Fractions

The goal of this session is to experience fractions in many different contexts and uses.

We started with Fraction Rummy, a card game designed to help students become comfortable with fractions. Played something like rummy, each player can either select the top card from the pile, or a card and all cards above it from the discard pile. A collection of cards that adds to 1 is a book, and wins the player one point. I found this game in an old issue of NCTM’s Arithmetic Teacher, and then lost it. Thanks, Halle, for helping me locate it again!

Fractions are both more complicated and simpler than we know, and the following questions are revealing!

  1. What is a fraction?
  2. It turns out this is a really tough question!

  3. If $latex \displaystyle \frac{a}{b}$ and $latex \displaystyle \frac{c}{d}$ are fractions, explain why $latex \displaystyle \frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$.
  4. Why not use the baseball method, of simply adding numerators and denominators?

  5. Make up a word problem that represents $latex 2 \frac{2}{5} \div \frac{4}{5} = 3$. Draw a diagram that represents your problem.
  6. Perhaps you should read my article, Elastic, Cottage Cheese, and Gasoline: Visualizing Division of Fractions!

  7. Reduce to lowest terms: $latex \displaystyle \frac{1357}{1403}$.
  8. There is more than one way to do this, and the Euclidean Algorithm is the most efficient in this case.

  9. If you drive the first 30 miles at 60 mph and the next 30 miles at 30 mph, what is your average speed for the full 60 miles?
  10. Hint: It’s not 45 mph. This is a good problem to check for meaning. It turns out to be the harmonic mean of 30 and 60, and follows from the definition of average speed.

  11. The picture shows two vertical poles AB of height 3m and CD of height 4m. Two wires are attached from the top of each pole to the bottom of the other pole, and intersect at P. What is the height h of P above the ground?
  12. Somehow, the answer is related to the harmonic mean.

    Crossing Wires Problem
    Crossing Wires Problem

  13. Which is bigger? $latex \displaystyle \frac{49}{108}$ or $latex \displaystyle \frac{54}{119}$. Find a fraction between them.
  14. The baseball method (also called the mediant) yields the fraction between them with the smallest denominator. This can lead to a discussion of Farey Fractions.

  15. Any rational number can be factored into prime powers. For example, $latex\displaystyle \frac{80}{99} = 2^4 3^{-2} 5^1 11^{-1}$. The 2-adic norm of a rational number is defined as the inverse of the power of 2 factor in the number’s prime power factorization. Thus $latex \left|\frac{80}{99}\right|_2 = 2^{-4}$, which is quite small, 2-adically.
    Explain why it makes sense 2-adically, that $latex 1 + 2 + 4 + 8 + 16 + 32 + \dots = -1.$

Conway’s Tangles
No session on fractions would be complete without learning about Conway’s Tangles. Four volunteers stand at the corners of square ABCD. Initially, the people at corners A and D hold one 10′ rope and likewise, the people at corners B and C hold another 10′ rope. Upon command, they perform three moves:

  • T for Twist. The volunteers at corners A and B switch places, with B going underneath the rope held by A.
  • R for Rotate. All four volunteers rotate positions, walking from A to B, from B to C, from C to D, and from D to A.
  • D for Display. Volunteers at positions C and D kneel down and hold their rope ends to the floor, while volunteers at positions A and B stand and hold their rope ends high. Everyone else in the room cheers and the photographer snaps away.

The initial position has a value of 0. Twisting adds 1 to the current value, while rotation changes the current value to its negative reciprocal. For Conway’s Tangles, the negative reciprocal of 0 is infinity, and vice-versa. Infinity plus 1 is still infinity.

The facilitator calls out Tangle moves, and once tangled, the participants try to untangle.

Find the fraction that results from the following choreographies:

  • $latex T^2R$ – that is starting with value 0, Twist (+1), Twist (+2) and then Rotate (-1/2)
  • $latex T^4RT^2$
  • $latex T^2RT^4RT^3R$

Find a choreography that untangles each of the following fractions:

  • $latex T^2RT^6 = \frac{11}{2}$
  • $latex T^2RT^2RT^2 = \frac{4}{3}$
  • $latex T^4RT^3RT^3 = \frac{29}{11}$.

Question: Can each position be untangled? Why?

Conway’s Prime Generator based on Fractions

In 1972 John Horton Conway presented a list of fractions that computes prime numbers.

$latex \frac{17}{91}, \frac{78}{85}, \frac{19}{51}, \frac{23}{38}, \frac{29}{33}, \frac{77}{29}, \frac{95}{23}, \frac{77}{19}, \frac{1}{17}, \frac{11}{13}, \frac{13}{11}, \frac{15}{14}, \frac{15}{2}, \frac{55}{1}.$

Starting with the number 2, multiply it by the first fraction in this sequence, so that the resulting product is an integer. Then go back to the start of the list, and multiply by the first fraction that again produces an integer product. Continuing in this manner, we come up with the following sequence of integers:

2, 15, 825, 725, …

What’s interesting is that powers of 2 occur infinitely often. In order of appearance, they are $latex 2^2, 2^3, 2^5, 2^7, 2^{11}, \dots$. This strange procedure produces exactly the prime powers of 2, in order!

Question: Does this really work? Why?

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