BRIJBOX · EVERY WAY TO MAKE A NUMBER · LADDER FOR 6

One across, one row.

Put all six counters in the left area, and write that first row: 6 and 0. Then move one counter across, and write the new row. The left box matches the left area, so a row is a picture of your table. No printer? Rule the same rows by hand — the left-to-left match is what matters.

Check the whole after every single move. When two numbers you have already used turn up again, that is the discovery — mark them as partners, and never rub a row out.

BRIJBOX · EVERY WAY TO MAKE A NUMBER · LADDER FOR 10

The same routine, a bigger whole.

Sitting 2. Start at 10 and 0 and run exactly the same one-across move. Stop where the pairs begin to repeat. Then ask: “Which row has the same number in both parts?”

BRIJBOX · EVERY WAY TO MAKE A NUMBER · THE 7-CARD

One way is missing.

Cut this card out before sitting 2. Someone made a ladder for seven, but one way is missing — and it is genuinely absent, not blanked out. Have seven counters ready for checking.

Ask: “Someone made a ladder for seven, but one way is missing. Which one?” If the answer arrives by guessing, ask “How did you know?” before deciding what you saw.

BRIJBOX · EVERY WAY TO MAKE A NUMBER · GROWN-UP GUIDE

How do you know we found every way?

Two sittings by design. Sitting 1: exactly six counters, two group areas, the 6-ladder sheet. Sitting 2: exactly ten counters, the 10-ladder sheet, the finished six-ladder to re-enter from, and the 7-card. App 2–3 min → table 11–15 min → app practice 2–3 min, per sitting. These sheets are the complete no-device fallback.

Sitting 1 — the whole of 6

1. Start at the zero pair. Six counters all in the left area: “Today we are finding every way to make six with two parts. Here is our first way.” Record the row together: 6 and 0.
2. “Move one across.” The child moves exactly one counter. “What are the parts now?” (5 and 1.) Record the row. “What is the whole now?” — checked by recount or by the unchanged set. Then demonstrate nothing further: the one-across move is now the child’s engine.
3. Keep going: “Move one across. What are the parts? What is the whole?” — 4 and 2, then 3 and 3, then 2 and 4… When 2-and-4 appears, ask “Look at your ladder. Have we seen these two numbers before?” The repeated pair is the discovery, not a mistake. Agree together where the ladder of new pairs ends.
4. Explain. “Why does every row make six?” … then the different question: “How do you know we found every way?” Echoing your phrasing back is not sufficient.
5. Stop here. Save the finished six-ladder flat and unedited; put the counters away. Ten fresh ones open sitting 2.

Sitting 2 — the whole of 10

6. Re-entry question, exactly: “Here is your ladder for six. What did we do each time to get the next row?” Only when the child states or shows the one-across move does ten begin. If it fails, replay two rows of six; if it fails again, rebuild six rather than pushing on.
7. Run the same routine on ten: start at 10 and 0, move one across, record, check the whole, and stop where pairs repeat (at 4 and 6). Prompt only from the hint ladder — and record any prompt you used; it does not count as independent.
8. The equal-parts question: “Which row has the same number in both parts?” (5 and 5.) Ask it this way round — on this shortened sheet, several rows have no partner visible, so “which row has no partner?” would not have one answer.
9. Transfer: the prepared 7-card — 7 and 0 · 6 and 1 · 4 and 3, with seven counters for checking. “Someone made a ladder for seven, but one way is missing. Which one?” ({5 and 2}). If they land on it by guessing, ask “How did you know?” before deciding what you saw.

The ladder is built by the child; never fill a row for them. Keep exactly the active whole on the table and spares away. Re-verify the whole after every one-across move. A repeated pair stays on the sheet, marked by the child as a partner — never crossed out. The 7-card must be genuinely missing its row, not blanked out visibly.

What did you notice?

Practice on screen is rehearsal of reading a ladder. The building itself — the one-across move, the recording, the checking — is the table’s, and it is what the rows below record.

• Builds the ordered ladder for 6: runs one-across and records unaided after one demonstration row. Needing hints to sustain the move is supported.
• Notices where pairs repeat: points out the partner rows unprompted, or on the notice question. Insisting swapped rows are different ways is a revisit.
• Transfers the routine to 10: starts at 10 and 0 and runs it unaided, including the same-number-in-both-parts row. Any structural prompt from the hint ladder is a genuine mathematical cue — record supported, even though it came from the authored ladder.
• Explains why every row totals the whole: cites the unchanged set or recounts a row.
• Finds the missing pair by order: steps down the left column, or builds 6-and-1 and moves one across. A lucky guess is uncertain until “how did you know?” is answered.

Mark each row independent / supported / revisit / uncertain. Leave anything you did not observe uncertain. Nothing is scored or stored.

Trial notes — leave blank until observed