Multiply and divide by 10, 100 and 1000

What this sequence teaches

The whole routine, start to finish. Students have met every step on its own: 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 and Write the number from the grid, and the middle three joined up. Here they do all five on one question, one line of working at a time.

The working, one click per line

• The question lands with an empty grid. Students copy the question.
• Show step 1: the number in the grid. The ones digit lights in the question and drops into the ones column; then the other digits, outwards, zeros included.
• Show step 2: the rule. Students say the rule for this question’s sign; the click shows it, all at once, in purple.
• Show step 3: 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 step 4: the fill. The ones column lights; the highlight moves through the empty places up to the digits; one zero is written in each. If nothing is empty, nothing is written.
• Show answer: write the number. The ones column lights, the highlight runs out to the last digit that isn’t a zero on each side, the zeros beyond fade (they’re left off), and the digits drop onto the answer line.

Every question takes five clicks, even when there is nothing to fill or nothing to leave off, so the button never gives anything away. The page never moves on by itself: you decide how long students have before each click.

The teaching sequence

Example 1 — 0.8 × 100. Row 1: 0 in the ones column, 8 in the tenths. The digits move two places left, the ones column is filled with a zero (0 8 0 .), and the 0 in front of the 8 is left off: 80.

Example 2 — 0.8 ÷ 100. Only the sign changes. The rule says right; the digits move the same two places the other way; the ones column and the tenths are filled (0 . 0 0 8). The 0 from 0.8 is now between the point and the 8, so this time it is kept: 0.008.

After the change, the previous answer stays on the answer line, dimmed and labelled, while students work. Ask: “How little can you change your answer from the last one?” On the last click it fades away before the new answer is written.

The rule and Why?

There is no Show rule button: the rule is the second line of the working. Why? explains the step just shown and points at what it used.

Where this sits

This is the last resource for multiplying and dividing by 10, 100 and 1000, after the five steps and the chain Recall the rule, move the digits, fill the empty places.

If you think a class is already secure, check with two questions first: 4.2 ÷ 100 and 0.05 × 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. Four have a zero inside the number. There is no scoring: the whiteboards are the data.

The line to check is the answer. If it’s wrong, the grid shows which step went wrong: row 1 (the number in the wrong columns), the rule (wrong way), the move (wrong number of places), the fill (missing or extra zeros) or the answer line (zeros kept or dropped).

What each item shows

• 3.7 × 1000 → 3700. A standard case to start well. 370: two places for 1000.
• 3.7 × 10 → 37. Two zeros fewer, and nothing to fill. 3.70 or 370: a zero added anyway.
• 6.4 ÷ 100 → 0.064. 640: the wrong way. 0.64: only the ones column filled. .064: no zero in the ones column.
• 50.8 ÷ 100 → 0.508. A zero inside the number. 0.58: the zero taken out. 5.08: one place.
• 50.8 × 100 → 5080. Only the sign changes. 508: the zero inside taken for the filled place. 5800: the zero moved to the end.
• 9200 ÷ 10 → 920. Row 2 reads 9 2 0 . 0: the 0 after the point is left off, the 0 before it is kept. 92: both left off.
• 9200 ÷ 100 → 92. One more zero, and now both zeros are after the point: both are left off. 92.00: the zeros left on.
• 450.7 ÷ 10 → 45.07. 45.7: the zero taken out.
• 2.6 × 1000 → 2600. 260: two places for 1000.
• 0.408 × 100 → 40.8. The hardest on the card. Row 2 reads 0 4 0 . 8: the front 0 is left off, but the 0 from the number lands in the ones column and is kept. 4.8 or 48: the wrong zero left off.

Reading the pattern

• Missing zeros after the point (0.64): the plutonium step. Go back to Fill the empty places with zeros.
• A zero inside the number lost (0.58, 508, 45.7, 4.8): ask “Which zeros came from the number, and which did you write?”
• Zeros left on (92.00, 040.8): go back to Write the number from the grid.
• Wrong only on the fresh questions: the student has been editing previous answers.

Discussion prompts

• “9200 ÷ 10 and 9200 ÷ 100: why is one zero kept in the first and none in the second?”
• “0.408 × 100: point to the zero you left off and the zero you kept.”

About the expansion sequence

The same five steps on questions that look different: five-digit answers, both edges of the grid, a whole number becoming a decimal, two zeros inside a number, and answers where most of the zeros are left off. Half the questions are × or ÷ 1000. No new teaching.

Item by item

1. 36 × 100 → 3600. Familiar ground to start.
2. 36 × 1000 → 36 000. Out to the first column: three empty places, three zeros, every one kept.
3. 36 ÷ 1000 → 0.036. Only the sign changes: out to the last column, and a whole number becomes a decimal.
4. 2040 ÷ 100 → 20.4. Row 2 reads 2 0 . 4 0: the 0 inside is kept, the 0 at the end after the point is left off.
5. 2040 ÷ 1000 → 2.04. One place further: the 0 inside now sits after the point, and is still kept.
6. 70.05 × 100 → 7005. Two zeros inside the number. Nothing to fill, nothing left off. 75 or 705: zeros taken out.
7. 0.03 × 100 → 3. Row 2 reads 0 0 3 .: every zero is in front, so all are left off.
8. 0.03 × 1000 → 30. One more zero: the ones column is filled, and that zero is kept, while the two in front are left off. The hardest item on the page.

Where classes stall

Items 4 and 5, and 8: deciding which zeros are kept. Ask: “Which digit is in the ones column? Where is the last digit that isn’t a zero on each side?”