1 Step 2 Step

This problem provides an opportunity to draw out from students an arsenal of techniques for problem solving.

A key to solving this problem is to simplify to smaller staircases, and use these earlier results to calculate later results. A perfect example of a powerful technique.

Students focus on the problem while the teacher focuses on the students' strategies.

"Here's a question I found on the Internet, what do you make of it?"

Students work on problem in rough. The teacher could eavesdrop, listening for key phrases to bring into the group discussion. Bring the group together and ask for comments. Write up some/all of these comments on the board, and ask students to suggest ways to resolve any problems.

Students get back to work; free to choose between continuing as they were or following up on something that has been suggested by the group. The teacher goes back to eavesdropping, keeping a record on the board of problems and good strategies - in the students' own words where possible.

At the end of the lesson, use a plenary to focus attention on the strategies on the board, and which ones were most useful to people in the class. A homework could be to make a poster about what to do with a problem that is too hard.

Pupils must be sure that their evidence is solid enough to justify the number pattern. They can be challenged to show how the list of possibilities for one staircase can be derived from the lists of the two previous ones. This leads to proof, rather than merely evidence.

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"

Possible responses:-

"Work it out for fewer steps", "Try to find a logical way to order the options", "Work together and check each other's work".

Pupils 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

eg change the '$1$ and $2$' to '$1$ and $3$' or '$1$, $2$ and $3$' or '$5$ and $7$' or...

Ask eager pupils to research into Fibonnacci.

Begin with a smaller number of steps, so it is still too big, but not quite so daunting.

Another problem that uses a similar idea is Colour Building