Ratio and Proportion Ideas
Ratio and proportion can be perceived as difficult but pupils often have an intuitive feel for the two if given time to model situations based on relationships between numbers, rather than being introduced to abstract notation and symbols before they have explored the ideas.
Even young pupils, using counters or other objects, can grasp the idea of keeping the same proportions by preserving the ratio, while changing the quantities.
Reinforce differences between ratio and proportion by using a range of entities as examples, including pupils; for example, find the ratio and proportion of people with blue and brown eyes in the class. There may of course be a wider range of colours rather than just blue and brown so embrace this and discuss what the ratios and proportions would be if we extended the range of eye colour. Also allow time to discover how preserving the proportion relationship relies on maintaining the ratio relationship between the entities being compared.
Encourage the expression of ratios in several ways. In a class of 30 pupils containing 18 boys and 12 girls, the ratio of boys to girls is 3:2 or 3 to 2 or 3 boys for every 2 girls or 3/2 (showing that the number of boys is 1.5 times the number of girls).
However, we express the proportion of boys as 3/5 of the total or 3 in every 5.
Lots of practical work helps such as: designing tiling patterns and then costing for a certain length of wall; cooking using recipes for different numbers of people; making drinks by combining ingredients and voting on which is the most popular, then working out how much would be needed for a class party.
Using paints to create shades of colour can illustrate why the numbers cannot just be swapped: if using blue to white in the ratio 6:1, if we actually use 6 parts white paint to 1 part blue, the resulting paint is a very different shade!
A former pupil recently reminded me about us exploring ratio and proportion while making caramel shortcake, despite the lessons taking place nine years ago! Another time, while studying Macbeth, we created potions where they had to decide on the ratios between different liquids such as juices, lemonade and water. Other groups then tested the potions to see if they could ascertain the ratio of the liquids in each one.
Scales can confuse pupils so get students to create their own scale and eventually move on to producing scale drawings of a part of the school or the classroom. Plan views of furniture and fittings, or new play equipment for the playground, can also be drawn to scale and then moved around to find the most suitable design according to given criteria. This could be linked to persuasive writing in English, so pupils decide what they would like in an area of the school and then produce a scale drawing to back up their ideas which could then be presented to the headteacher, governors or other pupils in the school. They could even take it further and cost their idea...
Maps are a great resource for working with scales - both ordnance survey and A to Z maps of the local area. Exploring scales is also a great link to measures since they can work out actual distances to various places from the school or their home to embed a better feel for longer distances.
Paintings can also be a useful resource: aspects of the painting can be scaled to full size so, for example, a Roman column in a painting can be created with chalk on the playground or in the hall on pieces of flipchart paper stuck together. It is also interesting to note the surprise if scaling a slope or triangle as life size, when pupils wonder what happens to the angle and then realise that they can use a normal size protractor because the angle stays the same size...
Allow time for pupils to model situations using materials such as counters, cubes or beads or cubes in two or three colours, depending on how many entities there are in the relationship. You could even try out some recipes for biscuits where the relationship is not maintained and allow the pupils to see the results.
Use relationship tables so that pupils can see how scaling one entity in the relationship needs to be matched by a scaling to the other entities if the relationship between the two is to remain in place.