Staircase

Summary

Staircase also appears on the Picture Puzzles Pattern & Algebra A menu where the problem is presented using one screen, two learners, concrete materials and a challenge. The extra challenges in this puzzle make a link to Task 18, Same or Different, and a template for creating new Triangle Numbers from smaller ones.

Also Staircase is the stimulus investigation for the DIY workshop video Aiming High? Dig Deeper which explores a pedagogy of achieving high aims in mathematics by digging deeper into fertile learning ground such as this task, rather than too quickly moving on to something new. The video highlights teaching craft - for example visual and kinaesthetic learning and journal writing in maths - and uncovers mathematics content from Year 1 to Year 12. Its introductory video Staircase Maths (in less than one minute) will give you an idea of the content (and teaching craft).

Materials

• 30 cubes
• Marker and cloth

Content

• basic operations with number
• patterns in tables
• quadratic algebra including:
  • concept of a variable
  • generalisation
  • substitution
  • solving equations
• visual representation of symbolic algebra
• graphical representation of functions
• concept of proof

Iceberg

This task can be extended using Task 61, Double Staircase.

The action of building the staircase is well within the grasp of quite young students. It is also possible for them to work out the total of blocks for a 10-step stair, and perhaps even a 100-step stair. The first they might do by building the staircase and counting, or by adding up the numbers from 1 to 10. For the second they might use a calculator. If the focus of the problem was the answer, then this could be all we required. However, if the focus is learning to work like a mathematician, then we need to ask the mathematician's question: Can I check it another way?

This may encourage students to look for connections within the set of numbers, rather than looking at each number as a separate object. Then, in adding 1 to 10 for instance, they might look for 10s and pair 1 with 9, 2 with 8, 3 with 7, 4 with 6 and realise that the total is therefore 5 tens plus the unpaired 5, which is 55.

However, visually-based approaches can also work and are to be encouraged:

• Some students might see that there is 'middle' step and that the first is as much below this middle as the last is above it. This is also true for the second and second last, and so on. This is more easily seen in staircases which have an odd number of steps. (For even staircases, the middle is the imaginary half-step). By shifting cubes from the 'big end' to the corresponding position at the 'small end', we can create a 'staircase' with steps all the same height as the middle. The total is therefore the middle step times the number of steps.
• The card suggests another approach. Using two copies of the same staircase and joining them as suggested, produces a rectangle with a known width, and a length just one cube longer than that. The total of cubes in the rectangle can therefore be calculated and it will be twice as much as the number in the staircase.

• If I tell you any number of steps, can you tell me the number of blocks I need to build the staircase?
• If I tell you the number of blocks I have, can you tell me the number of steps in the staircase I build?
• A mathematician sometimes uses a graph to investigate a problem. Make a graph which links the number of number of steps with the total number of blocks.
• What would you say to a student who said the dots on the graph should be joined up?
• What happens if each step in the staircase is 2 blocks higher than the one before it?
• If I tell you that each step is p blocks higher than the one before it, can you tell me the total of blocks needed to build a staircase with n steps?

Whole Class Investigation

To convert this task to a whole class investigation you only need a collection of cubes for each group. Alternatively, if you have Poly Plug you can introduce the investigation in a somewhat quieter, but equally concrete and visual way using the red board.

For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 115, Staircases, which includes an Investigation Guide.

Visit Staircase on Poly Plug & Tasks.