Decimals — place value, ordering, terminating & recurring

For an upper bound students wrote recurring decimals such as $28.4\dot{9}$ or $17.\dot{4}\dot{9}$ instead of the correct terminating bounds $28.5$ and $17.5$; the report also notes that terminating decimals like $28.499$ or $17.499$ were not accepted. The exact bound is the terminating value, not a string of 9s.

When a recurring decimal is turned into a fraction, students multiply by powers of $10$ at random (e.g. $x$ and $100x$ when one digit recurs) so the repeating tails do not line up and cannot be subtracted away.

Cutting a decimal short part-way through a calculation (premature rounding) loses accuracy and the final accuracy mark; the value should be kept to plenty of decimal places (or in the calculator memory) until the very end.

When ordering decimals, do not assume the value with more digits is larger: $0.4 > 0.39$ even though $0.39$ has more figures. Pad every number to the same number of places ($0.4 = 0.40$) and compare digit by digit from the left, starting with the whole-number part.

A hidden factor can cancel: $\tfrac{9}{60}$ seems to recur because $60 = 2^2 \times 3 \times 5$, but in lowest terms $\tfrac{9}{60} = \tfrac{3}{20}$ and $20 = 2^2 \times 5$, so it terminates ($0.15$). Always cancel down before inspecting the denominator's prime factors.

Only the dotted digits recur. In $0.1\dot{6}$ the $1$ does not repeat — the value is $0.1666\ldots$, not $0.161616\ldots$; a block needs a dot on both its first and last digit ($0.\dot{1}\dot{7} = 0.1717\ldots$), while a single dot marks one repeating digit only.