This activity gives you three collections, numbered counter buttons, a Make next pair action, and an answer check. Nine produces four pairs and one leftover; ten produces five pairs; zero has no pairs and no leftover. Students choose Odd or Even only after inspecting the collection, and a blank choice stays unanswered.
Make a pair mean exactly two counters
OpenStax describes divisibility through an integer multiple with no remainder and gives the last-digit rule for divisibility by two. This game turns that principle into original pairs and leftovers. OpenStax: Prime and Composite Numbers
A pair contains two distinct counters. Each counter belongs to at most one pair; a paired counter cannot also be counted as leftover. Before checking the classification, verify that twice the number of complete pairs plus the unpaired count equals the original collection.
An even count can be written as 2 × k with no leftover, where k is a whole number here. An odd count can be written as 2 × k + 1. Keep the original count visible so a disappearing counter cannot make an odd collection look even.
Round O1: Pair nine
Use counters O1-C01 through O1-C09. Four pairs use eight counters, leaving one. The equation is 9 = 2 × 4 + 1, so nine is odd.
Ask the learner to show the pairs and identify the leftover counter. A possible arrangement pairs C01/C02, C03/C04, C05/C06, and C07/C08, leaving C09. Different pairings are valid if all nine identities are retained and the final count is four pairs plus one.
Do not accept three-counter groups as pairs or split the last counter into halves. The collection uses whole counters. The learner’s reason should connect “odd” with one leftover after complete pairing, rather than treating the label as a property memorized from a list.
Round O3: Inspect the empty collection
The zero-count round contains no counters. It has zero pairs and zero leftover, and 0 = 2 × 0. Zero is even under the same rule; the game has not lost a counter or forgotten to load a positive collection.
Keep “Collection count: 0” visible. The collection value and learner answer are different pieces of state. A displayed count of zero does not mean that the learner has answered, and an empty response box is not a numerical zero.
Ask for an explicit zero-pair, zero-leftover explanation. A learner should be able to say that no unpaired counter remains and that the equation still has the even form. Never replace the empty collection with one counter to make the screen look populated.
Extend to 112 and 113 without drawing every counter
Use two fixed unscored examples to connect the pairing rule with decimal notation:
| Count | Pairing equation | Last digit | Classification |
|---|---|---|---|
| 112 | 112 = 2 × 56 | 2 | Even |
| 113 | 113 = 2 × 56 + 1 | 3 | Odd |
The tens part, 110, already forms pairs: 110 = 2 × 55. The units determine whether an extra unpaired counter remains. For 112, the two units form another pair. For 113, two of the three units form a pair and one remains.
More generally, a nonnegative whole number written in decimal can be separated into ten times its tens part plus its units digit. Ten times a whole number is divisible by two, so the units digit decides parity. The original examples make the reason visible without a picture containing over a hundred objects.
Keep this extension distinct from the three core rounds. It does not add scored attempts or require an arbitrary large-number calculator.
Use the pair builder
Choose the 9, 10, or 0 collection. Press Make next pair to group the next two unused counters, or activate two individual counters to pair them yourself. The text summary counts all complete pairs and remaining counters. Clear pairs restores the same collection without changing its size. Choose a classification and press Check proof; if pairs remain incomplete, finish grouping before checking the label.
Odd and prime answer different questions. Nine is odd but equals 3 × 3, so it is composite. Two is even and prime. Ask students to explain the leftover first, then discuss factors separately.
Copy this prompt
Create a classroom pairing game with three fixed collections: O1 nine counters, O2 ten counters, O3 zero counters. Every counter needs a stable numeric identity and may belong to only one two-counter pair. Display total, complete pairs and unpaired count together. Allow students to activate two free counters to make a pair and provide a Make next pair convenience button. Clear pairs restores the current collection. Keep separate records for each collection.
Provide Odd and Even buttons and Check proof. Nine is complete at four pairs plus one leftover: 9 = 2 × 4 + 1, odd. Ten is complete at five pairs and no leftover: 10 = 2 × 5, even. Zero is explicitly a zero-count collection, with no dummy counter: 0 = 2 × 0, even. A blank response is not zero or an answered classification. Require complete possible pairing before accepting the classification. Changes clear stale feedback. Include fixed unscored examples 112 = 2 × 56 and 113 = 2 × 56 + 1, showing why the final digit determines parity. Explain that nine is odd and composite while two is even and prime.
Return one complete self-contained HTML file with embedded CSS and JavaScript. Use native labeled controls with type="button" actions, readable text of at least 16px, strong contrast and visible focus. Make the layout responsive at 360px without horizontal scrolling. Keep supplied text literal. Use no remote dependencies or storage, and describe only actions the page actually performs.
Compare the empty and full collections
After the three rounds, ask students to explain zero using the same sentence pattern as ten: no unpaired counter remains. Keep their written explanations in your lesson notes. For a lesson walkthrough, NOS can render the prepared HTML pairing view alongside its written equations. Use the zero collection in the next walkthrough so an empty picture still has an explicit mathematical meaning.