Addition and subtraction strategies (age 10+)

Activities to try

1. 1

   Decimal exchange bench: aligning by the point and trading ten-for-one

   Why it works: Decimals are not a new arithmetic: the point just marks where the columns keep going, and the ten-for-one exchange across every boundary (hundredths→tenths, tenths→ones) is unchanged from whole numbers. You must line up tenths under tenths and hundredths under hundredths because you can only add like to like.

   1. Set up 3.4 + 2.78 in place-value columns with the decimal point fixed in a central column; notice that 3.4 has an empty hundredths slot (to the right of the 4, above the 8), and add the placeholder zero so it becomes 3.40 with tenths under tenths and hundredths under hundredths.
   2. Compute 3.45 + 2.78: the hundredths total 13, so fuse ten hundredth-pieces into a single tenth-piece and carry it one column left — exactly the ones-to-tens carry already known. Record the answer 6.23.
   3. Solve 5.03 − 2.67 by decomposing: exchange one whole into ten tenths and one tenth into ten hundredths so you can take 7 hundredths from 3 hundredths; record 2.36.
   4. Before each calculation, estimate roughly (3.45 + 2.78 is about 6) and use the estimate to check the point is in the right place afterward.
   5. Re-scale base-ten blocks so a flat = 1, a rod = 0.1, and a small cube = 0.01, and physically trade ten small cubes for one rod to mirror ten-ones-make-a-ten.
2. 2

   Confronting 'longer-is-larger' with ragged decimals

   Why it works: The digits after the point are not a separate whole number, so a 'longer' decimal is not automatically larger — 4.9 (nine tenths) is larger than 4.11 (eleven hundredths) despite having fewer digits, while 0.45 (forty-five hundredths) is larger than 0.4 (forty hundredths). You must compare the same place values. Trailing zeros do not change value because they add nothing to any place.

   1. Decide which is larger, 4.11 or 4.9. Many children wrongly pick 4.11 because it 'looks longer' — but 4.9 is larger. Rewrite with equal places (4.11 vs 4.90) and read them as eleven hundredths versus ninety hundredths to see why.
   2. Decide which is larger, 0.4 or 0.45, and whether 0.4 and 0.40 are equal or different. Then rewrite 0.4 as 0.40 (forty hundredths) to compare the same number of places directly (forty hundredths versus forty-five hundredths).
   3. Build or mark both numbers with hundredths blocks or on a number line, lining up equal place values, and state which is larger and why.
   4. Check the claim 'is 6.18 the same as 6.180?' and justify the answer from place value (appending a trailing zero adds zero in the next place, so the value is unchanged).
   5. Explain in your own words why 2.78 is smaller than 3.4 even though it has more digits and a larger part after the point (0.78 vs 0.40): compare the ones first (2 < 3). Explain why you cannot judge decimal size by counting digits.
3. 3

   Explain why the points must line up

   Why it works: You can only add tenths to tenths and hundredths to hundredths, so aligning by the point keeps like place values together; estimating first lets you catch a misplaced point. Full mastery means justifying the algorithm from place value, not just following steps.

   1. Answer the question 'why must the decimal points line up — what goes wrong if they don't?' in your own words, referring to place value.
   2. Work 5.03 − 2.67 and explain what you did to be able to take 7 hundredths from 3 hundredths (decomposing across the tenths and the zero).
   3. Estimate 3.45 + 2.78, 3.4 + 2.78, and 5.03 − 2.67 (about 6, 6, and 2) and describe how each estimate tells you where the point belongs in the exact answer.
   4. Transfer the reasoning to a fresh ragged or zero case you have not seen, setting it up, computing it, and justifying the alignment.