Cut, then decide if it rounds up

What this sequence teaches

How two steps students already know join up. Step 1 is Cut after the first decimal place. Step 2 is Does it round up? The new idea is the link: step 2 looks at the digit straight after the cut made in step 1. Neither step’s name appears on screen, so say as much or as little as you like. The only words on the stage are the cue “1 decimal place”, which ties the place to where the cut goes and sets up the expansion sequence, where the cue changes.

The two examples

Example 1 — 8.2476 → 8.2 | 476 → doesn’t round up. The two steps in order. The digit straight after the cut is 4.

Example 2 — 8.2576 → 8.2 | 576 → rounds up. One digit changes, the 4 to a 5, and nothing else. The cut stays exactly where it was, and the answer flips. That puts all the attention on the digit straight after the cut: it alone made the difference. The point doesn’t move anywhere in this sequence; that is saved for the testing sequence.

Running the sequence

• Nothing moves between the two steps until you click. Show step 1 plays the cut. The stage then waits for as long as you want to talk or check boards. Show step 2 lights the digit straight after the cut, pauses, then shows the answer.
• Moving to Example 2 is deliberately slow: the working dims, then the 4 fades out and the 5 fades in and pulses. The previous working stays on screen, dimmed and labelled, and each click updates one line of it. On Show step 1 only that one digit changes in the cut line.
• Replay shows the last thing again. Why? gives the reason for the step on screen.

Where this sits

This chains the first two atoms of rounding to one decimal place. Next come Recall the rounding rule and Add 0.1.

About the testing sequence

Seven numbers in a fixed order. Students write both lines on their whiteboards: the number with its cut, then “rounds up” or “doesn’t round up”. There is no scoring; the whiteboards are the data.

You choose when boards go up. For the first numbers you might check after each step: boards up, Show step 1, boards up again, Show step 2. Later, ask for both lines at once and click through.

Every number is chosen so that a wrong cut gives a wrong final answer. If the last line is right, the cut was almost certainly right too.

What each item catches

• 305.472 → 305.4 | 72 → rounds up. A cut after the first digit, 3 | 05.472, gives “doesn’t round up”.
• 30.5472 → 30.5 | 472 → doesn’t round up. The point has moved, for the first time on this page. Leaving the cut where it was gives 30.54 | 72 and “rounds up”.
• 0.0829 → 0.0 | 829 → rounds up. Skipping the zero gives 0.08 | 29 and “doesn’t round up”.
• 0.0429 → 0.0 | 429 → doesn’t round up. Rounding from the end (429 to 43 to 5) gives “rounds up”.
• 62.9538 → 62.9 | 538 → rounds up. Exactly 5 counts. A cut after two decimal places gives 62.95 | 38 and “doesn’t round up”.
• 49.3207 → 49.3 | 207 → doesn’t round up. The 9 just before the point tempts “rounds up”. Nothing else in this number does, so a wrong answer here points at the 9.
• 27.34998 → 27.3 | 4998 → doesn’t round up. “4998 is big” and rounding from the end both give “rounds up”.

Reading the boards

• Line 1 wrong: go back to cutting after the first decimal place.
• Line 1 right, line 2 wrong: go back to “Does it round up?”
• Wrong only on 49.3207: the student is looking at the 9, not at the digit straight after the cut.

About the expansion sequence

The same two steps for other decimal places, with no new teaching. The cue at the top of the stage has read “1 decimal place” until now. Here it changes, and the cut moves with it.

1. 5.0649, 2 decimal places → 5.06 | 49 → doesn’t round up.
2. The same number, 1 decimal place → 5.0 | 649 → rounds up. Only the cue changes, and the answer flips.
3. 81.3072, 2 decimal places → 81.30 | 72 → rounds up. The zero in the second place counts.
4. The same number, 3 decimal places → 81.307 | 2 → doesn’t round up.