Crossing Ten
To add past ten, move just enough from one part to fill a ten first — a ten-and-some number is the easiest kind to read. · 9 min
9 + 4 is hard to see in your head. 10 + 3 is easy. They make the same number. You are about to learn the move that turns one into the other.
Guess before you learn
9 + 4 makes the same number as which easy sum?
9 + 4 and 10 + 3 both make 13. The 4 lends 1 to the 9 to fill a ten, and keeps 3. If you picked 10 + 4, most people do — but the 1 that filled the ten had to come from somewhere.
K–2
3–5
To add 9 + 4, break the 4 into 1 + 3. Give the 1 to the 9. Now the sum reads 10 + 3, and ten-and-some numbers name themselves: thirteen. The move is called making a ten, and it works whenever one part sits close to ten.
Try 8 + 5. Eight needs 2 to reach ten, so the 5 lends 2 and keeps 3. The sum becomes 10 + 3 = 13 again. The pairs that make ten — 9 and 1, 8 and 2, 7 and 3 — tell you exactly how much to move.
6–8
The move is regrouping: you rewrite one addend so a ten appears. 9 + 4 = 9 + (1 + 3) = (9 + 1) + 3 = 10 + 3. Nothing was added and nothing removed — the parts were only regrouped, which is why the total cannot change.
Place value explains why the detour pays. Two-digit numbers are written as tens and ones, so a sum shaped like ten-plus-ones is already in its final written form: 10 + 3 is simply 13. Making a ten converts a sum into the notation itself.
9–12
The single law at work is associativity: (a + b) + c = a + (b + c). You choose to split 4 as 1 + 3 precisely because 9 + 1 = 10, and associativity licenses moving the parenthesis. Every mental-math shortcut of this kind is an associativity-and-commutativity argument in disguise.
The written algorithm you will meet later — column addition with carrying — is this same move performed in every column at once: whenever ones pile past nine, ten of them regroup into one ten. Crossing ten by hand now is the algorithm in slow motion.
K–2
A ten-frame is a box with ten holes. Put 9 counters in. One hole is empty. Now you want to add 4 more counters.
One counter jumps into the empty hole. The frame is full: that is ten. Three counters wait outside. Ten and three is 13.
Undergrad
In the Peano development, addition is defined recursively: a + 0 = a and a + S(b) = S(a + b). Associativity and commutativity are then theorems, each proved by induction. The making-ten computation 9 + 4 = (9 + 1) + 3 is a short equational derivation inside that theory, every step justified by a proved identity.
Base-ten numerals add a second layer: a numeral is a sum of digit-times-power terms, and carrying is the rewriting that restores every digit to the range 0–9. Correctness of the school algorithm is an induction on digit position, with associativity doing the regrouping at each step.
Postgrad
Carrying has honest cohomological standing. Working with residues, the carry function c(a, b) = ⌊(a + b)/10⌋ is a 2-cocycle on ℤ/10 with values in ℤ/10, and the group extension it classifies is ℤ/100 — the claim that tens-and-ones notation glues cyclic groups together, made precise in H²(ℤ/10, ℤ/10).
The child's move also generalizes cleanly: in positional notation over any base b, normal-form arithmetic is a confluent rewriting system whose one nontrivial rule is the carry. Making a ten is the b = 10 instance of reduction to canonical form — the first rewriting proof most humans ever perform.
make a ten
Move just enough from one part to fill a ten. 9 + 4 becomes 10 + 3.
Cross the ten: 8 + 6 — the steps fade as you master them
8 needs 2
6 = 2 + 4
8 + 2 = 10, with 4 left
10 + 4 = 14
Now try to see the pattern. Take 9, add 2, add 4, add 6. Where does each sum land? Put your guesses on the chart.
You own the move now: fill the ten, then read what is left. It works for 9, for 8, for 7 — any part that sits close to ten.
Practice — new ink and old, interleaved
1.Six pencils in the cup, then two more arrive. Count on: how many pencils?
2.7 + 6 = ?
3.Ten and five more. What number is that?
4.Match each sum to its ten-and-some twin.
5.Which pair makes ten?
6.Parts 4 and 2, whole 6. Which sentence belongs to this family?
7.What does the 4 do when you add 9 + 4?
It lends 1 to the 9 to fill a ten and keeps 3, so the sum reads 10 + 3 = 13.
How close were you? Grade yourself honestly — it sets your review date.
8.Which pair makes ten?
9.Match each number with its partner to make ten.