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Clearly if a, b and c are the lengths of the sides of an equilateral triangle then a^2 + b^2 + c^2 = ab + bc + ca. Is the converse true?

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Consecutive Squares

The squares of any 8 consecutive numbers can be arranged into two sets of four numbers with the same sum. True of false?

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Parabolic Patterns

The illustration shows the graphs of fifteen functions. Two of them have equations y=x^2 and y=-(x-4)^2. Find the equations of all the other graphs.

Alison's Mapping

Age 14 to 16 Challenge Level:

Why work on this project?

This month's NRICH site has been inspired by the way teachers at Kingsfield School in Bristol work with their students. Following an introduction to a potentially rich starting point, a considerable proportion of the lesson time at Kingsfield is dedicated to working on questions, ideas and conjectures generated by students.

Working on this project will encourage students to work together, discuss ideas, develop conjectures, suggest new lines of enquiry, solve problems and generally experience how a mathematical community functions within their own classrooms.

Possible approach

Alison's Mapping is one of several starting points on the site. Here are the sort of questions that might emerge:
What affects the direction and steepness of a graph?
Can I tell from a function where its graph will cross the axes?
Which functions give straight lines, and which give curves?
When will two quadratic functions intersect?
Can I tell from a quadratic function where its graph has a turning point?

Read the article Kingsfield School - Building on Rich Starting Points, which has links to a description of a Kingsfield teacher's first lesson on functions and graphs, and a video showing how these ideas are put into practice in the classroom.
For teachers who want to create their own alternatives to Alison's Mapping for use in the classroom, here is an introductory video explaining how to build, load and save your own examples.

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To read more about the Number Plumber, visit Grumplet's blog where you can comment on how you have used the number plumber and share links to files you have created. We are continuing to develop this resource so your feedback and ideas will be very useful.

Key questions

What do you think these function machines do?
What will happen if we input 5? 13? 100? 0.7? ...

Is it possible to get the same output from both machines using the same input number? Is there more than one way?

What other questions could we ask?
Can you make any predictions about what might happen when we change the function machines?
What's the same? What is different?
Can you explain?

Possible extension

Steve's Mapping provides a starting point based on rational functions.


Possible support

Charlie's Mapping provides a starting point based on linear functions.