Climbing Stairs

Introduction

In this activity you can climb the stairs one step at a time or two steps at a time.

Look at the diagram on the right. This staircase has 4 steps. The diagram shows one way to climb it.

Draw/ list all the ways of climbing this staircase (try to be systematic in the way that you try numbers). For example: 1,1,2

How many different ways are there?

Think like a mathematician, what questions could you now ask to further investigate? If you need some support look at the next section.

Support

Below are some ideas of how you can further investigate.

Step 1: predict what the next patterns may be. One good strategy is to draw out the next few patterns.

Draw/ list all the different ways of climbing a staircase with 2, 3, or 5 steps.

Step 2: organise your findings in a systematic way. One good strategy is to put your observations in a table.

Hint: label the first row of your table "number of steps" and the second row "number of different ways".

Step 3: use your table to come up with some general rules.

Can you describe how the number of different ways to climb the staircases goes up?

Step 4: verify (a.k.a. check) that your rules work. This can be achieved by comparing your drawings with the answers found using your rules.

Use your rules to find how many different ways there are to climbing a staircase with 6 steps.
Draw/ list all the different ways there are to climbing a staircase with 6 steps. Does the answer match the one found using your rules?

Step 5: justify (a.k.a explain) why the rule works.

Can you explain why the number of different ways to climb the staircases goes up this way?

Further Questions and Challenges

Suppose you can climb either one, two or three steps at a time. How many different ways of climbing staircases with 1 step, 2 steps, 3 steps are there? Can find a rule for how the numbers go up?

REMEMBER: start small, follow the steps and investigate in a systematic manner!

For more similar questions, check out this problem on nrich.org.

Extension

In this lesson, you came across a very special type of number. To get a brief introduction of what it is, look at this video.

Try Fibonacci Quest on Transum - a number of self marking quizzes based on the fascinating Fibonacci Sequence.

You may also wish to explore more surprising results looking at the worksheet below:

Fibonacci surprises.docx