Decimal place value

Activities to try

1. 1

   Move the digits, keep the point still (place-value conversion)

   Why it works: A conversion moves the SAME digits across a FIXED decimal point; the point does not move. Converting to a larger unit gives fewer of them (divide, digits move right); converting to a smaller unit gives more (multiply, digits move left). 3250 g and 3.25 kg are one quantity; 3.250 equals 3.25.

   1. Set up a place-value chart extended to thousandths. Write 3250 g in it, then convert to kilograms by sliding every digit three places to the RIGHT (dividing by 1000) while the decimal point stays fixed, landing on 3.25 kg; note that a holder zero is inserted only where place value demands it.
   2. Before moving, state the unit-size reason aloud: 'Kilograms are the bigger unit, so there will be FEWER of them — I divide.' Then check the direction of the digit shift matches.
   3. Convert the reverse case 0.45 km to metres (×1000), sliding digits three places LEFT to get 450 m; state that metres are smaller, so there are MORE.
   4. Convert 5 cm to 0.05 m (÷100), sliding each digit two places to the right — a shorter example of the same digit-movement method with decimal notation; say why metres are the larger unit so there are fewer.
   5. Confirm that 3.250 kg and 3.25 kg name the same quantity — the trailing zero after the last significant digit does not change the value.
2. 2

   Spot the false shortcut (error-spotting with decimals)

   Why it works: Adding or removing a zero is not the same as multiplying or dividing by a power of ten once decimals are involved. ×/÷ by 10, 100, 1000 moves the digits by one, two, or three places; the zero trick coincidentally works for some whole numbers but misvalues decimals.

   1. Judge true or false and explain: 'Is 0.45 ÷ 10 = .45?' Correct it to 0.045 by moving each digit one place right.
   2. Judge true or false: 'Is 25.34 × 10 = 25.340?' Explain that appending a zero after the last decimal does NOT multiply by ten; the correct answer is 253.4, with digits moving one place left.
   3. For each false statement, state the correct conversion and justify it with the value of each digit and the direction of movement.
3. 3

   Climb the powers-of-ten ladder (composing conversions)

   Why it works: The metric system is built on powers of ten, so every metric conversion is a single ×/÷ by a power of ten and mixed chains compose; knowing the structure removes the need to memorise each conversion separately. Time units are not decimal (60 s = 1 min, 60 min = 1 h, 24 h = 1 day), so time conversions use those factors instead of powers of ten.

   1. Build a ladder stating each relationship: 10 mm = 1 cm, 100 cm = 1 m, 1000 m = 1 km. For each rung, say whether you multiply or divide and by what power of ten.
   2. Hop along the ladder to compose a chain, e.g. mm → cm (÷10) → m (÷100) → km (÷1000), and recognise that any single conversion is one multiplication or division by a power of ten.
   3. Optionally use a Gattegno chart (rows for powers of ten) to see each conversion as a single ×/÷ by a power of ten — moving along a row, or jumping rows for the larger steps.
   4. Explain in words why 1 km is a thousand metres and not a hundred, using the powers-of-ten structure rather than reciting it as an isolated fact.
   5. Transfer the same structure to mass (g–kg) and volume (ml–l) to show the method is general across metric units, not tied to length. Then contrast with time (e.g. 2.5 h = 150 min, not 250 min) to see that time units use factors of 60 and 24, not powers of ten.