INforM – Interactive Notebooks for Mathematics



INforM – Interactive Notebooks for Mathematics

Book 18 – Video analysis

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Video analysis

This is the last of four `thematic’ Notebooks whose aim is to present some mathematics teaching ideas based on realistic problems and modelling which link to other subjects and which can be pursued in an open-ended and problem-solving manner.

The activities are related to work in Art & Design, PE, Sports and Science. The mathematics covers symmetry, shapes, linear functions, quadratic functions and transformation of function – so different pages may be omitted and the resulting Notebook saved for a particular year group.

Organisation of the materials

The Smart Notebook file is saved as `Book 18 Video analysis.notebook’.

It consists of 16 pages of which the first is the title page, shown above.

There are 13 pages to support the activity and its extension. Page 15 is a blank page.

Page 16 contains teacher notes which are amplified here.

The first activity

The title page shows a still image of a fountain and a frame from a video of a basketball game. Page 2 shows the fountain in greater detail, and is there to invite discussion. The first set of ideas suggested are geometric ones, as on Page 3.

Page 4 has an image of the graph of a parabola – which can be `flipped’, resized and relocated to explore whether parabola make good fits for the water spouts (as in theory they should)! The relation between the graph of quadratic function and a parabola is also established.

Page 5 uses a Flash application from the SMART Notebook Gallery.

You can use the sliders to see the effect of changing a, b or c. You can also drag the faint image of the fountain onto the graph paper choosing a suitable origin. While the sliders only change values by whole number increments, you can type in decimal values for the parameters,

such as a = 3.9 .

So now we have established some feeling about fitting quadratic functions to still images taken with a digital camera. On page 6 we start to look at how we can collect data from a video clip.

The Attachments tab shows a video clip called `Basket.avi’. For best results we suggest using the SMART Video Player from the Tools menu for your board. You can then slide to different frames and annotate the screen area to show successive positions of the ball. When you are happy with the results, use the floating camera tool to take a still image.

Page 7 shows the result of a previous capture just in case your own does not go as well as you would like.

We are now ready to start turning the crosses and blobs into numbers – we just need axes and a scale.

Page 8 shows nine clear blobs. The background is actually a flash video, so using a right click you should be able to access a limited menu of video tools. We know that the height of the basket is 3.05m from the floor of the sports hall, so by fixing a suitable origin and drawing axes as lines, you can use the rulers to measure the coordinates of each of the ball’s positions.

Page 9 shows how we can do the curve fitting in the MA/Intel Mathematical Toolkit which is part of the SMART Gallery. When the page opens, click on the Coordinates and Graphing button at the top. Enter -1 and 5 as the minimum and maximum values for the x-axis and 0, 6 for the y-axis. When you proceed you should see a graph screen with our data points displayed. You can use the mouse to click on or near the points. Although you cannot drag them, you can edit their coordinates in the table. There is a slight `glitch’ in running the software through Notebook – when you try to edit a number you don’t see a cursor – however you can use the Delete key and retype numbers. If you run the software outside Notebook, you will find that the video file is already built into the software and you can do all the analysis within the package. We hope that this will be available within Notebook in the near future.

Page 10 shows two screens from the Toolkit as first we collect the data, and then start to model it by drawing graphs of quadratic functions. For modelling purposes the usual form of the function y = ax2 + bx + c is not as instructive as the forms:

y = p ± q(x - r)2 where p, r relate to translations, and q to a stretch. The data and the function are given on this page but you need to drag the `screens’ away to reveal them.

Page 11 shows how easy the free US software called `Tracker’ makes the process of data collection from a video clip. It is a Java application and anyone can use it freely. The Attachments include a Word file describing how to use it to capture data from our Basket.avi clip. The missing element which Tracker provides is a way to collect data about the times which correspond to the ball’s positions. Each blob really has three pieces of data attached to it (t,x,y) – and Tracker lets you build up just such a data table. The software is available from:

.

Page 12 shows the data table which Tracker collects and which can be exported to other software. It is available from the Attachments tab in txt and csv file formats. It is also provided in Excel and TI InterActive! files with links on this page. The Fathom file is quite an extensive example of how this problem can modelled using some ideas of calculus. The coloured links are to files for different software which allow you to fit the equation to the data by dragging sliders to change the parameters p, q and r. The html file requires the installation of the free Cabri II Plus `plug-ins/ described in Book 17.

Page 13 invites students to undertake a range of further explorations based on the data collected. Some of these can be resolved using KS3 mathematics, others will require some knowledge of trigonometry and/or calculus.

Page 14 shows the `best-fit’ functions for the graphs of y against x, x against t and y against t, produced in TI InterActive! The values may be helpful in trying to answer some of the questions from the previous page!

Page 16 has the Teacher notes. While the photo and video clip for this example have been wrapped into the Notebook’s Attachments, it will obviously be more engaging for students to work with their own video clips – which they can take e.g. on a mobile phone, or locate in a book, magazine or on the Internet. Similarly the video clips can be of anything in motion – just provided that the video camcorder is kept still and the focus is not changed. Teachers TV have filmed an episode about mathematical topics which are difficult to teach at a Sports College where the maths, science, D&T and PE staff are working collaboratively at this kind of video analysis.

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