Brief Description

Pupils' reactions to a number of optical illusions give a rough measure of the power of each illusion using ordinal scales. Three of the illusions are followed up in detail. Optional sections include the magnitude of the deception measured on interval and ratio scales, and the differences between nominal, ordinal, interval and ratio scales.

Design time: 4-5 hours.

Aims and Objectives

On completion of this unit pupils should be able to calculate a mode for discrete ordinal and discrete interval data. They will have practised calculating the median and the mean of a frequency distribution on discretc data, using the mode, median and mean as representative measures: collecting and recording data in tables and bar charts and interpreting this data.

Those who complete the optional sections are introduced to a simple use of change of origin; the differences between nominal, ordinal, interval and ratio scales; and which of the mean, median and mode is appropriate for each scale.

Prerequisites

Pupils will need to be able to read a scale to the nearest 2 millimetres, carry out simple multiplication of decimals, divide by the number of pupils in thc class, correct to one place of decimals and (for those who do C6) add and multiply positive and negative numbers to one decimal place. Those who do Section D will need to be able to read flow charts. It will help if pupil~ have previously drawn simple bar charts, but they can refer to examples ia the text for help. The method of finding the median as the value assigned tc the middle item in a small ordered set of items should be known.

Equipment and Planning

The illusions are on page Rl and questions about them on page R2. The questions can be answered on page R2 or on a separate piece of paper. Class results can be entered in the tables on page R3.

Illusion 7 is not given on the page of illusions, since it depends on surprise and little study time. You will need to prepare it yourself either by drawing it on a large piece of paper, on the back of a roller blackboard (make sure it does not come round upside down) or on an overhead transparency. It is shown here. Take care to space the words out as shown.

Blank tables are given on sheet R3 for use with Sections B1 and C. Squared paper or graph paper will be needed for the bar charts. Calculators may be found useful in Sections C4 and C6.

Section C needs some simple apparatus made out of card, as follows. Allow yourself one hour to make this apparatus.

Experiment 1

The two part lines A and B must be parallel. Stick parts 1 and 2 on the base. The arrow on the slide, marked 'Read here', lines up with the scale on the 2mm square graph paper. This scale should be marked and numbered in centimetres. Zero on the scale should be opposite the arrow on the slide when the slide has not been moved. The part lines A and B should be at about 30^o with the vertical.

Experiment 2

Cut out the T-shaped hole. Stick the squared paper scale into the position shown. Stick the top layer on the base. The arrow on the slide should point to zero when C is level with D.

Experiment 3

Glue the spacers underneath the top, leaving room for the slide to fit between them. Mark the scale on the slide faintly in centimetres, subdivided into 2 mm intervals. Make sure the line and the arrow markings are bold. The model can be made more sturdy by including another 20 x 15 cm piece of card as a base.

Underlying the whole unit are two themes. One relates to the four types of scale - nominal, ordinal, interval and ratio - and the differences between them. These are brought out explicitly in optional Section D, but are implicit in the core work in which pupils work with both ordinal and interval scales. The second theme is increased sophistication in measurement, illustrated by the power of optical illusions. The first measure is based on the number of pupils with the right answer. The second approach measures the amount of error in the class answers.

Section A introduces the idea of optical illusions. Section B tests seven illusions and does a simple analysis using modes and medians on ordinal scales. Section C looks quantitatively at three of the illusions, using an interval scale as a more refined measure. Section D makes more explicit the differences between the four different types of scale. Optional Sections B7 and C4b are to reinforce techniques of calculation. They can be omitted by brighter pupils. Other optional sections (C3, C6 and D) are more difficult and need only be done by brighter pupils.

Detailed Notes

Section A

This section sets the scene for the later optical illusions. There is no 'correct' answer to the first two drawings. The wrong lines are joined in the two-pronged fork, giving the impression of three rods. It may be useful to extend the discussion slightly to discuss accuracy of scientific observation with problems of colour, measurement and parallax.

Section B

B1
It is advisable to outline the procedure to the pupils before showing them the two pages of illusions and questions. It is very important to stress that the aim is to test not the pupils, but the illusions. They should write down what appears to be the correct answer at first sight. Do all that you can to discourage cheating. An element of surprise is needed in illusion 7, which is described under 'Equipment and Planning'.

The correct answers are 1d, 2c, 3a, 4b, 5b, 6b, and 7 'A bird in the the hand (note the repeated 'the').

Predrawn tables on the blackboard are a convenient way to collect the class results quickly. With an even number of pupils in the class, add in your own results to make an odd number. This will help in Section B6.

B2
Pupils can work on question a while the class results of B1b are being collected. Here a simple measure of a successful illusion is used. The most successful illusion is the one that fools most people. We can thus use the class results to measure, for this particular class, which is the most successful illusion. It is the one with the fewest correct answers. This rough-and-ready measure of an illusion's success can be compared and contrasted with the more refined methods in Section C later.

B3
The mode is the particular answer given by more pupils than any other answer. This is not necessarily the correct answer.

B4
An ordinal scale is one which has an order inherent in the nature of the variable. In the illusions there are variables based on positions on a line (illusion 1), lengths of lines (illusion 6) and on size (illusions 2 to5). These all give ordinal scales. It would not be appropriate to use alphabetical order in illusion 6 and call this an ordinal scale, since the assigning of letters to the lines is essentially arbitrary and the question asked one to consider the lengths of the lines. With an ordinal scale more information is retained when keeping this order in tabulation and in drawing bar charts (B5). It would generally be considered incorrect not to put the variable on the horizontal axis in one of the two proper orders.

B6
Calculating the median involves putting the pupils in order. So medians can only be defined when the data are measured on at least an ordinal scale. The formula 'look at the value of the ¹/₂(n + 1)th pupil' is hinted at in the pupil notes. You can make this more explicit if you wish. The only problem arises when n is even and the two middle pupils disagree. For example, if n = 34, and the 17th pupil says 'line GH' and the 18th pupil says 'line CD', then there is no median. It is to avoid this problem that we suggest you make sure there is an odd number of answers to the illusions (by incorporating your own answers if necessary). The more general problem is then treated in C2. In cases where the ordinal scale is numerical, then conventionally the median is defined as the mean of the answers of the two middle pupils. But this can be misleading, and the convention is more useful with interval and ratio scales.

Reinforcement work on the median may well be needed here (and perhaps also for C2).

*B7
Further imaginary results for another class are included here for reinforcement, if thought necessary.

Two useful textbook references are:

1. Modern Mathematics for Schools by Scottish Mathematics Group, Book 4, pages 214-220, (Blackie/Chambers, 1973)
2. School Mathematics Project Book C, pages 149-161 (Cambridge University Press, 1970)

With brighter pupils you might like to take the opportunity of pointing out the difference between NOMINAL and ORDINAL scales. This is followed up in Section D. A nominal scale has no underlying order. For example, in 'Favourite sports' there is no reason to prefer the order (soccer, cricket, tennis) to (tennis, soccer cricket). In an ordinal scale there is an underlying order. In illusion 6 the lines EF, AB, GH and CD had lengths 5.5, 6.0, 6.2 and 6.5 cm, respectively. They were ordered by length, and the question was about length, so we have an ordinal scale. But the difference between EF and AB (0.5 cm) is not of the same magnitude as the difference between AB and GH (0.2 cm). An ordinal scale just shows the order, it does not have equal differences between successive values of the variables.

Section C

C1
The three experiments are refined versions of the illusions numbered 1 to 3. By using an interval scale measured in millimetres we can assess the degree of bias imparted by each illusion.. You will need to do some initial experimenting with your own models to find the appropriate figures to insert in the first columns of the tables on page R2. The experiments can be done while the class is completing Section B. One way of collecting the class results is to have the children enter their results in blank tables on the blackboard. These can then be copied by the pupils on to the R2 pages for future use.

Each pupil should do each experiment as it is passed round the class.

The problem with class intervals is avoided in the text, and can be similarly avoided in class. Otherwise, any problem can be overcome by (i) reducing the size of the model, or (ii) marking the scale:

or whatever is appropriate

in effect doing the class interval work for the pupils.

C2
The experimental results are measured on an interval scale (see notes under C5). An interval scale is ordered, so we can calculate medians as well as modes. As it stands, the first direct reference to the correct answers is in C5. You may wish to give them at this stage; they will be needed eventually. The method of measuring (to the nearest 2 mm) has effectively changed a continuous variable to a discrete variable, so the calculation of the median follows the same pattern as in Section B. If you think pupils do not need the practice, each pupil need do question c for only one of the experiments. You can arrange so that each experiment is analysed by a third of the pupils and the answers given to the other pupils for later use.

*C3
This optional section provides histograms of the results, although the term is not used in the pupil notes.

C4
With an interval scale we can find a mean as well as a median. The method described here leads to the later use of the formula:

Multiplication is shown as a quicker way of carrying out the required additions. Less able pupils may well require help here. Question c shows why it is not possible to get a mean for ordinal data. Clearly (1 x 'line P') + (9 x 'line Q'), etc., has no meaning. A common, but wrong, method is to give the numbers l, 2, 3 ... to ordinal data and use these to calculate a mean.

Reinforcement work may be needed here. For examples see School Mathernatics Project, Book F, pages 72-74, and most recent mathematics textbooks.

C5
This section attempts to compare the power of each illusion by means of the magnitude of the bias or error induced in the class results. The median and mean are taken as being 'representative' of the class. Each pupil needs to know the correct answers and class means and medians for the three experiments.

In Sections C1 to C5 all the measurements are on an interval scale. Section C6 moves on to a ratio scale. This is followed up in Section D but can be anticipated here if it helps brighter pupils. The answers 11.6 cm, 11.8 cm, etc., were all in centimetres. An interval of one centimetre from 11 to 12 is of exactly the same length as that from 12 to 13. An ordered scale with equal intervals is called an INTERVAL SCALE. But a reading of 0 cm does not mean that there is no error. A reading of 12 cm is closer than a reading of 2 cm, but it is not six times as good. In our experiment 12 cm is below the correct answer by 0.4 cm; 2 cm is below the correct answer by 10.4 cm. So in this case 2 cm is 26 times as bad as 12 cm. In an interval scale the zero does not mean 'nothing', and we cannot compare results by multiplication.

However, in this case and in many others, we can get these extra properties by a simple change of origin. This is done in Section C6 by measuring errors from the correct position. In this case 0 does mean no error, and an error of 2 cm is twice the error of 1 cm. We now have a RATIO SCALE. From data measured on interval and ratio scales we can calculate means as well as medians and modes. All this information is summarized in the flow chart in Section D.

For pupils who are not proceeding to Section D it is useful to summarize the main points made about calculating modes, medians and means with examples. The mode can always be calculated (although it is not always unique). The median can be calculated only when the scale is ordered, and hence not for nominal scales. The mean can only be calculated when there is at least an interval scale.

*C6
This optional section is especially for those who are to do Section D. It could also be done by older pupils too, if so desired.

The allocation of the signs + and to errors above and below the correct answer is arbritary. It is appropriate to attach signs to the errors here since a mean o f zero would mean that the illusion had no overall effect on the class. This would not show up if all the errors were treated as positive. Pupils may need help in working out the . It is probably easiest to add all the negative part totals, separately add all the positive part totals, and then combine these two totals to get the final answer. A convenient method is to use two columns.

*Section D

This optional section makes more explicit the differences between nominal, ordinal, interval and ratio scales. The notes under Sections B4, B6 and C5 are relevant here. The use of nominal, ordinal, interval and ratio scales cuts across the more usual discrete / continuous dichotomy. The following table gives examples:

Answers

A	a	Either two black faces or a vase.
	b	Most people see 9, some see 10.
	c	Most people see 10; those who saw 10 in b should now see 9.
	d	There appear to be 3 at the left, but only 2 at the right.
B2		The correct answers are:
		1 d, 2 c, 3 a, 4 b, 5 b, 6 b and 7 'A bird in the the hand'.
B4	b	P,Q,R,S (or S,R,Q,P)
	c	Illusions 2 and 3: AB is shorter than CD; AB and CD are the same length; AB is longer than CD (or the reverse order) Illusions 4 and 5: Answer a, answer b, answer c (or the reverse order)
B5	a	Figure 2
	b	Figure 2
	c	It makes it easier to answer questions like a and b. It shows the distribution more clearly.
B7	a	Line CD
	b	Class 2Z
C2	a	11.8 cm
C4	a	See detailed notes.
D	a	'Favourite games' is on a nominal scale.
	b	Could be correct.
	c	'Bed times' is on an interval scale.
	d	Median number of goals must be a whole number (or possibly ending in 0.5 if an even number of matches).
	e	Could be correct.
	f	Could be correct.
	g	Four is the middle pupil, not the median.
	h	Nominal
	j	Interval
	k	Nominal
	i	Ratio
	m	Ratio
	n	Ratio
	p	Interval
	q	Interval
	r	Nominal
	s	Ratio
	t	Ratio
	u	Interval

Test Questions

1. 25 children were playing in a field with a cricket ball to see who could throw it the greatest distance. The field was marked in coloured zones. Red was nearest, blue furthest away. Their results are shown in Table l.
   
   
   

   Zone	Number of children
   Blue	1
   Qrange	8
   Red	5
   White	2
   Yellow	9

   Table 1 - Class results

   1. How many balls landed in the zone nearest the start line?
   2. The order of zones in Table 1 is not the best to use. Why?
   3. Rewrite Table 1 with the zones in a better order.
   4. Find the median zone.
2. Table 2 shows the results obtained by a class of 29 pupils, who carried out the experiment to make the bars of the T the same length. The correct answer is 10.0 cm.
   
   

   Length in cm	Number of pupils
   9.2	4
   9.4	8
   9.6	7
   9.8	5
   10.0	3
   10.2	2
   Total	29

   Table 2
   

   1. How many pupils had the right answer?
   2. The mode and median can be used to represent the class answers. What else can be used?
   3. What is the mode of their answers?
   4. Copy and complete Table 3.
      
      

      Length in cm	Number of pupils	Pupils numbered
      9.2	4	1 to 4
      9.4	8	5 to __
      9.6	7	__ to __
      9.8	5	__ to __
      10.0	3	__ to __
      10.2	2	__ to 29

      Table 3
      

   5. What is the number of the 'middle' pupil? Find the median length of their answers.
   6. Use the median to measure the class error. (Correct length 10.0 cm)
      
      

      Length in cm	Number of pupils	Part totals
      9.2	4	36.8
      9.4	8
      9.6	7	67.2
      9.8	5	49.0
      10.0	3
      10.2	2
      Total	26

      Table 4
      

   7. Copy and complete Table 4.
   8. Find the mean length of the answers.

   Questions 3 and 4 are for those who completed Section D.
   

3. *What is wrong with each of these statements?
   1. Our class wrote down their favourite pets. The median was the cat.
   2. Our class lined up in order of height. The mode height was Jim Brown.
   3. I arrived home from school at 4 o'clock. Jane arrived home at 8 o'clock. We should have arrived at 3.30 pm. She was twice as late as I was.
   4. The mean subject in our class is English.
4. *For each of the following, do you have a nominal, ordinal, interval or ratio scale? (Use the flow chart to help you.)
   1. High jump heights
   2. Favourite football teams
   3. Weights of pupils in your class
   4. Time you arrive at school each morning

Answers

1	a	5
	b	Not in order of distance from the 'throw' line
	c	Table rewritten with order R, 0, Y, W, B or B, W, Y, 0, R
	d	Orange (zone of the 13th pupil)
2	a	3
	b	The mean
	c	9.4cm
	d	Complete third column: 1 to 4, 5 to 12, 13 to 19, 20 to 24, 25 to 27, 28 to 29
	e	Pupil number 15. Median length is 9.6 cm
	f	(Median) class error is 10.0 9.6 = 0.4 cm.
	g	Third column should read 36.8, 75.2, 67.2, 49.0, 30.0, 20.4. Total is 278.6.
	h	Mean length is 278.6/29cm = 9.61 cm (9.6 to 1 decimal place)
3*	a	'Pets' is nominal scale. No median with nominal scale.
	b	Mode could be Jim's height not Jim.
	c	'Time' is interval scale. Twice as late implies ratio scale. Actually nine times as late when measured from 3.30 on a ratio scale.
	d	Can't have mean with nominal scale.
4*	a	Ratio
	b	Nominal
	c	Ratio
	d	Interval

Connections with Other Published Units from the Project

Other Units at the Same Level (Level 2)

Authors Anonymous
On the Ball
Opinion Matters
Getting it Right
Fair Play

Units at Other Levels in the Same or Allied Areas of the Currlculum

Level 1

Practice makes Perfect
If at first...
Shaking a Six

Level 3

Net Catch
Cutting it Fine
Multiplying People
Phoney Figures

Level 4

Smoking and Health

This unit is particularly relevant to: Science, Mathematics.

Interconnections between Concepts and Techniques Used In these Units

These are detailed in the following table. The code number in the left-hand column refers to the items spelled out in more detail in Chapter 5 of Teaching Statistics 11-16.

An item mentioned under Statistical Prerequisites needs to be covered before this unit is taught. Units which introduce this idea or technique are listed alongside.

An item mentioned under Idea or Technique Used is not specifically introduced or necessarily pointed out as such in the unit. There may be one or more specific examples of a more general concept. No previous experience is necessary with these items before teaching the unit, but more practice can be obtained before or afterwatds by using the other units listed in the two columns alongside.

An item mentioned under Idea or Technique Introduced occurs specifically in the unit and, if a technique, there will be specific detailed instruction for carrying it out. Further practice and reinforcement can be carried out by using the other units listed alongside.

Page R1

Page R2

Illusion Answers

Tick the answer you think is correct. Do not use a ruler to help you;

1. The line X goes behind the black rectangle and comes out the other side. Which line is it?
   1. line P
   2. line Q
   3. line R
   4. line S
2. Which is true?
   1. AB is shorter than CD.
   2. AB and CD are the same length.
   3. AB is longer than CD.
3. Ignore the arrowheads. Which is true?
   1. AB is shorter than CD.
   2. AB and CD are the same length.
   3. AB is longer than CD.
4. Look at the two small squares. Which is true?
   1. The black square is smaller than the white square.
   2. The two squares are the same size.
   3. The black square is larger than the white square.
5. Look at the two centre circles. Which is true?
   1. The circle at the left is smaller than the circle on the right.
   2. The two circles are the same size.
   3. The circle on the left is larger than the circle on the right.
6. Which line is longest?
   1. AB
   2. CD
   3. EF
   4. GH
7. The words in the triangle were:

Page R3

Section B1 : Class Results

	Number of pupils
a
b
c
d

Table 1 and 6

Table 2, 3, 4, and 5

Table 7

Section C: Individual Results

Experiment 1 ____ cm

Expreiment 2 ____ cm

Experiment 3 ____ cm

Class Results

Reading	Number of pupils	Part totals
Total

Experiment 1, 2, and 3

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