Does it round up?

What this sequence teaches

How to decide whether a number, already cut after the first decimal place, rounds up: look only at the digit straight after the cut. If it’s 5 or more, it rounds up. Nothing else decides it — not the kept digit, not the last digit, not how big the tail looks.

Every number in the teaching and testing sequences is rounded to one decimal place, so the cut is always straight after the first decimal place. Only the expansion card moves it.

The teaching sequence

Example 1 — 72.3 | 4999 (doesn’t round up). The tail looks enormous. Reading it as “nearly 5000”, rounding from the end, and looking at the last digit all say it rounds up. Only the digit straight after the cut says no.

Example 2 — 72.3 | 5000 (rounds up). The number has gone up by the smallest possible step, and now it rounds up. This is the boundary, and it shows that exactly 5 counts — 5 is not “the middle, so it stays”.

Example 3 — 0.9 | 7 (rounds up). A tiny number with a one-digit tail. The 9 before the cut doesn’t stop it rounding up; what happens to that 9 is the next atom’s business.

Example 4 — 241.5 | 9786 (rounds up). A big number and the top of the range, 9. There’s a 5 before the cut, but it’s the 9 after the cut that decides.

Example 5 — 241.5 | 0786 (doesn’t round up). Only the digit after the cut has changed, and the verdict flips. The kept 5, the last digit 6, the big-looking 786, rounding from the end and skipping the 0 all still say it rounds up. The tail and the kept 5 stayed identical across Examples 4 and 5, so they can’t be what decides.

Why the sequence is shaped this way

• The two animated pairs are the lesson. In 4999 → 5000 and 9786 → 0786 the verdict flips while everything a wrong habit looks at stays put or points the wrong way.
• The three numbers in the middle are as different as possible: the digit after the cut runs 5, 7, 9; the whole part runs 72, 0, 241; the tail runs from one digit to four.
• The cut is always after the first decimal place, so it sits in the same column on every example, using the same layout as Cut after the first decimal place, so students’ eyes learn to look straight after the line.

What this sequence doesn’t address

Crude and sophisticated versions of each wrong habit are in the testing card. Cuts in other places — nearest whole number, ten, hundred, two or three decimal places — are in the expansion card. What rounding up does to the number belongs to the next atom.

The rule and the script

The rule, exactly as Show rule displays it: For the following examples: look only at the digit straight after the cut. If it’s 5 or more, it rounds up. “Only” is doing the work: it rules out the last digit and the tail.

The spoken justification for each example:

• 72.3 | 4999 — “It doesn’t round up. How do I know? Because the digit straight after the cut is 4, and 4 isn’t 5 or more. The digits after it don’t count.”
• 72.3 | 5000 — “It rounds up. How do I know? Because the digit straight after the cut is 5, and 5 is 5 or more.”
• 0.9 | 7 — “It rounds up. How do I know? Because the digit straight after the cut is 7, and 7 is 5 or more.”
• 241.5 | 9786 — “It rounds up. How do I know? Because the digit straight after the cut is 9, and 9 is 5 or more.”
• 241.5 | 0786 — “It doesn’t round up. How do I know? Because the digit straight after the cut is 0, and 0 isn’t 5 or more. The digits after it don’t count.”

Whether students see the words is your call. Nothing appears unless you click. Show rule puts the rule above the stage and stays until you turn it off. Reveal always points: the moment the ✓ or ✗ appears, the digit straight after the cut turns orange, pulses and is underlined, and it stays that way until you move on. Every verdict arrives with the only digit that decided it, so no student can miss which digit matters. Why? (the ? button) adds the words, and the digit pulses again as they appear.

Why the rule works

For you, or for a class that asks. 8.2 | 476 sits between 8.2 and 8.3, and the halfway point is 8.25. The digit straight after the cut says which side of halfway the number is: 5 or more means halfway or beyond. Everything after that digit moves the number by less than one step of that digit, so it can never carry it across halfway — which is why “the digits after it don’t count”.

Where this sits

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

If you think a class is already secure, check with two questions first: 0.6 | 49 (doesn’t round up) and 41.2 | 5 (rounds up).

Running the sequence

Run it straight through, in about a minute. Students show Yes or No on mini-whiteboards before you reveal.

Moving to Examples 2 and 5 is deliberately slow: the digits change in place, and the new digit straight after the cut pulses once. Replay shows the change again — after the reveal, it replays the change and then the answer. Ask: “What changed? Why did the answer change?” Examples 3 and 4 fade in fresh; they are new numbers, not changes. Before you reveal them, ask “Which digit will decide it?” — Reveal then confirms it.

About the testing sequence

Ten numbers in a new random order each time: five round up, five don’t. Every number is already cut after the first decimal place, so the rule decides every one. There’s no Show rule button here — students apply the rule without it on screen. As in the teaching sequence, the digit straight after the cut lights up and is underlined the moment each answer is marked; Why? adds the words.

The items are static. The animations in the teaching sequence mark the change between two examples; they don’t carry what “rounds up” means, so a still number is a fair test.

What each number shows

• 6.4 | 5. Says no: thinks 5 doesn’t count.
• 38.1 | 5024. Says no: thinks 5 doesn’t count (the digits after it look small), or is reading the last digit or the kept digit.
• 399.9 | 6. Says no: thinks a 9 can’t round up.
• 700.1 | 9. Says no: is reading the kept digit.
• 0.0 | 7. Says no: is reading the kept digit, or thinks the number is too small to round up.
• 69.9 | 3. Says yes: is reading the kept digit.
• 6.2 | 19. Says yes: is reading the last digit.
• 7.1 | 4998. Says yes: is reading the tail as a number, rounding from the end, or reading the last digit.
• 1.3 | 44478. Says yes: is rounding from the end (44478 → … → 5), or reading the last digit.
• 26.4 | 08. Says yes: skipped the 0.

Reading the pattern

• Wrong on 6.2 | 19 as well as the long tails: reading the last digit.
• Wrong on 7.1 | 4998 and 1.3 | 44478 but right on 6.2 | 19: rounding from the end. (19 rounds to 2, so that habit gets 6.2 | 19 right.)
• Wrong on 700.1 | 9, 0.0 | 7 and 69.9 | 3: reading the kept digit.
• Wrong only on 6.4 | 5 and 38.1 | 5024: “5 doesn’t count”.
• Wrong only on 26.4 | 08: skipping zeros — the same habit as on the cutting card, one place along.
• Wrong only on 399.9 | 6: worrying about what happens next. Reassure them: it still rounds up, and the next atom deals with the 9.

Discussion prompts

1. “Put your finger on the digit that decides. What is it?”
2. “Which digits could I change without changing the answer?”
3. “What’s the smallest change that would flip this one?”

Reading the summary

At the end, all ten numbers are shown together. Each cell has two channels: a green or red tint shows whether the student answered correctly; the ✓ or ✗ in the corner shows whether the number rounds up.

About the expansion sequence

On the teaching and testing cards, every cut was after the first decimal place. This is the only card where it moves. The rule never said “first decimal place”; it said “straight after the cut”. Here the cut lands somewhere new each time — nearest whole number, nearest ten, nearest hundred, two or three decimal places — and the question and the rule stay the same. It asks whether students learned after the cut, or only after the first decimal place. It works exactly like the testing sequence: random order, feedback, Why?, and a summary.

Every number here is one where the rule still works on an unfamiliar surface. There are none where it stops working: for positive numbers the rule is always right.

Number by number

• 4 | 3.8 (doesn’t round up). The cut is before the point. Students who look straight after the point read the 8.
• 12 | .7 (rounds up). The point sits straight after the cut, but it isn’t a digit: the 7 decides.
• 5.06 | 49 (doesn’t round up). Two decimal places. Rounding from the end gets it wrong.
• 0.004 | 5 (rounds up). Three decimal places, a tiny number, and exactly 5.
• 2 | 71 (rounds up). No point at all.
• 9 | 0.6 (doesn’t round up). A 0 after the cut, with a 6 past the point waiting to tempt them.

Where classes stall

The three numbers with the cut before the point: 4 | 3.8, 12 | .7 and 9 | 0.6. Students who stall there have tied the rule to the point, not the cut.

Discussion prompts

1. “What’s different about where the cut is? Did the rule change?”
2. For a strong class: “Does 8.2 | 4999…, with the 9s going on forever, round up?” It does: 8.24999… is exactly 8.25, because 0.999… = 1.

Reading the summary

The same two channels as the testing card: green or red tint for whether the student was right, ✓ or ✗ for whether the number rounds up.