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. |
|---|---|
| 3 min | Watch the film “One Multiplication Rule Draws Every Shape”. |
| 20 min | Worksheet; Starter students draw the ten-dot circle by hand. |
| 10 min | Discuss: the questions below. |
| 7 min | Live demo: times 2, 3, 4, then 101 with 200 dots. |
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?
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