Write the number from the grid

What this sequence teaches

How to write the number that a filled row of the place value grid shows. Start at the ones column: its digit is always written. Go out to the last digit that isn’t a zero on each side, and write every digit in between, zeros included. Zeros beyond that are left off, and the point is written only if a digit comes after it.

Each question is one row of the grid, exactly as the previous step (Fill the empty places with zeros) leaves it. The calculation isn’t shown: the only question here is which digits to write.

The danger is “leave off the zeros” applied too widely: 306 for 3060, 0.85 for 0.805, .064 for 0.064. The rule never touches a zero between digits, a zero at the end of a whole number, or the ones digit.

The teaching sequence

Three examples, because two things decide whether a zero is written: whether a zero at the end is before the point or after it, and whether a zero at the front is in the ones column. The same four digits slide right, so only where the point falls changes.

Example 1 — 0 6 3 0 . The ones digit is the 0 before the point, so it’s written; the 0 in front of the 6 is left off: 630.

Example 2 — 0 6 3 . 0 The row moves one place right. The end 0 is now after the point, with nothing after it, so it’s left off, and so is the point: 63. Students who write 63.0, or 63., go wrong here.

Example 3 — 0 . 6 3 0 The row moves two more places right. The front 0 is now in the ones column, so it’s written: 0.63. This is the zero students most often leave off (.63), so it comes last.

Running the sequence

• Run it straight through, in about a minute. Students write the number on mini-whiteboards 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; the highlight runs out to the left one place at a time and stops at the last digit that isn’t a zero; then out to the right, the same; the zeros beyond fade (they’re left off); then the digits drop to the answer line, left to right, with the point arriving from the grid just before the first digit after it.
• On Examples 2 and 3 the previous answer stays on the answer line, dimmed and labelled. Ask: “How little can you change your answer from the last one?” On Reveal, after the find, only what changed moves: digits that stay brighten and slide, digits that go fade, new ones drop from the grid.
• Moving to the next example is deliberately slow: the answer dims, the stage holds still, then the row moves. Replay shows it again.

The rule and the script

Show rule displays: Write the ones digit, and every digit out to the last one that isn’t a zero on each side. 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 places out to the last digit that isn’t a zero, then the answer:

• “The ones digit is the 0 just before the point, so it’s written. Out to the left, the last digit that isn’t a zero is the 6, so the 0 in front of it is left off: 630.”
• “The row has moved one place right, so the 3 is the ones digit now. The end 0 is after the point, with nothing after it, so it’s left off, and so is the point: 63.”
• “The row has moved two more places right. The front 0 is in the ones column now, so it’s written. After the point, the last digit that isn’t a zero is the 3, so the end 0 is left off: 0.63.”

Why the conventions

Numbers of five digits or more are written with a thin space (42 000), as in the rest of this routine. A board with the zeros left on (630.0, 0063) has the right value but isn’t how the number is written: say so, and point to the rule. Nothing is calculated.

Where this sits

The last 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; Write the number from the grid (this atom).

If you think a class is already secure, check with four rows first: 5 . (a 5 in the ones column, nothing else); 0 . 0 0 6; 4 5 . 0 0; 0 0 6 .

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. Half the rows have a zero between digits, on both sides of the point. There is no scoring: the whiteboards are the data.

After a small change, the previous answer stays on the answer line, dimmed and labelled. After a fresh row it is wiped.

What each item shows

• 0 4 7 . → 47. A standard case to start well. 047: the front 0 kept.
• 0 4 . 7 → 4.7. The row one place right: a digit after the point now, so the point is written. 47: the point missed. 04.7: the front 0 kept.
• 2 . 9 0 → 2.9. A fresh row. 2.90: the end 0 kept.
• 2 . 0 9 0 → 2.09. A 0 inserted between the point and the 9. 2.9: every 0 left off.
• 6 0 4 . 0 → 604. A fresh row. 64: the 0 between digits left off. 604. or 604.0: the point, or the end 0, kept.
• 6 0 4 0 . → 6040. The row one place left: the end 0 is now before the point, so it’s written, and nothing is left off. 604: an end zero left off before the point (the last board, or “leave off end zeros” applied too widely).
• 0 . 7 0 5 → 0.705. A fresh row. .705: the ones 0 left off. 0.75: the 0 between digits left off.
• 0 7 . 0 5 → 7.05. The row one place left: the 0 is no longer in the ones column, so it’s left off now. 07.05: the front 0 kept. 7.5: the 0 between digits left off.
• 0 0 2 . → 2. A fresh row: a single digit, every 0 left off. 02 or 002: front zeros kept.
• 0 0 0 2 . → 2. One more 0 at the front. It looks harder; the answer doesn’t change.

Reading the pattern

• Wrong on 6 0 4 0 . or 2 . 0 9 0 or 0 . 7 0 5: the student leaves off zeros wherever they are. Point to the rule: only zeros beyond the last digit that isn’t a zero are left off.
• Wrong on 0 . 7 0 5 only (.705): the student leaves off the ones 0. Ask: “Which digit is always written?”
• Zeros kept at the front or end (047, 2.90, 604.0): the student copies the row. Ask: “Where’s the last digit that isn’t a zero?”
• Wrong only on the fresh rows (the first, third, fifth, seventh and ninth items): the student has been editing previous answers.

Discussion prompts

• “Which zeros did you write, and which did you leave off? Why?”
• “6 0 4 . 0 and 6 0 4 0 .: the same digits. Why is one answer 604 and the other 6040?”

About the expansion sequence

The same rule on rows that look different: out to both edges of the grid, lots of zeros, and zeros either side of the point. No new teaching. Each new look arrives one change from a familiar row.

Item by item

1. 4 2 0 0 0 . → 42 000. Out to the left edge: every 0 is before the point, so every one is written.
2. 4 2 . 0 0 0 → 42. The same digits, three places right: now every 0 is after the point with nothing after it, so every one is left off, and so is the point.
3. 0 . 0 7 → 0.07. Familiar: the ones 0 and the 0 before the 7 are written.
4. 0 . 0 0 7 → 0.007. Out to the right edge. One more 0 between the point and the 7: written, like the other.
5. 0 0 9 0 . → 90. Zeros at the front left off; the 0 in the ones column written.
6. 0 0 9 . 0 → 9. The row one place right: every 0 left off, on both sides.
7. 4 0 . 5 0 → 40.5. Familiar: the ones 0 written, the end 0 after the 5 left off.
8. 4 0 . 0 0 → 40. The 5 becomes a 0. Every digit after the point is a 0 now, so they’re left off, and so is the point; the 0 in the ones column stays. The hardest item on the page: students who leave off every 0 write 4.

Where classes stall

The second item, where students who copy the row write 42.000, and the last, where students who leave off every zero write 4. In both, find the ones column first.

Discussion prompts

• “4 2 0 0 0 . and 4 2 . 0 0 0 have the same digits. Why does one keep every zero and the other none?”
• “In 4 0 . 0 0, which 0 did you write? Why that one?”