An idea for the classroom - clusters of counters
Students reduce ratios to their simplest forms in order to solve these puzzles, rather than for no apparent reason as in many textbook exercises.
Start by forming, on an interactive whiteboard or ordinary board, a clustered group of counters of two different colours.
For example, form a group of four green and six red counters:
Ask students to think what is the ratio of green to red counters. Allow them time to confer.
Having eventually established that the ratio is four to six, or, more simply, two to three, pose this problem:
The ratio green : red is presently 2 : 3. By adding counters to, or removing counters from, the group of red and green counters, obtain a group in which the ratio green : red is 1 : 2.
Students should imagine that they can add counters of either colour to the group or remove counters of either colour. Explain that adding a single counter is one move, and removing a single counter is one move. The aim is to solve the problem in the least number of moves.
They could work on this for a short time in pairs, using thought and talk only. Invite some pairs to demonstrate their solutions.
The best solution can be obtained in one move only, by removing one green counter. Some students might think of adding two red counters. This is also a solution, but, as it requires two moves, it is not as good as the previous solution. The most literal solution is to remove three green and four red counters. This is a very poor solution as it requires seven moves.
Students could record the number of moves required for their solutions. The pair with the lowest score at the end of the whole session wins.
Here are two more example problems:
- Assemble a group of three red and nine blue counters:
Establish that the ratio red : blue is 3 : 9, or, more simply 1 : 3.
Challenge students to change the ratio to 2 : 3.
(This can be done in four moves, by adding a red counter and removing three blue counters.)
- Assemble a group of three pink and twelve orange counters:
Establish that the ratio pink : orange is 3 : 12, or, more simply 1 : 4.
Challenge students to change the ratio to 1 : 5.
(This can be done in three moves in two ways: by removing one pink counter and two orange counters, or by just adding three orange counters.)
These are examples of simpler problems involving fewer counters:
- Assemble a group of one green and two red counters.
- change green : red from 1 : 2 to 1 : 3
- change green : red from 1 : 2 to 1 : 1
- Assemble a group of two red and three blue counters.
- change red : blue from 2 : 3 to 3 : 4
- change red : blue from 2 : 3 to 1 : 4
- Assemble a group of two pink and four orange counters.
- change pink : orange from 1 : 2 to 1 : 3
- change pink : orange from 1 : 2 to 1 : 1
Challenge students to make up problems for others to solve. Require them to find as many solutions to their own problems as possible, and ask them to put the solutions in order of ‘merit’.