Decimals: terminating and repeating

Practice 5 items →

All four decimal operations should be automatic, including mental multiplication and division by powers of ten. The new idea is that some fractions terminate as decimals and some repeat forever. After simplifying, a fraction terminates exactly when its denominator has no prime factors other than 2 and 5 — so ⅛ terminates (0.125) and ⅙ does not (0.1666…). Converting among fractions, decimals, and mixed numbers in both directions is the fluency goal. Two misconceptions to watch: writing 1 ÷ 3 as 0.3, and dismissing 0.999… as “not a real number.”

Specification · for a parent or teacher
Part no.
decimals-terminating-repeating
Course
Advanced Math 6 · Unit 3
Also in
High School Math 1
Realm
Foundations
Difficulty
3 of 4 — Grade 6–7 core. Multi-step, first symbolic work.
Qty
5 items

Part no. is the node's id; the realm is the kind of math; qty is how many practice items it carries.

Students can

four operations; mental × and ÷ by 10k10^k; convert among fractions, decimals, mixed numbers; name terminating vs repeating.

Ready when

they can predict which of 12,13,14,15,16\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5},\frac{1}{6} terminate and why (denominator’s primes after simplifying).

Formulas

Terminating test
ab terminates  ⟺  b=2m5n (after simplifying)\frac{a}{b} \text{ terminates} \iff b = 2^m 5^n \text{ (after simplifying)}
Repeating notation
13=0.3‾,14=0.25\frac{1}{3} = 0.\overline{3},\quad \frac{1}{4} = 0.25

Watch for

1÷3=0.31\div 3 = 0.3; 0.9‾0.\overline{9} dismissed as “not a number.”

Before · requires

After · used by

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decimals-terminating-repeating

Items: decimals-terminating-repeating-01 · decimals-terminating-repeating-02 · decimals-terminating-repeating-03 · decimals-terminating-repeating-04 · decimals-terminating-repeating-05