The ability to recognise situations where it is helpful to make an organised list can assist in a systematic analysis of data. Organised lists can be helpful for identifying a pattern in results or for generating graphs.
Teachers can:
- provide specific problems where this strategy is going to be helpful and efficient
- discuss the general features of the given problem that supports this strategy
- model the use of the strategy (HITS 4: Worked examples).
Understanding this strategy
When using this strategy, teachers should discuss how and why the use of an organised list is going to be helpful to represent the information needed to find a solution to a given problem. Use of an organised list should then be modelled, focusing students' attention on terminology in the problem that is helpful for organising the list (e.g. deciding on table headings, number of rows, etc.).
The steps teachers can follow to implement this strategy are:
- Ask students to read the problem and try to solve it in groups first. Discuss the strategies the student have used in class.
- Suggest the strategy 'use of an organised list' and ask students to identify features of the problem that suggest why using an organised list might be helpful for this problem.
- Discuss the type of list which is best to organise the data to solve the problem.
- Based on students' responses, demonstrate the use of the strategy.
- Discuss how the resulting organised list can help to answer the given problem.
The example below shows how this strategy can be used in a Year 9 lesson in a unit on Chance (VCMSP321). The strategy also supports the Mathematics proficiencies Reasoning ("prove that something is true or false") and Problem solving ("use mathematics to represent unfamiliar or meaningful situations") (VCAA, n.d.)
Example of using an ordered list to problem solve
Problem
The rules of a dice game are as follows. Two dice are rolled. If the product of the uppermost numbers is even then player A scores a point, and if odd, player B scores a point. The first player to score 20 points wins. Is this a fair game?
Student actions
Students undertake a close reading of the question . 'Product' and 'fair game' are key words.
Fair game suggests that there is a need to determine whether the probability of each person winning is equal. Playing a few games can help students to understand the game. Students can then discuss whether the game is "fair".
Students need to identify that the term 'product' means that they need to find all of the possible outcomes of throwing two dice and, in each case, multiply the pair of numbers.
Classroom discussion
Teacher leads a discussion to elicit possible methods of solving this problem
If students don't suggest the 'use of an organised list' method, the teacher suggests this method and explains why it is helpful.
The question "Is this game fair?" suggests that there is a need to determine whether the probability of each person winning is equal.
Students need to identify that the term 'product' means that they need to find all of the possible outcomes of throwing two dice and, in each case, multiply the pair of numbers. For a problem where there are defined outcomes such as this, it can be useful to list these outcomes
Ask students how they could make an organised list of the possible outcomes.
Reasoning a solution through an ordered list
Students can try to use a systematic approach to list all of the outcomes. The teacher should model each of the strategies based on students' descriptions of those strategies. An example of an organised list will contain 36 outcomes.
(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)
(2,1) (2,2) (2,3) (2,4) (2,5) (2,6) until (6,6) is reached.
As well as the list above, the data could be presented in a table (below). Students could complete the table below for the 6 x 6 alternatives: the entries in each cell are the products of the row and column headers (in blue).
In this case, students could be asked to make links between the worded problem, the organised list and the table cells.
Table listing outcomes on a die
Example
Dice
| 1
| 2
| 3
| 4
| 5
| 6
|
|---|
|
1 | 1
| 2
| 3
| 4
| 5
| 6
|
|---|
2
| 2
| 4
| 6
| 8
| 10
| 12
|
|---|
3
| 3
| 6
| 9
| 12
| 15
| 18 |
|---|
4
| 4
| 8
| 12
| 16
| 20
| 24
|
|---|
5
| 5
| 10
| 15
| 20
| 25
| 30
|
|---|
6
| 6
| 12
| 18
| 24
| 30
| 36
|
|---|
Solution
Guide students to understand and explain how their organised list can help them to solve the problem. Teachers might ask students:
- to consider whether the product of two numbers in each of these outcomes are evenly split between odd and even
- to consider whether there is a bias one way or the other
- to suggest a change to the rules to produce a fair game.