A classroom where we don't mimic. We think. We notice. We wonder. We argue. We revise. Welcome to the real math.
TASKS →
Before we begin — forget the textbook definition. What do you think a function actually does?
Turn to your group · whiteboards up · no one writes until everyone has spoken
PART 1 · SEEING THE PATTERN
The growing shape
~ a low-floor, high-ceiling puzzle ~
A shape is growing. Can you predict it?
Look carefully. Don't rush. Talk to your group before you write anything.
Here are the first 4 steps of a growing pattern:
Now use the slider to peek ahead — but FIRST, predict!
Predicted dots at this step: 1
Your group's mission:
Without counting one-by-one, can you find a rule — a recipe — that tells you how many dots are in any step? Even step 100? Even step 1,000?
✦ We notice...
✦ We wonder...
Nudge 1 · Look sideways
Instead of counting each step's total, look at the change between steps. How many dots are added each time? Is that number changing?
Nudge 2 · Build a table
Try making a table: step number in one column, total dots in the other. What do you see happening to the second column?
Nudge 3 · Two parts of a recipe
Every linear recipe has two ingredients: a starting amount and a growth amount. What's each one here? Can your rule tell a friend how to build step 50 without drawing it?
▸ Teacher note (don't peek, students)
Liljedahl move: Give groups vertical whiteboards, assign randomly. Do NOT confirm any rule — instead ask "How do you know?" and "Will your rule work for step 200?" Celebrate different forms: 2n−1, 1 + 2(n−1), n + (n−1). All equivalent. This surfaces the ideas of rate of change and initial value through pattern reasoning.
✦ Mindset moment
When a problem feels hard, that's your brain building new pathways. Slow, careful thinking beats fast, shallow answers every time.
— a note from your classroom
PART 1.5 · WHAT COUNTS AS A FUNCTION?
Inputs, outputs, and one big rule
~ definition through discovery ~
Is it actually a function?
Mathematicians have a strict rule about what counts as a function. But before we tell you the rule — let's see if you can figure it out by spotting the pattern.
The big definition:
A function is a relation in which every input has exactly one output. → The set of allowed inputs is called the domain. → The set of outputs that come out is called the range.
That sounds simple. But "exactly one output" is sneaky. Let's test it.
Round 1 · Which of these are functions?
For each scenario below, your group must decide: Function or not a function? Be ready to defend your answer.
Now argue:
What do all the functions have in common?
What do all the non-functions have in common?
Can your group write the rule in your own words — before checking the definition above?
Round 2 · The vertical line test
Here's a clever trick for graphs. Drag the red vertical line across each graph below. Count how many times it crosses the curve.
Drag the slider to move the red line. Watch what happens.
Graph A
Crosses: 0 times
Graph B
Crosses: 0 times
Graph C
Crosses: 0 times
Group challenge — figure it out:
→ On which graph does the red line ever cross the curve more than once?
→ What does that mean about that input having one output, or many?
→ Why does this prove something isn't a function?
Round 3 · Find the domain & range
Here's a linear function with a clear start and end. Drag the sliders to set what YOU think the domain and range are.
A function lives between its endpoints. What x-values does it accept? What y-values does it produce?
Drag the two dots on each number line. Where does the function start? Where does it end?
✦ What we now understand...
✦ Still confusing...
Real-life domain & range
Think of a vending machine. The buttons you can press are the domain — the allowed inputs. The snacks that can come out are the range. If pressing B4 sometimes gives chips and sometimes gives gum, is that machine a function? What if it always gives the same thing?
Why "exactly one" matters
Imagine a school where you type a student's ID number and it tells you their name. That's a function — one ID, one student. But if you type the name and it gives back the ID, is THAT still a function? What if two students have the same name?
▸ Teacher note
This task introduces the formal definition of a function (every input has exactly one output) without front-loading it as a rule to memorize. The vertical line test is presented as a discovery — students see it as a visual consequence of the definition, not a separate trick.
Domain & Range: Students often confuse these. The vending machine and student-ID metaphors in the hints are anchors worth returning to.
The non-functions in Round 1: the family-with-multiple-pets and the circle (sideways relation) both violate the rule. The phone-keypad example IS a function (each key maps to exactly one letter set), even though it feels strange.
~ from pattern to rule ~
Maya's mystery bike rental
Maya spots a bike-share stand at the park. The sign is frustratingly vague — it only shows three sample receipts:
The clues:
→ A 20-minute ride costs $3.50
→ A 35-minute ride costs $5.00
→ A 50-minute ride costs $6.50
Your job: figure out the hidden deal. There's a fixed unlock fee + a per-minute rate. What are they? And how did you KNOW?
✦ We notice...
✦ What if...
Three groups solved this. Whose thinking do you find most elegant?
Noor looked for change
"From 20 min to 35 min, that's 15 more minutes and $1.50 more. So the rate is $0.10/min. For 20 minutes of riding that's $2.00... so the unlock fee must be... hmm."
Can you finish Noor's thought?
Daniel drew it on a graph
"I plotted the three points. They were in a straight line. I extended the line backward until it hit the y-axis — that's where x = 0 minutes. That's the unlock fee."
Why does the y-intercept give the fee? Convince your group.
Priya used algebra
"Cost = fee + rate × minutes. I wrote 3.50 = fee + rate(20) and 5.00 = fee + rate(35). Two equations, two unknowns. Subtract!"
All three methods work. Which feels most natural to YOU — and why?
Build Maya's equation and see all three pieces come alive:
rate = $0.05/min
fee = $0.00
y = 0.05x + 0.00
Keep trying to match all 3 points...
▸ Teacher note
This is defronting in action. Three student voices, three methods. Do NOT rank them. Ask: "How are these the same underneath?" Students discover that rate-of-change, y-intercept, and simultaneous equations are the same idea in different clothes. Target: y = 0.10x + 1.50 (rate = $0.10/min, unlock fee = $1.50).
PART 2 · ONE IDEA, MANY FACES
A function in four outfits
~ the four representations ~
Which belong together?
Below are 12 cards: three functions, each shown four different ways (a story, a table, a graph, an equation). Work with your group. Group them. No peeking — this is about arguing, not guessing.
After you match them, argue:
Which representation do you trust most? Which is easiest to lie with? Which would you use to convince a skeptic?
▸ Teacher note
This is the heart of functions — fluency across representations (verbal, table, graph, equation). Let students disagree loudly about groupings; the productive struggle IS the learning.
~ play, then explain ~
The Slope & Y-Intercept Playground
Stop. Before you touch anything — predict. What will happen to the line when you make m bigger? What about b? Write your prediction in your group, THEN experiment.
Drag the sliders. What changes? What stays the same?
m = 1.0
b = 0
y = 1x + 0
Challenge your group:
→ Can you make a line that goes through (2, 5) AND (4, 11)?
→ Can you make a line with slope 0? What does it mean in real life?
→ What happens when m is negative? Tell a story that fits.
Slope = 0 means...
The line is flat. In life: your phone charge when it's 100% (stays at 100%, doesn't change with time). The temperature in a thermostat-controlled room. Can you find three more?
Negative slope means...
Something is DECREASING as time passes. Example: Adnan buys a laptop for $1,200 and it loses about $180 in value each year. That's m = −180. What other "going down" stories fit?
✦ Mindset moment
Not knowing something isn't the end of the story — it's the beginning of one. Add the word "yet" and watch it change.
— a note from your classroom
~ linear? or something else? ~
Not every relationship is a line
Three scenarios. Which are linear? Which are not? Justify with your group — NO calculations allowed yet. Use reasoning first.
Scenario A
You save $20 every month in a jar. How much is in the jar after x months?
Scenario B
The area of a square as you make its side longer. Side = x, Area = ?
Scenario C
A tank drains 3 liters per minute from a 100L start. Water left after x min?
Test your guesses:
Fill in a table for each. Look at the differences. Linear relationships have CONSTANT differences. Non-linear ones don't. Can you prove which is which without graphing?
A simulator. Watch what "linear" vs "non-linear" looks like:
▸ Teacher note
Distinguishing linear from non-linear. Scenario B (y = x²) produces non-constant differences (1, 3, 5, 7, 9...) — students often assume "growing" means "linear." Huge insight!
~ putting it all together ~
Your turn: create a story
Your group's final challenge. Choose one of these:
Invent a linear life
Create a real-world story for the function y = −5x + 50. What does x mean? y? Why is the slope negative? What does 50 mean? Share it out loud so your classmates can critique it.
Design a deceptive graph
Pick a linear function. Draw TWO graphs of the same data — one that makes the growth look dramatic, one that makes it look tiny. What did you change? (Hint: axes.) This is how news headlines lie!
Predict the future
A plant grows 2 cm per week. At week 3 it was 11 cm. Write the function. Use it to predict its height at week 20. Then ask: is it REASONABLE to trust your prediction? Why or why not?
Before you leave this classroom:
Write ONE sentence on your group's board that starts with: "The most important thing I now understand about functions is..." — and don't copy anyone else.
✦ Today I learned...
✦ I still wonder...
✦ Final mindset
Here, the right answer isn't the point. Thinking is — and so is defending what you think.