Counting forwards and backwards (age 9+)

Activities to try

1. 1

   Single-column counting on a place-value chart

   Why it works: Counting in a power of ten means repeatedly nudging a SINGLE place-value column up or down while every other digit stays frozen, as long as no boundary is crossed; the chosen step decides which column ticks. (When a column rolls over, e.g. 492,000 → 502,000, a neighbouring column changes too. This is covered in the boundary activity.)

   1. Choose a step that is a power of ten (10; 100; 1,000; 10,000; or 100,000) and a direction (forwards or backwards).
   2. Starting from a given number such as 462,000, count on in 10,000s aloud: 462,000 → 472,000 → 482,000 → 492,000, each time saying which single column changed.
   3. On a place-value chart (e.g. a Gattegno chart) or with place-value counters, add or remove ONE counter in the chosen column per step and read the whole number.
   4. Say the count structurally too, keeping the unchanged places fixed: '46 ten-thousands and 2 thousand, 47 ten-thousands and 2 thousand, 48 ten-thousands and 2 thousand...' so only the ten-thousands count ticks while the 2,000 stays put.
   5. Repeat for a different power of ten (e.g. count back in 100,000s from 800,000: 700,000 → 600,000 → 500,000) and name the column that now ticks.
2. 2

   Crossing the boundary: the roll-over cascade

   Why it works: When a column fills, the single-column rule does not break — it cascades: counting forwards, the full column rolls over (9→0) and a carry moves into the next column, sometimes through several columns up to 1,000,000; counting backwards, an empty column rolls back (0→9) and the next column decreases by one.

   1. Count on in 10,000s and deliberately approach a boundary: 472,000 → 482,000 → 492,000 → 502,000, pausing at the crossing.
   2. At 492,000, answer 'how many ten-thousands make the next hundred-thousand?' before stepping, then bridge to 502,000 and read the new hundred-thousands digit.
   3. Count on in 10,000s up from 990,000 and watch the ten-thousands roll 9→0 while a carry cascades into the hundred-thousands and then the millions, reaching 1,000,000.
   4. Count back in 10,000s across a boundary (522,000 → 512,000 → 502,000 → 492,000 → 482,000), noticing that at 502,000 → 492,000 the ten-thousands digit rolls back from 0 to 9 and the hundred-thousands digit decreases by one.
   5. Predict a crossing before taking the step, then check: does only the next column change, or does the carry cascade further?
3. 3

   Spot the step and predict the next terms

   Why it works: The counting pattern becomes reasoning: identifying a constant power-of-ten step lets you justify and predict any later term, including across boundaries.

   1. Look at an unlabelled sequence such as 375,000; 475,000; 575,000 and work out the step by comparing consecutive terms (here +100,000).
   2. State the step as a power of ten and which column changes, then write or say the next three terms (675,000; 775,000; 875,000).
   3. Try sequences that cross a boundary so a prediction requires a roll-over, e.g. continue counting on in 10,000s from 472,000 through 502,000.
   4. Make and swap sequences with a partner: hide the upcoming terms and have them predict, then articulate the rule in words.
   5. For a backwards sequence, confirm the terms decrease and name the changing column.