Patterns in Mathematics

Lots of them! Paying for fruits, vegetables and groceries (adding up the bill), calculating the speed of a vehicle, spotting the designs and patterns in buildings, and finding the area of a plot or our own home all use maths patterns.

There are tons more — try and think of one from your own day, like sharing biscuits equally or reading a bus timetable!

This is a class-discussion question — there's no single fixed answer, but here are ideas to talk about: patterns helped us run scientific experiments, build our economy and democracy (counting votes and money!), construct bridges, houses and complex buildings, and design TVs, mobile phones, computers, bicycles, trains, cars, planes, calendars and clocks.

Basically — almost nothing we use every day would exist without someone first spotting a pattern!

Start at 1. The rule: each time, add one more than you added last time.

Why "triangular"? Stack 1 dot, then a row of 2 below it, then a row of 3 — the dots make a real triangle! 🔺

Powers of 2 can be pictured as a row of dots that keeps doubling: 1 dot, then 2 dots, then 4, then 8, then 16 — each new group is exactly twice as long as the one before it.

Powers of 3 can be pictured as a growing square grid: start with 1 dot, then a 3-dot row (3¹), then a full 3×3 = 9 grid of dots (3²), and the next picture would be a 3×3×3 cube of 27 dots (3³) — each step multiplies the dots by 3.

Both sequences grow by multiplying instead of adding — that's why their pictures explode in size so much faster than Triangular or Square numbers!

Notice the pattern in the patterns: the first four sequences all grow by a simple, easy-to-draw step. The Koch Snowflake is different — its rule is just as simple to say, but following it forever makes a shape with infinitely many infinitely tiny bumps, so in practice you can only draw the first few steps by hand.

A "Complete Graph" joins every dot to every other dot, with exactly one line between each pair. Step 1 — work out how many NEW lines each new dot adds: a brand-new dot has to connect to every dot that's already there, so it adds one line per existing dot.

1, 3, 6, 10, 15… that's the Triangular numbers from Section 2! 🤯 It makes sense once you see it: adding the new lines from each new dot is exactly "add 1, then add 2, then add 3…" — the same growing-jump rule that builds triangular numbers.

Step 1 — find the rule from the numbers you already have, THEN apply it three more times. Here's the rule used for every row, and the working that gets you the next 3 numbers:

Notice the pattern across the whole table: every sequence has ONE rule you find once, and then you just repeat that same rule to get as many more numbers as you want — that's the whole point of spotting a pattern!

Step 1 — look at how many dots each picture added compared to the one before it. That "how many more dots" number is the rule you copy forward to draw the next picture.

The trick for every row is the same: spot the rule that turns picture n into picture (n+1), then draw one more step of it. You don’t need to memorise pictures — just the rule that grows them!

Triangular numbers can be arranged as dots in the shape of a triangle (row 1 has 1 dot, row 2 has 2, row 3 has 3…). Square numbers can be arranged as dots in a square grid (like 3 rows of 3 dots for 9). Cube numbers can be built as an actual 3D cube out of little cubes (like 3×3×3 = 27 little cubes).

As a square: 6 × 6 = 36, a 6-by-6 grid of dots.

As a triangular number: it's the 8th one in the list — 1, 3, 6, 10, 15, 21, 28, 36. Count the jumps: 2+3+4+5+6+7+8 added to 1 gets you to 36.

36 dots can be arranged perfectly as EITHER shape!

Look at the jumps between numbers: 7-1=6, 19-7=12, 37-19=18. The jumps are 6, 12, 18 — each one is 6 more than the last!

Next jump = 18+6=24, so the next number is 37+24=61.

Step 1 — think about what each rule actually does to a picture. Powers of 2 doubles the count every time; Powers of 3 triples it. A good picture should show that doubling or tripling happening.

Powers of 2 → draw a row that doubles in length each step. Powers of 3 → draw a square grid that grows to 3×3, then imagine a 3×3×3 cube for the next step. (Full picture and dot-count walkthrough is in Section 2 above — “🖼️ Picturing Powers of 2 AND Powers of 3”.)

Step 1 — just add each row up first, small to big, and check what number you land on:

Step 2 — ask WHY. Regroup each up-and-down row like this: the middle number stands alone, and every other number pairs up with its twin on the other side — 1+2+3+2+1 = 3 + (1+2) + (1+2). That's just the same "1, 3, 5, 7…" odd-number split in disguise: the middle number plus two matching triangular halves always adds up to an odd number of dots at each layer, wrapping into a bigger and bigger square.

It's the SAME L-shaped layers as the odd-number pattern — just drawn as a diamond instead of a corner! Picture it as a diamond of dots: row lengths grow 1,2,3… up to the middle, then shrink back down — and a diamond of dots is really just a square turned 45°.

Step 1 — recognise the pattern type. This is exactly the up-and-down counting from Q1, just with a much bigger middle number: it goes up to 100 and back down to 1 again.

Step 2 — reuse the rule you just proved in Q1: counting up and down to "n" always totals n × n. Here the peak number is n = 100, so imagine (or partially draw) the same diamond-of-dots picture, but 100 dots wide instead of 4 — it's still one full square, just a much bigger one.

= 10,000! Without the pattern you'd have to add 199 separate numbers by hand — spotting the pattern turns that into one multiplication.

Step 1 — add the All 1's sequence (1,1,1,1…) going up, one more 1 at a time:

Every step you're literally just counting how many 1's you've added so far — so the running total IS the counting numbers.

Step 2 — now add All 1's up-and-down (like Q1's diamond, but every row is worth exactly 1):

Adding 1's going up gives the Counting numbers, because the running total tracks "how many terms so far". Adding 1's up-and-down gives the Odd numbers, because each up-and-down step adds exactly 2 more 1's than the step before it (one more on the way up, one more on the way down)!

Step 1 — keep a running total of the Counting numbers (1,2,3,4…):

Step 2 — picture it small: put 1 dot in row 1, 2 dots in row 2 below it, 3 dots in row 3… stack the rows and you get a triangle shape — exactly the small triangle picture from Section 2.

Adding up the Counting numbers gives you the Triangular numbers — because "row lengths 1,2,3,4… stacked up" is the literal definition of a triangular number!

Step 1 — take the Triangular numbers (1,3,6,10,15…) and add each one to its neighbour:

Step 2 — ask why. Picture a triangle of side "n" (like a triangle made of 3 rows) sitting next to a triangle of side "n−1" turned upside-down, so its slanted edge fits snugly against the first triangle's slanted edge. Together the two triangles tile a perfect n × n square with no gaps and no overlaps — that's WHY two neighbouring triangular numbers always add to a square number.

4, 9, 16, 25… the Squares sequence! Two triangles of the right sizes fit together perfectly to make one square — that's the picture explanation.

Step 1 — add up Powers of 2 starting from 1, then add 1 more to the total:

Step 2 — ask WHY. Think of Powers of 2 as doubling dots: 1, then 2, then 4, then 8… each new group is bigger than ALL the earlier groups put together (because doubling means the newest group alone already equals the sum of every group before it). So the running total is always just one short of double the last term — i.e. one short of the next power of 2. Adding that missing 1 lands you exactly on it.

You always land on the NEXT Power of 2! Every power of 2 is exactly one more than the sum of every power before it.

7, 19, 37, 61… the Hexagonal numbers from Q4! Multiplying triangular numbers by 6 and adding 1 always gives a hexagonal number.

1, 8, 27, 64… the CUBE numbers! You can picture this as building a cube one hexagonal "shell" at a time — each new shell of hexagonal-shaped dots adds up to the next cube number.

This is an open-ended, explore-it-yourself question — the book doesn't give one fixed answer here, so don't worry if yours looks different from a friend's!

Here's one idea to get you started: add two consecutive Powers of 2 together — 1+2=3, 2+4=6, 4+8=12, 8+16=24. Each answer is exactly 1.5 times the bigger number you added — can you see why, with a picture of doubling dots?

Try your own! Pick any two sequences from Table 1, add / subtract / multiply terms from them, and see if you land on a sequence you recognise. That's exactly what mathematicians do for a living. 🔎

Sides: 3, 4, 5, 6, 7, 8, 9, 10. Corners: 3, 4, 5, 6, 7, 8, 9, 10. Exactly the same!

Why? Every side has one corner at each end, and every corner is where exactly two sides meet — the count can never be different.

Step 1 — count the lines shape by shape, remembering every dot joins to every OTHER dot exactly once:

Step 2 — explain why: each time you add one new dot, it must connect to every dot that already exists, so it contributes exactly that many brand-new lines. Going from 4 dots to 5 dots, the new dot adds 4 new lines (6→10); going from 5 to 6, the new dot adds 5 new lines (10→15). Adding a steadily growing number of new lines each step is exactly how triangular numbers are built.

1, 3, 6, 10, 15… the Triangular numbers! You already found this in Section 4.

Step 1 — count the little squares inside each grid by multiplying rows × columns (they're always equal, since it's a stacked SQUARE):

Step 2 — why this sequence? An n×n grid of little squares contains exactly n rows with n little squares in each row, so the total is always n×n — and "a number multiplied by itself" is the exact definition of a square number.

1, 4, 9, 16… the Squares sequence — makes sense, since an n×n grid literally is the definition of a square number!

Hint from the book: count how many triangles are in each row. Row 1 has 1 triangle. Row 2 has 3. Row 3 has 5. Add them up per shape: shape 2 has 1+3=4, shape 3 has 1+3+5=9, shape 4 has 1+3+5+7=16.

1, 4, 9, 16… ALSO the Squares sequence! Different shape, same secret number pattern — exactly the sneaky trap from Section 4.

Each straight line gets replaced by 4 smaller lines (the "speed bump" shape). Starting from a triangle (3 sides):

3, 12, 48, 192… each number is 4 times the one before — the book calls this "3 times Powers of 4".

(b) 25 — this is the Squares sequence, and 5×5=25.

(b) Cubes — each is a number multiplied by itself twice more.

Step 1 — recognise the question type. "Sum of the first n odd numbers" is exactly the pattern from Section 3: adding odd numbers 1+3+5+7+… always builds a perfect square, because each odd number is one more L-shaped layer wrapped around the last square.

Step 2 — apply the shortcut n × n, using n = 12 (since we want the first 12 odd numbers):

12 × 12 = 144 — no need to actually add 1+3+5+…+23 one by one!

Step 1 — recall the rule: the Virahānka sequence starts with 1, 1 (or is often shown starting 1, 2), and every next number is the sum of the two numbers right before it.

Step 2 — build the list term by term:

1, 2, 3, 5, 8, 13. Rule: each number is the sum of the two numbers just before it (1+2=3, 2+3=5, 3+5=8, 5+8=13).

Start with one dot in a corner. Each odd number of dots (1, then 3, then 5…) can be arranged as an L-shaped layer wrapped around the previous square. After wrapping n layers, the dots form a complete n×n square, which has exactly n² dots. So the sum of the first n odd numbers is always n².

Step 1 — identify the sequence. 1, 3, 6, 10… are the Triangular numbers: the jump from one term to the next grows by 1 each time (jump of 2, then 3, then 4…).

Step 2 — keep applying that rule until you reach the 10th term:

The 10th term is 55. (Jumps used: 2,3,4,5,6,7,8,9,10.)

This is really a Triangular-numbers question in disguise.

(a) The total after 6 rows is the 6th triangular number: 1+2+3+4+5+6=21 trees.

(b) Keep building the list until you hit 45: 1,3,6,10,15,21,28,36,45 — that's the 9th triangular number, so she needs 9 rows.