Key Ideas - Chapter 3: Ratio and proportional reasoning
A list of resources to support the book and website "Key Ideas in the Teaching of Mathematics"
Ratio Makes Sense
Cooking numbers - developing understanding of ratio by solving problems in the context of cooking recipes is on pages 10-11.
Students have to work out different quantities of ingredients for different numbers of people. Some can be counted, such as eggs, but others have to be measured, such as flour. The smallest amount is the amount that can be made with one egg, so that could serve as a unit to construct a mini-recipe which can then be scaled for other quantities.
To prevent the task deteriorating into a grid-filling exercise, there needs to be discussion using the words ‘ratio’ and ‘proportion’ when it is completed. Quantities down columns and/or across rows can be compared multiplicatively, and the scaling between them discussed. Proportionality is embedded in the task because for any final amount internal ratios stay the same.
For this task, it is not so much the completion of the table as the reflective discussion which can move learners towards an understanding of the underlying reasoning.
Zin Obelisk
This NRICH activity is a group task requiring some preparation in printing and assembling sets of information. Much of the information is about using rates: work rates, material rates, price rates.
When groups complete this task their reasoning is often ad hoc, especially in the excitement of a group task. If there is a requirement to record their reasoning in order to keep track of their thinking, these work records can be used later as raw material for presenting more formal ways of setting out their work.
It is likely that some additive reasoning will be used, and doubling and halving, but all of these can be re-expressed as scaling – as multiplications – and learners can compare these notations and methods with the ways they reached their group conclusions.
In the ancient city of Atlantis a monument called a Zin Obelisk was built in honour of the goddess Tina. The structure (a cuboid) took less than two weeks to complete. The task is to determine on which day of the week the monument was completed.
For the information cards print off this Word document or this pdf
Ratios and Dilutions
This NRICH task uses realistic data about dilutions to pose questions about adjusting proportions. You have mixtures of given proportions, and can tinker with them to add quantities and adjust to new dilutions, or to get back the original dilutions.
The context prevents any easy ad hoc methods and therefore helps towards an understanding of how to identify units which can be used as the basis for several cases. The numbers involved are realistic, and therefore do not offer exact integer arithmetic.
Product Wars
This Bowland Maths task is one of the Case Studies.
Students assume the role of apprentices in a soft drinks company and are invited to create the ultimate range of smoothies. Different mixtures with different tastes and nutritional values can be tested using market research. Spreadsheets are used as tools to keep track of varying taste and nutrition. Ratio of ingredients within each recipe can be recorded and compared, and scaling up for production or scaling down for testing purposes can be carried out.
As with many ratio and proportion tasks, it is possible to maintain an ad hoc approach and learners left to themselves may not make links to other ratio and proportion tasks and the underlying concepts.
Balancing 2
This NRICH task requires students to find an unknown number of marbles by using a known number on a balance beam and to look for a relationship between position and quantity. There are four numbers involved, three known and one unknown, and trials can be carried out until a balance is achieved. If players do not know what the conditions for balancing are, they can find out first before using it to find the unknown number.
You may want to set up a real balance to enable them to explore balancing more generally. The task also includes an interactive Flash activity in order to explore further.
Oh! Harry!
This NRICH task requires students to find the relations between quantities expressed as fractions or multiples of each other, so that quantities are worked out by scaling rather than by adding. It is an example of how a relatively simple context can be used to extend learners’ understanding of multiplicative relationships.
The questioning could be extended to include inverses such as one quantity being two-thirds of another means that the second is 3/2 times the first. The mixture of units, litres and millilitres, allows for the relation between them to be seen as also one of scaling. The context can be returned to again and again for more complex questions to be devised.