Maths - Unravelling the problem

23rd November 2012 at 00:00
Little Bo Peep experiences a woolly dilemma with fractions

Little Bo Peep's business was booming. Sheep were in demand and, when it came to shepherdesses, there was none finer. She had been successful enough to build up a vast flock of sheep. This gave her a headache: how to count them? She had tried putting 1, 2, 3, 4 ... on the side of each sheep in turn. After a long time, during which she had become incredibly quick at painting a number on the flank of a sheep, she had managed to number them all.

This did not, however, completely solve her problem. "Little Bo Peep," said an exasperated local farmer. "We also paint 1, 2, 3, 4 ... on the sides of our sheep. How can we tell yours and ours apart?" Little Bo Peep's forehead creased as she grappled with this conundrum. Then it came to her in a flash: "I'll paint fractions on the sides of my sheep!"

The next morning she set to work. "Every sheep must have a fraction - and every fraction must correspond to a sheep," she thought. "And I will count 11, 22, 33 as the same fraction - equivalent fractions must only occur once, in their simplest form!" she cried. "In addition, I must be able to put my sheep in order."

Immediately, she was struck by a dilemma: would she try to arrange her sheep in order of size? "Between 35 and 712 lies 1017," she thought. "More generally, between any two sheep ab and cd comes the sheep (a+c)(b+d)," she realised, despairingly. "So I can never put them all in order of size."

Little Bo Peep went for a hike to clear her head. As she looked down on her flock from the top of a nearby fell, it came to her suddenly. "A fraction can give birth to two others, like this!" she cried (see Figure 1, right).

"Now if I start with 11, I get this." (See Figure 2, right.)

"Every fraction occurs just once, in its simplest form. Now if I put the rows together in a long line, I have ordered my sheep!"

That afternoon, she set to with a paint brush. Two hours later a passing shepherd found her, fast asleep.

Jonny Griffiths teaches maths at a sixth-form college. With thanks to Neil Calkin and Herbert S. Wilf: bit.lygHxqzD

WHAT ELSE?

Help lower ability students get to grips with fractions using Rushtini's simple Bo Peep problem. bit.lyBoPeepMaths

Check out TES maths adviser Craig Barton's top 10 resources on fractions, decimals and percentages. bit.lyFractionsTop10.

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