Decimal place value

For ages 10–11: multiply numbers with up to two decimal places by a whole number (e.g. 3.47 × 6) by computing as whole numbers and rescaling by the powers of ten removed, supported by the area/expanded model and estimation to place the decimal point by reasoning rather than a rote 'count the decimal places' rule.

• Blindly 'counting decimal places' without understanding, then mishandling trailing zeros (3.2×5 → thinking 16.0 is wrong, or 1.60).
• Importing 'line up the decimal points' from decimal addition into multiplication.
• Multiplying only the whole-number part or only one place (3.47×6 done as 3×6 = 18).
• Misplacing the decimal point (2.082 or 208.2) because the point was not reasoned via scaling or estimate.
• Forgetting to rescale after computing the whole-number product (leaving 2082 instead of 20.82).

• Area model / grid: rows for 3, 0.4, 0.07 (or expanded 3.47) × the whole number, each cell a partial product to be summed.
• 'Scale out, scale back' chain: 3.47 → ×100 → 347 → ×6 → 2082 → ÷100 → 20.82, linking to powers-of-ten shifting.
• Place-value columns showing the partial products aligned by place (ones, tenths, hundredths).
• Estimate bracket: 3.47×6 sitting between 3×6 = 18 and 4×6 = 24, anchoring the point at 20.82.

• Teach 'compute as whole numbers, then rescale by the powers of ten': 3.47 = 347 ÷ 100, so 3.47×6 = (347×6) ÷ 100 — connecting to the ×÷-by-powers-of-ten work (mt_wPgpMJ0-PA) and giving a reason for the decimal placement.
• Always pair the calculation with an ESTIMATE (3.47×6 ≈ 21) so the decimal point is placed by sense, not by a mechanical digit-count that fails learners when trailing zeros vanish.
• Use the area model / expanded form: partition the decimal into whole, tenths and hundredths, multiply each part by the whole number, and sum — making the distributive structure and each place's contribution visible.
• Confront the 'line up the decimal points' import from addition: it does NOT apply to this multiplication — the answer's places come from the scaling, and children should predict then check with an estimate.
• Address trailing zeros explicitly (3.2×5 = 16.0 = 16): the 'count decimal places' rule seems to give 16.0, and children must understand equivalence rather than think a place has been lost.