This particular session focused on using a foreign number system with a base 4. It was a wonderfully rich example of hands on maths, stretching the children’s understanding of place value and helping them see numbers as flexible ideas rather than fixed symbols.
Please excuse the terrible photos. The children were in the middle of rehearsing a play they had written and planned to perform for my birthday. The living room had been transformed into a full theatre, complete with scenery, stage, and movable curtains! It was wonderfully creative… but rather dark when I tried to take pictures.
They were so bad, I almost didn’t include them. In the end, I decided to put them all into a gallary – if you’d like to see any of them enlarged, just click on the image:
Despite the theatrical chaos, it turned out to be the perfect setting for one of our favourite kinds of lessons: living maths.
Starting with What They Already Knew
Before introducing something new, we talked about number systems they were already familiar with.
We discussed:
- Base 10 – our everyday decimal system
- Base 60 – used by the ancient Mesopotamians
- Base 20 – used by the Mayans
The children already understood that different cultures used different systems to represent numbers. This helped them see mathematics as a human invention rather than a rigid set of rules.
Then I posed a challenge:
“Could you work with any number system if I invented one?”
They were confident they could.
So I chose base four.
Introducing Base Four
In a base-4 system, there are only four symbols representing values:
0, 1, 2, and 3.
After 3, we don’t create a new symbol like “4”. Instead, we move to the next place value—just like we do in base 10 when we go from 9 to 10.
The children already understood from a previous lesson that numbers represent values, not just shapes. To make the activity more interesting, we replaced the usual numerals with simple shapes representing the values 0–3.
This helped reinforce the idea that symbols themselves don’t matter—the value does.
We then discussed an important question:
Why don’t we need any numbers beyond three in base four?
Once they realised that the next number simply becomes 10 (one group of four and zero ones), the concept clicked quickly.
Building Numbers in Base Four
The children began experimenting with creating numbers larger than three.
For example:
- 10 (base 4) = 4 in base 10
- 11 (base 4) = 5 in base 10
- 12 (base 4) = 6 in base 10
- 13 (base 4) = 7 in base 10
- 20 (base 4) = 8 in base 10
This was the easy part. They quickly grasped how place value worked.
It’s amazing how quickly children can adapt when they haven’t yet become rigid thinkers.
Addition and Subtraction in Base Four
Next, I asked them to try addition and subtraction in base four.
Only T11 managed this immediately without help. The girls needed a bit of guidance, but once they understood when to “carry” or “borrow,” they managed perfectly well.
For example:
Addition in base 4
13
+ 12
-----
31
(You regroup whenever the total reaches four.)
Subtraction in base 4 followed the same logic, borrowing from the next column when needed.
By this point the children were confidently working within the system.
When Things Got Tricky: Multiplication
Then we attempted multiplication.
This is where everything came to a halt.
The children were stuck.
So I did what any responsible homeschooling parent would do.
I confidently told them I would show them how to do it.
Unfortunately… I couldn’t.
It took me half an hour to figure it out myself. By that point the children had completely lost interest!
We revisited the problem later, and I slowly walked them through the logic.
What fascinated me most was how strongly the base 10 decimal system is ingrained in our thinking. Even the children instinctively tried to solve the multiplication using base ten thinking rather than staying inside base four.
In many ways, it was strangely satisfying to see this happen. It showed just how powerful our everyday number system is in shaping our mathematical instincts.
Working Together as a Team
One of the reasons I include T11 in these lessons is to strengthen his ability to work in a team.
He is a wonderfully independent and motivated child. However, that independence sometimes comes with impatience if others aren’t doing things exactly his way.
(Perhaps he inherited that trait from his mother…)
So part of our living maths lesson included something deeper than mathematics.
I shared with the children that my goal was:
- for T11 to learn gentle leadership by example
- for the girls to support that leadership while still expressing their thoughts
Every member of our family has a strong personality. While they usually get on beautifully, teamwork sometimes requires intentional effort.
I left them to solve the problems together, with one instruction:
“Please try not to kill each other.”
What Happened Next
To my great relief, there were no casualties.
There were a few raised voices, and I heard T11 take some very deep breaths before speaking to his sisters. But I could also hear genuine effort from all three children to work together.
And they succeeded.
By the end of the lesson they had:
- a solid understanding of base systems
- confidence working in base four
- experience solving problems as a team
Both mathematically and relationally, it was a success.
Continuing the Learning
Over the next few weeks I plan to give the children five maths questions a day, building on what we’ve covered this summer.
This simple routine helps me see whether:
- concepts are truly understood
- more practice is needed
- we’re ready to move on
Sometimes the most powerful maths learning happens through short, consistent practice.
Reflection Questions for Homeschool Parents
- What assumptions about numbers might your children hold without realising it?
- How might exploring foreign number systems deepen their understanding of place value?
- Do your children see maths as fixed rules or as flexible patterns?
- What opportunities do maths lessons provide for teaching character and teamwork?
- How comfortable are you letting your children struggle productively with a problem?
Hands On Maths Activities to Try
Here are some ways to extend this lesson using hands on maths.
1. Create Your Own Number System
Let children invent their own base system (base 3, base 5, base 7).
Ask them to:
- design symbols
- write numbers
- teach the system to someone else.
2. Build Base Systems with Objects
Use manipulatives such as:
- LEGO
- beads
- counters
- buttons
Group them according to the base system to physically show place value.
Example for base 4:
- 4 ones = 1 four
- 4 fours = 1 sixteen
3. Base Conversion Challenge
Give children numbers in base 4 and ask them to convert them to base 10.
Example:
- 21 (base 4) = ?
Let them use drawings or blocks.
4. Base Four Board Game
Create a simple board game where players move using base-4 dice results instead of base 10.
Children must translate the numbers before moving.
5. Real World Number Systems
Research together:
- Babylonian base 60
- Mayan base 20
- Roman numerals
Discuss why different cultures created different systems.
Final Thoughts
Exploring using a foreign number system with a base 4 reminded me why I love living maths.
When maths becomes:
- exploratory
- relational
- hands-on
- and occasionally chaotic
…it becomes something children experience rather than simply complete.
And sometimes the best lessons happen in a dark living room theatre, while children rehearse a birthday play.
Even if the photos are terrible.
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