Unit rates, the constant of proportionality in a table, a graph and an equation, and percent in the real world — tax, tip, discount, percent change and simple interest. Study cards, hints, a practice quiz at three levels, and four workshop activities that check themselves.
Hours worked on the left. Dollars earned on the right. Three different days.
That 12 is the constant of proportionality. It is called k. Every row is the same equation: y = 12x. A straight line through (0, 0) is what proportional looks like.
Four things people get wrong
⚠️People often think…
If both numbers go up together, it is proportional.
Going up together is not enough: every row has to divide to the same k, and the graph has to start at (0, 0). A $5 fee plus $12 an hour goes up together and is not proportional.
Same k in every row, and it starts at zero.
⚠️People often think…
For percent, you move the decimal two places.
Two places, but the direction depends on which way you are going: 25% becomes 0.25 to calculate with it, and 0.25 becomes 25% to name it.
To calculate, left. To name it, right.
⚠️People often think…
For percent change, you divide by the new number.
Divide the change by the ORIGINAL number, because the original is what changed. $40 to $50 is 10 ÷ 40 = 25% up, not 10 ÷ 50.
Change over original. Always.
⚠️People often think…
25% off a $40 shirt means the answer is 25% of 40.
25% of 40 is $10, which is the discount, not the price. The price is $30. Read whether the question wants the part taken off or the amount you pay.
Discount, or price? Underline which one.
Find k, then use it
1Watch one
3 hours of work pays $36. What does 7 hours pay?
Find the pay for one hour.
36 ÷ 3 = 12.
So k = 12.
7 × 12 = $84.
2Do one with me
4 tickets cost $52. What do 9 tickets cost?
One ticket: 52 ÷ 4 =
So k =
Nine tickets: 9 × 13 =
💬One sentence, then you move on
Why do you divide first instead of multiplying first?
3Try one
A $40 shirt is 25% off. What do you pay? Write the number of dollars.
I want a hint first
25% off means you pay 75%. That makes k = 0.75. Multiply.
💬Last one — then you're done here
What does k tell you?
Where this goes
Where this lives
A pay stub, a tip, sales tax and a sale tag are all one number times another. An hourly wage is a k.
What this feeds
Next unit is inequalities — the same equations, but the answer is a range instead of one number.
Write one k from your own life: pay per hour, cost per game, minutes per page.
One card at a time — tap “Show me” to check yourself, then Next. Start at Foundation; when those feel easy, climb.
Helpful Hints
🪜 From a ratio to a percent — the ladder
Ratio → rate → unit rate → the same unit rate every time (proportional) → call it k → y = kx → a percent is just k written out of 100.
Compare, divide, repeat, name it, write it.
🤝 The whole topic in one idea
Everything here is one multiplication: y = k × x. Pay = 12 × hours. Tax = 0.06 × price. Tip = 0.20 × bill. Find the k, and the rest is one step.
⚠️ Traps the test loves
A straight line is NOT enough. Proportional means straight AND through (0, 0).
“Goes up by the same amount” is a line. Proportional needs the same RATIO — check y ÷ x.
8% is 0.08, not 0.8. Two places left.
Read what the question wants: the tax or the total? The discount or the sale price?
Percent change divides by the ORIGINAL, never the new number.
20% off then 10% off is not 30% off. The second one is taken from the new price.
📐 Three places to find k — the chart
Where
What to do
Proportional if…
Table
divide y by x in every row
you get the same number every time
Graph
find the point above x = 1
the line is straight and passes through (0, 0)
Equation
read the number in front of x
it looks like y = kx, with nothing added
Words
listen for per, each, for every
there is no starting fee
🎯 How the test will ask
A table — “is this proportional? what is the constant?” (divide every row).
A graph with a point marked — “what does (1, 12) mean?” (12 dollars for 1 hour: the unit rate).
A receipt — price, then discount, then tax. Do them in that order, one at a time.
Before and after — “what is the percent change?” (change ÷ original).
Two fractions — “what is the unit rate?” (divide the first by the second; flip and multiply).
✅ Before the test, can you…
Find a unit rate, even when both numbers are fractions?
Tell whether a table, a graph, an equation or a story is proportional — and say why?
Write y = kx for a situation and say what k means in words?
Find a tip, a tax, a discount and the total, without mixing them up?
Find a percent increase or decrease, dividing by the original?
Work backward from a sale price to the original price?
Pick your level
Look back at anything you missed — the hint that appeared is exactly what to reread tonight.
How sure did you feel?
Workshop
Work like a mathematician: sort it, graph it, put the steps in order, then argue from the numbers. Every activity checks itself, and hints are free.
Your practice never leaves this device. There is no account and no sign-in. Your work is saved in this browser only, and you can erase it whenever you want.
Your practice record — saved on this device
This is your record of the module on screen — it stays here and goes nowhere. Independent means you got it right on the first tap; supported means you got it after the explain-and-retry, or marked ‘I had it’ on a revealed answer. Both count, and neither is a grade. If your teacher asks, copy the row or show them this screen.
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