 = 8  →  Check: 16 ÷ 2 = 8 ✅
y = 8x    or    earnings = 8 × hours
⚠️ Common Mistake: The rate (k) must be the SAME for every row. If it's different, the relationship is NOT proportional and y = kx doesn't apply!

🧩 Matching — Drag Into the Right Box

Each card shows a table or equation. The highlighted part is what you need to classify.
Drag into the correct box. You'll get instant feedback! ✅❌
🚗 A car goes 60 mph. Table shows hours → miles.
→ The equation is: y = 60x
💵 A babysitter earns $8/hr. Hours worked → money earned.
→ The rate (k) is: 8
🍕 Each pizza costs $12. Number of pizzas → total cost.
→ The equation is: y = 12x
📚 Table: x=1,y=5 / x=2,y=10 / x=3,y=15
→ The rate (k) is: 5
🌱 A plant grows 3 cm per week. Weeks → height.
→ The equation is: y = 3x
🎮 A gamer earns 50 pts per level. Levels → total points.
→ The rate (k) is: 50

⚖️ This IS the Equation (y = kx)
The highlighted part is the correct equation

⚡ This IS the Rate (k)
The highlighted part is the unit rate

🗂️ Sort It!

Each card shows a scenario. The highlighted part is what you need to classify.
Sort into: Proportional (y = kx) or NOT Proportional — based on whether the table shows a constant rate. Instant feedback! ✅❌
Table: (1,4)(2,8)(3,12) → Rate = 4 each time
Table: (1,3)(2,7)(3,12) → Rate changes: 3, 3.5, 4
🚗 A car goes 55 mph every hour → y = 55x
🌡️ Temperature rises differently each hour → No constant rate
💵 $7 earned per hour worked → k = 7
Table: (2,10)(4,18)(6,28) → 10÷2=5 but 18÷4=4.5

✅ Proportional (y = kx)
Constant rate — can write y = kx

❌ NOT Proportional
Rate changes — y = kx does NOT apply

📋 Find the Rate & Write the Equation

Look at each table. Find the constant rate (k = y ÷ x), then write the equation!

🚗 Scenario 1 — Road Trip: A car travels at a constant speed. Every hour, it covers the same number of miles. Look at the table and figure out the rate per hour, then write the equation.
Hours (x) Miles (y)
1 65
2 130
3 195
Rate k = y ÷ x =
Equation: y =
💵 Scenario 2 — Babysitting: A babysitter earns the same amount each hour. Find the hourly rate and write the equation for total earnings.
Hours (x) Earnings $ (y)
1 9
3 27
5 45
Rate k =
Equation: y =
🍕 Scenario 3 — Pizza Party: Each pizza costs the same amount. Find the cost per pizza and write the equation for total cost.
Pizzas (x) Cost $ (y)
2 24
4 48
6 72
Rate k =
Equation: y =
🌱 Scenario 4 — Plant Growth: A plant grows at a constant rate each week. Find the growth rate per week and write the equation.
Weeks (x) Height cm (y)
1 4
3 12
7 28
Rate k =
Equation: y =

🔢 Complete the Table

Use the equation given to fill in the missing y-values. Type your answer and press Enter or click Check!

📏 Table 1 — Ribbon Cost: Ribbon costs $1.50 per foot. The equation is y = 1.5x (x = feet of ribbon, y = total cost in dollars). Fill in each missing cost!
Feet of Ribbon (x) Total Cost $ (y)
1
2
4
6
10
🎟️ Table 2 — Raffle Tickets: Each raffle ticket costs $2.50. The equation is y = 2.5x (x = tickets, y = total cost). Fill in the missing values!
Tickets (x) Total Cost $ (y)
1
2
3
5
8
🪙 Table 3 — Piggy Bank: You save $0.50 every day. The equation is y = 0.5x (x = days, y = total saved). Fill in the missing amounts!
Days (x) Total Saved $ (y)
1
4
6
10
20

✏️ Fill in the Blank

Read each scenario. The highlighted part is exactly what you need to identify. Choose the correct answer!

🔵 KEY CONCEPT: Identifying the Independent Variable
✏️ A student answers 3 questions each minute while taking a test.
Let m stand for the number of minutes and q stand for the number of questions answered.
⬇️ REMEMBER THIS ⬇️

The independent variable is the one you choose or control.
The dependent variable depends on it (it's the result).

Minutes is what you pick or measure first → it causes the number of questions to change!
The number of questions depends on how many minutes have passed.

🚗 A car travels 55 miles per hour. The equation is y = 55x. The 55 in this equation represents how many miles are traveled for every 1 hour.
The 55 is the
📚 A table shows: (1, 7), (2, 14), (3, 21). Every time x goes up by 1, y goes up by 7. The constant rate in this table stays the same at every row.
The rate (k) in this table is
💵 A worker earns $12 per hour. After 3 hours, they've earned $36. The equation is y = 12x. x = 3 is the number of hours — it's what the worker controls.
In this equation, x = 3 (hours) is the
💵 Same worker, y = 12x. After 3 hours, y = 36 (dollars earned) is the result — it's calculated from the hours worked.
In this equation, y = 36 (dollars) is the
🎮 A gamer earns 50 points per level. At level 4, they have 200 points. Which equation correctly shows this relationship? The correct equation uses levels as x and points as y.
The correct equation is
🌱 A plant grows 3 cm per week. After 5 weeks, how tall is it? Use the equation y = 3x where x = 5 weeks. Substitute in and calculate.
When x = 5, y =
📊 A table has a constant rate of change — every time x increases by 1, y increases by the same amount. This means the relationship IS proportional.
If the rate is constant, the relationship
🍕 Each pizza costs $14. You want to find the total cost for 6 pizzas. The equation is y = 14x. You plug in x = 6.
Total cost = y =
📊 Consider a set of data involving independent and dependent variables.
Which are the advantages of displaying the data in a TABLE instead of a graph?
Choose ALL that apply, then click Submit.

📈 Reading Graphs

Use each graph description to identify the rate and equation. The highlighted part is what to focus on!

📈 Graph 1: A line passes through (0,0), (1,5), (2,10), (3,15). The line goes up 5 units for every 1 unit right — that's the slope/rate.
The rate (k) is
The equation is y =
📈 Graph 2: A line passes through (0,0) and (4,20). The point (4, 20) means when x = 4, y = 20. Use this to find k.
k = y ÷ x = 20 ÷ 4 =
The equation is y =
📈 Graph 3: A graph shows hours worked on the horizontal axis and dollars earned on the vertical axis. The line passes through (1,11), (2,22), (3,33). The vertical axis label tells you which variable is dependent.
The dependent variable is
The equation is y =
🔑 On a graph: rate = rise ÷ run = how much y goes up for every 1 unit x increases. If the line passes through the origin (0,0), the equation is y = kx!