Ray's New Higher Arithmetic · Arts. 167–171 · Unit 10: Circulating Decimals
44. Circulating decimals: what they are
Goal: Your student explains why some fractions make repeating decimals, finds the repetend, and uses the words circulate, repetend, pure and mixed.
You'll need
- The book
- A slate, small whiteboard or scrap paper
- A notebook and pencil
What's in the book
Why some decimals never end (two principles), the example 1/7 = .142857 repeating, the definitions of circulate and repetend (pure, mixed, simple, compound, similar, perfect), and remarks on extending repetends and making them similar.
The lesson, step by step
Warm-up
Divide 1 by 3, then 1 by 7, as far as your student likes. What happens?
Why they repeat
Read:
In reducing common fractions to decimals, the process, in some cases, does not terminate.
The book explains: every remainder is less than the divisor, so sooner or later a remainder comes back, and then the figures repeat. Follow the division of 1/7 on the page: .142857, then 1 again.
The words
Read:
A Circulate or Circulating Decimal has one or more figures constantly repeated in the same order.
A Repetend is the figure or set of figures repeated
The book marks the repetend with dots over its first and last figures.
Making repetends alike
Read:
Dissimilar repetends can be made similar, by carrying the dots forward
Show: .3 repeating can be written .33 repeating, and a repetend can start later: 5.1836 (with 836 repeating) = 5.183683 (with 683 repeating).
Practice
Your student turns 1/11, 2/3, 5/6 and 3/13 into circulates and marks each repetend.
Wrap-up
Read:
Circulating decimals may be added, subtracted, multiplied, and divided as other fractions.
Tip
Circulating decimals are now a side topic. One or two sessions are plenty; the main idea is that every fraction makes either an ending or a repeating decimal.
Answers to the book's problems
Practice: 1/11 = .09 repeating (09); 2/3 = .6 repeating; 5/6 = .83 with 3 repeating; 3/13 = .230769 repeating (230769).
Worked out for this site from the scan. If a number in your copy differs, trust the book.