Generic examples - March 2012, Stage 4&5

Sometimes it is possible to state what is general by perceiving the structure in just one example. Vivid proofs of generalisations can be located in a particular, well-chosen example. This month we invite you to explore and notice what is generalisable in carefully chosen generic examples.

Problems

Route to Infinity

Stage: 3 and 4 Challenge Level:

Can you describe this route to infinity? Where will the arrows take you next?

Seven Squares

Stage: 3 and 4 Challenge Level:

Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?

Slick Summing

Stage: 4 Challenge Level:

Watch the video to see how Charlie works out the sum. Can you adapt his method?

Speedy Summations

Stage: 5 Challenge Level:

Watch the video to see how to add together an arithmetic sequence of numbers efficiently.

Polynomial Interpolation

Stage: 5 Challenge Level:

Can you fit polynomials through these points?

Particularly General

Stage: 5 Challenge Level:

By proving these particular identities, prove the existence of general cases.