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# Back Fitter

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Age 14 to 18

Challenge Level

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- Getting Started
- Student Solutions
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This problem offers an opportunity to reflect on the very important concept of fitting a curve to experimental data. Along the way, students will utilise their skills of transforming graphs in order to find a close fit, and consider ways of deciding how close their fit is. The problem is marked as challenge level 1 as it is a straightforward task to begin, but to find a complete solution for all 10 graphs is rather more challenging!

Although this problem stands alone, it could also be done as a follow-up to work on transformations of graphs based on the problem Parabolic Patterns.

Students will need access to computers or graphical calculators to get the best out of this task. Familiarity with spreadsheet software is assumed.

Part of the challenge of this problem is to identify which graphs are easiest to fit, as they are not presented in any particular order. One approach is to start by displaying the graphs and discussing as a class or in pairs which have recognisable shapes, such as straight lines, quadratics, trig graphs and exponential graphs.

If students haven't met graphs such as $y=a^x$ and $y=a^{-x}$ it might be fruitful to give them some time to experiment with graphical calculators to see what these graphs look like for different values of the constant $a$.

Once students have some preliminary ideas about graphs which might fit, small groups could start to work on the spreadsheet, entering a possible equation and seeing how closely it matches the given data, then using their knowledge of transformations of graphs to tweak their equation to get a closer match. If the groups have graphical calculators but not spreadsheets, they could use graphical calculators instead to find graphs with the right shape. Each group should agree on the best equation for each graph, so they need to convince each other that their equation is a good fit.

After students have had time to choose equations for some or most of the graphs, let them know that the whole class will come together to choose the best equations. Ideally, different groups will come up with slightly different suggestions for functions, and this can stimulate discussion about how to decide which function most closely matches the data. Each group could argue the case for their equation, and the class could decide together which is the best fit.

What clues can we find from the axes and the points given to help us to guess a likely function?

How can we modify our guess once we've seen how closely it fits?

Does joining the points in order of increasing time help?

How do we decide when the fit is close enough?

Graphs 1, 3 and 5 are the most straightforward functions to fit, so this is a good place to start.

Students could investigate and discuss the benefits of a least squares method of determining how close the fit is.

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?