Add 0.1

What this sequence teaches

Adding 0.1 to a number with one decimal place, and writing the answer with one decimal place. The sum is set out in columns, so the 1 sits directly under the digit it is added to. Students write only the answer.

The teaching sequence

Example 1 — 4.8 + 0.1 = 4.9. One digit goes up by one. The 4 stays.

Example 2 — 4.9 + 0.1 = 5.0. The answer to Example 1 is now the number, so students are counting on in tenths: 4.8, 4.9, 5.0. There is no digit after 9, so both digits of the answer change: the 9 becomes 0 and the 4 goes up to 5. The 0 is written. Without this example, the second test item produces 27.10 and 28.

Running the sequence

• Students write each answer on mini-whiteboards before Reveal. The two reveals take about 13 and 16 seconds, so the sequence runs in about a minute.
• Reveal is deliberately slow, so there is time to talk over it. The decimal point goes into the answer first, lined up under the other two. Then it works one column at a time, from the right, with a soft band marking the column. In each column the digits being added light up one by one with the + between them (“8, plus 1”), there is a pause to think, and then the answer digit appears under the line. When a column makes ten, a 0 appears, then a small 1 is carried to the top of the next column, and that column adds it last: “4, plus 0, plus the 1, is 5”. The small 1 disappears when the reveal ends.
• On Example 2, the previous answer stays on screen, dimmed and labelled. It is the same number as the one now in the question. Ask: “We just got to 4.9. What comes next?”
• Moving to Example 2 is deliberately slow: the answer dims, then the 8 fades out and the 9 fades in and pulses. Replay shows the change again — before Reveal, just the change to the question; after it, the change to the question and then the answer.

The rule and the script

Show rule displays: Add 0.1 to the number. The answer has one decimal place. It’s off by default; whether to show it is your call.

Why? gives each example’s reason, and the pointer steps through the digits that changed, from right to left:

• “I added 0.1, so the 8 goes up to 9. The 4 stays the same.”
• “The 9 can’t go up to 10 in one place, so it becomes 0 and the 4 goes up to 5. The 0 is written: the answer has one decimal place.”

Three things to know about this atom

• It works the answer out. Other transformation atoms stop at the set-up and never calculate. This one is the calculation.
• The columns are for looking, not for copying. They show which digit the 1 goes with, and why there is a 0 in the answer. Students don’t need to set the sum out: this should end up as a quick mental step, counting on by one tenth.
• Why 5.0 and not 5. As a sum, 4.9 + 0.1 = 5 is correct. In columns the 0 has to be written, and it is insisted on here because of where the atom is used: when rounding to one decimal place, 5.0 says the number has been rounded to one decimal place and 5 doesn’t. If you use this atom somewhere the 0 doesn’t matter, relax it.

Where this sits

This is the last atom in rounding to one decimal place: Cut after the first decimal place; Does it round up?; Recall the rounding rule; Add 0.1 (this atom). In that routine, the rounding rule gives students the sum, written on one line (3.9 + 0.1), and this atom works it out. It has no cut in it, so it can be used in any routine that needs this sum.

If you think a class is already secure, check with two questions first: 59.9 + 0.1 (60.0) and 52.6 + 0.1 (52.7).

About the testing sequence

Six items in a fixed order. The first is a completely new number, so the teaching answers can’t be adapted. The next three are small changes from it; then a fresh number, which catches students who are editing their last board, and one more small change. There is no scoring — the whiteboards are the data.

After a small change, the previous answer stays on screen, dimmed and labelled. Reveal clears it and works the new sum from the right. After a fresh number the previous answer is wiped.

What each item shows

• 27.3 + 0.1 = 27.4. A standard one to open. 37.3 or 28.3: added to the wrong digit.
• 27.9 + 0.1 = 28.0. Only the 3 has changed, to a 9. 27.10: wrote ten in one place. 28: dropped the 0.
• 29.9 + 0.1 = 30.0. 29.10 or 210.0: a 9 turned into “10”. 20.0: forgot to pass the one along. 30: dropped the zeros.
• 29.3 + 0.1 = 29.4. 30.4 or 20.4: changed the 9 because “9s become 0”. Only the digit being added to matters.
• 0.9 + 0.1 = 1.0. 0.10: ten in one place. 1: dropped the 0.
• 0.0 + 0.1 = 0.1. 1.0: edited the last answer. 0.01: put the 1 in the wrong place.

Reading the pattern

• A “10” written inside the number: the student is adding 1 to a digit, not adding 0.1 to a number.
• Right digits, missing zero (28, 30, 1): the student is simplifying. Go back to “the answer has one decimal place”.
• Wrong only on 29.3 + 0.1: the student has learned “9s become 0” as a rule of its own.

Discussion prompts

• “Which digits changed? Which stayed the same?”
• “Why is the 0 written?”

About the expansion sequence

The first two items make the run of 9s longer. From the fourth item the number has two or three decimal places, and the amount added changes with it: 0.01, then 0.001. The 1 is always under the last digit. There is no new teaching. The question is whether students see that they are always adding one in the last decimal place, and that the answer keeps as many decimal places as the number had. This prepares rounding to two and three decimal places.

Item by item

1. 399.9 + 0.1 = 400.0. Three 9s in a row. 399.10 or 3910.0: ten in one place. 400: dropped the 0. In Does it round up?, 399.9 | 6 came with the promise that a later step deals with the 9s. This is that step.
2. 99.9 + 0.1 = 100.0. The same number without its 3. Every digit is a 9, so a new 1 appears in front.
3. 2.9 + 0.1 = 3.0. A fresh number. It sets up the next two items.
4. 2.98 + 0.01 = 2.99. One more digit, and the amount added has changed to match. Students who write 3.08 are still adding 0.1.
5. 2.987 + 0.001 = 2.988. One more digit again. Ask: “Three sums that start with the same digits, three different answers. What decided each one?”
6. 3.99 + 0.01 = 4.00. Two 9s become 0, and both zeros are written. 4 and 4.0 are the boards to look for.
7. 0.049 + 0.001 = 0.050. 0.05 drops the last 0; 0.0410 writes ten in one place.

Where classes stall

The last two items. Students who accepted 5.0 will still write 4 for 4.00, because two zeros after the point look more wrong than one.

Discussion prompts

• “What changed in the amount you’re adding? Which digit does it go with?”
• “How many decimal places does your answer have? How do you know that’s right?”