Statistics

4.8 STATISTICS

STATISTICS LESSON IDEA 1

TITLE: INVESTIGATING THE PROPERTIES OF PIE CHARTS USING A SPREADSHEET
TOPIC: STATISTICS
AIM: To encourage the students to experiment with graphing pie charts and to see the relationship between changing data and the graphical representation of the data.

RESOURCES:
Computers with spreadsheet software installed.

METHOD:

1. Students enter data into the spreadsheet.
2. Students are instructed to use the standard spreadsheet graphing tools to create a pie chart.
   The advantage of the spreadsheet is that changes made to the table will be automatically reflected on the pie chart.
3. Now students can investigate the answers to questions such as the following (some suggested answers are given).

Q. What happens to the pie chart if I double the number of students who chose blue as their favourite?
A. The blue section increases in size (but it does not double).
Q. What happens to the pie chart if I double the number of students who chose each colour?
A. Nothing happens. (Why?) Because the relative size of the numbers stays the same. 10 out of 30 people is the same fraction as 20 out of 60 people.
Q. What happens to the pie chart if I halve the number of students who chose each colour?
A. Nothing happens. (Why?) Because the relative size of the numbers stays the same.

CLASSROOM MANAGEMENT IMPLICATIONS:
If a reasonable number of computers is not available for use by mathematics students, the above ideas could be demonstrated by the teacher to groups of students gathered around a single computer, or with the aid of a data projector, if available. Students could invent their own data sets or use a data set gathered by means of a simple classroom survey that might be part of a project in a subject area such as CSPE. Students will probably also have their own suggestions for "What would happen if..." questions.

STATISTICS LESSON IDEA 2

TITLE: THE MEAN: WHAT DOES IT MEAN?
TOPIC: MEASURES OF CENTRAL TENDENCY
AIM: To develop an understanding of the mean and the mode, and of suitable uses for each of them.

RESOURCES:
No special ones.
It is assumed that the students have already met both the mode and the mean. Alternatively, some of the ideas here could be used in introductory lessons for either concept, and some could be incorporated in later lessons, as appropriate.

METHOD:

1. Recall: the mode is the "most fashionable" (most frequently occurring) number.
2. So what is the mean?
   - Take as an example the marks scored by a group of twelve children in a test:
   10 5 6 8 9 5
   5 11 7 9 9 12
   - What is the mean? 96/12 = 8.
   - So, suppose the twelve people shared the marks out equally among themselves; they would each get 8.
   - Also, imagine the marks sitting on a number line, and that the number line forms a seesaw pivoted at some point: 8 is the balance-point: the point such that the seesaw balances.
   - What happens if we change one mark? Suppose the test was out of 20 and the last person got 20, not 12; the seesaw will tip ... and the balance-point will be more to the right. Or suppose the person scoring 7 got 11; again the seesaw will tip a bit, and a new balance-point will be needed.
   - In general, changing any mark changes the mean; this is not usually the case for the mode.
3. Consider an example: shoe sizes for the class; collect data (or use previously collected data) and find the mode and the mean. If the mean turns out to be a natural number, add the teacher's shoe size to the collection; hopefully the answer is no longer a natural number!
   - What can we say about the mean shoe size? Is it the most usual shoe size? Is it a shoe size at all? Where might the mode be more useful ... less useful? [Students can discuss in pairs, say, and record their opinions - there are many acceptable answers.]
4. Repeat for some of the following (again, perhaps, using previously collected data, and utilising calculators where appropriate for the calculations): height of students; number of children in the family; age (to the nearest day); distance travelled to school (to nearest half-kilometre). [For height, age and distance, nonintegral values are "possible values"; moreover, there may be no modal value, or the mode may be no guide to where the data cluster.]
5. For homework, students write up an explanation of uses for the mode and the mean.

CLASSROOM MANAGEMENT IMPLICATIONS:
None

STATISTICS LESSON IDEA 3

TITLE: FROM "ADD UP AND DIVIDE" TO "FORMULA FOR MEAN OF A FREQUENCY DISTRIBUTION"
TOPIC: STATISTICS: MEAN OF A FREQUENCY DISTRIBUTION
AIM: To establish the formula for and a method for calculating the mean of a frequency distribution.

RESOURCES:
No special ones.

METHOD:
1. Consider a frequency distribution (perhaps using class data, but numbers which might be marks from a test are provided here by way of an example - choose easier ones for Foundation students):

Can we find the mean? Guess possible / likely values (could it be 10?).

2. Relate to an already-known method: the numbers are
0 1 3 3 4 5 5 6 6 6 6 6 7 7 7 7 7 8 8 8 8 8 8 9 9 9 9 9 10 10
... and there are 30 numbers. So the mean is
(0 + 1 + 3 + 3 + 4 + 5 + 5 + 6 + 6 + 6 + 6 + 6 + 7 + 7 + 7 + 7 + 7 + 8 + 8 + 8+ 8 + 8 + 8 + 9 + 9 + 9 + 9 + 9 + 10 + 10) ÷ 30, or 199 ÷ 30.

3. This is not a nice sum, and it would be worse if there were 1000 numbers. (Even with a calculator, one might easily make a data-entry error.) Is there a better way?

4. Yes; note it reduces to (indeed, came from):
(0 + 1 + 3×2 + 4 + 5×2 + 6×5 + 7×5 + 8×6 + 9×5 + 10×2) ÷ 30
which is probably easier to handle....

5. ... and in fact we have an easy way of setting out the calculation by using the frequency table (and so ending with vertical rather than horizontal additions) [bright students might jump from step 1 almost directly to here]:

So the mean is the sum of the "fX"s divided by the sum of the "f"s:
SfX/Sf
... work it out, and check the result for plausibility!
Yes, 6.66... is reasonable.

6. Note that the formula can also be read as a sequence of instructions:

• Add the "f"s (to get the total number of scores)
• Multiply each "X" by the corresponding "f"
• Add the "fX"s
• Divide

7. Practise other examples (checking each answer for plausibility).

CLASSROOM MANAGEMENT IMPLICATIONS:
None

NOTE ON DATA HANDLING

The remaining lesson ideas in this section focus in particular on the "data handling" section of the Foundation level syllabus (see syllabus page 31). They suggest ways in which students might move between the following activities:

• looking at graphs - perhaps initally at sketch-graphs, in order to study the general shape and salient features rather than the precise details;
• looking at tables of data;
• listening to or creating stories round the data presented in either form.

The problems which these lesson ideas present are "realistic", in that they are based on contexts with which students can identify. Some are real-life contexts; others involve games which the students enjoy. In all cases they may promote discussion and involve the students in mathematics that in some way is personal to them. This can help in developing a positive attitude to mathematics.

It is suggested in the mathematical education literature that work of this type should precede the formal introduction of algebraic notation. For example, lesson ideas 2 and 3 point to ways in which the work can lead naturally to the introduction of variables and/or the idea of a function. Hopefully, students will then see some good reason (in their own terms) for devising the corresponding terminology and conventions.

The approach to "data handling" described here was included specifically in the Foundation level syllabus because students at this level have such great difficulties with basic algebra. The placing of the relevant section of the syllabus before the sections on algebra and functions is intended to suggest a corresponding teaching sequence. In fact students other than those working at Foundation level might also benefit from such sequencing.

STATISTICS AND DATA HANDLING LESSON IDEA 1

TITLE: SKETCHING THE GRAPH, TELLING THE STORY
TOPIC: INTRODUCTION TO DATA HANDLING; PREPARATION FOR TREND GRAPHS AND FUNCTIONS
AIM: To develop a feel for the way in which the shape of a graph "tells a story".

RESOURCES:
No special ones are necessary, but it may be helpful to have prepared a set of cards containing appropriate graphs or stories (related to the students' interests).

METHOD:

1. The graph below shows attendance at a football match against time. The teacher displays the graph and asks the students to tell a story accounting for the shape of the graph.
2. The second graph also shows attendance at a football match against time. Once more the students are asked to tell a story accounting for the shape of the graph. They can then compare their stories with their neighbours' versions.
3. The students now make up another story about attendance at a match, and see if their neighbour can draw a graph to represent it.
4. Stories and graphs can be shared around the class (or the teacher can use the set of cards suggested above, some showing graphs and some telling stories, so as to provide an element of more controlled reinforcement).

CLASSROOM MANAGEMENT IMPLICATIONS:
It is helpful if the students are used to working in groups, and ideally to negotiating and sorting out problems themselves before appealing to the teacher as "referee".

NOTE:
The students can tell their own stories and so "buy in" to the activity, debating with friends over the merits of their various versions.

STATISTICS AND DATA HANDLING LESSON IDEA 2

TITLE: SLEEP PATTERNS
TOPIC: DATA HANDLING
AIM: To link verbal rules with tables of data and graphs.

RESOURCES:
No special ones.

METHOD:

1. The teacher suggests the following rule: "The number of hours sleep you need per night is given by the rule: Sixteen minus half your age."
2. The students are invited to discuss this; is it a good rule? Do individual students feel that it represents their sleep patterns over the years?
3. The students draw up a table for ages from babyhood to adulthood and corresponding numbers of hours of sleep, using the rule. They consider: how would it work for their parents - or for the teacher(!)? How about their grandparents? (Answers may differ, depending on whether the students think that their grandparents sleep for more hours or less hours than do their parents.)
4. The students are invited to draw up another table which might be better for the older age-groups, and to try and describe the modified rule in words.
5. The students draw or sketch a graph for an extended version of this table.
6. The class can discuss different graphs and rules.

CLASSROOM MANAGEMENT IMPLICATIONS:
Students may need to learn to discuss in mathematics class, and to put forward and argue for their own ideas. Hopefully, once they get used to the idea, they take ownership of their results and feel personally involved.

STATISTICS AND DATA HANDLING LESSON IDEA 3

TITLE: SHOE SIZES
TOPIC: DATA HANDLING; A BASIS FOR ALGEBRA
AIM: To link verbal rules with tables of data and graphs; to introduce the idea of a variable and/or a function.

RESOURCES:
A bus (or other transport) timetable, showing fare stages and prices.

METHOD:

1. The teacher displays data from a page of a (fictional and old) bus timetable supposedly found in some interesting circumstances recently:
   Students are invited to consider the table below. Does the fare go up or go down as the journey gets longer? Does it go up as much for long journeys as for short ones? Which is "better value," a long journey or a short one? How much might you pay for 14 stages?
2. The students consider: do Bus Éireann (or other relevant local) fares follow this pattern? If the "timetable page" considered initially is one for the same company, how long ago might the timetable on the page have been in operation?
3. The students are asked: can you sketch a graph showing price against number of stages (for the given data, or for current data)? How might the graph go for a greater number of fare stages?

CLASSROOM MANAGEMENT IMPLICATIONS:
None

STATISTICS AND DATA HANDLING LESSON IDEA 5

TITLE: SUNSET
TOPIC: DATA HANDLING; A BASIS FOR ALGEBRA AND FUNCTIONS
AIM: To link tables of data with graphs

• It is envisaged that this lesson would follow the one using lesson idea 4. For more able classes, the material might be covered in one period.

RESOURCES:
None necessary, but tables of sunset and sunrise times (perhaps from different parts of the world) would be useful

METHOD:

Consider the sunset times shown in the table. What time of year is it? [The times are those for sunsets in Ireland from September 17th.] How did you know?
What time do you think the sun would set on day 5? ... on day 6?
Why are the time intervals not always the same between successive days? [Answers can relate to approximation as well as to geographical features.]
Can you sketch a graph showing time against day?
How does the shape of this graph compare with the shape of the previous one?
Can you sketch the graph for a one-year period? [For students with appropriate geographical knowledge]
What differences might be found in other countries [especially any that class members have visited or with which they are familiar from television]?

CLASSROOM MANAGEMENT IMPLICATIONS:
None