Pair things two by two to meet even and odd — then read the rhythm of a sequence.
Some piles split into two equal halves with nothing left behind; others always have one stubborn piece sitting alone. That single fact splits the whole counting world into two families — even numbers and odd numbers — and it does it by the oldest tool we have: matching, the same pairing you used to count in Lesson 1.1. Once you can pair two by two you will see that the families take turns in a perfect back-and-forth beat, and that you never even need to do the pairing — the very last digit gives the answer away. From there it is a short step to the bigger idea behind every pattern: find the steady step a list grows by, and you can keep it going forever.
Imagine a handful of cookies and two plates, and you deal them out one to each plate, around and around. There are only two ways the dealing can end. Either the two plates come out exactly equal with the box empty — or there is one last cookie with no partner, leaving one plate with a single extra. That is the whole idea.
We say a number is even when its things can be split into pairs with nothing left over, and odd when, after making every pair you can, one is left alone. Let us pair up 7 and watch.
Seven makes three pairs, and one disc has no partner — so 7 is odd. Try 8 instead and the leftover disappears: four clean pairs, none alone, so 8 is even. The rule is exact, with no in-between: every whole number is one or the other, never both.
Pair the things two by two. Pairs exactly, nothing left over → even. One left alone → odd. The leftover disc shows up only when the number is odd — that is the test.
And zero? Zero things make zero pairs with nothing left over, so 0 is even — it fits the rule perfectly. Drive the pairer below and watch the lone red disc blink in and out: it appears for 1, 3, 5, 7… and vanishes for 0, 2, 4, 6….
Pairing 7 discs is easy. Pairing 128 of them would be a chore — and you would never do it for 4,596. The wonderful shortcut is that you do not have to. Only the last digit decides. Look at the ones place and ignore everything to its left.
Why does that work? Every full ten is itself two fives — a ten splits perfectly into five pairs, so any number of complete tens is already paired with nothing left over. All the hundreds, all the tens, contribute no leftover at all. Whatever extra cannot be paired must come from the ones alone. So the parity of the whole number is exactly the parity of its ones digit.
A ones digit of 0, 2, 4, 6, or 8 means the number is even. A ones digit of 1, 3, 5, 7, or 9 means it is odd. The other digits do not matter at all.
Is 128 even or odd? Its ones digit is 8, and 8 is even — so 128 is even. Is 4,596 even or odd? It ends in 6, so it too is even. Is 4,591 even or odd? It ends in 1, so it is odd. We never paired a single disc.
The detector below reads a number, circles its ones digit, and checks the verdict against actually pairing the ones — the two answers always agree, because that is exactly why the shortcut is true.
Parity is decided by the last digit only. 128 is even because it ends in 8 — never mind that it starts with an odd-looking 1. And "even" does not mean a number "looks even" or is "a big round number"; it means it pairs up exactly. The digit in front does no work here.
Walk along the counting numbers and the two families take strict turns: 1 is odd, 2 is even, 3 is odd, 4 is even, on and on with no skips and no repeats. Odd, even, odd, even — a steady alternating beat that never breaks.
That makes sense from pairing: each time you add one more thing, you either give the lonely leftover a partner (odd becomes even) or you create a brand-new leftover (even becomes odd). So every single step flips the family. The strip below paints each number by its parity — read it left to right and you can almost tap the rhythm.
Count the even numbers in order and you skip every other one: 2, 4, 6, 8, 10, 12… The odd numbers do the same in between: 1, 3, 5, 7, 9, 11…. Each list jumps by 2 — which is the next idea.
A sequence is just a list of numbers in order: 2, 4, 6, 8, … . The secret to continuing one is to find the steady step — the same amount you add to get from each number to the next. In 2, 4, 6, 8 the step is always +2: from 2 to 4 is +2, from 4 to 6 is +2, and so on. Once you know the step is constant, the next term is no mystery — just add the step again.
The step does not have to be 2. Count by fives — 5, 10, 15, 20 — and the step is +5; count by tens — 10, 20, 30 — and it is +10. Whatever the step, the trick is the same: find it once, then repeat it. Pick a step and a starting number below; the widget draws the first terms and then reveals the next two in green so you can check your prediction.
To continue a sequence, find the constant step between consecutive terms, then add it again. Counting by 2 gives the evens or the odds; counting by any fixed amount builds a steady, predictable pattern.
Four small rules, and they all come from one picture — pairing two by two.
| Idea | The rule | Quick read |
|---|---|---|
| Even | pairs exactly, nothing left over | 0, 2, 4, 6, 8 … |
| Odd | one left alone after pairing | 1, 3, 5, 7, 9 … |
| The shortcut | the last digit decides parity | ends in 0/2/4/6/8 → even |
| The rhythm | add 1 and the family flips | odd, even, odd, even … |
| A sequence | find the constant step, then repeat it | 2, 4, 6, 8 … (+2) |
Six questions to lock it in. Tap the answer you think is right.
This lesson builds the even/odd distinction from one-to-one pairing rather than memorized digit lists, which is the foundation of 2.OA.C.3 (determine whether a group of objects up to 20 has an odd or even number of members; pair objects or count by twos). The last-digit shortcut is justified, not just stated — every full ten pairs evenly, so only the ones place can leave a partner unmatched. The closing work on the constant step in 2, 4, 6, 8 … introduces analyzing patterns and generating a sequence from a rule, supporting 4.OA.C.5 (generate a number pattern that follows a given rule and notice features of the pattern). At home, deal cards or snacks onto two plates and ask "did it come out even?" — the leftover card is the whole idea.