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Lesson card · Probability · ages 15–18 · 45 min The Galton board: order from random bounces
Goals
- Students count paths with Pascal’s triangle and binomial coefficients.
- Students turn path counts into probabilities.
- Students see why many random events give a predictable shape.
Before the film: predict
1,000 balls fall through 10 rows of pegs. At every peg a ball goes left or right with equal chance. Sketch where you think they pile up.
The film
Random Balls Build a Perfect Bell Curve — Hypnotic Galton Board · hypnoticpatterns.doner.cloud/video/random-balls-build-a-perfect-bell-curve
The lesson, minute by minute
| 5 min | Predict: each student sketches the pile; collect two or three different sketches on the board. |
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| 3 min | Watch the film; compare it with the sketches. |
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| 20 min | Worksheet in pairs, each pair at its level: Starter, Core or Challenge. |
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| 10 min | Discuss: the questions below. |
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| 7 min | Live demo on the board: tilt the board to 60 % and add rows; check the predictions. |
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The worksheet
Starter
A board has 3 rows. Draw every left/right path a ball can take. How many are there?
Answer: 2 × 2 × 2 = 8 paths.
How many of the 8 paths end in each of the 4 bins?
Answer: 1, 3, 3, 1: one path to each edge, three to each middle bin.
What is the chance that a ball lands in one of the two middle bins?
Answer: 6 ÷ 8 = 75 %.
Core
With 10 rows, how many different paths can one ball take?
Answer: 2¹⁰ = 1,024.
How many of them end in the middle bin (exactly 5 bounces to the right)?
Answer: C(10, 5) = 252.
What is the chance of the middle bin?
Answer: 252 ÷ 1,024 ≈ 24.6 %.
What is the chance of one of the two outer bins?
Answer: 2 ÷ 1,024 ≈ 0.2 %.
Out of 1,000 balls, about how many land in bins 4, 5 or 6?
Answer: (210 + 252 + 210) ÷ 1,024 ≈ 65.6 %, so about 656 balls.
Challenge
Every peg now sends 60 % of balls to the right. Which bin of the 10 is most likely, and how likely?
Answer: Bin 6: C(10, 6) · 0.6⁶ · 0.4⁴ ≈ 25.1 %. The centre of the bell moves to 10 × 0.6 = 6.
With an even number of rows n, the middle bin gets C(n, n/2) ÷ 2ⁿ of the balls. From how many rows is that below 10 %?
Answer: From 64 rows: C(64, 32) ÷ 2⁶⁴ ≈ 9.9 %. The bell spreads, so the middle bin gets a smaller share.
Discuss
- One ball tells you almost nothing. Why do 1,000 balls tell you almost everything?
- Where else do you meet a bell curve? (heights, measurement errors, test scores)
- What would change if every peg sent 60 % of balls to the right?
Free to use with your class. If it helped, tell another teacher, or Support the channel →
Before the film: predict
1,000 balls fall through 10 rows of pegs. At every peg a ball goes left or right with equal chance. Sketch where you think they pile up.
Starter
A board has 3 rows. Draw every left/right path a ball can take. How many are there?
How many of the 8 paths end in each of the 4 bins?
What is the chance that a ball lands in one of the two middle bins?
Core
With 10 rows, how many different paths can one ball take?
How many of them end in the middle bin (exactly 5 bounces to the right)?
What is the chance of the middle bin?
What is the chance of one of the two outer bins?
Out of 1,000 balls, about how many land in bins 4, 5 or 6?
Challenge
Every peg now sends 60 % of balls to the right. Which bin of the 10 is most likely, and how likely?
With an even number of rows n, the middle bin gets C(n, n/2) ÷ 2ⁿ of the balls. From how many rows is that below 10 %?
Answers · for the teacher The Galton board: order from random bounces
Starter
A board has 3 rows. Draw every left/right path a ball can take. How many are there?
2 × 2 × 2 = 8 paths.
How many of the 8 paths end in each of the 4 bins?
1, 3, 3, 1: one path to each edge, three to each middle bin.
What is the chance that a ball lands in one of the two middle bins?
6 ÷ 8 = 75 %.
Core
With 10 rows, how many different paths can one ball take?
2¹⁰ = 1,024.
How many of them end in the middle bin (exactly 5 bounces to the right)?
C(10, 5) = 252.
What is the chance of the middle bin?
252 ÷ 1,024 ≈ 24.6 %.
What is the chance of one of the two outer bins?
2 ÷ 1,024 ≈ 0.2 %.
Out of 1,000 balls, about how many land in bins 4, 5 or 6?
(210 + 252 + 210) ÷ 1,024 ≈ 65.6 %, so about 656 balls.
Challenge
Every peg now sends 60 % of balls to the right. Which bin of the 10 is most likely, and how likely?
Bin 6: C(10, 6) · 0.6⁶ · 0.4⁴ ≈ 25.1 %. The centre of the bell moves to 10 × 0.6 = 6.
With an even number of rows n, the middle bin gets C(n, n/2) ÷ 2ⁿ of the balls. From how many rows is that below 10 %?
From 64 rows: C(64, 32) ÷ 2⁶⁴ ≈ 9.9 %. The bell spreads, so the middle bin gets a smaller share.
Lesson card · Numbers and geometry · ages 15–18 · 45 min Fibonacci in a sunflower
Goals
- Students see how a fraction of a turn makes spokes.
- Students meet the golden ratio as the limit of Fibonacci ratios.
- Students explain why the golden angle fills a flower evenly.
Before the film: predict
Seeds are placed one by one, each turned by the same angle from the one before. What shape do you get with half a turn (180°)? With a third of a turn (120°)?
The film
Why Does a Sunflower Have 21 and 34 Spirals? · hypnoticpatterns.doner.cloud/short/why-does-a-sunflower-have-21-and-34-spirals
The lesson, minute by minute
| 5 min | Predict: sketches for 180° and 120°; ask for a guess at a “best” angle. |
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| 3 min | Watch the Short “Why do sunflowers turn every seed 137.5°?”. |
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| 20 min | Worksheet at three levels. |
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| 10 min | Discuss: the questions below. |
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| 7 min | Live demo: 137.5°, then 144° and 135° on the board; count the spokes together. |
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The worksheet
Starter
What fraction of a full turn is 180°? 120°? 144°? 135°?
Answer: 1/2, 1/3, 2/5, 3/8.
How many straight spokes does each of these angles give?
Answer: 2, 3, 5 and 8: the bottom number of the fraction.
Why does a fraction of a turn always make spokes?
Answer: After as many seeds as the bottom number, the turns add up to a whole number of turns, so the next seed lands on the first spoke again.
Core
Work out 8/5, 13/8, 21/13, 34/21 and 55/34 to three decimals. What do you notice?
Answer: 1.600, 1.625, 1.615, 1.619, 1.618: the ratios close in on the golden ratio φ ≈ 1.618 from both sides.
Work out 360° ÷ φ², with φ = 1.618034.
Answer: 360 ÷ 2.618034 ≈ 137.5°.
The golden angle is 0.381966 of a turn. Multiply by 13, 21 and 34. How far is each from a whole number?
Answer: 4.9656, 8.0213, 12.9868: 0.034, 0.021 and 0.013 of a turn away. Seeds 21 or 34 apart nearly line up, which makes the spirals.
Challenge
Show that 360° × (1 − 1/φ) = 360° ÷ φ², using φ² = φ + 1.
Answer: From φ² = φ + 1, dividing by φ: φ = 1 + 1/φ, so φ − 1 = 1/φ. Then 1 − 1/φ = (φ − 1)/φ = (1/φ)/φ = 1/φ².
Why do sunflowers show 21 and 34 spirals rather than 20 and 30?
Answer: Spirals appear between seeds that nearly line up, and for the golden angle those are a Fibonacci number apart: 13, 21, 34, 55…
Discuss
- Why would a plant gain from leaving no gaps between its seeds?
- Why can no fraction of a turn do as well as the golden angle?
- Where else have you met the Fibonacci numbers?
Free to use with your class. If it helped, tell another teacher, or Support the channel →
Before the film: predict
Seeds are placed one by one, each turned by the same angle from the one before. What shape do you get with half a turn (180°)? With a third of a turn (120°)?
Starter
What fraction of a full turn is 180°? 120°? 144°? 135°?
How many straight spokes does each of these angles give?
Why does a fraction of a turn always make spokes?
Core
Work out 8/5, 13/8, 21/13, 34/21 and 55/34 to three decimals. What do you notice?
Work out 360° ÷ φ², with φ = 1.618034.
The golden angle is 0.381966 of a turn. Multiply by 13, 21 and 34. How far is each from a whole number?
Challenge
Show that 360° × (1 − 1/φ) = 360° ÷ φ², using φ² = φ + 1.
Why do sunflowers show 21 and 34 spirals rather than 20 and 30?
Answers · for the teacher Fibonacci in a sunflower
Starter
What fraction of a full turn is 180°? 120°? 144°? 135°?
1/2, 1/3, 2/5, 3/8.
How many straight spokes does each of these angles give?
2, 3, 5 and 8: the bottom number of the fraction.
Why does a fraction of a turn always make spokes?
After as many seeds as the bottom number, the turns add up to a whole number of turns, so the next seed lands on the first spoke again.
Core
Work out 8/5, 13/8, 21/13, 34/21 and 55/34 to three decimals. What do you notice?
1.600, 1.625, 1.615, 1.619, 1.618: the ratios close in on the golden ratio φ ≈ 1.618 from both sides.
Work out 360° ÷ φ², with φ = 1.618034.
360 ÷ 2.618034 ≈ 137.5°.
The golden angle is 0.381966 of a turn. Multiply by 13, 21 and 34. How far is each from a whole number?
4.9656, 8.0213, 12.9868: 0.034, 0.021 and 0.013 of a turn away. Seeds 21 or 34 apart nearly line up, which makes the spirals.
Challenge
Show that 360° × (1 − 1/φ) = 360° ÷ φ², using φ² = φ + 1.
From φ² = φ + 1, dividing by φ: φ = 1 + 1/φ, so φ − 1 = 1/φ. Then 1 − 1/φ = (φ − 1)/φ = (1/φ)/φ = 1/φ².
Why do sunflowers show 21 and 34 spirals rather than 20 and 30?
Spirals appear between seeds that nearly line up, and for the golden angle those are a Fibonacci number apart: 13, 21, 34, 55…
Lesson card · Modular arithmetic · ages 13–18 · 45 min Times tables on a circle
Goals
- Students calculate with remainders (modular arithmetic).
- Students draw a cardioid as the envelope of straight lines.
- Students link the multiplier to the number of cusps.
Before the film: predict
Ten dots on a circle are numbered 0 to 9. Each dot n is joined to dot 2n, counting past 9 back to 0. Which dot does 7 join?
The film
One Multiplication Rule Draws Every Shape — 200 Dots on a Circle · hypnoticpatterns.doner.cloud/video/one-multiplication-rule-draws-every-shape
The lesson, minute by minute
| 5 min | Predict on the board with ten dots; agree that 7 joins 4, as 14 − 10 = 4. |
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| 3 min | Watch the film “One Multiplication Rule Draws Every Shape”. |
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| 20 min | Worksheet; Starter students draw the ten-dot circle by hand. |
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| 10 min | Discuss: the questions below. |
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| 7 min | Live demo: times 2, 3, 4, then 101 with 200 dots. |
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The worksheet
Starter
With 10 dots and times 2, list where every dot 0 to 9 goes.
Answer: 0→0, 1→2, 2→4, 3→6, 4→8, 5→0, 6→2, 7→4, 8→6, 9→8.
Draw the ten lines. Which dots get two lines arriving?
Answer: 2, 4, 6 and 8 get two lines each: for example 1 and 6 both go to 2. Dot 0 gets one, from 5, as 0 goes to itself.
Core
With 200 dots and times 2, where do dots 150 and 199 go?
Answer: 300 − 200 = 100, and 398 − 200 = 198.
With 200 dots, which dots go to themselves under times 2? Under times 3?
Answer: Times 2: only dot 0, as 2n = n needs n = 0. Times 3: dots 0 and 100, as 2n must be 0 or 200.
How many cusps do times 2, times 3 and times 4 give?
Answer: 1, 2 and 3: the cardioid, the nephroid and a curve with three cusps, always k − 1.
Challenge
With 200 dots, show that times 101 sends every even dot to itself and every odd dot to the opposite side.
Answer: 101n = 100n + n. For even n, 100n is a multiple of 200, so the dot stays n. For odd n, 100n leaves 100, so the dot is n + 100: half way round.
With 200 dots, what does times 201 draw, and why?
Answer: Nothing: 201n = 200n + n, so every dot goes to itself and every line has length zero.
Discuss
- Why do the straight lines look like a curve?
- Why does only the remainder after dividing by 200 matter?
- What would times 202 draw?
Free to use with your class. If it helped, tell another teacher, or Support the channel →
Before the film: predict
Ten dots on a circle are numbered 0 to 9. Each dot n is joined to dot 2n, counting past 9 back to 0. Which dot does 7 join?
Starter
With 10 dots and times 2, list where every dot 0 to 9 goes.
Draw the ten lines. Which dots get two lines arriving?
Core
With 200 dots and times 2, where do dots 150 and 199 go?
With 200 dots, which dots go to themselves under times 2? Under times 3?
How many cusps do times 2, times 3 and times 4 give?
Challenge
With 200 dots, show that times 101 sends every even dot to itself and every odd dot to the opposite side.
With 200 dots, what does times 201 draw, and why?
Answers · for the teacher Times tables on a circle
Starter
With 10 dots and times 2, list where every dot 0 to 9 goes.
0→0, 1→2, 2→4, 3→6, 4→8, 5→0, 6→2, 7→4, 8→6, 9→8.
Draw the ten lines. Which dots get two lines arriving?
2, 4, 6 and 8 get two lines each: for example 1 and 6 both go to 2. Dot 0 gets one, from 5, as 0 goes to itself.
Core
With 200 dots and times 2, where do dots 150 and 199 go?
300 − 200 = 100, and 398 − 200 = 198.
With 200 dots, which dots go to themselves under times 2? Under times 3?
Times 2: only dot 0, as 2n = n needs n = 0. Times 3: dots 0 and 100, as 2n must be 0 or 200.
How many cusps do times 2, times 3 and times 4 give?
1, 2 and 3: the cardioid, the nephroid and a curve with three cusps, always k − 1.
Challenge
With 200 dots, show that times 101 sends every even dot to itself and every odd dot to the opposite side.
101n = 100n + n. For even n, 100n is a multiple of 200, so the dot stays n. For odd n, 100n leaves 100, so the dot is n + 100: half way round.
With 200 dots, what does times 201 draw, and why?
Nothing: 201n = 200n + n, so every dot goes to itself and every line has length zero.
Lesson card · Geometry and probability · ages 14–18 · 45 min The chaos game: a fractal from chance
Goals
- Students find midpoints with coordinates.
- Students describe the Sierpiński triangle and its self-similarity.
- Students see how a random rule can still give an exact shape.
Before the film: predict
Pick one of a triangle’s corners at random, jump halfway towards it, mark the spot and repeat thousands of times. Will you get a random cloud, or something else?
The film
Pure Randomness Draws a Perfect Fractal — The Chaos Game · hypnoticpatterns.doner.cloud/video/pure-randomness-draws-a-perfect-fractal
The lesson, minute by minute
| 5 min | Predict; then everyone does five jumps by hand with a die on a printed triangle. |
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| 3 min | Watch the film “Pure Randomness Draws a Perfect Fractal”. |
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| 20 min | Worksheet at three levels. |
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| 10 min | Discuss: the questions below. |
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| 7 min | Live demo: the square with and without the restriction, then the pentagon at 0.618. |
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The worksheet
Starter
The corners are A (0, 0), B (8, 0) and C (4, 8). Start at (4, 4) and jump halfway to A, then C, then B, then C. Write down each point.
Answer: (2, 2), (3, 5), (5.5, 2.5), (4.75, 5.25).
How do you find the point halfway between two points?
Answer: Take the average of the x values and the average of the y values.
Core
Each stage of the Sierpiński triangle removes the middle quarter of every triangle. What share of the area is left after 1, 2 and 5 stages?
Answer: 3/4 = 75 %, 9/16 ≈ 56.3 %, (3/4)⁵ ≈ 23.7 %.
How many small triangles are there after 5 stages?
Answer: 3⁵ = 243.
Each jump halves the distance to the fractal. After 10 jumps, what part of the starting distance is left?
Answer: (1/2)¹⁰ = 1/1,024, about a thousandth.
Challenge
The Sierpiński triangle is made of 3 copies of itself at half the size. Its dimension d solves 2ᵈ = 3. Find d.
Answer: d = log 3 ÷ log 2 ≈ 1.585: more than a line (1), less than a filled triangle (2).
Play the game in a square with jumps of one half. Why do the points just fill the square?
Answer: Four half-size copies of a square fill the square exactly, with no gaps, so the shape the rule leaves unchanged is the whole square.
Discuss
- How can random choices give a shape that is exactly the same every time?
- What changes if you never pick the same corner twice in a row?
- Where in nature do you see shapes made of smaller copies of themselves?
Free to use with your class. If it helped, tell another teacher, or Support the channel →
Before the film: predict
Pick one of a triangle’s corners at random, jump halfway towards it, mark the spot and repeat thousands of times. Will you get a random cloud, or something else?
Starter
The corners are A (0, 0), B (8, 0) and C (4, 8). Start at (4, 4) and jump halfway to A, then C, then B, then C. Write down each point.
How do you find the point halfway between two points?
Core
Each stage of the Sierpiński triangle removes the middle quarter of every triangle. What share of the area is left after 1, 2 and 5 stages?
How many small triangles are there after 5 stages?
Each jump halves the distance to the fractal. After 10 jumps, what part of the starting distance is left?
Challenge
The Sierpiński triangle is made of 3 copies of itself at half the size. Its dimension d solves 2ᵈ = 3. Find d.
Play the game in a square with jumps of one half. Why do the points just fill the square?
Answers · for the teacher The chaos game: a fractal from chance
Starter
The corners are A (0, 0), B (8, 0) and C (4, 8). Start at (4, 4) and jump halfway to A, then C, then B, then C. Write down each point.
(2, 2), (3, 5), (5.5, 2.5), (4.75, 5.25).
How do you find the point halfway between two points?
Take the average of the x values and the average of the y values.
Core
Each stage of the Sierpiński triangle removes the middle quarter of every triangle. What share of the area is left after 1, 2 and 5 stages?
3/4 = 75 %, 9/16 ≈ 56.3 %, (3/4)⁵ ≈ 23.7 %.
How many small triangles are there after 5 stages?
3⁵ = 243.
Each jump halves the distance to the fractal. After 10 jumps, what part of the starting distance is left?
(1/2)¹⁰ = 1/1,024, about a thousandth.
Challenge
The Sierpiński triangle is made of 3 copies of itself at half the size. Its dimension d solves 2ᵈ = 3. Find d.
d = log 3 ÷ log 2 ≈ 1.585: more than a line (1), less than a filled triangle (2).
Play the game in a square with jumps of one half. Why do the points just fill the square?
Four half-size copies of a square fill the square exactly, with no gaps, so the shape the rule leaves unchanged is the whole square.