# Average Looking

Purpose

In this unit we use measurements of our classmates to find the average (means and modes) of our features. We use our findings to create a 3-dimensional "class head".

Achievement Objectives
GM4-1: Use appropriate scales, devices, and metric units for length, area, volume and capacity, weight (mass), temperature, angle, and time.
GM5-1: Select and use appropriate metric units for length, area, volume and capacity, weight (mass), temperature, angle, and time, with awareness that measurements are approximate.
Specific Learning Outcomes
• plan an investigation
• discuss features of data displays using mean and mode
• state implications from investigation
Description of Mathematics

In this unit we plan an investigation to find out what the average student in our class might look like. In doing this we consider whether use two different measures of central tendency: mean and mode. The mode is the number or event that appears most often in a set of data. Suppose that the eye colours of the students in the class are blue, blue, blue, brown, brown, brown, brown, green, green. Then the most common eye colour is brown. This is the mode.

The number that all the numbers in a data set cluster equally around is the mean. This is calculated by adding all the numbers together and dividing by the number of numbers. It is important that students understand that the mean of a group of numbers (or measurements) in our unit represents what would happen if we equally redistributed our measurements so that everyone had the same measure.

Although we don't use it in this investigation the other way to measure the central tendency is by finding the "middle" number in a set of data. The number that comes in the middle of a set of numbers when they are arranged in order is the median. If we had the following set of numbers (4, 4, 4, 8, 9, 10, 10), then the middle number will be the fourth one. This number is 8, so the median of the numbers is 8. Now here we are lucky that there are an odd number of numbers. Otherwise there wouldn't be a precise "middle" number. If the numbers had been 4, 4, 4, 8, 9, 10, then we have to take the "three and a half" number as the middle number. As this is halfway between 4 and 8 we take the mean of 4 and 8 to be the median. So the median in this case is 6. The three concepts of mode, mean and median, measure central tendency in some way. That is, they give some idea of the "middle" number in a set. However, they are often different numbers. The point of the mode is that its central tendency is the sameness of data, what is the most common "same" number. As for the mean, that tells which number is as close as possible to all numbers. When you add all the differences, both positive and negative, between the mean and the other numbers in the set, the result is zero. Finally the median is literally in the middle. When the data set is put in order, the actual number that is at the halfway point of the list (or as close as we can get in the even case).

Although the focus of the unit is on creating the "average head" we also get to practice our measuring skills.

Required Resource Materials
• Centimetre cubes
• Tape measures
• Butcher paper (or other strong paper)
Key Vocabulary

mean, mode, median, average, circumference

Activity

#### Getting Started

1. We begin the week by recording the average head circumference and the most common eye colour of the students in our class.
2. Put up two large sheets of paper and get the students to record the circumference of their head on one sheet (you may like to record this as a stem and leaf graph) and their eye colour on a second sheet.
3. Tell the students to use this data to find the average head circumference and the most common eye colour. Students are to work with a partner, recording their findings so that the way a solution was reached is clearly evident.
4. When the students have had time to get the mean or mode discuss the terms average/mean, and mode (most common response).
5. As a class discuss the mean and mode they determined and how they reached that point. To help facilitate the discussion ask questions like
Does everyone share the same view?
Convince the class that your view is correct.
6. Ask: Is the average student a boy or girl?
7. This should lead to a lively discussion about the "usefulness" of such a question.

#### Exploring

Over the next 2-3 days the students work in small groups to create their own "average looking" head.

1. With the class brainstorm the features that you need in order to create the "average head".
- Length of ear
- Length of nose
- Distance between eyes
- Width of closed mouth
- Distance from eyes to ear
- Hair colour
- Eye colour
- Length of hair
- Is hair curly, straight?
2. With the class decide how the data is best collected. One way is to get the students to each record their details on large sheets of paper– one for each feature being used.
3. The students work in pairs to gather these measurements and record on the class sheets.
4. Put the students into groups. Each group is to tally the data and decide whether they need to find the mean and mode. The groups will need to decide whether to look at the average boy, the average girl or the average student.
5. At the end of each maths lesson ask the groups to reflect on the progress they are making. Is everyone in the group sharing the tasks? How are they determining the mean or mode from the data? Why are they choosing either the mode or mean to analyse? Do your average measurements make sense? How are you making sure your analysis is accurate?
6. Get each group to draw their head onto paper with the measurements, use the head circumference and eye colour as an example.
7. Make a model of the head, either as a life-size poster or a 3-D model.
8. Display the average measurements on a chart beside the model.
9. If possible share your average measurement with another class with the same age students. Get the students to discuss the different measurements. Where are the averages the same? Where are the averages different? Why do you think this is?

#### Reflecting

Get the students to discuss with a partner how they are alike and different to the model of the average head. They write a letter to their parents explaining how they are alike and different to the model of the average head. Encourage the use of the terms mean and mode.

During the year; leave your model on display. Re-do some of the measurements to see if any of the average measurements have changed. Ask why things have changed, haven’t changed? What things will stay the same? Why? When we measure next month what things might change? Why?