Mathematics from zero
Addition
You have 3 apples in one hand and 2 in the other. You tip them all into one basket and count: five. You did not invent a new number — you just joined two amounts into one. That joining is the whole of addition.
After this lesson you can say what addition really means, name the parts of an addition, add a small number by counting on, explain why the order of the two numbers does not matter, and add larger numbers by working one place at a time.
Addition combines two amounts into one total. When you push two groups together
and ask “how many now?”, that is addition. We write it with the plus sign, +,
which means and, and the equals sign, =, which means makes. So 3 + 2 = 5
is read “three and two makes five”. The amounts you start with do not change — you are
only counting them together.
The parts of an addition have names. In 3 + 2 = 5, the numbers being joined — the
3 and the 2 — are the addends. The total they make — the 5 — is the sum.
A useful picture: two parts come together to form one whole. The addends are the
parts; the sum is the whole. Every addition is just two parts and the whole they make.
You can add by counting on. To work out 3 + 4, you do not need to start over at
one. Start at the first addend, 3, and count up four more steps: 4, 5, 6, 7. You
land on 7, so 3 + 4 = 7. Picture a number line — a straight row of numbers in
order. Adding is a jump to the right: stand on 3, jump 4 places, read where you
land.
The order of the addends does not matter. 3 + 4 and 4 + 3 both make 7. This
makes sense: you are joining the same two groups either way, so the total is the same.
Because of this you may always start counting on from the larger addend — for
2 + 9 it is far less work to start at 9 and count on 2 than the other way round.
Same answer, fewer steps.
For larger numbers, add one place at a time. You already know that a digit’s worth depends on its place — ones, tens, hundreds. To add two whole numbers, line them up by place and add each place separately: ones with ones, tens with tens. If one place adds up to more than 9, it will not fit in a single digit. Then you regroup: ten ones become one ten, and that one ten is carried into the tens place. Carrying is not a trick — it is just trading ten small units for one of the next size up.
Add 27 + 15.
Line the numbers up by place: 27 has 2 tens and 7 ones; 15 has 1 ten and 5 ones.
Start with the ones: 7 + 5 = 12. Twelve does not fit in the ones place — it is
1 ten and 2 ones. Write the 2 in the ones place and carry the 1 ten over to the
tens place.
Now the tens: 2 tens + 1 ten, plus the 1 ten you carried, makes 4 tens. Write 4
in the tens place.
The sum is 42. Check it by counting on if you like: 27, then 15 more, lands on 42.
Why this works
Why does carrying work? Because ten ones and one ten are the same amount — place value (the previous lesson) is exactly the rule that one ten is worth ten ones. Carrying just rewrites an amount that grew too big for its place into the place above, where it fits. Nothing is added or lost; the total is untouched.
Common mistake
The most common mistake is forgetting the carry: writing 7 + 5 as a bare 12 under the
ones, or dropping the extra ten entirely. Always ask, for each place: “does this fit in
one digit?” If the place total is 10 or more, the leftover ten must move up. A second
mistake is adding a ones digit to a tens digit — only ever add a place to the same
place.
Add by counting on: 6 + 3. Type the sum.
7 + 2 makes 9. Without re-counting, what does 2 + 7 make? Type the sum.
Add: 9 + 4. Type the sum.
Add: 28 + 5. Type the sum.
Add: 36 + 27. Type the sum.
When you add 8 + 5, the ones place comes to 13. What do you do?
Addition joins two amounts into one total. The numbers joined are the addends; the total is the sum. You can add small numbers by counting on from one addend, and the order of the addends never changes the sum — so count on from the larger one. For larger numbers, add one place at a time; when a place comes to 10 or more, regroup it and carry the extra ten into the next place. Carrying works because ten ones and one ten are the same amount.