# Average Looking

The Ministry is migrating nzmaths content to Tāhurangi.
Relevant and up-to-date teaching resources are being moved to Tāhūrangi (tahurangi.education.govt.nz).
When all identified resources have been successfully moved, this website will close. We expect this to be in June 2024.
e-ako maths, e-ako Pāngarau, and e-ako PLD 360 will continue to be available.

Purpose

In this unit we use measurements of our classmates to find the average (mean and mode) 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.
• Calculate the mean and mode of a data set.
• Discuss features of data displays using mean and mode.
• State implications from an investigation.
Description of Mathematics

See Planning a Statistical Investigation (Level 3) for general information related to planning a statistical enquiry/.  Data detective posters showing the PPDAC (problem, plan, data, analysis, conclusion) cycle are available to download from Census At School in English and te reo Māori.

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 (mode) is brown.

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 finding the "middle" number in a set of data when all the numbers are arranged in order (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. 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.

Mode, mean and median all 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 median is literally in the middle. However, the point of the mode is that its central tendency is the sameness of data (what is the most common "same" number). The mean states 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.

Although the focus of the unit is on creating the "average head" it also provides an opportunity for the practising measuring skills.

The learning opportunities in this unit can be differentiated by providing or removing supports to students and by varying the task requirments. Ways to support students include:

• adjusting expectations regarding the level and depth of data analysis to be carried out
• providing pre-prepared data collection tables, surveys, and data display templates (e.g. graphs)
• grouping your students strategically to encourage tuakana-teina (peer learning) and mahi-tahi (collaboration)
• providing teacher support and explicit teaching at all stages of the unit.

The context for this unit could be adpated to further suit the interests and experiences of your students. For example, instead of measuring head size and eye colour, you might choose to measure hand size and hair colour. Alternatively, you might be able to investigate variables related to students' favourite animals.

Te reo Māori kupu such as toharite (average, mean), tau tānui (mode), ine (measure), taura ine (tape measure), tūhuratanga tauanga (statistical investigation), and kauwhata (graph) could be introduced in this unit and used throughout other mathematical learning

Required Resource Materials
• Centimetre cubes
• Tape measures and other measurement tools
• Butcher paper (or other strong paper)
• Data collection tools (e.g. paper, graphic organisers, survey tables, spreadsheets)
Activity

#### Introduction to Statistical Investigations

This session provides an introduction and purpose to statistical investigations. The teacher will need to provide the students with plenty of magazines, newspapers and websites that have some good examples of how data can be presented effectively and perhaps some examples of poorly displayed data. This could be collated into a chart or slideshow. Prior to the session, ask the students to spend some time at home looking through magazines and newspapers to find examples of statistics to bring in for the session.

1. Start the session with a class discussion to get the students thinking about whether or not we have a need for statistical investigations, and who uses the information?
What is a statistical investigation?
Can you think of an example when we might need to carry out a statistical investigation?
2. Organise the students into groups of two or three. Give out magazines, newspapers and website links and ask the students to find some examples of statistics.
3. Ask the students to look closely at the examples they have selected. Ask them to consider the following questions;
Who has done the research for/carried out this investigation?
Who will benefit from the results of this investigation?
Is it clear to you what the purpose of the investigation is?
What do you like about the way that the information is presented?
Does it help you in any way to understand the information better?
Do you think the information could have been presented in a different way to help the audience understand the findings? If so, what would have made it better?
4. Use a class discussion to share ideas from each group. Have the students all come up with the same ideas? Try and steer the students towards the conclusion that the best way to present the information depends on the information itself. They might notice that category data is displayed differently to numerical data.

#### Getting Started

1. Tell the students that, as a class, you are going to investigate the average student in our class might look like. The first step to this is recording the average head circumference and the most common eye colour of the students in our class.
Our investigative question: What does the average student in our class look like?
2. Put up two large sheets of paper. Get the students, in pairs, to use the tape measures and  record the circumference of their head. Model this process before hand and draw attention to the scale. Often tape measures shown inches on one side and metric measurements on the other. You might discuss why this is, as a class, and confirm your use of metric measurements for this task. Record the circumference measurements on one sheet (you may like to record this as a stem and leaf graph) and students' eye colours on a second sheet.
3. Tell the students to use this data to find the average head circumference and the most common eye colour. You might need to provide explicit teaching around how to find each of these measures of central tendency, and might need to structure the unit around learning about each of these measures. Students should work with a partner, recording their findings so that the way a solution was reached is clearly evident.
4. When the students have found one of more measures of central tendency, gather the class together and share strategies and thinking. Use questions to support students' in developing their ideas:
Does everyone share the same view?
Convince the class that your view is correct.
5. Ask: Is the average student a boy or girl?
6. This should lead to a lively discussion about the "usefulness" of such a question.

#### Exploring

Over the next 2-3 days the students will 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. Alternatively, you might get students to use a spreadsheet or survey form. Ensure you model whichever process is chosen.
3. Have students work in pairs to gather these measurements and record them in the decided-upon manner.
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 session, ask the groups to reflect on the progress they are making. Is everyone in the group contributing? Does everyone understanding the meaning of mean and mode? How are they determining the mean or mode from the data? Why are they choosing either the mode or mean to analyse? Do the group's 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 could 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?