What quantity did you change?
Labs / Mathematics
Pascal's Triangle
Every number is just the sum of the two above it, yet that tiny rule hides binomial coefficients, powers of two, and a fractal. Click any cell to light up the paths that build it, then colour the triangle by divisibility to watch a pattern emerge.
Labs / Mathematics / Pascal's Triangle
Pascal's Triangle
Every number is just the sum of the two above it, yet that tiny rule hides binomial coefficients, powers of two, and a fractal. Click any cell to light up the paths that build it, then colour the triangle by divisibility to watch a pattern emerge.
Why it matters
Why explore this?
Mathematics helps us describe patterns precisely and predict what comes next. This model turns an abstract relationship into something you can change and observe.
Explore
Try the model
C(5, 2) = 10row sum 2⁵ = 32term 10·a³b²
Investigate
Questions to test
Predict first, try it in the model, then use what you observe to answer.
What to try
- Click a cell and count the highlighted paths from the tip. Does the number of routes match the value inside?
- Add up a whole row in your head. Why does every row total land on a power of two?
- Switch on Mod 2. Which cells stay coloured, and why does the leftover shape look like a Sierpinski triangle?
- Compare the Mod 3 and Mod 5 patterns. Does a bigger divisor make the coloured triangles bigger or smaller?
- Pick a cell near the edge versus one in the middle. Where do the largest coefficients — and the most paths — live?
Explain
Make sense of what happened
State the pattern in words, test it with another example, and then connect it to the mathematical rule or representation shown in the model.
Check
Before you leave
What pattern did you observe?
Does the pattern work for another example?
If you can answer all three using evidence from the model, you understand the main idea. If not, return to Explore and change one setting at a time.