# Matching Faces

### Task 223 ... Years 4 - 10

#### Summary

The task is constructed around the story shell of a student tricking a new teacher by randomly placing name cards in front of a group of classmates. Given the placement is random, how many correct matches of name card to person would you expect. Students at almost any level can design an experiment to gather data about the focus question. Given the guidance on the card, more experienced students might break the problem into parts, collect and organise data and discover a pattern or two.

• 6 face cards
• 6 name cards

#### Content

• combination theory / ordered arrangements
• mathematical modelling
• multiplication, multiplicative thinking, multiplication principle of counting
• patterns, number
• recording mathematics
• statistics, analysing data
• statistics, collecting & organising data

#### Iceberg

A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.

In this task our student mathematicians are faced with the question of the number of matches (name to person) to expect for different size groups. The suggested approach is to predict an answer and then carry out experiments to get an idea of what might be expected. Four students and four names is the first experiment because it is just enough to provide a response that might not be expected. It is not unusual for students to reason that if there are more faces receiving names there will be more matches.

If four faces are involved then initial reasoning suggests that for any trial there could be 4, 3, 2, 1 or 0 matches. Here is a partial record of one set of student results:

 Trial No. No. of Matches 1 4 2 0 3 2 4 2 5 1 6 0 7 1 8 1 9 1 10 1 11 1 12 0 13 2 14 1 15 0 16 0 17 0 18 1 19 1 20 0
This record shows:
• 4 matches = 1
• 3 matches = 0
• 2 matches = 3
• 1 match    = 9
• 0 matches = 7
On this evidence the expected number of matches would be 0 or 1, with 1 being a 'bit more expected'. But is this enough evidence? Perhaps students will want to do another 20 experiments to decide whether 0 or 1 matches is the expected number. You could also draw their attention to the 3 matches = 0 result, especially if they do another 20 trials. Three matches can never happen with four faces. Why?

Questions 2 and 3 ask the students to carry out similar experiments for 3 and 5 faces. In those cases too, depending on the number of trials, it will be clear that one match is the expected result. The stage is now set for breaking the problem into smaller parts and examining the all the possibilities in each case. So, if children A, B, C, D have names a, b, c, d we get:

 One Face - Perfect Match! Two Faces - It is or it isn't Three Faces - Easy enough Four Faces - A bit of effort Five Faces - Over to you!
Given two faces is a special case, the evidence is strong that the expected number of matches for any number of faces will be one. However, it is a challenge to test this hypothesis by writing out all 120 possibilities for Five Faces - tedious might be a better word. If only there was a piece of software to do it for us. Then we could quickly get data to complete the fifth row of this table - and lots more. There could be patterns lurking here!

 Matches Faces 0 1 2 3 4 5 1 0 1 _ _ _ _ 2 1 0 1 _ _ _ 3 2 3 0 1 _ _ 4 9 8 6 0 1 _ 5 ? ? ? ? ? ?

#### Whole Class Investigation

Tasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works.

As a whole class investigation, this problem cries out to be acted as an introduction. You will also need 5 blocks or counters in different colours for each pair to be the 'faces'.

Choose four students to be 'the class' and one to be the monitor. Write the students' names on pieces of paper and tell the story as the monitor acts it out, randomly placing the names face down in front of the students. Dramatically pause and take a straw vote of the expected number of matches. Record on the board. Carry out the experiment and record the result of this first trial. Repeat the experiment another two or three times, modelling recording of the outcomes.

Invite students to begin their journal entry for the problem by describing the experiment, copying the results so far and including their prediction of the expected number of matches. As students complete their entry they collect a set of 'faces' and carry out their own five experiments. Results are added to the class data. The evidence will point to one match as the expected number.

• Suppose there were 100 students in the class and 100 name cards were given out. What would you predict would be the likely number of matches then?
Suggest that the result might be surprising and ask how we might investigate the question. Draw attention to the Working Mathematically Process and in particular the strategy of breaking a problem into smaller parts. Continue from here guided by the card questions and the strategy above of listing every possible case.

Round off by discussing the evidence which suggests that even for 100 students and names the expected number of matches is just one.

At this stage, Matching Faces does not have a matching lesson on Maths300.

#### Is it in Maths With Attitude?

Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner.

The Matching Faces task is an integral part of:

• MWA Chance & Measurement Years 5 & 6
• MWA Chance & Measurement Years 7 & 8