Recall the rule, move the digits, fill the empty places

What this sequence teaches

How the three steps join up. Students have met each one on its own: Recall the rule for multiplying and dividing by 10, 100 and 1000, Move the digits and Fill the empty places with zeros. Here they do all three on one question, one line of working at a time. The chain stops on the filled grid; writing the number out is the next step of the routine.

The working, one click per line

• The question lands with the number already in row 1 of the grid. Students copy it onto their grid.
• Show step 1: the rule. Students say the rule for this question’s sign; the click shows it, all at once, in purple.
• Show step 2: the move. The sign and then the zeros of the 10, 100 or 1000 light up; the digits drop into row 2 and move one place for each zero, with an arrow showing which way.
• Show answer: the fill. The ones column lights; if it is empty, the highlight moves through the empty places up to the digits; then one zero is written in each. If the ones column already has a digit, nothing is written.

Every question takes three clicks, even when there is nothing to fill, so the button never tells students whether a zero is coming. The page never moves on by itself: you decide how long students have before each click.

The teaching sequence

Example 1 — 5.9 × 100. Left, two places; the ones column is left empty and gets a zero: 5 9 0 .

Example 2 — 5.9 ÷ 100. Only the sign changes, and every line of working changes with it: the rule says right, the digits move the same two places the other way, and the empty places are now after the point. Two zeros this time: 0 . 0 5 9. The number in row 1 stays exactly where it was.

After the change, the previous answer drops into the faint row underneath, dimmed and labelled, and stays there until Next. Ask: “How little can you change your answer from the last one?” When a zero is written in the ones column, the previous answer’s zero in that column lights too.

The rule and Why?

There is no Show rule button: the rule is the first line of the working. Why? explains the step just shown and points at what it used: the sign for the rule; the sign and the zeros of the 10, 100 or 1000 for the move; the ones column and the zeros written for the fill.

Where this sits

Multiplying and dividing by 10, 100 and 1000: Put the number in the grid; Recall the rule for multiplying and dividing by 10, 100 and 1000; Move the digits; Fill the empty places with zeros; Write the number from the grid. This chain joins the middle three and is taught straight after Fill the empty places with zeros.

If you think a class is already secure, check with two questions first: 4.2 ÷ 100 and 7.3 × 1000.

About the testing sequence

Ten questions in a fixed order. The first is a fresh question, so no teaching answer can be adapted. Small changes alternate with fresh questions, which catch students who edit their last board instead of working from the question. There is no scoring: the whiteboards are the data. Students can write all three lines before any click, or one line per click: your call.

The last line to check is the filled row 2. If it’s wrong, the earlier lines show which step went wrong: the rule (wrong way), the move (wrong number of places) or the fill (missing or extra zeros).

What each item shows

• 3.7 × 100 → 3 7 0 . A standard case to start well. 3 7 . with an empty ones column: no zero written (37).
• 3.7 × 10 → 3 7 . One zero fewer, and now there is nothing to fill: the 7 lands in the ones column. 3 7 0 . or 3 . 7 0: a zero written anyway.
• 6.4 ÷ 100 → 0 . 0 6 4. A fresh question. 6 4 0 .: the rule for multiplying (wrong way). 0 . 6 4: only the ones column filled. . 0 6 4: no zero in the ones column.
• 6.4 × 100 → 6 4 0 . Only the sign changes. 0 . 0 6 4: the last board left as it was.
• 8.5 ÷ 10 → 0 . 8 5. A fresh question. . 8 5: no zero in the ones column.
• 8.05 ÷ 10 → 0 . 8 0 5. A zero appears inside the number. It moves with the other digits, and the ones column still gets its own zero. 0 . 8 5: took the zero out. . 8 0 5: thought the zero inside was enough.
• 450.7 ÷ 10 → 4 5 . 0 7. A zero inside the number again, and this time nothing to fill. 4 5 . 7: took the zero out.
• 2.6 × 1000 → 2 6 0 0 . Two empty places before the point. 2 6 0 .: two places for 1000.
• 3.06 × 1000 → 3 0 6 0 . A zero inside the number, right next to an empty ones column. 3 0 6 .: treated the zero inside as if it filled the gap. 3 6 0 0 .: moved the zero to the end.
• 3.068 × 1000 → 3 0 6 8 . It looks the hardest on the card, but only the length has changed: still three places, the 3, 0 and 6 land where they did, and the 8 lands in the ones column, so there is nothing to fill.

Reading the pattern

• Right rule, wrong row 2 on the divides: the student says the rule but doesn’t use it. Ask them to point to the word “right” before they move.
• Missing zeros after the point (0.64, 0.85): the plutonium step. Go back to Fill the empty places with zeros.
• A zero inside the number lost or misused (0.85 for 0.805, 45.7, 306 for 3060): the student is treating the number’s own zero as a gap or as a filled place. Ask: “Which zero came from the number, and which did you write?”
• Zeros written when there is nothing to fill (3.70, 30680): “always add a zero”. Ask: “Which places are empty?”
• Wrong only on the fresh questions (the first, third, fifth, seventh, eighth and ninth items): the student has been editing previous answers.

Discussion prompts

• “Which line of your working tells you which way the digits go?”
• “Point to every empty place from the ones column up to the digits.”

About the expansion sequence

The same three steps on boards that look different: both edges of the grid, a whole number becoming a decimal, a zero from the number beside the empty places, and zeros left at the front or the end. Half the questions are × or ÷ 1000. No new teaching. A zero in front or at the end looks wrong, but it’s right as the rule gives it: the next step of the routine tidies it.

Item by item

1. 45 × 10 → 4 5 0 . Familiar ground to start.
2. 45 × 1000 → 4 5 0 0 0 . Out to the first column; three empty places, three zeros.
3. 45 ÷ 1000 → 0 . 0 4 5. Only the sign changes: out to the last column, and a whole number becomes a decimal.
4. 4500 ÷ 100 → 4 5 . 0 0. The two zeros move to the end, after the point. Nothing to fill.
5. 40.5 × 1000 → 4 0 5 0 0 . Out to the first column with a zero inside the number: three zeros in the answer, but only two are written. 4 0 5 0 .: counted the zero inside as one of the gaps.
6. 0.7 ÷ 10 → 0 . 0 7. The 0 in 0.7 moves too and lands after the point, so only the ones column is empty: one zero written, not two.
7. 0.06 × 100 → 0 0 6 . Two zeros in front, nothing to fill.
8. 0.06 × 1000 → 0 0 6 0 . One more zero: the digits move one place further, the ones column is left empty and gets a zero, and the zeros in front stay. The hardest item on the page.

Where classes stall

Items 5 and 6: with a zero from the number already in the row, students write too many zeros or too few. Ask: “Which zero came from the number, and which did the fill write?” Items 7 and 8: students want to tidy the zeros in front, or don’t believe 0 0 6 0 . can be right.

Discussion prompts

• “40.5 × 1000: point to the zero that came from the number, and the zeros you wrote.”
• “45 × 1000 and 45 ÷ 1000: what’s the same, and what’s different?”