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Farey Sequences

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Paving Paths

How many different ways can I lay 10 paving slabs, each 2 foot by 1 foot, to make a path 2 foot wide and 10 foot long from my back door into my garden, without cutting any of the paving slabs?

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1 Step 2 Step

Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?

Seven Squares

Age 11 to 14 Challenge Level:

How did Phoebe group the matchsticks that she drew?
How did the others, Alice and Luke, group their matchsticks?

 

In the follow-up activities draw them out for yourself and notice how YOUR drawings develop. Always begin with simple cases and try to PREDICT what will happen.

 

Look for patterns.
How can you describe the lines? Horizontal? Vertical?
Try to understand why the patterns develop in the ways that they do.