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Preface Welcome to the joys of Investigating Mathematics through Patterns. Hewlett-Packard Calculators is pleased to introduce you to “Investigating Mathematics through Patterns”, a discovery and investigation based learning resource that will help your students build new mathematical knowledge through the guided exploration of patterns in problems which are solidly based on the real world. We have taken great care to ensure this resource will make learning realistic and fun for the students. This material again demonstrates Hewlett-Packard’s dedication to the calculator and its educational base. The HP 30S and “Investigating Mathematics through Patterns” simply take this dedication to a new level. We know this is just the beginning of the support we hope to provide to educators and students and hope this will grow to the levels our other product categories have achieved. To find out more about this and the many other products and services we have to offer please visit our web site at www.hp.com/calculators. Again thank you for your support and use of this resource in your classroom. If you have any questions or comments on how to improve or change this or any of our products or materials, please contact us at the address above. We look forward to hearing from you. Hewlett-Packard Calculators
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How to Use This Resource The activities in this book have been carefully designed to follow an investigative approach to the teaching of mathematics. Through introductions which relate the activity to the student’s world, they may help to answer the question “When are we ever gonna use this?”. By following this approach and by encouraging the students to develop their own conjectures, compare their results, and communicate their conclusions, these activities address a multiplicity of the objectives of the Australian curriculum. The Activities Teachers may choose to use an activity as an adjunct to concepts being explored, as an enrichment activity, or as a vehicle to introduce a new topic in a way that is meaningful to the students. Teacher Notes The Teacher Notes at the beginning of each activity provide the following resources:
• a summary of the activity, including an outline of the material covered; • the expected outcomes; • any required skills and knowledge; • any required materials; • any natural follow-up teaching points;
Student Worksheets Each activity comprises a worksheet to be photocopied for the student. In almost all cases this worksheet includes space for the student to record their answers.
iii
Table of Contents Discovery Activities As any educator knows, the best understanding of a concept is achieved by a student who is encouraged to discover the principles and concepts for themselves. The emphasis in all of these activities is the guided investigation of patterns in various mathematical topics, grounded in concrete, real-world situations. The student is encouraged to share results, and to communicate conclusions verbally and in writing. Activity 1: Planning the Paths
Through a realistic introduction, this guided lesson explores the number of paths through a network, carefully guiding the student to discover a pattern based on Pascal’s Triangle. Suitable for: Year 10 upward, any upper school courses involving
networks Required knowledge: Addition of positive integers. Familiarity with Pascal’s
Triangle is helpful but not essential. Activity 2: Infinite Series
The mowing of lawns is used as a geometric base through which the student is introduced to the concepts of an infinite series, the algebraic calculation of a sum to infinity and the conversion of recurring decimals to their fractional equivalents. Suitable for: Year 10 (mid to upper ability), any upper school courses
involving sequences and series. Required knowledge: Addition of fractions, simple percentages, linear algebra,
familiarity with the concept and notation of recurring decimals.
Activity 3: Lines and Regions
A fascinating investigation into the patterns formed by intersecting lines. Through arranging lines in patterns that maximize the number of intersections, the student is encouraged to discover rules involving quadratic equations and triangular numbers.
Suitable for: Year 9 & upward, courses for mid ability upper school.
Required knowledge: Basic properties of parallel lines, deduction of quadratic rules from sequences of numbers, use of mathematical notation for formulae.
Activity 4: Games for Charity
A realistic scenario is used to introduce the student to the basic concepts of expected outcomes and their application to the expected profit or loss for simple games of chance. The student is encouraged to apply this knowledge to the design of their own game. Suitable for: Year 9 & upward, any course involving probability and
beginning principles of expectations. Required knowledge: Simple probability (cards, area and random draw with
replacement), profit and loss.
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Activity 5: Transforming Data Through a realistic situation, the student is guide to investigate the effects of change of scale and origin on the mean and standard deviation of univariate statistical data. Suitable for: Year 10 & upward, any upper school course involving
change of scale and origin for univariate data. Required knowledge: Use of a calculator to find the mean and standard
deviation of ungrouped univariate data. Activity 6: How Large is “Large”?
The student is encouraged to investigate the change in volume of an open top box, made by cutting squares out of the corners of a piece of paper and folding up the sides. By investigating the change in volume as the size of the square cut-out increases, the student discovers the concept of optimization. Suitable for: Year 10 & upward, any pre-calculus or early calculus
course. Required knowledge: Volume of a cuboid, algebra (derivation and use of
simple formulae), trial and adjustment solution of equations.
Activity 7: Moths to a Candle
A simple model for the flight of a moth around a candle involving similar triangles leads the student to the investigation of limits to infinity and tangential orbits. Suitable for: Any upper school course involving advanced
trigonometry. A good introduction to, or reinforcement of the concept of, limits.
Required knowledge: Similar triangles, Sine Rule, some understanding of the basic concept of a limit as x →∞ of k
x .
1 1 1 1 1A
Z
1
1
1
2
3
4
3 4
10
106
20
5
15
35
21
56
6
A
Z
1
1
1
2
3
2 2 2 2
5 7 9 11
5 14 25
(no value above = 0)
(no valueto left = 0)
Teacher Notes Activity Summary Suitable for: Year 10 upward, any upper school courses involving networks Planning the Paths provides a carefully guided introduction to the skill of finding the number of paths through a rectangular network. The resulting algorithm is very simple and easily deduced by students. As can be seen right, the numbers resulting from the algorithm developed in the activity are directly related to Pascal’s Triangle. The algorithm used is to add together the values immediately above and left of the vertex being worked upon. This is easily extended to networks having missing links or joins. A requirement not to pass through a vertex is handled by forcing its value to be zero. Note: The final page can be omitted for low ability students. Expected outcomes The student will deduce the algorithm for finding the number of paths through a regular rectangular network, assuming no backtracking is allowed. Required skills and knowledge Some familiarity with the language of networks might be an advantage, although the teacher may prefer to use this worksheet as an introduction to the general topic based on a concrete example. The most common error made by students in this activity is to confuse the length of a path with the number of paths. This should be watched for and corrected by the teacher. Required materials None. Follow-up teaching points The activity ends with an invitation to the students to try to use the nCr button to find a rule giving the same result. For regular rectangular networks this is simple, as it is also for networks linked at a corner (see above), for which answer is 2 2
24 2
2+ +×b g b gC C or 2 2
24 2
4+ +×b g b gC C .
The teacher may also wish to follow this investigation with other network problems such as the Shortest Path through a network, many of which have quite simple algorithms that can be easily understood by students of all levels. These problems are regarded well by students because of their clear basis in the real world.
A
Z
1 - 1
A good start to solving any problem is to simplify it.
Planning the Paths Janet is a television reporter and wants to cover the route taken by the Australian Cricket team during a victory parade. Her problem is that the exact route will only be announced on the day of the parade, although she knows that they will be starting at A and finishing at B on the map below right. Janet would like to check out the route in advance to find the best positions for her camera crews. “Hey!” said Jim, her assistant, “How hard can it be? We have a map of the streets they’ll be going through. There must be only a few paths they could take.” “I don’t know,” said Janet, “I suspect that you might find that there are more paths than you expect. Why don’t you spend a bit of time working it out and let me know?” Jim wondered if there was a simple pattern or rule which he could use to find out the number of paths.
Your task is to find out for him.
Through this sheet you will be investigating the number
of paths that can be taken through a network.
We will begin with a very simple network and try to find a pattern for that. From there we will gradually work up to more complex networks until we can solve Jim’s problem. In order to make your task easier we will restrict the directions in which you are allowed to move and make all the networks right angled grids. In the network shown right (and all those that follow) you are only allowed to move in the direction of the arrows. Thus it is all right to move from A to B but not backwards from B to A. The other thing that you need to know before you start is exactly what is meant by a “path”. For example, if we were to travel from A to G by following the path A – B – D – E – F – G then that would only be one path. No matter how long a path is, it still only counts as one path. In this investigation we will find a way to work out how many paths there are from (for example) A to G without having to list them all and count them.
A
GF
EDC
B
OK
to m
ove
this
way
OK to move this way
1 - 2
Careful! Don’t confuse the number of blocks with the number of paths.
1. Use the diagram on the right to answer the questions that follow. How many paths are there from: (a) A to B? ________ (b) A to C? ________ (c) A to D? ________ 2. Use the diagram below right to find numbers of paths for the following questions. As
you work out each one, write it next to the letter in the diagram. You may find that this helps to find a way of working out subsequent letters.
(a) A to B? ________ (b) A to C? ________ (c) A to E? ________ (d) A to I? ________ (e) A to F? ________ (f) A to G? ________ 3. In working out how many paths there were from A to G, did you need to take into
account the number of paths from A to K? Why? 4. Now that you know how many paths there are from A to I and from A to F, explain
how this can be used to find the number of paths from A to J.
A
D
B
C
A
GFE
DCB
LKJI
H
1 - 3
5. Continue this process to find the number of paths from A to each of the remaining 4 vertices (D, H, K and L).
D: ___________ H: ___________ K: ___________ L: ___________
Check: Don’t proceed past this point if you are on the wrong track. The answer you should have obtained for the number of paths from A to K is 6 and the number of paths from A to L is 10.
If you did not get those two answers then check with your teacher to find out where you have gone wrong.
6. Explain in words the rule which you have found. Make sure you include a diagram to
illustrate your explanation. Show your explanation to another student who has reached this point. Discuss any differences and amend your explanation or diagram accordingly.
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7. On this page are a number of networks of increasing difficulty. For each one the usual rule applies as to which directions you may move. In each case you are to find the number of paths from A to Z unless there are other instructions.
(g) Find the number of paths through the network from
question (f) which do pass through the point M. The diagram is shown again on the right.
A
Z
(a)
A
Z
(b)
A
Z
(e) (f)
A
Z
M
Find the number of paths from A to Zif you cannot pass through M·
A
Z
(c)A
Z
(d)
A
Z
M
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(h) In the network right, find the number of paths which pass through either M or N but not both.
You may need to redraw the grid as part of the
working. Ask for graph paper if you need it. 8. It’s now time to go back and solve Jim’s original problem
for him. The road map is shown right. (a) On graph paper, draw a network which is equivalent to
the set of roads leading from A to B. Remember that it is not necessary to include any roads falling outside the rectangle with corners at A and B.
(b) Calculate the number of paths from A to B using the rule which you worked out earlier.
9. When Jim reported his results to Janet she was a bit
discouraged but told him that she had some additional information which might narrow the number of possibilities.
She had surveyed the area and had found that there were
road works at point X which would not allow the official parade through and, more importantly, that there was a particularly good “photo spot” at point Y at which she was sure the team would want to stop to sign autographs.
How many paths does this reduce the total to?
A
Z
N
M
1 - 6
10. Although Janet was pleased with Jim’s work, she was sure that there had to be an easier way to calculate the results.
She pointed out that the numbers he was coming up with were clearly related to Pascal’s Triangle (shown below). In Pascal’s Triangle, the value in each cell is obtained by adding together the two cells above it.
For example… 3 3
6
Pascal’s Triangle 1 Row 0 1 1 Row 1 1 2 1 Row 2 1 3 3 1 Row 3 1 4 6 4 1 Row 4 1 5 10 10 5 1 Row 5 1 6 15 20 15 6 1 Row 6 1 7 21 35 35 21 7 1 Row 7 Position # 0 1 2 3 4 5 6 7
Your calculator can find these numbers for you. For example if you type in [ 7 ] [ nCr ] [ 3 ] your calculator will give you the answer 35. Janet suggested to Jim that he should check to see if the nCr function could be used in finding the number of paths through a network. Use the networks below to find a way to apply the nCr function to Jim’s problem. It will help to calculate the next few rows in Pascal’s triangle. Explain what you have found:
Position # 3 in row 7 is 35
A
Z
(a) (b)
A
Z
Teacher Notes Activity Summary Suitable for: Year 10 (mid to upper ability), any upper school courses In Infinite Series, Jacob has been told to mow the lawn but chooses to start by mowing ½ the lawn, then ½ of what remains, then…. By examining a physical model of mowing halves of a lawn the student is guided to the idea of a finite sum for an infinite series. The reasoning shown in the panel on the right is used to by Jacob to prove to his mother that he will in fact mow the whole lawn, and this is extended to infinite series using other ratios. This concept is then extended to recurring decimals to show that a number like 0.121212… is equivalent to12
99 . Using similar methods the student will be led to prove that 0 9 1. = . Expected outcomes The student will be able to find the sum to infinity of a GP by algebraic means, and will be able to convert any recurring decimal to its fractional equivalent. Required skills and knowledge The student will need sufficient algebraic skills to understand and apply the method of proof used in the box above right. In addition, they will need to be familiar with place value and its application to decimals such as 0.22222…, and with the notation of recurring decimals. Required materials None. Follow-up teaching points; The teacher may choose to extend the algebraic method used in order to derive the formulae for the sum to infinity and the sum to n terms of a GP.
Let = the total area mowed.
Since 1 = 100%, I will mow the whole lawn!
x
So x
x
x
x xx
.. .....
.....
.....
= + + + +
= × + + + +FHG
IKJ
= + + + +
= +=∴
12
14
18
116
2 2 12
14
18
116
2 1 12
14
18
2 11
2 - 1
Infinite Series Poor Jacob! His mother finally cornered him, switched off the television, and insisted that he mow the lawn “Now!”. Reluctantly, Jacob went out to the shed and took out the lawn-mower.
After half an hour Jacob’s mother checked on his progress. She found Jacob stretched out under a tree reading a book. “What’s going on?”, she exclaimed. “You’ve only mowed half the lawn!”
“Ah!”, said Jacob, “That’s true. But I was just about to read a book when I began. So when I’d mowed half the lawn, I stopped and read chapter 1 of my book. Now I’ll mow half of what’s left to be done, and read chapter 2 as a reward. Then I’ll mow half of what was left to do after chapter 2 and read chapter 3. And so on.” 1. The diagram right is shaded to show the amount that Jacob will
have mowed after he finishes reading chapter 2. Shade in the amount that he will have mowed after reading chapter 3.
2. Complete the table below showing Jacob’s progress.
Chapter
read
Fraction of lawn mowed after that
chapter
Total fraction
of lawn mowed
1
12
12
2
14
12
14
34
+ =
3
12
14
+ + =
4
5
3. What pattern is visible in the denominators of the fractions 12
, 14
etc?
Mownarea
2 - 2
4. Write algebraic rules which will show the fraction mowed after each chapter (m), and the total mowed (t), from the number of the chapter read (c).
m = ? t = ? 5. If the book has 12 chapters, what fraction and what percentage of the lawn will be mowed
when Jacob has finished his book? 6. If the lawn is 30 metres by 30 metres, what area will Jacob mow immediately before
reading the 12th chapter? Give your answer in square metres and in square centimetres.
Jacob’s mother was not very happy with his method. She pointed out that he would never finish the job. Jacob, being a good math student disagreed and proceeded to prove that he would finish. Jacob’s demonstration went as follows…
Let = the total area mowed.
Since 1 = 100%, I will mow the whole lawn!
x
So x
x
x
x xx
.. .....
.....
.....
= + + + +
= × + + + +FHG
IKJ
= + + + +
= +=∴
12
14
18
116
2 2 12
14
18
116
2 1 12
14
18
2 11
7. Jacob’s proof above assumes that the series of additions goes on forever. Is this true in
the real world or only in his math proof? Is it still valid in the sense of being useful?
Hint: When looking for a rule for t, compare the numerator and denominator.
2 - 3
8. Suppose Jacob had chosen to mow 13
of the area remaining instead of 12
. Use a similar
method to Jacob’s to show that even if he continued for infinite time he would only complete half the lawn.
9. Find the values of each of the following series when summed to infinity.
(a) 14
116
164
1256
+ + + +.....
(b) 15
125
1125
1625
+ + + +.....
(c) 110
1100
11000
110 000
+ + + +,
.....
(d) 310
3100
31000
310 000
+ + + +,
.....
10. We can also apply this method to recurring decimals as follows:
Let (ie. 0 2 recurring or 0 2)
then (by place values)
x
x
= ⋅ ⋅ ⋅
= + + +
0 2222222
102
1002
1000
....
....
(a) Show that 0 2 0 2222 29
⋅ ⋅ =( or ....)
(b) Find the value of 0 7 0 7777⋅ ⋅( or ....)
Hint: Multiply by 3 instead of 2
2 - 4
11. Complete the table below:
What is the final line of the table saying about the value of 0 99999⋅ ....? How do you feel about this result?
12. A similar method to this can be used for any recurring decimal.
Let (ie. 0 27 recurring or 0 27)
then (by place values)
So 100
x
x
x
x
x xx
x
= ⋅ ⋅ ⋅
= + + + +
= × + + + +FHG
IKJ
= + + + +
= +=
= =
0 2727272
107
1002
10007
10 000
100 210
7100
21000
710 000
100 20 7 210
7100
100 2799 27
2799
311
....
,....
,....
....
Use this method to find the fractional equivalents to: (a) 0 12121212⋅ .... (b) 0 18181818⋅ .... (c) 0 25252525⋅ .... (d) 3 09090909⋅ .... (e) 0 123123123123⋅ ....
0 11111⋅ ....
0 2⋅ 29
0 3⋅ 39
13
=
0 4⋅
0 5⋅
0 6⋅
0 7⋅
0 8⋅
0 99999⋅ ...
Hint: Find the value of 0.090909 and add 3.
Hint: Multiply by 1000.
Teacher Notes Activity Summary Suitable for: Year 9 & upward, courses for mid ability upper school. Lines and Regions is a fascinating investigation into the patterns formed by intersecting lines. Based upon a geometrical design to be used as a company logo, the student is guided through the process of arranging lines in patterns that maximize the number of intersections. Patterns which appear in the number of intersections, the number of interior regions and the number of exterior regions are studied with a view to deducing both iterative and non iterative quadratic rules based upon the set of triangular numbers. Expected outcomes Through consideration of the process of arranging lines in patterns which maximize the number of intersections, the student will appreciate properties of lines, parallel lines and the division of the plane into bounded and unbounded regions. Examination of data derived from these patterns will lead to conjectures which are discussed and compared, and which lead to the deduction of algebraic rules in both iterative and non iterative forms. Required skills and knowledge The student should be familiar with the concept that lines extent to infinity, and the basic properties of parallel lines. A familiarity with quadratic equations is advantageous but not essential, as the pattern in the triangular numbers is relatively simple. Some familiarity with iterative rules may also be an advantage but this activity would equally serve as an excellent introduction to this topic of mathematics. Required materials None. Follow-up teaching points If the student has not encountered triangular numbers then further work with similar patterns will be rewarding. The need to find the number of lines which will produce at least a given number of intersections could also be used as a vehicle to introduce the idea of the need for truncation rather than rounding in solving situations such as these. Another topic which is richly rewarding is the study of iterative equations in general, particularly the transformation of iterative equations into non iterative equivalents.
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N E
W S
T O D A Y
TO
AY
Lines extend foreverin both directions
I2
I1
I3
E1
E2
E3
E4
Intersections
Lines and Regions
Eduardo has developed the design shown on the left as a possible logo for his new web based news service. In the process of developing this logo Eduardo used a multiple reflection of a single smaller design using only four lines:
Eduardo enjoys investigating mathematical patterns and noticed that there were patterns developing in his designs, particularly as he added more and more lines to them.
Diagram A Diagram B Diagram C
In diagram C, the three interior regions (I1, I2 & I3) have been marked, as well as some of the external regions (E1, E2…). Note: It is assumed in each diagram that the lines continue on forever, thereby dividing the
plane (the page) into separate regions. 1. In each of Eduardo’s diagrams above, the lines have been placed so as
to obtain the maximum number of intersections. This need not be the case, of course. For example, the diagram right shows three lines and only one intersection. In the space below, draw three lines with
(i) no intersections. (ii) two intersections.
Remember - lines extend forever. This shows 2 intersections but there are really 3!
3 - 2
2. Challenge: You have seen in diagram C that the maximum number of intersections for four lines is ______. In the space below, draw diagrams showing four lines intersecting in zero, one, two… places. Not all may be possible!
3. Constructing any other diagrams necessary, use diagrams A, B & C on the previous page,
to fill in the table below:
Diagram
No. of lines (n)
No. of interior regions
(In)
No. of
intersections (Xn)
No. of external regions
(En)
1
A
2
B
3
C
4
6
D
5
4. In the space below, write about any patterns which you can see in the table above.
Note: If you have not seen it before, Xn lets you easily refer to rows in the table. For example, you could say “the no. of intersections in one row plus 3 gives the no. of intersections in the next row” by writing 31 +=+ nn XX .
Make sure your diagram for 5 lines shows the maximum number of intersections.
Next value of XCurrent X and add 3
3 - 3
5. Use the patterns you found in question 4 to predict the values for 6 lines and 7 lines.
My predictions:
No. of lines (n)
No. of interior regions
(In)
No. of
intersections (Xn)
No. of external regions
(En)
6
7
On a separate piece of paper, check your answers. Remember: (i) use a ruler! (ii) make sure your diagram shows the maximum number of intersections.
6. Most people find iterative rules when they look for rules giving In and Xn in the table in
question 4. An iterative rule is one where the value in one row comes from a formula which uses the value from the previous row. Such as 31 +=+ nn XX , seen earlier. The problem with this type of rule is that if you want to know the number of intersections for 100 lines, you have to know the value for 99 lines, which means you need the value for 98 lines, which means…. well, you get the idea! If you haven’t already, find rules for In and Xn which are not iterative and let you go straight from the number of lines n to the number of intersections Xn.
Hint: Both rules are quadratics and may involve consecutive numbers. There are many different valid rules which are equivalent.
7. Use your rules to find the number of intersections for:
(i) 20 lines. ________________
(ii) 25 lines. ________________
(iii) 200 lines. ________________ 8. How many lines would you need to draw to have:
(i) 1035 intersections. ________________
(ii) at least 10,000 intersections. ________________
Is it possible to get an answer which is not a whole number? If not, why not?
Why not two solutions to a quadratic equation?
Teacher Notes Activity Summary Suitable for: Year 9 & upward, any course involving probability and beginning
principles of expectations. In Games for Charity, examination of a simple game of chance involving a spinner is used to guide the student through the exploration of expected outcomes. Expected profit and loss are considered in view of the fee to play the game, weighed against the expected payment of prizes based on simple probabilities. When the first proposal ends with an expected loss, students are guided to consider methods of accomplishing a profit by adjusting the fee and the prizes. The final stage of this activity is the construction of a similar game of chance by the student, with a report required which includes the physical equipment needed, the rules, the fee and any prizes, and the expected profit per turn. Expected outcomes The student will understand the application of simple probabilities and expectations to games of chance, and be able to calculate the expected profit or loss for games involving simple probabilities. Required skills and knowledge An understanding of simple probabilities based on finite, listable sample spaces is required, as is an understanding of the meanings of revenue, profit and loss. Required materials Although no material is strictly required for this activity, the teacher may care to have on hand some of the common tools used when designing games of chance. These might include packs of cards, dice, and spinners or material which could be used to construct them. Follow-up teaching points Because of the intuitive appeal of this approach, the activity takes the path of examining the expected results of n repetitions of the game, where n is a large number, and then averaging this to find the expected profit or loss per game. Clearly the use of n cancels out and this can be shown algebraically as an exercise for the class or the individual. Another possible area for further investigation is the concept of ‘expected outcomes’. Experimental research can be used to validate the theoretical probabilities used in this activity.
4 - 1
1
2
3
45
6
7
8
Games for Charity Salome and her class have been split into groups and given the task of designing a game which can be used at a school function to raise money for charity. Before they start, the teacher uses a very simple game to demonstrate the mathematical ideas to the class. “Our game will consist of one turn on the spinner shown here, and it will cost the players 50 cents for a spin”, said Ms Cartwright. “If they land on an 8, then they win $5.” Ms Cartwright explained, “If they land on a 4 or a 5 then we give them 25 cents as a consolation prize. Otherwise, they just lose their money.”
There was a thoughtful silence from the class for a while, and then Salome said “I think the main prize is too big. It seems to me that we’ll end up losing money if we do it that way.” “Let’s find out for sure.”, said Ms Cartwright.
1. Assuming that all the segments of the circle are the same size, what is the probability of
spinning:
(a) an eight? _____________
(b) a four or a five? _____________
(c) any other number than these? _____________ 2. If the wheel were to be spun 1000 times, then:
(a) how much money would be collected in 50c playing fees? $____________
(b) how many times would you expect the spinner to land on an eight? ____________ times.
(c) how much money would you expect to be paid out for the wins in (ii)? $____________
(d) how much money would you expect to be paid out for the 25 cent wins? $____________
4 - 2
3. The profit (or loss) for our 1000 spins is calculated from the revenue minus the costs. Revenue (total of 50¢ fees) = $____________ Costs (total expected payout) = $____________ Therefore, profit/loss = $____________ Average profit/loss per spin = $__________ = __________¢ Salome pointed out smugly that she had been right. The money paid out was quite a bit larger than the money coming in. “If we don’t adjust the prizes or the fee to play we’ll end up losing money every time we play.”, she said. 4. Keeping the prizes the same, what is the smallest fee that could be charged,
instead of 50¢, if the game is to make a profit? 5. Keeping the 50¢ fee the same, what is the best prize that could be given instead of the $5
if the game is to at least break even? 6. Ms Cartwright has estimated that the class will be able to play the game for a total of 4
hours. She asked the class to assume a fee to play of 50¢, an average time per spin of 30 seconds, and the usual consolation prize of 25 cents for a 4 or 5. If Ms Cartwright wants to make at least $100 profit, what is the best prize that can be offered for the win on an 8? Your answer must be a multiple of 50¢.
4 - 3
1
234
7. Frances suggests an alternative game which uses an ordinary pack of cards (no jokers). In this game people are charged 50¢ to purchase a card at random. If they choose an Ace they win $2. A royal card (King, Queen or Jack) wins 25¢. Any other card has no prize. Find the average profit for the game in cents/card.
8. Helen suggests charging $1 to throw three coins in the air. If they land with three heads or three tails then the person wins $5. Averaged per throw, will this game make a profit or a loss, and how much?
9. If I spin the counter shown on the right, charging $2 per spin and
paying $5 for a 4, what will be the expected average profit or loss per spin?
10. Ten counters are placed into a bag. Two of them are white and labelled ‘Win $1’. One
counter is gold and labelled ‘Win $5’. The rest are black and labelled ‘Sorry – not a winner’. It costs $2 to play by reaching into the bag and drawing out one counter at random. What is the expected profit earned when the game is played 100 times?
11. Design your own game which could be used to raise money. Explain the rules of your
game thoroughly, including in your report: (a) the physical equipment needed.
(b) the rules.
(c) the fee and any prizes.
(d) the expected profit per turn.
Teacher Notes Activity Summary Suitable for: Year 10 & upward, any upper school course involving change of
scale and origin for univariate data. Through a concrete physical situation, Transforming Data guides the student to investigate the effect on the mean and standard deviation of adding, subtracting, multiplying or dividing by a constant. The student is encouraged to form hypotheses, write them down, compare results with others and to test their hypotheses with a variety of values including decimals and negative. At the end of the investigation the student is required to apply their deductions to the original problem. Note: The method used to introduce this concept is the conversion of centimetres to inches. To facilitate this, the work sheet takes the point of view of a farmer in the United States of America who is using research performed in Australia. See required skills. Expected outcomes The student will deduce the effects of change of scale and origin on the mean and population standard deviation, and develop an appreciation of the need for such tools when applied to research data which uses different units of measurement. Required skills and knowledge Conversion of units of measurement centimetres to inches and vice versa, and feet to inches. Use of a calculator to find the mean and standard deviation of ungrouped univariate data. Required materials A calculator capable of evaluating the mean and population standard deviation of univariate sets of data. The HP30S is recommended but others may suffice. Follow-up teaching points If the student has covered the summary statistics of bivariate data then a follow-up could be the investigation of the effects of change of scale and origin on the covariance, correlation and line of regression.
5 - 1
Transforming Data Juan and his family have run a successful farm in California, U.S.A. for many years. Now, to earn extra money, they have decided to try planting Australian Jarrah trees on some land which is not being used. Jarrah is a hard wood which can be made into beautiful furniture. Juan has researched the growth rate of Jarrah trees so that he can tell how long it will be before he can expect to harvest the timber. He thinks that it may be his children who benefit from his work. The data he found is shown below. It was recorded by a botanist in Western Australia. Juan’s problem is that the mean and standard deviation are given in units of centimetres (cm) rather than the feet and inches that Juan and his family are familiar with. Unfortunately he doesn’t have the original measurements to work with.
Years since planted
Mean height (cm)
Standard deviation (cm)
5 995 135 10 1890 245 15 2660 190 20 3290 155
Juan looked up an old maths textbook and found that the conversion from inches to centimetres was:
1 inch = 2.54 cm Skill check – Can you convert as needed? Convert: (a) 3 inches to cm. (b) 5 feet 11 inches to cm Hint: Convert to inches first. __________________________ __________________________ (c) 300 cm to inches (d) 2.5 metres to feet and inches. Hint: 1 metre = 100 cm. __________________________ __________________________
5 - 2
These are the values to which you will compare all the others. Check them with your teacher to ensure that they are correct!
Juan wondered if he could convert the means and standard deviations from the table on the previous page from centimetres to inches as easily as he could actual measurements.
Your task is to find out for him.
Through this sheet you will be investigating the effect of multiplying
and adding on the mean and standard deviation.
We will begin by working out the mean and standard deviation for a simple set of data. Once we have this starting point, we will investigate the effects on the mean and standard deviation of transforming the data by multiplying, dividing, adding and subtracting. 1. Using your calculator, find the mean and standard
deviation for data set A below. Round the standard deviation to 1 decimal place.
A = { 0, 5 6⋅ , 7, 1 6⋅ , -3, 7 9⋅ , 7, 0, 1 1⋅ , 2 8⋅ } Mean (A): ___________ Standard Deviation (A): ___________ (to 1 dec. place) 2. Multiply each data value from set A by 2 and record the results as a new set B in the
space below. Calculate the new mean and standard deviation. B = { ____, ____, ____, ____, ____, ____, ____, ____, ____, ____ } Mean (B): ___________ Standard Deviation (B): ___________ (to 1 dec. place) 3. Add 7 to each data value from set A (not set B!) and record the results as a new set C
in the space below. Calculate the new mean and standard deviation. C = { ____, ____, ____, ____, ____, ____, ____, ____, ____, ____ } Mean (C): ___________ Standard Deviation (C): ___________ (to 1 dec. place) 4. Divide each data value from set A by 10 and record the results as a new set D in the
space below. Calculate the new mean and standard deviation. D = { ____, ____, ____, ____, ____, ____, ____, ____, ____, ____ } Mean (D): ___________ Standard Deviation (D): ___________ (to 2 dec. places)
Note!
5 - 3
If you are unable to find a connection, see your teacher for a hint...
5. Subtract 1.5 from each data value from set A and record the results as a new set E in the space below. Calculate the new mean and standard deviation.
E = { ____, ____, ____, ____, ____, ____, ____, ____, ____, ____ } Mean (E): ___________ Standard Deviation (E): ___________ (to 1 dec. place) 6. Record your results from questions 1 to 5 in the summary table below. The values for
A are given so that you can check your accuracy.
Data Set
Operation Performed
Mean
Standard Deviation
A
original data set:
B
Set A × 2
C
Set A + 7
D
Set A ÷10
E
Set A − ⋅1 5
7. Explain, in your own words, the connection between the operation performed (x and +) and the mean of the new data set produced. ie. When the whole set of data is multiplied by a constant, then the mean.......
5 - 4
Remember... the aim in this section is to try to make your model fail! That way you can modify it and so make it stronger.
To save time, we will check only for multiplication and addition.
8. Explain, in your own words, the connection between the operation performed and the standard deviation of the new data set produced. Check with your teacher or the students around you to ensure that you are on the right track.
Hopefully you have now developed a model which seems to explain what happens to the mean and the standard deviation when the data set is changed by multiplying or adding (or dividing or subtracting).
The next step is to test your model very carefully to ensure that it works in all conditions.
9. Many models fail because they work only for whole numbers. Check your model by
transforming data set A using a decimal value of your choice (ie 7.4 or 132.76). First predict the result you should find, then check that you are correct. Multiply by: ____________ Predicted mean: _____________ Actual: _____________ Predicted standard deviation: _____________ Actual: _____________ Add: ____________ Predicted mean: _____________ Actual: _____________ Predicted standard deviation: _____________ Actual: _____________ Write your conclusions on the next page.....
5 - 5
Conclusion – did you need to modify the model in any way? 10. You must also ensure that your model correctly accounts for the
behaviour of negative numbers. Check your model by transforming data set A using a negative value of your choice.
Predict the result you should find, then check that you are
correct. Multiply by: ____________ Predicted mean: _____________ Actual: _____________ Predicted standard deviation: _____________ Actual: _____________ Add: ____________ Predicted mean: _____________ Actual: _____________ Predicted standard deviation: _____________ Actual: _____________ Conclusion – what is your final model?
HINT: Check this one very carefully!
5 - 6
Finishing the job..... The last job which you must complete in this investigation is to return to Juan’s original problem and apply your new knowledge to it.
Juan knows that the trees can be harvested when they reach a height of 18 m. The graph below shows the mean height as the centre line and a band on either side which has a width of 1 standard deviation. The large majority of trees can be expected to fall within the banding. Use the graph to estimate the range of years within which Juan can expect to harvest his trees. My estimate: ___________________ years
Years since
planted
Mean height
(inches)
Standard deviation (inches)
5
10
15
20
Complete your investigation of his problem by filling in Juan’s table with all measurements converted to inches.
5 - 7
Developing a deeper understanding... Why does multiplying affect both the mean and the standard deviation, while adding only affects the mean? Let’s examine a set of data using a dot frequency graph. The diagram shows the mean (marked with a pointer) and the range.
spread
mean 5 10 15 200
Now let’s transform the data by adding 10 to all scores....
Add 10 to all scores
spread
mean
old
old
spread
mean
new
new 5 10 15 200
Finally, we will transform the data by doubling all scores....
Double all scores
spread
mean
old
old0
new
new spread
mean
The data is moved as a whole block. The spread is not affected by this...
... but the mean moves with the data.
Although the range is not the same as the standard deviation it is a measure of spread and behaves in a similar fashion.
The mean is also doubled with the data...
The data is stretched by the doubling. The spread is affected by this.
16 cm
9 cm
Teacher Notes Activity Summary Suitable for: Year 10 & upward, any pre-calculus or early calculus course. In the activity “How Large is ‘Large’?”, the student investigates the change in volume of an open top box formed by cutting out squares at the corner of a sheet of cardboard and folding up the resulting tabs to form sides. By experimenting with a tabular list of possible values for the size of the square the student is guided to discover that there is a value which gives a maximum volume, and hence to attempt to discover, refining by trial and adjustment, what that maximum value is. Boundary values for possible solutions are also addressed by consideration of what values for the size of the square cut-out make sense physically in the context of the original problem. Expected outcomes The student will appreciate the concept of maximizing or minimizing a dependent quantity by manipulation of an independent variable, and that not all solutions to a mathematical model of a physical problem are feasible. Required skills and knowledge The student should be familiar with the formula for the volume of a cuboid, and be able to deduce the formula V x l x w x= − −( )( )2 2 from the physical situation, where V = volume, x = size of cut-out, l = length of sheet and w = width of sheet. Some experience with the solution of equations by refining with trial and adjustment methods would be advantageous. Required materials A calculator which will allow the entering and evaluation of formulae would be advantageous. Instructions applicable to the HP30S are given but others may suffice. The teacher may choose to introduce the problem by giving the student a sheet of paper or light cardboard with which they can physically make a box in the manner shown. The materials required are, sheets of paper or light cardboard 16cm by 9cm, scissors, and tape. Follow-up teaching points This worksheet is primarily designed to lead into the concepts and practice of differentiation and calculus. Other possible teaching points are to investigate the methods of refining by various techniques, and to look at the behavior of the formula for volume outside the bounds imposed by the physical problem. A graphical calculator could also be used to explore extrema of functions.
6 - 1
Cut out the squaresat the corners
Fold up tabs
16 cm
9 cm
How large is “large”? Alanna’s mother has asked her to help with the organisation of her younger sister’s birthday party. Alanna has been given the job of preparing ‘baskets’ of candy which will be given to the guests. She was asked to make them as large as possible. Alanna’s mother has some rectangles of coloured cardboard left over from another birthday party. Each piece is exactly 16 centimetres long by 9 centimetres wide. Alanna decides to cut squares out of the corners and then to fold up the sides. ie. As can be seen in the picture on the right, the result is an open-top box which Alanna can use to hold the candy for her sister’s guests. Alanna wondered if there was a ‘best size’ for the square she was cutting out at the corners. She wanted the box to hold as much candy as possible.
Your task is to find out for her.
Through this sheet you will be investigating the maximum volume for a box created as shown above.
6 - 2
1. It is probably obvious that the smallest possible size for the square cut-out is 0 cm. Is there a largest possible size? What is it and why is it the largest possible?
2. What is your first estimate for the best size? (Or, do you think that it will not matter?) 3. If Alanna cuts out a square of size 1 cm, what will be the height, length and width of
the box, and the resulting volume? Height = ________ cm. Width = ________ cm. Length = ________ cm. Volume of box = ___________ cm3. 4. Fill out the table below. The row for 1 cm has been filled in so that you can check
your answers.
Cutout (x cm) Width (cm) Length (cm) Height (cm) Volume (cm3.)
0
0.5
1
7
14
1
98
1.5
2
2.5
3
3.5
4
4.5
5. From the table above, what is the size that Alanna should use in order to have the
largest possible volume for her box? Size = ________ cm Volume of box = ___________ cm3.
6 - 3
6. On graph paper, plot the points on axes showing Cutout (horizontally) against Volume (vertically). • Carefully draw a smooth curve through the points you have
plotted. (Don’t simply join them with straight lines!)
(i) What volume will result from a cutout size of 2.7 cm.? ________ cm3.
(ii) What cutout size is needed to give a volume of 80 cm3.? ________ cm and ________ cm 7. Alanna originally did not expect the volume to vary so much. She was now quite
interested in the problem and suspected that the answer she had obtained from the table was only approximately correct. She decided to try to find a more accurate result by investigating the values near her earlier answer. Using scrap paper to work on, help her to find an answer accurate to one decimal place.
Size = __________ cm Volume of box = ___________ cm3.
Tired of repeated calculations, Alanna decided to see if her calculator could find the answer more efficiently...
8. If the size of the cutout is x, then find algebraic expressions in terms of x for: Length: L=__________________ Width: W=__________________ Height: H=__________________ Volume: V=________________________________________
Check your answers with your partner or teacher before proceeding.
9. On your HP30S, enter the expression for the volume that you worked out in the
previous question (using the X key). When you have done this, store the equation in the EQN variable for use in the next stage. To do this, press ?, then / and \. Now check your expression by evaluating it for X=2. To do this, press « / o . You will be asked for a value for X. Type 2 and \. You should obtain an answer of 120. If not, ask for help.
10. Using the EQN you have stored, find the best size for the cutout, accurate to 3 decimal places.
Vol (cm )3
Cutout (cm)
<-------Flight path
R metres
55°o
R metres
55°
<-------Flight Path
R
metr
es
R
m
etre
s
55°55°
R metres
Light sourceA
B
C
L
2
1
34
Teacher Notes Activity Summary Suitable for: Any upper school course involving advanced trigonometry. A good
introduction to, or reinforcement of the concept of, limits. In Moths to a Candle, a relatively simple model (see below right) for the flight of a moth around a candle involving similar triangles leads the student to the investigation of limits to infinity and tangential orbits. The student is guided to investigate the change in flight path as the angle and the number of adjustments per orbit change, and then to consider the implications to these results as the number of adjustments per orbit tends to infinity. Expected outcomes The student will use their knowledge of trigonometry and similar triangles to make deductions regarding the path of the moth as various assumptions are adjusted. The student will make and communicate conjectures regarding the behavior of the resulting equation as parameters tend to infinity. Required skills and knowledge Recognition of, and calculations using similar triangles, use of the Sine Rule, and some understanding of the basic concept of a limit as x →∞ of k
x . Required materials None. Follow-up teaching points
In question 9, the formula R R
p
n
n
= ×
− −FHG
IKJ
F
H
GGGGG
I
K
JJJJJ0
180 360
sin
sin
θ
θ
o
ois used to deduce the fact that the
moth’s path will be circular for θ = −90 180oo
p. This means that as p →∞ , θ → 90o , implying
that to circle the flame, the moth flies at a tangent to the circle. This can be used to stimulate discussion regarding vectors of velocity and acceleration for circular motion.
7 - 1
A
B
L 30° 55°
B
C
L
55°
30°
Moths to a Candle One theory about “moth navigation” is that moths fly at a constant angle relative to the source of the light. Obviously the truth is more complex but this does offer some interesting ideas. Consider the following mathematical model of a moth’s flight path.
A particular moth starts its travels Ro metres from a light source L, flying in the horizontal plane only. Starting at A, it flies in the direction of the arrow at an angle of 55o to a line AL drawn from the light source to its current position, ending at B, which is R1 metres from the light source. From there it flies on an angle of 55o again to C, and so on. 1. Assuming that the moth adjusts its path 12 times during each revolution of 360o, find the
size of ∠ALB . ∠ALB = __________ 2. In view of your answer to question 1, what is the mathematical relationship between the
successive triangles in the diagram above?
<-------Flight path
R metres
55°o
R metres
55°
<-------Flight PathR
m
etres
R
m
etre
s
55°55°
R metres
Light sourceA
B
C
L
2
1
34
7 - 2
A
B
L 30° 55°Ro
R1
3. Using the Sine Rule, find the value of:
(a) R1 in terms of R0.
(b) R2 in terms of R1.
(c) R2 in terms of R0. Hint: Use (a) and (b) above.
(d) R3 in terms of R0.
(e) R10 in terms of R0. 4. Produce a formula for the value of Rn (the nth distance from L) in terms of R0.
7 - 3
5. Using your calculator, explore the behaviour of Rn as the value of n increases. What part of the formula causes this? What does this represent in terms of the moth’s flight path?
As n →∞ , the value of Rn ….. 6. Our first moth recalculated its flight path 12 times per 360o. If a different species of moth
recalculated its flight path p times per revolution, then what is the formula for Rn? Note: Assume that it still flies at an angle of 55o to a line drawn to the light source.
7. Generalise your formula to apply to a moth which flies at a constant bearing of θ to the
light source L, adjusting its path p times per revolution.
7 - 4
8. On a separate sheet of paper, investigate the flight path of the original moth for different values of θ. Describe its possible paths, with the aid of diagrams, ensuring that you answer the following questions…
For what values of θ will the moth’s flight path… (a) spiral into the light source? (b) move away from the light source? (c) circle the light source?
9. Investigate how your answers to question 8 change
as the value of p changes. What happens for very large values of p? Derive a formula, in terms of p, which gives the value of θ for which the moth will circle the light source.
i
Solutions Activity 1: Planning the Paths 1. a) 1 b) 1 c) 2 2. a) 1 b) 1 c) 1 d) 1 e) 2 f) 3 3. No. Can’t go backwards 4. Add them together. 5. D. 1 H. 4 K. 6 L. 10 6. To get the number of paths you add the
numbers vertically above and to the left of the intersection.
7. a) 120 b) 65 c) 110 d) 90 e) 36 f) 29 g) 64 h) 24 8. 143 9. 36 10. If h is the number of blocks horizontally (not roads, blocks) and v is the number vertically then the number of paths through a simple rectangle is (h+v)Cv or (h+v)Ch. In example a) h=7 & v=3 so use 10C3 or 10C7. When two blocks are joined at the corner like example b), just do each separately and multiply. Hence b) is 7C3 x 4C2 Activity 2: Infinite Series 1. no response required 2.
18
116
132
78
1516
3132
, ,
, ,
3. fractions halve each time.
4. m tn
n
n= =−1
22 1
2,
5. 40954096
or 99.98%
6. 0.2197 m2 or 2197 cm2 7. Not true in real world. Still an increasingly
good approximation, particularly over longer periods.
8.
x
x
x xx
x
= + + +
= + + +
= +=
=
13
19
127
3 1 13
19
3 12 1
12
....
....
9. a) 13
b) 14
c) 19
d) 39
13
=
10. a) check with your teacher b) 79
11. The final line shows that 0 99999 1. ...= . Most students don’t like this the first time they see it, but it’s true!
12. a) 433
b) 211
c) 2599
d) 3 111
e) 41333
Activity 3: Lines and Regions 1.
2.
Zero
One
Three
Four
Five
Other arrangements possible·Two impossible
3.
n In Xn En 1 0 0 2 2 0 1 4 3 1 3 6 4 3 6 8 5 6 10 10
4. There are many patterns possible. Ask the teacher if yours is valid.
5. n In Xn En 6 10 15 12 7 15 21 14
6. ( )
( )( )2
212
12
1−−
==
−=
=
−nnEI
nnX
nE
nn
n
n
7. a) 171 b) 276 c) 19701 8. a) 47 b) 143
ii
Activity 4: Games for Charity
1. a) 18
b) 28
c) 58
2. a) $500 b) 125 c) $625 d) $62.50 3. a) $500- $687.50 loss of 18.75 cents per
spin 4. 69 cents 5. $3.50 6. $1.50 ($1.83 rounded) 7. profit of 29 cents/game (28.8) 8. loss of 25 cents/game 9. profit of 75 cents/game 10. profit of $130 Activity 5: Transforming Data 0. a) 7.62 b) 180.34 c) 118.11 d) 98.43 1. 3, 3.5 2. 6, 7.0 3. 10, 3.5 4. 0.3, 0.35 5. 1.5, 3.5 6. Table values are given above 7. … has the same operation performed on it. 8. Adding and subtracting: no effect. Multiplying
and dividing: same operation performed. 9. No modification needed. 10. Multiplying or dividing by negatives, drop
sign. 11. 391, 53, 744, 96, 1047, 75, 1295, 61, 8-11
years. Activity 6: How Large is “Large”? 1. Largest is 4.5 cm. Any larger would be more
than the width. 2. Many answers possible 3. 1, 7, 14, 98 4. Volumes are
0,60,98,117,120,110,… 90,63,32,0
5. 2, 120 6. i) 103.032 ii) 0.7, 3.2 See graph above. 7. 1.9, 120.536
8. L x W x H xV x x x= − = − == − × − ×
16 2 9 216 2 9 2
, , ,( ) ( )
10. Other methods are possible 11. 1.851 cm.
Activity 7: Moths to a Candle 1. 30o 2. They are similar triangles.
3. a) R R1 05595
= ×FHG
IKJ
sinsin
o
o b) R R2 1
5595
= ×FHG
IKJ
sinsin
o
o
c) R R2 0
25595
= ×FHG
IKJ
sinsin
o
od)
R R3 0
35595
= ×FHG
IKJ
sinsin
o
o e)
R R10 0
105595
= ×FHG
IKJ
sinsin
o
o
4. R Rn
n
= ×FHG
IKJ0
5595
sinsin
o
o
5. As n →∞ , Rn → 0 because the ratio sinsin
5595
o
o
between successive triangles is < 1. This means the moth will spiral in to the candle flame .
6. R R
p
n
n
= ×
−FHG
IKJ
F
H
GGGGG
I
K
JJJJJ0
55
125 360
sin
sin
o
o
7. R R
p
n
n
= ×
− −FHG
IKJ
F
H
GGGGG
I
K
JJJJJ0
180 360
sin
sin
θ
θ
o
o
8. a) spiral in for θ < 75o b) spiral out for θ > 75o c) circle for θ = 75o .
9. The moth’s path will be circular for
θ = −90 180oo
p. This means that as p →∞ ,
θ → 90o . ie. to circle the flame, the moth flies at a tangent to the circle.
i
Preface Welcome to the joys of Investigating Mathematics through Patterns. Hewlett-Packard Calculators is pleased to introduce you to “Investigating Mathematics through Patterns”, a discovery and investigation based learning resource that will help your students build new mathematical knowledge through the guided exploration of patterns in problems which are solidly based on the real world. We have taken great care to ensure this resource will make learning realistic and fun for the students. This material again demonstrates Hewlett-Packard’s dedication to the calculator and its educational base. The HP 30S and “Investigating Mathematics through Patterns” simply take this dedication to a new level. We know this is just the beginning of the support we hope to provide to educators and students and hope this will grow to the levels our other product categories have achieved. To find out more about this and the many other products and services we have to offer please visit our web site at www.hp.com/calculators. Again thank you for your support and use of this resource in your classroom. If you have any questions or comments on how to improve or change this or any of our products or materials, please contact us at the address above. We look forward to hearing from you. Hewlett-Packard Calculators
ii
How to Use This Resource The activities in this book have been carefully designed to follow an investigative approach to the teaching of mathematics. Through introductions which relate the activity to the student’s world, they may help to answer the question “When are we ever gonna use this?”. By following this approach and by encouraging the students to develop their own conjectures, compare their results, and communicate their conclusions, these activities address a multiplicity of the objectives of the Australian curriculum. The Activities Teachers may choose to use an activity as an adjunct to concepts being explored, as an enrichment activity, or as a vehicle to introduce a new topic in a way that is meaningful to the students. Teacher Notes The Teacher Notes at the beginning of each activity provide the following resources:
• a summary of the activity, including an outline of the material covered; • the expected outcomes; • any required skills and knowledge; • any required materials; • any natural follow-up teaching points;
Student Worksheets Each activity comprises a worksheet to be photocopied for the student. In almost all cases this worksheet includes space for the student to record their answers.
iii
Table of Contents Discovery Activities As any educator knows, the best understanding of a concept is achieved by a student who is encouraged to discover the principles and concepts for themselves. The emphasis in all of these activities is the guided investigation of patterns in various mathematical topics, grounded in concrete, real-world situations. The student is encouraged to share results, and to communicate conclusions verbally and in writing. Activity 1: SEDs and DEDs
How could you find the units digit of a number that is so large that the calculator will not display it? Through this activity the student is led to investigate patterns in the single and double end digits of numbers, resulting in the discovery of recurring patterns which can be used to answer this question. An ideal way to lay the groundwork for the concept of modulo numbers in a fashion that is meaningful to the student. Suitable for: Year 7 to 10
Required knowledge: Use of the calculator to find powers of integers. Activity 2: Congruent Triangles
What is the minimum required information to ensure that two people are guaranteed to draw identical triangles? This is the question which is posed in order to encourage the student to deduce the rules for congruency in triangles. Through this carefully guided investigation students are encouraged to share their deductions and to communicate results with their peers and their teacher. Suitable for: Year 8 to 10
Required knowledge: An understanding of the basic concept of congruency, detailed knowledge of the classification and properties of triangles.
Activity 3: Going Around in Circles
The investigation of patterns in circles produces a fascinating investigation into factors and geometric shapes. The student is guided to the deduction of an interesting rule which uses the highest common factor. Suitable for: Year 7 or 8
Required knowledge: Naming of geometric shapes, familiarity with factors and highest common factors.
iv
Activity 4: Covering Up the Facts When teaching the interpretation of different styles of graphs, educators will find this activity an interesting demonstration of the use of a nomogram. In this lesson the student will use a formula and a nomogram to calculate the area of a person’s skin, as a function of their height and weight. See also “A Burning Question” Suitable for: Year 8 & upward, any non-tertiary bound upper school
courses. Required knowledge: Use of a calculator to find powers of decimals,
conversion of units of mass and length, including metric units, solution of equations by trial and adjustment.
Activity 5: A Burning Question
A follow-up activity to “Covering Up the Facts” one, this lesson also uses a nomogram to investigate the changing mortality risk to burn patients as a function of age and percentage of skin area burned. Suitable for: Year 8 & upward, any non-tertiary bound upper school
courses. Required knowledge: Percentages, ratios, elementary probability.
Activity 6: Farey Fractions
The investigation of interesting sequences of fractions known as ‘Farey sequences’ is used as a vehicle to give students practice in basic operations on fractions. Suitable for: Any course involving operations on fractions. Required knowledge: Addition and subtraction of fractions, ordering of fractions.
Activity 7: Locker Patterns
This investigation of a seemingly simple pattern leads to the examination of the factors of perfect squares. Suitable for: Years 7 to 9 Required knowledge: Factors, recognition of simple patterns.
Teacher Notes Activity Summary Suitable for: Year 7 to 10 If we evaluate 712 to obtain 13841287201, then we can see that the Single End Digit (SED) is 1 and the Double End Digit is 01. But how could we find the last two digits of a number that is so large that the calculator will not display it? In the activity SEDs and DEDs, the student is led to investigate patterns in the SEDs and DEDs of numbers, resulting in the discovery of recurring patterns which can be used to answer this type of question. Each digit from 0 to 9 has a pattern, some more complex than others. Expected outcomes The student will make and test conjectures regarding patterns in powers of integer bases. Required skills and knowledge The student will need to be able to find integer powers of integer bases. Required materials None. Follow-up teaching points; This activity is an ideal way to lay the groundwork for the concept of modulo numbers in a fashion that is meaningful to the student.
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SEDs and DEDs Using a calculator, we can easily find the value of 177 and obtain an exact answer of 6975757441. From this we can see that the Single End Digit (SED) of 177 is 1 and the Double End Digits (DED) are 41. But… can you work out what the SED and DED of 1729 are? If you try, you will quickly discover that any normal calculator will display the answer in standard form as 4 819685721 1035⋅ × . The problem with this is that the digits displayed are only the first dozen or so starting from the left. The ones we want are at the other end, and have been rounded off and dropped! How can we find them? The answer, of course, is that we need to investigate in order to find patterns that we can use to predict the missing digits. 1. Let’s start with an easy one. Using your calculator, evaluate each of the powers of five in
the table below.
2. Using the patterns you found in question 1, find the SEDs and DEDs for:
SED DED
(a) 512 _________ ___________
(b) 5126 _________ ___________ 3. Does this pattern for 5 also work for 15, 25 & 35? Check with your calculator whether
the SEDs and DEDs patterns for 15, 25 & 35 match those for 5.
Conclusion:
Power Value SED DED
51
5
5
05
52
25
53
125
25
54
55
56
Try this!
What patterns can you see in the SEDs and DEDs?
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4. Now try a harder one. Evaluate each of the powers of seven in the table below. What are the patterns in the SEDs and DEDs for 7?
Patterns:
5. Using the patterns you found in question 4, find the SEDs and DEDs for:
SED DED
(a) 713 _________ ___________
(b) 726 _________ ___________
(c) 7500 _________ ___________ 6. Do your patterns for 7 also work for 17? i.e. is the SED for 1710 the same as the SED for
710? What about the DEDs? 7. Find the SEDs for:
(a) 171000 ___________
(b) 27511 ___________
Power Value SED DED
71
7
7
07
72
49
73
74
75
76
77
78
79
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8. A common way of writing recurring decimals such as 3 1616161⋅ .... is as 3 16⋅ . Using this notation as a model, we write the pattern in the DEDs for 7 as 07 49 43 01, , , starting at 71 . Find the SED and DED patterns for the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. Note: In each case, check that if the pattern works for (say) 6, then it also works for numbers such as 16, 26 and 36.
Base Patterns 0 SED: __________________________________ DED: __________________________________ 1 SED: __________________________________ DED: __________________________________ 2 SED: _________________________________________________________ DED: _________________________________________________________ 3 SED: _________________________________________________________ DED: _________________________________________________________ 4 SED: _________________________________________________________ DED: _________________________________________________________ 5 SED: _________________________________________________________ DED: _________________________________________________________ 6 SED: _________________________________________________________ DED: _________________________________________________________ 7 SED: _________________________________________________________ DED: _________________________________________________________ 8 SED: _________________________________________________________ DED: _________________________________________________________ 9 SED: _________________________________________________________ DED: _________________________________________________________
Warning: The patterns for 2, 4 and 8 will need care!
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9. Using the patterns you have discovered, find the SED and DED (where possible) for each of the following.
SED DED
(a) 1011 _________ ___________
(b) 226 _________ ___________
(c) 3333 _________ ___________
(d) 18100 _________
(e) 1955 _________
(f) 2251 _________
(g) 474474 _________ 10. If you look carefully at the SED and DED patterns for 2 and 4 you will see that
they are related, as are the patterns for 2 and 8. How are they related and why? What other pair would you expect to be related in this way?
Teacher Notes Activity Summary Suitable for: Year 8 to 10 Congruent Triangles forms an excellent introduction to the topic of the conditions for congruency in triangles. By posing the question of how much information is the minimum necessary to ensure that two people must draw the same triangle, the student is encouraged to develop conjectures leading to the conditions of congruency. The activity encourages the forming of conjectures, comparison and communication of results in both written and oral form, and an appreciation of the need for generalization. The approach used often has unexpected and interesting results beyond those expected. For example, “extra” conditions are often deduced along the lines of “knowing the length of one side and that they are equilateral”. Class discussion after the activity will quickly narrow the results to those desired by the teacher if the activity has not done so. Expected outcomes The student will deduce and communicate the conditions for congruency in triangles. Required skills and knowledge An understanding of the basic concept of congruency, and detailed knowledge of the classification and properties of triangles. Required materials Drawing and measuring instruments, a substantial supply of scrap paper. Follow-up teaching points As mentioned, unexpected extra conditions are often deduced in addition to the classic conditions for congruency. Further exploration and discussion of these ‘extras’ can be rewarding.
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7 cm
7 cm
2 cm2 cm
Congruent Triangles Beatrix and Margaret had finished drawing their triangles and stopped to compare results. “Hang on,” said Beatrix, “I thought we had agreed to draw the same triangle? Yours looks nothing like mine.” “Well,” replied Margaret thoughtfully, “I guess only specifying that two of the sides were 2 cm and 7 cm was not enough information to be sure we ended up with identical triangles.” “It’s called congruent in maths, not identical.”, said Beatrix absently. “Well then, I wonder just how much information you do need to ensure that two triangles must be congruent? Surely you don’t have to know all three sides and all three angles?” “That would certainly be more than enough,” commented Margaret dubiously, “but you would surely only want the minimum information necessary wouldn’t you? I wonder what that ‘minimum information’ is?” Beatrix and Margaret need to know what minimum information is required in order to be absolutely sure that two people must draw the same triangle.
Your task is to find out for them.
Through this sheet you will be investigating the conditions for congruency in triangles.
1. It is probably obvious to everyone that knowing all three sides is enough information to
ensure that two triangles are congruent (identical). What about knowing all three angles? Is this enough to ensure that two people must draw exactly identical triangles? If so, explain why you think this. If not, give an example to show why it is not enough.
Beatrix pointed out quickly that so far they had only considered two possibilities - three sides (SSS) and three angles (AAA). “Surely we should also consider combinations of these?”, she pointed out. “And,” said Margaret, getting interested in spite of herself, “we should also consider other possibilities. For example, what about knowing only one side, but also that it was an equilateral triangle? Surely that would be enough.”
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A counter example is an example which demon-strates that an idea or theory is not true.
2. On a separate sheet of paper, write down a list of possible sets of information which you think might be enough to ensure that two people must draw congruent triangles. When you have finished, compare your list with two neighbours and add any of their ideas to yours.
Hint: SAS (an angle between two sides) is not the same as having SSA (two sides and an
angle, but the angle not between them). You should check both, and consider this in other combinations. Ask your teacher if you are not sure you understand this hint. Also: remember properties of triangles, such as that the sum of the angles in a triangle is 180o, and those of isosceles triangles.
3. Drawing diagrams as needed, carefully investigate
each of the possibilities on your list. Keep a list of successes and failures, including counter examples for the failures.
Remember, when testing theories in mathematics and science, one important method is to try to prove yourself wrong. If you can’t then your theory may be true.
4. When you have finished testing, sort your successes into two groups. The first group should contain only those which are combinations of sides and angles, such as the SSS combination. This group will be fairly small. The second group should contain all other combinations, such as the ‘one side + equilateral triangle’ combination. If you see a sensible reason to do it, you may wish to also split this second group into sub-groups.
5. Compare your results with your neighbour, if he or she has finished. If there are any
disagreements between your lists then resolve them by discussion and examples (or counter examples).
6. Working alone or with a partner, write a concise report on your findings. Include in your
report:
(i) a short introduction explaining what you investigated.
(ii) your findings, grouped sensibly, including any diagrams, examples and counter examples.
(iii) a short conclusion.
Teacher Notes Activity Summary Suitable for: Year 7 or 8 Going Around in Circles examines patterns in circles, producing a fascinating investigation into factors and geometric shapes. It guides the student to examine patterns in the shapes formed and the number of points of contact around the circle. The resulting formula uses the highest common factor of the jump size and the number of points around the circle. Expected outcomes The student will form conjectures regarding geometrical and numerical patterns. Required skills and knowledge Naming of geometric shapes, familiarity with factors and highest common factors. Required materials There may be a need for extra copies of the final page of the activity, containing multiple copies of the 12-point circles required in question 6. Follow-up teaching points; The questions in the activity concentrate primarily on the numerical patterns but there are many possible discussion points stemming from the geometrical patterns.
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Going Around in Circles The diagram on the right shows a 12-point circle. On it has been marked the first few moves in a 12 7,b g pattern. 1. Continue the pattern shown here, stopping when you
land on zero. Include the arrows. A pattern of 12 7,b g means that on a 12-point circle you are to make jumps of seven points, starting at zero, in a clockwise direction.
2. At how many points does your resulting shape touch the
circle? ______________ 3. On the 12-point circle right, draw a 12 4,b g pattern.
(a) How many points are there this time? ____
(b) What differences do you notice between the patterns? 4. Predict in your own mind what you would expect to see for 12 3,b g and 12 8,b gpatterns.
Use the special 12-point circle paper supplied by your teacher to check your predictions. 5. Explain briefly why the 12 0,b g and 12 12,b g patterns are uninteresting.
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6. Using the 12-point circle paper, complete any remaining diagrams for the patterns 12 0,b g through to 12 12,b g .
Number of
points in pattern. Type of pattern
(shape etc.)
12 0,b g
12 1,b g
12 2,b g
12 3,b g
12 4,b g
12 5,b g
12 6,b g
12 7,b g
12 8,b g
12 9,b g
12 10,b g
12 11,b g
12 12,b g
7. What patterns can you see in the table? 8. Find a rule which will give the number of points of contact with the circle (p) from the
step size (s) used. p = ? 9. Test your rule by predicting the number of points in the patterns below, using the circles
provided below the table:
Hint: Think about factors.
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Predicted no. of points
Predicted shape
Actual no. of points Actual shape
(i) 9 3,b g
(ii) 9 6,b g
(iii) 9 2,b g
(iv) 10 2,b g
(v) 10 5,b g
(vi) 24 6,b g
(vii) 24 9,b g
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1011121314
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How did you do?
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12-Point Circles
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Teacher Notes Activity Summary Suitable for: Year 8 & upward, any non-tertiary bound upper school courses. When teaching the interpretation of different styles of graphs, educators will find Covering Up the Facts an interesting demonstration of the use of a nomogram, a style of graph consisting of three vertical lines and used when a dependent quantity is derived via a non-linear formula from two independent variables, in this case skin area from height and weight. In this lesson the student will use a formula and a nomogram to calculate the area of a person’s skin, as a function of their height and weight. It is an example of the application of relatively straight-forward mathematics to an innately practical problem. See also the following activity: “A Burning Question” Expected outcomes The student will appreciate the use of different styles of graphs as an alternative to formulae, and their applicability to real-world situations. Required skills and knowledge A basic understanding of percentages is required, as well as the ability to use a calculator to evaluate decimal powers of decimal bases. An understanding of the process of solving equations by refinement by trial and adjustment would be advantageous. Required materials None. Follow-up teaching points Activity 5: “A Burning Question” follows this activity with the use of another nomogram to calculate mortality risk for burn patients as a function of age and percentage of skin area burned. There are numerous other applications of mathematics to the skills of medicine and, in particular, those of nursing are often well within the grasp of high school students. Example are the experimental determination of the volume of a drop of water, leading to calculations of drip rates for intravenous administration of liquids, and the calculation of medicinal dosages for injections by the use of ratio and proportion. Any first year nursing handbook is a source of endless materials.
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Covering Up the Facts What do you think is the largest organ in your body? Your brain? In fact, the largest organ and the toughest is the skin! Your skin has to put up with an incredible amount of daily abuse from the environment. For some medications, and also in evaluating the extent of injuries to the skin, it is sometimes necessary to be able to estimate the surface area of a person’s skin. As you can imagine, this is very difficult to measure directly. What would you guess the surface area of your body to be in square metres? My estimate is: ________________ m2. There are two common methods of estimating the area of a person’s skin. Method 1 – Using a formula Through medical research, a formula was developed which is shown below: S W H= × ×0 007184 0 425 0 725. . .
where S is the surface area in square metres (m2) W is the weight in kilograms (kg) H is the height in centimetres (cm.)
Use the yx button on your calculator to find the surface area of skin for a person who is 168 cm tall and who weighs 65 kilograms.
S x xy y= 0 007184 64 8648 0 167 64 0 725. * . .425 * . . You should obtain an answer of 1.7386 m2.
If you did not get this answer, see your teacher to find out what you did
wrong.
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You should now use the formula to check the estimate you made earlier of your skin area, using your weight and height in the formula. My estimate was: ________________ m2. The formula value is: ________________ m2. My error was: ________________ m2. My percentage error was: ________________ %.
How good was your estimate compared to your friends’?
1. Determine the body surface area for the following people: (a) James, who is 200cm tall and weighs 105kg S = ________ m2. (b) Antonia, who is 173 cm tall and weighs 50 kg. S = ________ m2. (c) Jeremy, who is 143 cm tall and weighs 63 kg. S = ________ m2. (d) Gillian, who is 185 cm tall and weighs 55 kg. S = ________ m2.
2. Challenge: (a) Charles is 143cm tall and has a skin surface area of 1.5 m2 . How much does Charles weigh, to the nearest 0.5 kg?
W = ___________ kg. (b) Jeanette has a skin area of 1.2 m2 and
weighs 50kg. How tall is she, to the nearest centimetre?
W = ___________ kg. 3. If a person who is 120cm tall increases in weight from 50 to 55 kg, by what
percentage will their skin surface area increase?
Original area: __________ m2 New area: __________ m2
Percentage increase: ________ %
The difference between the two
values.
% Error = error x 100 true value
Hint: You may need to use ‘guess, check and improve’ techniques.
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Method 2 – Using a Nomogram Although the formula gives the most accurate results, a calculator may not be available or it may not be convenient to use one. A special graph called a nomogram was developed for use when two quantities (height & weight) are used to find a third (skin area). A nomogram designed for the calculation of skin area is shown below. It has been set up to allow both metric and non-metric values. Use it to answer the questions which follow.
From Geigy Scientific Tables; 1990, 8th ed., Vol 5, p. 105, Novartis
Use the nomogram to answer the questions on the following page:
Height (cm) (inches)
Body Surface Area(m2)
Weight (kg) (lb)
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4. Find the body surface area of the following people: (a) Angela, who is 162cm tall and weighs 55kg S = ________ m2. (b) Beata, who is 165 cm tall and weighs 41 kg. S = ________ m2. (c) Frances, who is 137 cm tall and weighs 53 kg. S = ________ m2. (d) Gillian, who is 180 cm tall and weighs 58 kg. S = ________ m2. 5. (a) In question 2 earlier you were set the challenge of finding the weight of
Charles, who was 143cm tall and had a skin surface area of 1.5 m2. Use the nomogram to check your earlier answer.
Note: There are slight discrepancies between the formula and the nomogram
for some values. (b) You were also asked to find the height of Jeanette, who had a skin area of 1.2
m2 and weighed 50kg. Use the nomogram to check your earlier answer. 6. Find the weight (in the units indicated) of people with the following heights and body
surface areas: (a) 150 cm and 1.70 m2 Weight: ___________ kg (b) 120 cm and 1.25 m2 Weight: ___________ kg (c) 170 cm and 1.90 m2 Weight: ___________ kg (d) 144 cm and 1.35 m2 Weight: ___________ kg 7. (For experts!) For a person who is 165cm tall, use graph paper to draw a graph
showing how his body surface area will change as his weight increases from 50kg to 120kg.
Teacher Notes Activity Summary Suitable for: Year 8 & upward, any non-tertiary bound upper school courses. Note: This activity follows on from activity 4: “Covering Up the Facts”.
Activity 4 must be completed first. In addition, some questions in this activity refer to the Skin Area nomogram in Activity 4.
A Burning Questions uses a nomogram to investigate the effects on mortality risk of the twin factors of age and percentage of skin area burned. Extensive use is made of percentages and the student is also required to produce a graph of mortality risk vs. age when percentage of burns is held constant. It is an example of the application of relatively straight-forward mathematics to an innately practical problem. Note: This activity refers to burn patients and mortality risks. As such, it may
be disturbing to some students. Expected outcomes The student will appreciate the use of different styles of graphs as an alternative to formulae, and their applicability to real-world situations. Required skills and knowledge Understanding of percentage and ratio. Required materials None. Follow-up teaching points There are numerous other applications of mathematics to the skills of medicine and, in particular, those of nursing are often well within the grasp of high school students. Example are the experimental determination of the volume of a drop of water, leading to calculations of drip rates for intravenous administration of liquids, and the calculation of medicinal dosages for injections by the use of ratio and proportion. Any first year nursing handbook is a source of endless materials.
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A Burning Question Note: You will need the Body Surface Area nomogram from the activity
“Covering Up the Facts” to complete questions in this activity. Most forms of skin injuries can be treated fairly easily by modern medicines. Unfortunately, while new advances are being made all the time, burn injuries are still slow to heal and very painful to the victim. When they are first admitted to hospital (or perhaps during the ambulance trip) a fast judgement of how much skin is damaged can be very helpful in determining treatment. This quick estimate can be made using the “Rule of Nines”. Using this rule, the human body is divided up into 11 blocks of 9% each, plus a 12th section containing the missing 1%. A diagram is of this is shown below
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Use the “Rule of Nines” diagram to estimate the extent of burns for the following patients. 1. Mrs Davis, whose face, arms and the front of her body above the waist were burned in
a car accident. _____________ % 2. Ben, whose right arm, face and the right half of his chest (front and back) were burned
by removing the radiator cap of his car too quickly. _____________ % 3. Carlotta, who is a volunteer fire-fighter and was caught by a fire. She lay face down
to protect herself and is burned on the whole of the rear of her body, as well as over the whole surface of her arms.
_____________ % 4. If Carlotta, from the question above, is 191cm tall and weighs 87 kg then use the Skin
Area nomogram from the previous activity to determine her total body surface area. Using this and your answer to the previous question, calculate the burned area in m2.
Body S.Area: _____________ m2 Percentage burned: _____________ % Area burned: _____________ m2 5. Gregor is 195cm tall, weights 76kg and is scalded (burned by water) from the bottom
of his ribs downwards (front only). Calculate the burned area of skin in m2. Body S.Area: _____________ m2 Percentage burned: _____________ % Area burned: _____________ m2 6. Jenny is one of the victims of an aircraft accident and is burned on the entire rear of
her body and on the whole of her arms and legs. If she is 160cm tall and weighs 56kg then estimate the area burned in m2.
Body S.Area: _____________ m2 Percentage burned: _____________ % Area burned: _____________ m2
Hint: The face is half of the head.
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7. (Challenge!) Alberto has burns of 1.02 m2 on his body, which the paramedic has estimated at 60% of his total body surface area. If Fred weighs 70kg then how tall is he?
Height:_____________ cm 8. Jemima is burned on 55% of her body surface area. She is 150cm tall and weighs
56kg. Use the nomogram to find her body surface area in m2 and use this to calculate the area burned in m2.
Area burned: _____________ m2. 9. Bruce is 185cm tall, weighs 65kg and has been burned by lightning on his entire right
side and the whole of his back. Calculate the area of burned skin in m2. Body S.Area: _____________ m2 Percentage burned: _____________ % Area burned: _____________ m2 10. Lee has burns of 0.37m2, which is estimated by a nurse to be 27% of her body surface
area. If she is 140cm tall, what does she weigh? Weight: _____________ kg
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The greater the percentage of skin which is destroyed, the greater is the risk that the patient will die of their injuries. This chance is known as the “mortality risk” and can be estimated using a nomogram by knowing the patient’s age and the amount of skin burned. A nomogram which can be used to estimate mortality risk is given on the next page. Use it to answer the questions which follow. 11. A mortality risk of 0.75 means that the patient has a 75% chance of dying. What is the
chance that the patient will live? ________________% 12. Estimate the mortality risk of Stephanie, who is 25 years old and has burns to 70% of
her body. What is the chance of her surviving? ________________% 13. If Mrs Davis (from question #1) is 37 years old, estimate her mortality risk. ________________% 14. Which patient has the higher mortality risk? A 40 year old with burns to 40% of his
body or a 6 year old with the same burns? Why do you think this is so? 15. At age 45, what percentage of a person’s skin could be burned before their mortality
risk rose to more than 50%? ________________% 16. Frank’s mother has been told that he has only a 30% chance of survival. If Frank is 10
years old, what percentage of his skin is burned? ________________% 17. For a set amount of burns, is it possible to have the same mortality risk at two different
ages? Give an example. 18. Using the nomogram, estimate the mortality risks at ages 5, 10, 15, 20,….. for a
person with burns to 50% of their skin. • Use these estimates to produce a line graph showing how mortality risk changes
with age for a patient with 50% burns. • Why do you think that the mortality risk starts high, lowers and then rises again?
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Nomogram showing Mortality Risk
as a function of Age (years) and Percentage of Body Surface Burned.
(adapted from “A Severity Grading Chart for the Burned Patient”, by Larry D. Roi, Head, Biostatistics Dept., Michigan Cancer Foundation)
Teacher Notes Activity Summary Suitable for: Any course involving operations on fractions. John Farey (1766 – 1826) was a geologist who occasionally dabbled in mathematics and discovered an interesting sequence of fractions which was named the Farey Sequence after him. The Farey Sequence of order 7 is created by forming all the possible fractions of size at most one from the integers 1 to 7 and arranging them into increasing order of magnitude, adding zero as the starting element. ie. 0 1
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1, , , , , , , , , , , , , , , , , ,
This sequence has a number of interesting properties, amongst which are:
• the difference between any two consecutive fractions always has a numerator of 1. • the sequence is palindromic by addition: adding any two fractions which are the
same distance from respective ends of the sequence always gives a total of 1. Later mathematicians who investigated Farey’s sequence subsequently proved this result for all integers n, and deduced a number of results which were later to prove usefully applicable to Pick's Area Theorem. Expected outcomes The student will gain practice in the skills of addition, subtraction and ordering of fractions. The student is also encouraged to form mathematical conjectures, and to extend their mathematical knowledge by communicating hypotheses and considering the work of others. Required skills and knowledge It is assumed that the student is able to order, add, and subtract proper fractions. Required materials None. Follow-up teaching points More information on John Farey and on other mathematicians can be found at many web sites. In particular see http://www.cut-the-knot.com/ctk/Farey.html
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Farey Fractions Although the idea may seem a little bizarre to you, mathematicians often spend valuable TV time just playing around with numbers! Sometimes they discover some interesting things. John Farey (1766 – 1826) was one of those who found a small but interesting fact about fractions. He began by writing down all the possible fractions, no larger than 1, that could be made using the numbers 1 to 4 only. This set of fractions are shown right. Next, he went through the list and removed any fractions which cancelled down to one that appeared in a simpler form earlier in the list. For example, he crossed out because it cancels down to , which appears earlier in the list. 1. Cross out any other fractions which cancel down to ones that
appear earlier in the list. Hint: There should be six left.
2. In the space below, start with zero and write the list of fractions in ascending order.
Hint: If in doubt about whether one fraction is larger than another, you might try converting them both to a decimal. For example, on your HP30S enter by pressing 2 V 3 and then pressing $ [_] \ to convert to decimal.
0, _____, _____, _____, _____, _____, _____ 3. Choose any pair of consecutive fractions from your list (ones that are next to each
other) and find their difference. Repeat this for another two pairs of consecutive fractions, recording your results below.
Pair #1 Pair #2 Pair #3
____ - ____ = ____
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4. Compare your results to those of the person next you. See if you can spot the pattern which John Farey saw. Explain the pattern in words below:
5. Look at your list of fractions again, and add the first and last. Now add the second and
the second last. Keep going in to the centre. What do you notice? John Farey wondered if these patterns worked only for the fractions in his first list, or whether it might work for any such list. This type of question is characteristic of mathematicians. 6. This special sequence of ascending fractions was named the Farey sequence after John
Farey. The one you worked out using the numbers 1 to 4 is called the Farey sequence of order 4. Write down the Farey sequence of order 7. Don’t forget zero.
7. Test Farey’s conjectures (theories) using the Farey sequence of order 7. Test at least
three fractions from it when subtracting. Choose a different set of three from your neighbour so that you can compare results. Check also that the palindromic pattern works for your new set of fractions.
Results:
This is known as a palindomic pattern. A palindrome reads the same backwards as forwards. For example: “Able was I, ere I saw Elba”
Hint: Look at the numerators. Don’t expect too much: it’s a very simple pattern.
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So far you have only shown that this is true for fractions using the numbers 1 to 4 and 1 to 7. John Farey wondered if he could prove that this would happen for any list of fractions like this, no matter how large it was. This was not at all easy but after a great deal of work by many people it was successfully proven. In the process, a number of mathematical tools and methods were developed that were very useful in other areas.
This fascination with patterns in numbers and geometry has been a characteristic of mathematicians throughout history. You may be looking at this result and thinking to yourself that John Farey must have been a bit strange to have thought that this was interesting.
On the other hand, one of the first mathematicians to play with interesting patterns was Pythagoras…… We all know that he didn’t discover anything useful don’t we?
c a
b
a + b = c2 2 2
Teacher Notes Activity Summary Suitable for: Years 7 to 9 In Locker Patterns, the investigation of a seemingly simple numerical pattern resulting from a physical situation, leads to the examination of the factors of perfect squares. Expected outcomes The student will examine patterns in numbers, making conjectures regarding the governing rules and the mathematical reasons for them. Required skills and knowledge Knowledge of factors, recognition of simple patterns, particularly the square numbers. Required materials None. Follow-up teaching points The teacher may choose to follow this activity with further work with factors, factor pairs, factor trees, and prime numbers.
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Locker Patterns Isaac Newton Senior High School has just been renovated and contains exactly 1000 brand new lockers available for the students. The lockers are numbered 1 to 1000. Mr Jackson, the new mathematics teacher, decided to try an experiment with his maths class. He lined up the students in his class at the beginning of the corridor of lockers. Each student is to walk down the corridor one after the other. The first student is to open all the locker doors. He then goes to the end of the line to wait for his next turn. The second student follows and closes all the locker doors which have even numbers. The third student then follows and changes the state of every third locker door. ie. any locker with a number which is a multiple of three. Mr Jackson explained that by changing the state, she is to open any door that is closed and close any door that is open. The fourth student now changes the state of all the locker doors that are numbered with a multiple of four during his trip down the thousand locker doors. This continues until ten students have made their trip down the corridor of lockers. After the first ten students have made their trip, Mr Jackson posed the following question: “If we keep on doing this until this trip has been made by 1000 students, which locker doors will be open?”
Your task is to find out for him.
Through this sheet you will be investigating patterns in numbers and factors.
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A good start to solving any problem is to simplify it.
If you can’t see a pattern, check that you have the right numbers.
1. If you have a theory already on what the answer might be, record it below. 2. You may well be looking at “1000 lockers” and thinking that there is
no way you can solve this problem. If so, the best thing to do is to start with a smaller number. In this case, we will try just 17 lockers.
In the table below, O = Open and C = Closed. The state of the 17 lockers are shown for the first two students and for part of the third. Complete the table.
Locker No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Student 1 O O O O O O O O O O O O O O O O O Student 2 O C O C O C O C O C O C O C O C O Student 3 O C C Student 4 Student 5 Student 6 Student 7 Student 8 Student 9
Student 10 Student 11 Student 12 Student 13 Student 14 Student 15 Student 16 Student 17
3. List the numbers of the lockers which are open at the end of the
17th student’s run. Can you now see a pattern in the ones that are open? If so, record it.
Open: Pattern:
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4. If you now know the pattern then what is the answer to Mr Jackson’s original question? How many of the 1000 lockers will be open when the 1000th student has passed through?
5. The question you should now try to answer is: “Why does it happen?”
Hint: It has something to do with the number of factors that the different numbers have. In one column, try writing down all the factors, in order, for a couple of the locker numbers which were closed. Now, in a second column, do the same for some of the lockers that were open. Look at how many factors there are in each. You should try to explain in a logical fashion why this causes the pattern to occur, in terms of opening and closing.
i
Solutions Activity 1: SEDs and DEDs 1. SED always 5, DED always 25 except for 51. 2. a) 5, 25 b) 5, 25 3. Yes, the pattern works for them also. 4. SED pattern 7, 9, 3, 1 and repeat. DED pattern is
07, 49, 43, 01 and repeat. 5. a) 7, 07 b) 9, 49 c) 1, 01 6. SEDs are the same, DEDs are not. 7. a) 1 b) 3 c) 3 8. SED & DED patterns:
0 SEDs: all 0 DEDs: all 00 1 SEDs: all 1 DEDs: all 01 2 SEDs: 2 4 8 6, , ,
DEDs: 02 04 08 16 32 64 28 56 12 24 48
96 92 84 68 36 72 44 88 76 52
, , , , , , , , , , ,...
... , , , , , , , , ,
Note: the 02 does not repeat. 3 SEDs: 3 9 7 1, , ,
DEDs: 03 09 27 81 43 29 87 6183 49
47 41 23 69 07 21 63 89 67 01
, , , , , , , , , ...
... , , , , , , , , ,
4 SEDs: 4 6, DEDs: 04 16 64 56 24 96 84 36 44 76, , , , , , , , , 5 SEDs: all 5 DEDs: 05 25, 6 SEDs: all 6 DEDs: 06 36 16 96 76 56, , , , , 7 SEDs: 7 9 3 1, , , DEDs: 07 49 43 01, , , 8 SEDs: 8 4 2 6, , ,
DEDs: 08 64 12 96 68 44 52 16 28 24
92 36 88 04 32 56 48 84 72 76
, , , , , , , , , ,...
... , , , , , , , , ,
9 SEDs: 9 1, DEDs: 09 81 29 61 49 41 69 21 89 01, , , , , , , , ,
9. a) 0, 00 b) 4, 64 c) 3, 23 d) 6 e) 9 f) 8 g) 6 10. Because 22 = 4, every second entry in the patterns
for 2 is an entry in the pattern for 4.
i.e. 02 04 08 16 32 64 28 56 12 24
04 16 64 56 24, , , , , , , , , ...
, , , , ...
Similarly, because 23 = 8, every third entry in the pattern for 2 is an entry in the pattern for 8. Another pair of numbers which behave similarly is 3 and 9.
Activity 2: Congruent Triangles 1. Three angles is not enough.
Triangles may be congruent but may only be similar.
Subsequent questions have many possible answers. Consult your teacher if you are unsure of your results. Activity 3: Going Around in Circles 1. Check with your teacher if unsure 2. 12 3. a) 3 b) No contact with some points. No crossed
lines. 4. (12,3) is a square with 4 points of contact. (12,8)
is a triangle like (12,4) but with arrows the opposite direction.
5. They are a single point at zero. 6. Number of points column: 1, 12, 6, 4, 3, 12, 2,
12, 3, 4, 6, 12, 1 7. Symmetrical. Other patterns also valid.
8. No. points p = 1212H C F s. . .( , )
, where H.C.F.
means Highest Common Factor. 9. Actual points column: 3, 3, 9, 5, 2, 4, 8 Activity 4: Covering Up the Facts 1. a) 2.42 b) 1.59 c) 1.53 d) 1.74 2. a) 60.5 b) 117 3. 1.2185, 1.2689, 4.1% 4. a) 1.62 b) 1.49 c) 1.38 d) 1.80 5. Answers may be slightly different on the
nomogram 6. a) 75 b) 54 c) 75 d) 44 (your answers may vary
slightly)
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Activity 5: A Burning Question 1. 40.5% 2. 22.5% 3. 58.5% 4. 2.21, 58.5%, 1.29 5. 2.14, 28%, 0.599 6. 1.61, 76.5%, 1.23 7. 154.5 8. 0.84 9. 1.93, 50%, 0.965 10. 60 11. 25% 12. 47% 13. 15% 14. The 40 yr old. The six yr old is still growing. 15. 51% 16. 82% 17. Many possible. eg. Age 5 and 28 Activity 6: Farey Fractions 1. See next answer for list.
2. 0 14
13
12
23
34
1, , , , , ,
3. varying answers 4. The numerator is always one. There are others. 5. Pairs always add to 1.
6. 0 1
716
15
14
27
13
25
37
12
47
35
23
57
34
45
56
67
1
, , , , , , , , ,...
... , , , , , , , , ,
7. Patterns from 4 & 5 should still work.
Activity 7: Locker Patterns 1. Many possible answers 2. Compare with your neighbor. 3. Open: 1, 4, 9, 16. All others closed. 4. 31 lockers are open. ( 1000 rounded down) 5. All the perfect squares have an odd number of
factors. Each factor means a student changes state of that locker. Every two changes of state mean an open then a close, so this means that the perfect squares all have one change of state ‘left over’ and remain open.
