Patterns that lead to algebra and proof - October 2010, Stage 4&5

Mathematicians are fascinated by pattern - one writer has even described 'pattern sniffing' as a mathematical behaviour. In this month's activities and problems we invite you to discover patterns, describe them in different ways and begin to think about how you can convince yourself, and others that you have sniffed out all there is to find!

Problems

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Quadratic Transformations

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Explore the two quadratic functions and find out how their graphs are related.

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Robert's Spreadsheet

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Robert noticed some interesting patterns when he highlighted square numbers in a spreadsheet. Can you prove that the patterns will continue?

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More Quadratic Transformations

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Here are some more quadratic functions to explore. How are their graphs related?

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Weekly Challenges

Stage: 4 and 5 Challenge Level: Challenge Level:2 Challenge Level:2

The NRICH Stage 5 weekly challenges are shorter problems aimed at Post-16 students or enthusiastic younger students. There are 52 of them.

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Patterns of Inflection

Stage: 5 Challenge Level: Challenge Level:1

Find the relationship between the locations of points of inflection, maxima and minima of functions.

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Calculus Analogies

Stage: 5 Challenge Level: Challenge Level:1

Consider these analogies for helping to understand key concepts in calculus.

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Agile Algebra

Stage: 5 Challenge Level: Challenge Level:1

Observe symmetries and engage the power of substitution to solve complicated equations.

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Interactive Workout - Mathmo

Stage: 5 Short Challenge Level: Challenge Level:1

Mathmo is a revision tool for post-16 mathematics. It's great installed as a smartphone app, but it works well in pads and desktops and notebooks too. Give yourself a mathematical workout!

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Tens

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

When is $7^n + 3^n$ a multiple of 10? Can you prove the result by two different methods?