# Maths - Unravelling the problem

Little Bo Peep's business was booming. She had built up a vast flock of sheep, but how could she count them? She had tried putting 1, 2, 3, 4... on the side of each sheep. But this did not solve her problem, writes Jonny Griffiths.

"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?"

"I'll paint fractions on the sides of mine," she decided. "Every sheep must have a fraction - and every fraction must correspond to a sheep. And I will count 11, 22, 33 as the same fraction - equivalent fractions must only occur once, in their simplest form."

In addition, she had to put her sheep in order, but how - by 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.

Then it came to her: "A fraction can give birth to two others, like this!" she cried (see Figure 1).

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

"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. bit.lygHxqzD

What else?

Help lower-ability students 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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