Why We Count by 10 but Keep Coming Back to 12
Two numbers that organize the world in very different ways
Ten seems to run the world of numbers.
We count:
1, 2, 3, …, 9, 10
and then begin again:
11, 12, …, 19, 20.
Ten ones become one ten.
Ten tens become one hundred.
Ten hundreds become one thousand.
Our entire place-value system is built around powers of 10.
And yet, for a number that supposedly lost the contest, 12 refuses to disappear.
There are 12 months in a year.
Clocks divide the day into familiar sets of 12 hours.
A dozen means 12.
A foot contains 12 inches.
Eggs are still commonly packaged by the dozen.
Children traditionally learn multiplication tables through the 12s.
So what is going on?
Why did 10 become the number we count with, while 12 remains one of the numbers we organize things with?
The answer reveals something wonderful about number sense.
Ten and twelve are useful for very different reasons.
10 Has a Huge Advantage: We Built Our Number System Around It
Look at your hands.
There is a very convenient counting tool attached to them:
10 fingers.
It is not difficult to imagine how counting in groups of ten became natural across many cultures.
Once a number system is organized around 10, an extraordinary amount of arithmetic becomes easier to write and understand.
10 100 1,000 10,000
The position of a digit tells us how many ones, tens, hundreds, thousands, and so forth we have.
The number
347
means
3(100) + 4(10) + 7.
Ten is therefore much more than one number among many.
It is the organizing benchmark of our decimal number system.
That is one reason children who really understand 10 have such an advantage in later arithmetic.
What Does It Mean to Really Understand 10?
Knowing that the number after 9 is 10 is only the beginning.
A child can begin to see 10 as something that can be built and rebuilt.
10 = 1 + 9
10 = 2 + 8
10 = 3 + 7
10 = 4 + 6
10 = 5 + 5
Suddenly, 10 is not a fixed symbol.
It has structure.
A child who sees
8 + 2 = 10
can later use that relationship to think about
8 + 7.
Instead of counting everything:
8, 9, 10, 11, 12, 13, 14, 15
the child can break 7 into 2 + 5:
8 + 7
= 8 + 2 + 5
= 10 + 5
= 15.
That is a major change in mathematical thinking.
The child has stopped treating every calculation as something that must be started from scratch.
A known relationship is being used to build a new one.
This Is Why “Make 10” Matters
Making 10 is sometimes taught as though it were simply another addition trick.
It is much more than that.
It teaches children to look at a number and ask:
What could I do to make this number friendlier?
For example,
9 + 6
can become
10 + 5.
And
18 + 7
can become
20 + 5.
Later, the same habit of mind appears in multiplication.
Instead of treating
9 × 7
as an isolated fact, a child can eventually see:
9 × 7 = 10 × 7 − 7.
The numbers have changed.
The mathematical habit has not.
Use something friendly that you already understand to work out something less familiar.
That is one of the central ideas behind Multiplication Flip.
But Then There Is 12
Now let us play the same game with 12.
Break 12 apart.
One obvious possibility is:
12 = 10 + 2.
That alone is useful because it connects 12 to our familiar benchmark of 10.
But 12 has many other personalities.
12 = 6 + 6
12 = 8 + 4
12 = 9 + 3
12 = 7 + 5.
Nothing too surprising yet.
Now try breaking 12 into equal parts.
12 = 6 + 6
12 = 4 + 4 + 4
12 = 3 + 3 + 3 + 3
12 = 2 + 2 + 2 + 2 + 2 + 2.
Something interesting has happened.
The decompositions are beginning to look like multiplication.
12 Loves Equal Groups
Take
4 + 4 + 4 = 12.
We can describe that as:
3 groups of 4
which leads naturally to
3 × 4 = 12.
Or look at
3 + 3 + 3 + 3 = 12.
That is:
4 groups of 3
so
4 × 3 = 12.
And
6 + 6 = 12
becomes
2 × 6 = 12.
So 12 can organize itself into several whole-number rectangles:
1 × 12
2 × 6
3 × 4
Ten has fewer such possibilities:
1 × 10
2 × 5.
This is one reason 12 has historically been so convenient for dividing and grouping things. It can be divided evenly by 2, 3, 4, and 6, whereas 10 divides evenly by 2 and 5.
Twelve is unusually cooperative.
Try Dividing 10 and 12
Imagine that you have 12 cookies.
Two children?
Easy.
12 ÷ 2 = 6.
Three children?
12 ÷ 3 = 4.
Four children?
12 ÷ 4 = 3.
Six children?
12 ÷ 6 = 2.
Now try the same with 10 cookies.
Two children works beautifully:
10 ÷ 2 = 5.
Five children also works:
10 ÷ 5 = 2.
But sharing 10 equally among 3 or 4 children immediately requires fractions.
Neither number is “better.”
They are simply good at different jobs.
10 Is a Brilliant Counter. 12 Is a Brilliant Grouper.
That may be the simplest way to think about the difference.
10 is wonderfully suited to our place-value system.
12 is wonderfully suited to equal grouping.
And this is visible in ordinary life.
We still encounter dozens, 12-hour clock divisions, and measurement systems containing 12 units because grouping by 12 can be remarkably convenient.
So perhaps the story is not:
10 defeated 12.
It is closer to:
10 became the language of our number system.
12 kept proving useful whenever people needed to divide and group things.
Now Let a Child Discover This
This is where the mathematics becomes more interesting than the history.
Give a child 12 cubes, counters, beans, coins, or buttons.
Do not begin with:
“What is 3 times 4?”
Instead ask:
“How many different ways can you organize 12?”
You may see:
- 10 and 2
- 6 and 6
- 8 and 4
- 3 groups of 4
- 4 groups of 3
- 2 groups of 6
- 6 groups of 2
- a 3 × 4 rectangle
- a 2 × 6 rectangle
- perhaps something you did not expect
Now mathematics becomes an investigation.
The child is not merely being asked to produce the answer 12.
The child is studying the structure of 12.
And That Changes the Meaning of Multiplication
Consider the difference between these two experiences.
A child sees:
3 × 4 = 12
and memorizes it.
Or a child already knows:
12 = 4 + 4 + 4,
sees three equal groups of 4, builds a 3 × 4 array, and then writes:
3 × 4 = 12.
The equation is identical.
The mathematical experience is not.
In the second case, the multiplication fact belongs to a network of relationships.
That network gives the child somewhere to go if the fact is forgotten.
Forgotten Facts Do Not Have to Be Lost Facts
Suppose a child forgets 3 × 4.
If multiplication has only been memorized, there may be nowhere to go except:
“I can’t remember.”
But if the child understands 12 structurally, there are many ways back.
Three groups of four:
4 + 4 + 4 = 12.
Four groups of three:
3 + 3 + 3 + 3 = 12.
Double six:
6 + 6 = 12.
An array:
3 × 4.
The answer can be reconstructed.
That is a very different kind of mathematical security.
What 10 and 12 Teach Us About Number Sense
Strong number sense is not simply knowing more answers.
It is seeing more relationships.
Ten teaches children about:
- benchmarks
- complements
- place value
- composing and decomposing
- friendly-number strategies
Twelve adds:
- equal groups
- factors
- arrays
- repeated addition
- different ways to organize the same quantity
Together, they make a beautiful pair.
One helps children navigate our number system.
The other opens a doorway into multiplication.
A Bigger Mathematical Habit
There is also a broader lesson here.
When a child encounters a number, we want the question eventually to become more than:
What number is this?
We want questions such as:
What can I make from it?
How can I break it apart?
What is it close to?
Can I make equal groups?
Can I make a rectangle?
What other number do I already know that could help me?
Those questions are the beginning of flexible mathematical thinking.
And they matter far beyond 10 and 12.
The same habit eventually allows a student to look at
7 × 8
and see not one fact, but many possible routes:
7 × 8 = 5 × 8 + 2 × 8
or
7 × 8 = 8 × 8 − 8
or
7 × 8 = 10 × 8 − 3 × 8.
One multiplication fact can be seen many ways.
That is precisely the kind of flexibility we want to begin building long before multiplication becomes a table to memorize.
Explore 10 With Your Child
Our Make 10 Activities help children build and break apart 10, discover missing parts, and begin using 10 as a friendly benchmark for calculation.
The point is not to race through combinations.
It is to help the child become so familiar with the structure of 10 that relationships such as
8 + 2, 7 + 3, 6 + 4
begin to feel useful.
Explore the Make 10 Activities
Then Try Something Different: Break 12
Once children realize that one number can have many structures, 12 offers a wonderful next investigation.
The Break 12 activity moves from
12 = 10 + 2
to equal parts,
12 = 4 + 4 + 4,
to equal groups,
3 × 4 = 12,
and finally to arrays and factors.
It is designed to help children discover the multiplication hiding inside one familiar number.
Get Break 12
One Number. Many Structures.
Enter your name and email and we will send you the complete Break 12 PMG activity.
The Point Is Not 10 Versus 12
There is no winner.
That is actually the most interesting conclusion.
Ten shows us the power of a benchmark.
Twelve shows us the power of structure.
A mathematically flexible child learns to use both.
And perhaps that is the larger lesson we want children to discover:
Numbers are not merely answers waiting to be remembered.
They are structures waiting to be explored.