Practising Number Bonds Using Pascal’s Triangle

I decided recently to plan a deliberately fun and slightly frivolous homeschool maths lesson. Both L10 and C10 had been handling every number bond question I threw at them with ease, and it felt like the perfect moment to introduce something new, something intriguing. Enter learning number bonds using Pascal’s triangle:

Pascal’s triangle is a triangular arrangement of numbers built entirely from number bonds, and it has fascinated me for years. I know it will feature heavily in our future maths studies, so this felt like the ideal time for a gentle introduction. This was never meant to be an in-depth exploration of all its patterns, that will come later. Instead, the aim was simple: to showcase the playful, curious side of maths.

Find out why we decided to learn maths without text books for a while

Introducing Number Bonds and Pascal’s Triangle Through Discovery

The version of Pascal’s triangle I gave the girls was much larger than the standard introductory one, extending well into three-digit numbers. My plan was for them to look at a partially completed triangle and uncover the number bond pattern for themselves.

What actually happened was even better.

They immediately began spotting other patterns, diagonals, repetitions, and sequences, and became quite convinced they had cracked the “rule.” Unfortunately, it wasn’t quite right. That gave me the perfect opening to ask whether they could find any number bonds hidden within the triangle.

It didn’t take long.

Once they spotted that each number was formed by adding the two numbers above it, everything clicked. I then challenged them to disprove their original theory by continuing to fill in the triangle themselves. As the numbers grew larger, I noticed something interesting.

L10 started counting on her fingers.

I hadn’t seen her do that all summer.

This post describes how we began learning number bonds and our second lesson on them

Using Number Bonds and Pascal’s Triangle to Teach Mental Arithmetic

When I asked why she’d reverted to finger counting, she explained that the larger numbers were becoming harder to manage mentally. This was the perfect opportunity to pause the activity and explicitly teach some mental arithmetic strategies, some familiar, some new.

Mental Arithmetic Strategies We Practised

1. Add from largest to smallest
Instead of working right to left as we do on paper, we added hundreds first, then tens, then units.

2. Think in terms of place value and regrouping
For example:
367 + 596 becomes
800 + 150 + 13 → 963

3. Break down and rebuild using number bonds of ten
Using the same example:

367 + 596
= 300 + 500 + 60 + 40 + 50 + 7 + 3 + 3
= 800 + 100 + 50 + 10 + 3
= 900 + 60 + 3
= 963

On paper, this looks long-winded. But with practice, these steps begin to happen almost invisibly in the mind. As a visual learner, I actually see the numbers split apart and reassemble, like a small mathematical dance.

The goal wasn’t instant mastery. It was to begin training the brain to calculate without relying on fingers or written methods.

C10 seems to do this naturally. For L10, maths continues to be hard work, but compared to just five weeks ago, her confidence with smaller numbers has grown enormously. That alone tells me we’re heading in the right direction.

Practising Number Bonds with the “Magic Triangle”

To reinforce everything we’d discussed, we moved into a hands on maths activity often known as the Magic Triangle.

An equilateral triangle is outlined, with counters placed:

• One at each corner
• One in the middle of each side

The challenge is to arrange the numbers (we started with 1–6) so that each side of the triangle adds up to the same total.

Larger triangles can be introduced later to increase difficulty.

We kept it simple: wooden pencils for triangle outlines and equate tiles for numbers and I was delighted by how confidently the girls tackled it. This kind of tactile problem-solving is invaluable in homeschool maths, especially for children who benefit from seeing and touching their thinking.

Reinforcing Number Bonds Through Games

One thing we’ve learned as a family is that maths practice is far more effective when it feels like play—especially when parents join in.

Totally Tut

A gentle game for 2–4 players that practises all four operations. For now, we’re using it purely for addition and subtraction.

Equate

Think number Scrabble. This game allows for whole numbers, fractions, decimals, and all four operations. We’re currently limiting it to whole numbers with addition and subtraction, making it perfect for reinforcing number bonds.

21 (Card Game)

A simple card game where players must make a total of 21 using two or more cards. Brilliant for practising addition facts in a low-pressure way.

Sudoku

While Sudoku doesn’t involve addition or subtraction, it builds logical thinking and numerical confidence. We’ve started printing child-friendly puzzles for early finishers, and they’ve been surprisingly popular.

Reflection Questions on Number Bonds and Pascal’s Triangles

1. What patterns did you notice first in Pascal’s triangle?
2. How are number bonds used to build the triangle?
3. Which mental maths strategy felt easiest for you?
4. Why do you think finger counting becomes harder with larger numbers?
5. How did using physical objects help you think about the maths?

Hands-On Maths Activities to Try Next

• Build Pascal’s Triangle with counters or LEGO bricks
• Create Your Own Magic Triangle using larger numbers
• Daily Mental Maths Challenge using one Pascal’s triangle row
• Explain the Pattern: have your child teach Pascal’s triangle to someone else
• Design a Maths Game that practises number bonds

We’ve now moved on to place value, which we are thoroughly enjoying, and I’ll be sharing that next. But this gentle introduction to Pascal’s triangle has reminded me once again how powerful curiosity-led, hands on maths can be in a homeschool setting.

For all of my living hands-on maths posts, click here