Cut after the first decimal place

What this sequence teaches

Where the cut goes when rounding to one decimal place: straight after the first digit after the point. Every digit is written down and nothing is rounded yet. The cut sets up the next step, deciding whether it rounds up. That step reads the digit just after the cut, so the tail has to stay on the board.

The teaching sequence

Example 1 — 36.2791 → 36.2 | 791. The two-digit whole part means “cut after the first digit” gives a visibly wrong board from the start.

Example 2 — 3.62791 → 3.6 | 2791. The same digits with the point moved one place. The cut moves with the point. Watch the answer: the point and the cut stay put while the digits slide past, and the 2 crosses into the tail. Without this example, both answers would cut after the third digit of the number, and some students would take that as the rule.

Example 3 — 3.02791 → 3.0 | 2791. Only the first decimal place changes, to 0, and the cut doesn’t move. A 0 straight after the point still counts as the first decimal place. Students who know significant figures often skip it. It comes last because it’s the mistake most likely to follow them into rounding.

Running the sequence

• Run it straight through, in about thirty seconds. Students write each answer on mini-whiteboards before Reveal.
• Reveal follows the rule in order: the number copies down whole (every digit stays), the point pulses, the digit after it pulses, then the cut draws down into the gap after it. No digit moves: the spaces between digits are wide enough for the cut, so the answer is visibly the same number with a line in it.
• 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?”
• Moving to Example 2 is deliberately slow: the answer dims, the point hops left over the 6, then the number slides so the point is back where it always sits. Replay shows the change again for anyone who missed it — before Reveal, just the change to the question; after it, the change to the question and then the change to the answer.

The rule and the script

Show rule displays: Find the point. Cut straight after the first digit after it. Every digit stays. It’s off by default; whether to show it is your call.

Why? gives each example’s reason, and the pointer steps through the point, the digit after it, and the cut:

• “I found the point and cut straight after the first digit after it, the 2. Every digit stays.”
• “The point has moved, so the cut moves with it: straight after the 6. Every digit stays.”
• “The first digit after the point is 0, and it still counts. I cut straight after it. Every digit stays.”

Why the conventions

Any clear vertical line counts, and spacing doesn’t matter. Don’t let students round or drop digits yet. Deciding whether it rounds up is the next atom’s job. Keeping each step separate means a wrong board shows which step went wrong.

Where this sits

This is the first atom in rounding to one decimal place: Cut after the first decimal place (this atom); Does it round up? (it reads the digit straight after the cut); Recall the rounding rule; Evaluate.

If you think a class is already secure, check with two questions first: 417.635 and 0.0829.

About the testing sequence

Six 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. The fresh numbers 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 question is slow on purpose, and the changed part pulses once when it lands. Replay shows it again.

What each item shows

• 748.5163 → 748.5 | 163. 7 | 48.5163: cut after the first digit of the number. 748.51 | 63: two decimal places.
• 74.85163 → 74.8 | 5163. 74.85 | 163: moved the point but not the cut.
• 74.05163 → 74.0 | 5163. 74.05 | 163: skipped the 0.
• 0.6284 → 0.6 | 284. 0 | .6284: counted the 0 before the point as the first digit.
• 0.0284 → 0.0 | 284. 0.02 | 84: significant-figures habit, skipping zeros.
• 305.972 → 305.9 | 72. 30 | 5.972, or a cut after the 0: a whole-part 0 has thrown them.

Reading the pattern

• Wrong only on the two items with a 0 straight after the point: the student is skipping zeros.
• Wrong on every item, with the cut one digit too far: the student is cutting after two decimal places.
• Wrong only on the fresh numbers (the first, fourth and sixth items): the student has been editing previous answers.
• Any rounding, or a missing tail: go back to “every digit stays”.

Discussion prompts

• “Say the rule, then point to where it tells you to cut on this number.”
• “What does the 0 straight after the point count as?”

About the expansion sequence

The instruction changes to the second or third decimal place, with no new teaching. The question is whether the rule transfers: find the point, count along, cut. The spoken version to offer is “Find the point. Count along to the second digit. Cut straight after it.”

Item by item

1. 305.972, second place → 305.97 | 2. The last test number, with the instruction changed. Students still cutting after the first place haven’t read the instruction.
2. 30.5972 → 30.59 | 72. The point moves, and the cut moves with it.
3. 81.40637, second place → 81.40 | 637. A 0 in the second place counts.
4. The same number, third place → 81.406 | 37. Ask: “Why is the answer some of you wrote last time right now?”
5. 6.28057, third place → 6.280 | 57. A 0 in the third place counts.
6. The same number, first place → 6.2 | 8057. Back to the routine’s own rule.

Where classes stall

The third and fifth items, the zeros. These are the same mistake as skipping the zero on the testing card, in a new place.

Discussion prompts

• “What changed in the instruction? What does that change in your answer?”
• “Point to the digit you counted to.”