Copyright © University of Cambridge. All rights reserved.
Why do this problem ?
provides an opportunity to draw out from students techniques for problem solving.
A key to solving this problem is to simplify to smaller staircases, and use these earlier results explain what is happening and hence calculate later results.
The Article Go Forth and Generalise
may be of interest.
"Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at a time. For example: He could go down 1 step, then 1 step, then 2 steps, then 2, 2, 1, 1, 1, 1.
In how many different ways can Liam go down the 12 steps, taking one or two steps at a time?"
Give students some time to explore the problem. As they work, wander round the classroom, listening for any particularly useful insights. In particular, listen out for comments like:
"There are going to be loads of different ways"
"How are we going to be able to make sure we don't miss any?"
Then bring the group together to share those insights.
If no-one has suggested it: "Perhaps we could work on a simpler version of the problem to see if that helps? Let's see how many ways there are of going down three, four, five or six steps."
Students could all work on these or they could be shared out with different groups working on different sizes of staircase.
Then bring the class together and invite students out to the board to list the number of different ways they found, while the rest of the class make sure they have worked in a way that makes sure they haven't missed any.
Once the number of ways for three, four, five and six steps have been worked out:
"With your partner, look at the number of ways for each different size of staircase, and see if you notice any patterns. In a short while we will return to the original problem of a staircase with twelve steps. Can you use what we have found out about smaller staircases to make predictions about the answer for twelve steps? Or any number of steps?"
Give students some time to work with their partner, and circulate, listening in for any who make connections. If they are struggling to make progress, here are some prompts you could use:
"How does the number of ways of descending 5 steps compare to the number of ways of descending 3 and 4 steps? What about the number of ways of descending 6 steps compared to the number of ways of descending 4 and 5 steps?"
"If I want to go down 5 steps and I start with a one-step, how many ways can I descend the remaining four steps?"
"If I want to go down 5 steps and I start with a two-step, how many ways can I descend the remaining three steps?"
Finally, bring the class together and discuss their findings. They could present their answer to the twelve-step problem or an explanation of the general case on a poster.
What is making it difficult?
What have you found out so far?
Key comments students may make:-
"There'll be too many", "I can't keep track", "I might have done some twice"
"Work it out for fewer steps", "Try to find a logical way to order the options", "Work together and check each other's work".
Students will generate lots of numbers and have to find a way to keep track of what they've done, ensuring that each set of results is complete
Take a look at Walking Down the Stairs
for suitable follow-up questions.
Begin with a smaller number of steps, so it is still too big, but not quite so daunting.