This unit introduces students to combinations as a situation type that multiplicative thinking can be applied to.
- Use models such as tree diagrams and two-way tables to find all the combinations for a simple pairing situation.
- Use multiplication to count all the possible combinations (and permutations).
- Use models of all the possible outcomes, and experimental results, to compare the chances of different outcomes.
Cartesian products are known to be one of the most difficult situations to which students apply multiplicative thinking (Greer, 1992). A Cartesian product (named after Rene Descartes) is needed when the objects in one set are mapped in one-to-one correspondence with the objects in another set. For example, in this unit students consider the number of different outfits that Lucy can make from a set of shorts matched with a set of t-shirts. Suppose Lucy has these clothing items:
Each arrow represents a possible pairing of shorts and t-shirts. There are 3 x 2 = 6 pairings from the number of pairs of shorts multiplied by the number of t-shirts.
Cartesian products are also involved in arrays, arrangements of objects in rows and columns. Arrays are fundamental to measurement of area and volume, and two-way tables are very important in statistics, logic and algebra. Each pairing above can be represented as a cell in this two-way table:
As with the arrow diagram the number of shorts and t-shirts make up the dimensions of the array, 3 x 2 = 6.
Specific Teaching Points
Students are likely to need support in systematically finding all the combinations. Acting out the pairings physically is one way to help students see that representations such as tree diagrams are anticipations of the combinations that can be made. For example, in the outfit situation above drawing a pair of shorts and a t-shirt from a pool of objects and representing that combination as one arm of a tree diagram helps students to see that each arm represents a unique combination that may occur. Finding all the combinations is part of a theoretical approach to probability. Experimenting is also very important. Students need to understand that the results of an experiment are seldom an exact match to what might be expected from the theoretical model. In the two t-shirt with three shorts scenario above it might be predicted that a same colour pair occurs twice in trials. Experimental results from selecting one t-shirt and one pair of shorts randomly are likely to vary considerable from the predicted frequencies, especially in the short term. For example, six trails may result in no same colour pairings. An important understanding is that samples will vary, even when the theoretical probabilities are the same.
Observations of students during this unit can be used to inform judgments in relation to the Learning Progression Frameworks. Click for tables of guidelines.
It is expected that students have applied multiplication to other situation types such as equal sets. They should also have some experience in ordering everyday events by likelihood and using words such as “likely”, and “unlikely”, to describe chance.
In this session students learn about systematically organising pairings of objects, to find all the possible combinations. They consider efficient ways to count the number of combinations.
Play PowerPoint One, slides 1 and 2, about Lucy going on holiday. The scenario is that Lucy wants to wear a different outfit everyday of her 24 day summer holiday. She has packed 5 different t-shirts (tees) and 5 different pairs of shorts. Surely that should be enough!
Ask the students to explore the different outfits Lucy could make in small groups. You can provide copies of Copymaster One so students can either, mix and match the t-shirts and shorts, or create separate graphics for each combination they find. Students can use colouring pens or pencils to illustrate the combinations or you can provide coloured copies that are laminated for later use using Copymaster Two.
After a suitable period of investigation bring the class together to discuss their strategies.
Look for students to show a system for finding combinations, like:
“I started with the pink shorts and matched the different tees one-by-one to it. There are five combinations with that pair of shorts…”
A strategy like that might be shown as a tree-diagram (see slides 3 and 4 of PowerPoint One).
What ways can the students find to efficiently count the number of combinations?
They may count in fives, for each grouping in the tree diagram of use repeated addition. Some students may know 5 x 5 = 25. Record each method using symbols, e.g. 5, 10, 15. 20, 25; 5 + 5 + 5 + 5 + 5 = 25; 5 x 5 = 25.
Another systematic way to find all of the combinations is to use a two-way table. In the case of pairings this is a clear representation.
So Lucy is able to create 5 x 5 = 25 different outfits for her 24 day holiday.
Pose scenarios to see if students can work with the tree diagram and two-way table representations.
If Lucy ripped one of her pairs of shorts and could not wear them, how many outfits could she make? (4 x 5 = 20) Relate this to losing one arm of the tree diagram or a row of the table.
So Lucy goes shopping and buys two new pairs of shorts. How many outfits can she make now? (6 x 5 = 30) Relate this to adding two arms to the tree diagram or two rows to the table.
Lucy buys a red t-shirt on sale. How many outfits can she make?
Just as the combinations can be made more complicated they can also be simplified by reducing the options for shorts and t-shirts.
In this session students explore the relationship between probability and the set of possible outcomes (sample space). Chance is involved if the selection of shorts and t-shirts is random, that is each item has an equal chance of being selected so any predictions are uncertain.
Show PowerPoint Two about Lucy’s untidy chest of drawers. Since most of her clothes are in the wash she only has two pairs of shorts and two t-shirts to choose from. Point out that the clothes are messed up inside the drawer so Lucy could get any pair of shorts and any t-shirt.
What are the chances of her getting both items of the same colour?
Let the students discuss the chances of a same colour outfit occurring, in small groups. Remind them that they need to justify their ideas. Bring the class together to share ideas.
Do the children recognise that only four (2 x 2) combinations are possible?
Can they list all the possibilities?
Can they organise the possibilities in a diagram, e.g. tree diagram or two-way table?
So Lucy washes a green pair of shorts and a green t-shirt and adds those items to the drawers. Now she has three different pairs of shorts and three different t-shirts.
Have her chances of getting a same colour outfit improved now?
Ask the students to act out Lucy taking two items to wear. They will need two opaque containers (e.g. plastic icecream containers) and paper copies of the shorts and t-shirts from yesterday. Without looking they reach into the containers one at a time and remove an item. Tell them to record what happened and place the items back in the containers for the next turn.
Ask each group to carry out nine trials. Bring the class together to discuss the findings.
Are your results what you expected? Explain
Let’s see how many times out of nine the colours matched. Collate the results of all groups.
What do you notice? Students should see that the sample results vary.
Put the ‘Same Colour’ frequencies in a dot plot on the whiteboard, like this:
Do students notice that three is the centre of the frequencies? Do they realise that 3 out of 9 equals 1 out of 3, or one-third?
Think about the number of possible outcomes that can occur with three t-shirts and three pairs of shorts. Add to the two way table or tree diagram from Part One. There are nine possible outcomes.
Why would three out of nine by the most common? There is a one third chance of getting a colour match.
Only three of the nine outcomes are colour matches (marked with x).
If time permits pose other scenarios:
- How can Lucy improve her chances of a colour match?
- Could she ever get a 100% (certain) chance of a colour match? How?
- Could she ever have no chance, zero, of a colour match? How?
- If she had all five t-shirts and all five shorts available, what would be her chances of getting a colour match?
In this session students consider combinations in a different context, Build Your Own Sandwich. They move beyond pairings to consider what happens with three or four event combinations.
PowerPoint Three introduces the scenario. The Daily Bread Sandwich bar has an option where you can choose two fillings to go in your sandwich, one from the main selection of meats and one from the extras selection. The options are:
Mains: Ham, Chicken, Beef
Extras: Salad, Cheese, Tomato, Avocado, Peppers
Without any discussion, ask the students to work in groups to answer Lucy’s question:
How many different sandwiches can be made with these choices?
Copymaster Three contains images of the fillings that students might use to find all the sandwiches that are possible. Note that extra fillings are included in the bottom row of the Copymaster for extension. Look for:
- Do the students use a systematic strategy to find all the possibilities?, e.g. organised list, tree diagram, two-way table.
- Do the students use multiplication to count the number of possible sandwiches or do they rely on one by one counting, skip counting or addition?
- Do they consider the impact of changes to the fillings? e.g. Adding another extras topping would increase the number of combinations to 3 x 6 = 18. After a suitable period of investigation gather the class to discuss the points above.
Useful extension questions might be:
- If Daily Bread wanted more sandwich combinations are they better to provide another choice of main or another choice of extras? (Another combines with 5 extras, another extra only combines with 3 mains).
- Suppose Daily Bread wanted to advertise that you can make 28 different sandwiches. What could they add to their fillings so that is possible? Note that the extra fillings on Copymaster Three make this possible.
- If you were allowed two extras with the main filling instead of one, would that increase the number of possible sandwiches? By how much?
This is a complex problem since duplications need to be avoided, e.g. Ham with lettuce and tomato is the same sandwich as Ham with tomato and lettuce. The number of possible combinations doubles to 30.
In this session students explore a calculator number game called Odds and Evens. The students consider what combinations are possible in the game and, therefore, what is each player’s chance of winning. They then consider how that game is different to a similar game where numbers are not replaced.
Tell the students that Lucy has a couple of favourite holiday games for times when the weather turns bad. Play the Video 4a for this lesson then get two students to play the game in front of the class. Then ask the students to a play the game in pairs. They need to play the game at least ten times and record who wins each time.
The question you want them to answer is: Is the game fair to both players?
Remind them that they will need to justify their ideas and that they will need to think about the game, not just play it. Look for students to:
- Record the results of their games systematically.
- Use strategies they used in the previous sessions to find all the combinations, e.g. table, tree diagram.
- Knowing all the possible combinations, make decisions about which player, odd or even, has the best chance of winning the game.
After a suitable time invite ideas about the fairness of the game. Some views may be based on experimental results. Players do tend to win about the same number of times. However if you pool the results across the whole class you might find slightly more wins for the even than the odd players. Expect some students to use systematic ways to find all the possible outcomes.
“I thought there must be 9 x 9 = 81 possible combinations. It is like the shorts and t-shirts problem.”
A complication in this game, that does not occur in the Lucy’s outfits scenario, is that what appear to be the same combination are different. Player A choosing 6 and Player B choosing 5 is not the same outcome as Player A choosing 5 and Player B choosing 6, although both outcomes have a 6-5 combination. Look for systems to find all the possible outcomes. A two way table is a tidy strategy, in which the even and odd outcomes can be shaded differently. Use PowerPoint Four to stimulate discussion.
Seeing patterns in the table helps speed up the shading process, e.g. Every second cell is shaded. Ask the students if there is a quick way to count the number of shaded (even) cells. They might notice that number of shaded cells in each row going down is 5 + 4 + 5 + 4 + …
Can they use properties of multiplication to find the total? (5 x 5 + 4 x 4 = 41)
So how many non-shaded (odd) cells are there?
Students might suggest 81 – 41 = 40, the total number of cells less the even cells. Others might notice the pattern in the number of blank cells going down by row, 4 + 5 + 4 + 5 +… which gives 5 x 4 + 4 x 5 = 40.
So 41 out of 81 possible outcomes are even, and 40 out of 81 are odd. That makes the game close to fair. Do the students recognise that fairness? To realise that 41 out of 81 is very close to 50% requires proportional thinking.
If time permits contrast the Odds and Evens calculator game with a variation that Lucy has made up using a pack of cards. Play the Video 4b for this lesson to see how the other game is played.
Ask the students to investigate:
How is the game the same and different to the calculator game?
Is this game fair or unfair? Why?
Students can cut up five pieces of paper and label them with digits 1-5. They can act out the game many times. Student may notice that the results tend to favour the odd player. Why?
How could the game be made fair?
This session gives you an opportunity to see if the students can apply multiplication to problems involving combinations, and to apply their knowledge of possible outcomes to think about chance.
Begin by showing the students PowerPoint Five. Lucy’s family travel from Tauranga to Palmerston North, via Taupo. There are four possible routes leading from Tauranga to Taupo and five possible routes from Taupo to Palmerston North. So in total there are 4 x 5 = 20 different ways for the family to drive home. Pose the problem and let the students explore it independently using Copymaster Four. Look for them to:
- Identify that multiplication can be used to count the number of different ways.
- Systematically organise the routes using a strategy like a two-way table or tree diagram.
The second part of the problem explores probability. The family place the Highway numbers, 2, 28, 29 and 33, in a sunhat and randomly select one of those numbers as the route from Tauranga to Taupo. Similarly, they put the Highway numbers, 1, 5, 30, 38, and 43, in the sunhat to select the last leg from Taupo to Palmerston North. Three questions are asked:
- What are the chances that the route home will take the family through Taupo? (100% Certain)
- What are the chances that the family will travel through Napier? (2 out of 20 or 1/10)
- What are the chances that the trip from Taupo to Palmerston North will take them by the sea? It is easier to think about which trips are not by the sea at any point. Only the trip down Highway 1 does not touch coastline. So the chances of a coastal route are four out of five or 4/5.
A harder problem is to consider all 20 routes from Tauranga to Palmerston North and how many of these routes encounter coastline. That problem can be set as extension.