Put the number in the grid

What this sequence teaches

How to put a number in a place value grid: find the ones digit, put it in the ones column, and put every other digit either side, one to a column, zeros included. The grid is the one the whole routine uses: the headings and the point never move, and the digits do the moving later. Nothing is calculated.

The teaching sequence

Example 1 — 452.7. The general case: digits both sides of the point, no zeros. The ones digit is the 2, just before the point.

Example 2 — 4527. The same digits with the point taken away. With no point, the ones digit is the last digit, so every digit moves one column left; the point and the headings stay put. Students who start every number in the ones column, or who think the first digit always goes in the same place, go wrong here.

Zeros aren’t taught separately: a zero is a digit and goes in its column like any other. The testing card checks it, first with 3680.

Running the sequence

• Run it straight through, in about thirty seconds. Students write each answer on mini-whiteboards before Reveal. A blank grid in a plastic sleeve saves drawing eight columns every time: print one from Print blank grid.
• Reveal follows the rule in order: the ones digit pulses in the number, and so does the point in the grid; it drops into the ones column; then the digits to its left, one at a time, moving outwards; then the digits after the point.
• On Example 2, the previous answer stays on screen, dimmed and labelled. Ask: “How little can you change your answer from the last one?”
• Moving to the next example is deliberately slow: the answer dims, the stage holds still, then the number changes. Replay shows the change again — before Reveal, just the change to the number; after it, the change to the number and then the change to the answer.

The rule and the script

Show rule displays: Put the ones digit in the ones column, then every other digit either side, one to a column. It’s off by default; whether to show it is your call.

Why? gives each example’s reason, and the pointer marks the ones digit in the number and in the grid, then the digits either side:

• “The ones digit is the 2, just before the point, so it goes in the ones column. The 5 and the 4 go to its left, the 7 after the point.”
• “There’s no point now, so the ones digit is the last digit, the 7. Every digit is one column further left.”

Why the conventions

Every digit goes in, zeros included: the zeros are in the number, so they go in the grid. Columns the number doesn’t reach stay empty; don’t let students fill them with zeros. Filling the gaps is a later step in the routine, and it’s the hard one. Students don’t write the point: the grid has one.

Where this sits

This is the first atom in multiplying and dividing by 10, 100 and 1000: Put the number in the grid (this atom); 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.

If you think a class is already secure, check with two questions first: 4500 and 0.06.

About the testing sequence

Seven items in a fixed order. The first is a completely new number, so no teaching answer can be adapted. After that, small changes alternate with fresh numbers, 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, and Reveal changes only what needs changing. After a fresh number the previous answer is wiped. The change to the number is slow on purpose, and the changed part pulses once when it lands. Replay shows it again.

What each item shows

• 36.8 → 3 T, 6 O, 8 t. A standard case to start well.
• 368 → 3 H, 6 T, 8 O. 3 O, 6 t, 8 h: started a whole number in the ones column.
• 3680 → 3 Th, 6 H, 8 T, 0 O. 3 H, 6 T, 8 O, 0 t: took the last non-zero digit as the ones digit.
• 0.06 → 0 O, 0 t, 6 h. 6 in O, or 0 O and 6 t: skipped the zeros in front. 0 t, 0 h, 6 th: started at the point.
• 0.6 → 0 O, 6 t. 0 O, 6 h: edited the last board and left the 6 where it was.
• 506.007 → 5 H, 0 T, 6 O, 0 t, 0 h, 7 th. Gaps where the zeros should be: left the zeros out.
• 40 506.007 → every column. It looks like the hardest number on the card, but nothing already there moves: only the 4 and the 0 at the front are new. A student who starts at the left of the grid gets the earlier items wrong too.

Reading the pattern

• Wrong on the whole numbers only: the student doesn’t yet know that the last digit is the ones digit when there’s no point.
• Wrong on 3680 only: the student treats zeros at the end as not counting.
• Wrong on 0.06: the student skips zeros in front, as with significant figures.
• Wrong only on the fresh numbers (the first, fourth and sixth items): the student has been editing previous answers.

Discussion prompts

• “Point to the ones digit in the number. Where does it go?”
• “Which digit did you place first? Why that one?”

About the expansion sequence

The same rule on numbers that look different: out to both edges of the grid, mostly zeros, zeros written at the end, and a whole number written with a point. No new teaching.

Item by item

1. 0.006 → 0 O, 0 t, 0 h, 6 th. Out to the last column.
2. 40 000 → 4 TTh, then four zeros. Out to the first column. Count from the ones digit, not from the left.
3. 40 000.006 → every column. The whole part hasn’t changed, so it stays put.
4. 2.60 → 2 O, 6 t, 0 h. A zero at the end of a decimal is in the number, so it goes in.
5. 2.600 → one more 0, in th. Nothing else moves.
6. 45.0 → 4 T, 5 O, 0 t. A whole number written with a point and a zero.
7. 45 → 4 T, 5 O. Ask: “Why are the 4 and the 5 in the same columns as last time?”

Where classes stall

The second item: students who count from the left of the grid put the 4 in the wrong place. Send them back to the ones digit.

Discussion prompts

• “Point to the ones digit. Where did it go?”
• “45.0 and 45: what’s the same on your board, and what’s different?”