let’s use decimals to describe increases and decreases.\nwarm-up: notice and wonder: fractions to…

let’s use decimals to describe increases and decreases.\nwarm-up: notice and wonder: fractions to decimals\na calculator gives the following decimal representations for some unit fractions:\n\\(\\frac{1}{2} = 0.5\\) \\(\\frac{1}{7} = 0.142857143\\)\n\\(\\frac{1}{3} = 0.3333333\\) \\(\\frac{1}{8} = 0.125\\)\n\\(\\frac{1}{4} = 0.25\\) \\(\\frac{1}{9} = 0.1111111\\)\n\\(\\frac{1}{5} = 0.2\\) \\(\\frac{1}{10} = 0.1\\)\n\\(\\frac{1}{6} = 0.1666667\\) \\(\\frac{1}{11} = 0.0909091\\)\nwhat do you notice? what do you wonder?

let’s use decimals to describe increases and decreases.\nwarm-up: notice and wonder: fractions to decimals\na calculator gives the following decimal representations for some unit fractions:\n\\(\\frac{1}{2} = 0.5\\) \\(\\frac{1}{7} = 0.142857143\\)\n\\(\\frac{1}{3} = 0.3333333\\) \\(\\frac{1}{8} = 0.125\\)\n\\(\\frac{1}{4} = 0.25\\) \\(\\frac{1}{9} = 0.1111111\\)\n\\(\\frac{1}{5} = 0.2\\) \\(\\frac{1}{10} = 0.1\\)\n\\(\\frac{1}{6} = 0.1666667\\) \\(\\frac{1}{11} = 0.0909091\\)\nwhat do you notice? what do you wonder?

Answer

Brief Explanations:

Observations (Notice):

  1. Unit fractions with denominators that are powers of 2 ($2, 4, 8$) or powers of 5 ($5, 10$) terminate after a finite number of decimal places.
  2. Unit fractions with denominators that are primes other than 2 or 5 ($3, 7, 11$) or composites with such prime factors ($6, 9$) produce repeating decimal expansions (the calculator rounds the final digit for some, e.g., $\frac{1}{6}$ and $\frac{1}{7}$).
  3. As the denominator increases (for positive integers greater than 1), the decimal value of the unit fraction decreases.
  4. The repeating pattern length varies: $\frac{1}{3}$ and $\frac{1}{9}$ have a 1-digit repeat, $\frac{1}{11}$ has a 2-digit repeat, and $\frac{1}{7}$ has a 6-digit repeating cycle (truncated/rounded by the calculator).

Questions (Wonder):

  1. Why do some unit fractions terminate and others repeat?
  2. What determines the length of the repeating cycle in a repeating decimal?
  3. How can we predict if a unit fraction will have a terminating or repeating decimal expansion without calculating it?
  4. Why does the calculator round the final digit for fractions like $\frac{1}{6}$ and $\frac{1}{7}$ instead of showing the full infinite repeat?

Answer:

Notices:

  • Unit fractions with denominators that are powers of 2 or 5 (or products of these) have terminating decimals (e.g., $\frac{1}{2}=0.5$, $\frac{1}{5}=0.2$, $\frac{1}{10}=0.1$).
  • Unit fractions with denominators containing prime factors other than 2 or 5 have repeating decimals (e.g., $\frac{1}{3}=0.\overline{3}$, $\frac{1}{7}=0.\overline{142857}$, $\frac{1}{11}=0.\overline{09}$; the calculator rounds these for display).
  • Larger denominators result in smaller decimal values for unit fractions.

Wonders:

  • What rule determines if a fraction's decimal expansion terminates or repeats?
  • How is the length of a repeating decimal cycle related to the denominator?
  • Can we write the exact infinite repeating decimal for fractions like $\frac{1}{7}$ without relying on a calculator's rounded output?
  • Why do denominators that are powers of 2 or 5 always produce terminating decimals?