Move the digits

What this sequence teaches

How to move the digits on the place value grid to multiply or divide by 10, 100 or 1000. Count the zeros in the 10, 100 or 1000; every digit moves one place for each zero, to the left to multiply and to the right to divide. The point and the column headings never move. Nothing is calculated, and no new zeros are written: the empty places the move opens are filled in the next step of the routine.

The teaching sequence

Three examples, because two things decide the answer: how many zeros, and which way. Each changes on its own, on the same number.

Example 1 — 47.5 × 10. The general case. One zero, so every digit moves one place left. Every place between the ones column and the digits is still taken.

Example 2 — 47.5 × 100. Only the number of zeros changes. One more zero, so every digit moves one more place. The ones column is left empty, and stays empty.

Example 3 — 47.5 ÷ 100. Only the sign changes. The same two places, the other way. This is the mistake the routine most needs to prevent (6.4 ÷ 100 = 640), so it comes last.

Running the sequence

• Run it straight through, in about a minute. Row 1 is filled in for you; students copy it and write row 2 on mini-whiteboards before Reveal. A blank grid in a plastic sleeve saves drawing eight columns: print one from Print blank grid.
• Reveal follows the rule in order: the sign pulses, and a pause; then the zeros of the 10, 100 or 1000 pulse one at a time, and a pause. Then the digits drop straight down from row 1 into row 2; the sign flashes again with a short arrow (left to multiply, right to divide); and each zero lights again in turn as every digit moves one place sideways together, with a pause after each place. The digits stay blue until the last place, so the dropped copy never reads as the answer. The digits in row 1 stay where they are.
• On Examples 2 and 3 the previous answer stays on screen, dimmed and labelled. Ask: “How little can you change your answer from the last one?” Reveal then clears the old answer and plays the whole move again from 47.5 in row 1, every time: on Example 2 the digits move two places, not one more than before, because 100 has two zeros.
• Moving to the next example is deliberately slow: the answer dims, the stage holds still, then only the changed part of the question changes and pulses. Replay shows it again.

The rule and the script

Show rule displays the fact students learned in Recall the rule for multiplying and dividing by 10, 100 and 1000, word for word, both sentences:

• To multiply by 10, 100 or 1000, move every digit to the left, one place for each zero.
• To divide by 10, 100 or 1000, move every digit to the right, one place for each zero.

Both sentences show on every example, so students still have to match the operation to its direction. It’s off by default; whether to show it is your call.

Why? gives each example’s reason, and the pointer marks the sign and the zeros in the question, then the digits in row 2:

• “10 has one zero, so every digit moves one place to the left. The point stays where it is.”
• “100 has two zeros, so every digit moves two places to the left: one more than last time. Nothing has moved into the ones column, so it stays empty.”
• “Now it’s divide, so every digit moves to the right. 100 still has two zeros, so it’s still two places.”

Why the conventions

Every digit moves, including any zeros in the number. Only the zeros in the 10, 100 or 1000 are counted. Students write no new zeros: an empty place is left empty, and filling it is the next step, the hard one. Students don’t write the point: the grid has one, and it never moves. A student who writes a zero in an empty place is ahead of the routine, not wrong; a student who adds a zero on the end (3.70) has not moved anything.

Where this sits

The third atom in 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 (this atom); Fill the empty places with zeros; Write the number from the grid.

If you think a class is already secure, check with two questions first: 5.8 ÷ 100 and 2.6 × 1000.

About the testing sequence

Ten items in a fixed order. The first is a completely new question, so no teaching answer can be adapted. After that, small changes alternate with fresh questions, which catch students who are editing their last board rather than using the rule. There is no scoring: the whiteboards are the data.

After a small change, the previous answer stays on screen, dimmed and labelled. Reveal clears it and plays the whole move from row 1 every time, so the digits always move from the number itself, never from the last answer. After a fresh question the previous answer is wiped.

What each item shows

• 3.8 × 10 → 3 T, 8 O. A standard case to start well. 3 O, 8 t, 0 h: added a zero (3.80) instead of moving the digits.
• 3.8 ÷ 10 → 3 t, 8 h. Only the sign changes. 3 T, 8 O: moved the wrong way, or left the last board as it was.
• 45 ÷ 10 → 4 O, 5 t. A whole number divided.
• 4500 ÷ 10 → 4 H, 5 T, 0 O, 0 t. Two zeros added to the number. 4 T, 5 O, 0 t, 0 h: counted the zeros in 4500 instead of the zero in 10. The 0 after the point is right as it stands; it is tidied in the last step.
• 6.09 × 100 → 6 H, 0 T, 9 O. A zero inside the number. 6 T, 9 O: dropped the 0 and closed the gap (69).
• 6.09 × 1000 → 6 Th, 0 H, 9 T. One more zero. 6 H, 0 T, 9 O: moved two places for 1000 (the last board unchanged). 6 TTh, 0 Th, 9 H: counted the 1 as well.
• 70.4 × 100 → 7 Th, 0 H, 4 T. 7 H, 0 T, 4 O: counted the zero in 70.4 instead of the zeros in 100 (704).
• 70.4 ÷ 100 → 7 t, 0 h, 4 th. Only the sign changes. 7 t, 4 h: took the zero off and closed the gap (0.74), the commonest mistake with a zero inside the number.
• 9.8 × 1000 → 9 Th, 8 H. A fresh question to set up the last.
• 9.876 × 1000 → 9 Th, 8 H, 7 T, 6 O. It looks the hardest on the card, but only the length has changed, and the rule ignores length: still three places, and the 9 and the 8 land where they did.

Reading the pattern

• Wrong on the divides only: the student hasn’t linked divide with right. Go back to the fact.
• Wrong on the 1000s only: the student counts the places wrongly. Ask them to point to each zero as they move.
• Wrong on 4500 ÷ 10 and 70.4 × 100: the student counts the number’s own zeros. Ask: “Which zeros tell you how far?”
• Wrong on 6.09 and 70.4 ÷ 100: the student drops a zero inside the number. Ask: “Where did the 0 go? Every digit moves.”
• Wrong only on the fresh questions (the first, third, fifth, seventh and ninth items): the student has been editing previous answers.

Discussion prompts

• “Which zeros did you count? Point to them.”
• “How do you know which way the digits go?”

About the expansion sequence

The same rule on boards that look different: a zero in front, undoing a move, and a zero inside the number carried right across the grid, out to both edges. No new teaching.

Item by item

1. 5.3 × 10 → 5 T, 3 O. Familiar ground to start.
2. 0.53 × 10 → 0 T, 5 O, 3 t. The 0 in front moves too, so the board reads 05.3. That’s right as the rule gives it; the 0 in front is tidied in the last step.
3. 4.6 × 100 → 4 H, 6 T. Familiar again: it sets up the next item.
4. 460 ÷ 100 → 4 O, 6 t, 0 h. 460 is the last answer once its gap is filled. Dividing by 100 moves every digit back, so the 4 and the 6 return to where they started, and the 0 comes too: 4.60. Ask: “Did we just add a zero to 4.6?”
5. 80.2 ÷ 10 → 8 O, 0 t, 2 h. A zero inside the number, divided. The 0 moves with the others.
6. 80.2 ÷ 100 → 8 t, 0 h, 2 th. Out to the last column, with nothing left in front of the point. That’s fine: the ones column is filled in the next step.
7. 80.2 × 100 → 8 Th, 0 H, 2 T. Only the sign changes: the same two places, the other way.
8. 80.2 × 1000 → 8 TTh, 0 Th, 2 H. Out to the first column, with the 0 still between the 8 and the 2. Two places stay empty. The hardest item on the page.

Where classes stall

The fourth item: students expect 460 ÷ 100 to give 4.6 and are surprised by the 0. Let them see the 0 came from 460; nothing was added. On the 80.2 items, watch for students who drop the 0 and close the gap.

Discussion prompts

• “4.6 × 100, then ÷ 100: where did the 0 come from?”
• “80.2 ÷ 100 and 80.2 × 100: what’s the same, and what’s different?”