Fill the empty places with zeros

What this sequence teaches

Which empty places on the place value grid get a zero. Start at the ones column and go towards the digits: every empty place on the way gets a zero, and nothing beyond the digits does. If a digit is already in the ones column, there is nothing to fill. This is the step students most often get wrong when they multiply or divide by 10, 100 or 1000 (37 for 370, 0.64 for 0.064, .42 for 0.42).

Each question is one row of the grid, as the previous step (Move the digits) leaves it. The calculation isn’t shown: the only question here is which places need a zero.

The teaching sequence

Three examples, because two things decide the answer: which side of the point the digits are on, and how many places are empty. Each changes on its own, and every example writes a zero.

Example 1 — 5 8 _ . The ones column is empty, between the digits and the point, so it gets a zero: 580.

Example 2 — _ . 5 8. Only the side changes: the digits are after the point. The ones column is still empty, so it still gets a zero: 0.58. This is the zero students most often leave out (.58).

Example 3 — _ . _ 5 8. Only the count changes: one more empty place, so two zeros: 0.058. This is the board behind the commonest error of all (0.58 for 0.058), so it comes last.

A board with nothing to fill comes in the testing sequence, one change from a board that needs a zero.

Running the sequence

• Run it straight through, in about a minute. Students copy the row onto a mini-whiteboard grid and write in the zeros before Reveal. Print blank grid gives twelve grids to a page, to cut out and put in plastic sleeves.
• Reveal follows the rule in order: the ones column lights up, and a pause; then, if more places are empty, the highlight moves outwards one empty place at a time (left when the digits are in front of the point, right when they are after it) and stops when the next place has a digit, and a pause; then a zero goes into each empty place, from the ones column outwards, with a pause after each. When there is nothing to fill, the ones column lights up, it already has a digit, and nothing is written.
• On Examples 2 and 3 the previous answer stays on screen in the faint row underneath the grid, dimmed and labelled, in its columns. Ask: “How little can you change your answer from the last one?” On Reveal, a zero in the same place as one in the previous answer lights that one up too. The previous answer stays until Next, so the two can still be compared.
• Moving to the next example is deliberately slow: the answer is copied into the lower row, the zeros clear, the stage holds still, and only then do the digits change. Replay shows it again.

The rule and the script

Show rule displays: Put a zero in every empty place from the ones column up to the digits. It’s off by default; whether to show it is your call.

Why? gives each example’s reason, and the pointer marks the ones column, then the zeros written (or the digit already in the ones column):

• “The ones column is empty, and it’s between the digits and the point, so it gets a zero.”
• “The digits are after the point now, but the ones column is still empty, so it still gets a zero.”
• “Two empty places now, the ones column and the tenths, so two zeros. The highlight stops at the 5.”

Why the conventions

Students write the whole row with its zeros, because that is the number the next step reads. When there is nothing to fill, the answer is the row copied as it is. Zeros that are already among the digits (045, 26.0) stay: tidying them is the next step, so a student who writes 45 or 26 is ahead, not wrong. Nothing is calculated.

Where this sits

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

If you think a class is already secure, check with two questions first: 3 7 _ . and _ . _ 6 4.

About the testing sequence

Ten items in a fixed order. The first is a completely new row, so no teaching answer can be adapted. After that, small changes alternate with fresh rows, 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 in the faint row underneath, dimmed and labelled. After a fresh row it is wiped.

What each item shows

• 6 3 _ . → 630. A standard case to start well. 63: no zero in the empty ones column.
• 6 3 _ _ . → 6300. The digits one place further left, so two empty places. 630 with a gap: one zero only (the last board).
• _ . 4 2 → 0.42. The digits after the point. .42: no zero in the ones column.
• _ . _ 4 2 → 0.042. One more empty place, after the point. 0.42 with a gap: only the ones column filled. .042: only the tenths.
• 7 2 _ . → 720. A fresh row, to set up the next.
• 7 2 8 . → 728. One change: an 8 in the ones column, so nothing to fill. 7280 or 728.0: a zero written anyway.
• 4 5 _ . → 450. A fresh row, to set up the next.
• 4 0 5 _ . → 4050. A 0 among the digits. 405: “there’s a zero already”, so the ones column was left empty.
• _ . 3 8 → 0.38. A fresh row, to set up the last.
• _ . 3 0 8 → 0.308. A 0 among the digits, after the point. .308: “there’s a zero already”, so the ones column was left empty. 0.0308 or 00.308: the 0 taken for an empty place.

Reading the pattern

• Wrong whenever the digits are after the point: the student doesn’t fill the ones column. Ask: “Which column always has a digit in a number?”
• Wrong on 7 2 8 .: the student adds a zero whatever the board shows. Ask: “Is anything empty between the ones column and the digits?”
• Wrong on 4 0 5 _ . or _ . 3 0 8: the student treats any zero as the placeholder. Point to the empty ones column: a 0 among the digits is a digit, not an empty place.
• Wrong only on the fresh rows (the first, third, fifth, seventh and ninth items): the student has been editing previous answers.

Discussion prompts

• “Where did you start? Where did you stop?”
• “In _ . 3 0 8, is the 0 a digit or an empty place?”

About the expansion sequence

The same rule on rows where zeros are already among the digits: right next to the empty places, at the end, and at the front. No new teaching. Each new look arrives one change from a familiar row.

Item by item

1. 7 5 _ . → 750. Familiar ground to start.
2. 7 0 _ . → 700. The 5 becomes a 0, right next to the empty ones column. The row already seems to end in a zero, so students write 70. The 0 is a digit; the ones column is still empty and still gets its zero.
3. _ . _ _ 6 → 0.006. Familiar: three empty places.
4. _ . _ 0 6 → 0.006. The same number, but a 0 is already in the hundredths, so only two zeros are written. Ask: “Why only two zeros this time?”
5. 2 6 . → 26. Nothing to fill.
6. 2 6 . 0 → 26.0. A 0 at the end, after the point. Still nothing to fill; the 0 is tidied in the last step.
7. 4 5 . → 45. Nothing to fill.
8. 0 4 5 . → 045. A 0 at the front. It is a digit, not an empty place: still nothing to fill. 045 is right as this step gives it; the front zero is tidied in the last step.
9. 0 4 5 _ . → 0450. The digits one place left: a front zero and an empty place together. Only the ones column is filled. The hardest item on the page.

Where classes stall

The second item: 7 0 _ . looks finished. And the fourth: students who count zeros write three. In both, the 0 already there is a digit; the rule only fills empty places.

Discussion prompts

• “7 0 _ . — is the 0 a digit or an empty place? How do you know?”
• “_ . _ _ 6 and _ . _ 0 6 give the same number. Why do we write fewer zeros in one?”
• “In 0 4 5 _ ., which zero did you write, and which was there already?”