pupils can then use the Colour and Fill options to improve their picture before clicking on the Tiling button. On the first mouse-click, the pupil’s image is replicated 4 times; the next click shows 16 images and so on until the pupils have produced wallpaper or wrapping paper with their personal design (see Figure 9.2).
Using Powerpoint for a maths quiz
Based on the Word activity in Figure 9.1, teachers can create a maths quiz using power-point to ‘test’ the pupils’ ability to recognise shapes when the properties are listed.
Initially, the answers are covered. However, a mouse click can be used to remove the image, revealing the correct answer (see Figure 9.3).
Figure 9.3 Partial revelation in a
Powerpoint quiz
Using a digital camera for demonstrations
A digital camera can be used to investigate symmetry and the geometric properties of shapes in real life. For example, the symmetry of bridges in the local area—foot bridges, road bridges and railway bridges, symmetry in a person’s face, the repetition of shapes in the appearance of the school building, symmetry when fruit is cut in half, tessellations on wooden floors or brick driveways, and many more. Also, by zooming in on the head of a sunflower and looking at the arrangements of seeds in the centre, or counting the number of florets in a spiral on a cauliflower or pine cone, or cutting an apple horizontally through the middle instead of lengthwise will produce instances of the Fibonacci numbers in real life.
It is not always possible to bring the real example into the classroom, so a picture taken by a digital camera allows all pupils to view an enlargement of the item under investigation and, using a data projector, it also facilitates whole-class demonstrations and discussions. This type of work can be integrated into the normal classroom activity and may be used to capture the pupils’ attention by looking at:
• the presence of the Golden Ratio in the natural world;
• the use of rectangles with Fibonacci dimensions in art and architecture;
• the equation of curves that best fit the shape of the bridges;
• the equation of motion of an object being thrown across a playground or sports field;
• line symmetry in people’s faces and how looks change if you reflect different sides of a person’s face;
• the occurrences of reflections, rotations, translations and enlargements in the structure and architecture of the school building or famous buildings around the world;
• types of symmetry in different fruit—using pictures of different cross-sections;
• identifying the two-dimensional shapes in wooden floors, brick driveways and patios.
In many cases, the use of pictures from a digital camera can be used to engage the interest of the pupils of any age and the novel context can be used to teach the necessary mathematical skills. The pupils can then work on similar real-life tasks, but in a different context to reinforce the concepts.
Using spreadsheets for problem-solving
Spreadsheets for primary schools
Another package from Black Cat software is Number Box 2. This offers a spreadsheet facility with a number of colour-coded levels of sophistication. Depending on the age group and ability level within the primary classroom, different settings can be selected for different levels of functionality. For example, pupils can define the type of data in a cell to be words, numbers, decimals, dates or time; in the graph options, pupils can select between bar charts and pie charts, two-dimensional or three-dimensional representations of the data, and pupils can rotate the direction in which the chart is displayed. The style of both the spreadsheet and graphs can also be changed when the font and font size options are available to the pupils. For the high-ability pupil, simple maths functions and the facility to enter your own formula are also present. It is clear that the functionality of
this spreadsheet package can be increased over time to make the step into Excel or MS Works more accessible for the pupils. Quicksheets containing data sets are pre-loaded to illustrate the various uses of the spreadsheets at each predefined level. These also act as a good starting point for the less experienced user.
Using Number Box 2, pupils can analyse and interpret existing data or enter their own data. Graphs can be drawn and the data can be sorted if appropriate. Calculations can be completed by the more able students using Autosum or by entering their own formulae.
Number patterns
All aspects of problem-solving require the production of a general formula that summarises the results. By investigating patterns of numbers, pupils are more likely to recognise a sequence of numbers if it occurs again in the course of their mathematical work. Using spreadsheets, it is possible to generate sequences of numbers and to discover their general formula. The most common number patterns are the natural numbers (n), the even (2n) and odd numbers (2n−l), followed by the multiples and then square numbers (n^2, where n is the cell reference for the sequence of natural numbers). Building on these simple examples, pupils can be encouraged to create formulae for the triangular, pentagonal, hexagonal,…numbers and the Fibonacci series.
Problem-solving
Once the pupils can enter formulae accurately into a spreadsheet, they can progress to
‘What would happen if…?’ scenarios where a situation is simulated and then one (or more) of the conditions changes. For example, in the next task the pupils set up a spreadsheet that stores information on sales of items of a school uniform. The pupils can investigate how the total cost changes as they purchase different combination of items and also what happens if the cost price increases by X per cent.
The worksheet shown in Figure 9.4 can be adapted to suit different age groups by investigating the cost of running a tuck shop, organising a disco, hiring a sports hall, and more business-orientated activities for post-primary school pupils such as school fêtes and fundraising activities spanning a financial year.
Spreadsheets can also be used to encourage pupils to investigate monetary systems such as investments and mortgages. Using the example shown in Figure 9.5, pupils can make an initial decision on which option they would prefer before using the spreadsheet to do the long-term calculations. Similar studies on the difference between simple and compound interest can be illustrated via a spreadsheet.
Problems that match the types of coursework currently used at GCSE level can also be completed using spreadsheets. A typical pair of examples of this would be the Sheep Pen, looking for the maximum area which can be enclosed with the minimum amount of fencing, and the Strawberry Box, where the maximum volume is required from the minimum area of cardboard or other box material (CCEA, 2002). Variations on each theme can be added, such as the presence of a wall for Sheep Pen or an open box (no lid) for the Strawberry Box problem. Line graphs can be drawn in the spreadsheet package of the key dimension against the area or volume enclosed. From the graph, the pupils can
also zoom in on a smaller range of values and use the spreadsheet to evaluate the answer to