Math 11 Academic : Principles of Mathematics 11

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Teac Teacher: ___________________

Student: ___________________

Math 11 Principles of Mathematics 1 1

Student Workbook 3x + 3 − x = 1

Sharpe Mathematics  2017

FOREWORD

This Student Student Workbook for Mathematics 220 2201 student-centered 201 was written as a studentresource for the course. In doing so, this workbook can be used by students to develop and practice their skills in an easily followed manner. This workbook can be used as a portfolio of student work, containing many practice questions and review material, and will enable students to keep their work work organized as opposed to having countless worksheets to supplement the notes and extra practice used to ensure mastery of skills. It will hopefully allow students to gain a better grasp of the material by seeing complete examples worked out in full, as well well as being able to practice the concepts with additional exercises. Where suitable, there are also hints, suggestions, and thought questions to keep students on the right track and to hopefully avoid common pitfalls along the way. The use of graphing calculators calculators enables students to realize the importance and applicability of technology in mathematics. Where necessary, the appropriate keystrokes have been included to aid with this objective. In addition, students considering taking Advanced Mathematics 3201 201 are encouraged to try the more challenging problems for further enrichment enrichment and extension of the topic. I would like to extend my sincere gratitude to my wife, Debbie, and my children, Joshua and Eric, for their nevernever-ending support and constant encouragement to complete this massive undertaking. undertaking. I would also like like to thank my students for thei heir feedback and their suggestions for improvement. improvement. Thank you! I hope you find this to be a useful resource for both students and teachers alike. I welcome your your feedback. Todd Sharpe (B.Sc., B.Ed., M.Ed.) e-mail: [email protected] webpage: https://www.facebook.com/sharpemathematics

This book is copyright. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the author. Sharpe Mathematics  2017

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Mathematics 2201 Mathematics 1201

TABLE ABLE OF CONTE ONTEN TENTS CHAPTER 1: INDUCTIVE & DEDUCTIVE REASONING 1.1 1.2 1.3 1.4 1.5 1.6

MAKING CONJECTURES: INDUCTIVE REASONING ............................... EXPLORING THE VALIDITY OF CONJECTURES ..................................... USING COUNTEREXAMPLES TO CONJECTURES .................................. PROVING CONJECTURES: DEDUCTIVE REASONING .............................. PROOFS THAT ARE NOT VALID ......................................................... REASONING TO SOLVE PROBLEMS AND PUZZLES ..............................

1 6 7 8 12 14

CHAPTER 2: PROPERTIES OF ANGLES AND TRIANGLES 2.0 2.1 2.2 2.3 2.4 2.5 2.6

REVIEW ........................................................................................... EXPLORING PARALLEL LINES ........................................................... ANGLES FORMED BY PARALLEL LINES .............................................. ANGLE PROPERTIES IN TRIANGLES .................................................... ANGLE PROPERTIES IN POLYGONS ..................................................... EXPLORING CONGRUENT TRIANGLES .................................................. PROVING CONGRUENT TRIANGLES .....................................................

18 20 24 25 30 36 40

CHAPTER 3: ACUTE ANGLE TRIGONOMETRY 3.1

3.2 3.3 3.4

A. THE PYTHAGOREAN THEOREM (REVIEW) .................................... 45 B. LABELLING TRIANGLES (REVIEW) ........... ................................... 47 C. THE PRIMARY TRIGONOMETRIC RATIOS (REVIEW) ...................... 48 D. FINDING ANGLES USING TRIGONOMETRY ..................................... 51 E. FINDING SIDES USING TRIGONOMETRY ....................................... 53 THE SINE LAW ................................................................................ 60 THE COSINE LAW ............................................................................ 64 A. APPLICATIONS OF THE SINE AND COSINE LAW .......................... 68 B. SOLVING PROBLEMS INVOLVING MORE THAN ONE TRIANGLE ....... 70

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CHAPTER 4: RADICALS 4.1

4.2 4.3 4.4 4.5

WORKING WITH RADICALS .................................................................... 80 A. ORDERING RADICALS ...................................................................... 80 B. RESTRICTIONS ON VARIABLES ......................................................... 81 C. PERFECT CUBES AND CUBE ROOTS .................................................. 82 D. MIXED AND ENTIRE RADICALS .......................................................... 84 E. ORDERING RADICALS ....................................................................... 87 ADDING AND SUBTRACTING RADICAL EXPRESSIONS ............................... 90 A. MULTIPLYING RADICAL EXPRESSIONS ............................................... 92 B. DIVIDING RADICAL EXPRESSIONS ...................................................... 95 SIMPLIFYING ALGEBRAIC EXPRESSIONS INVOLVING RADICALS ................ 98 SOLVING RADICAL EQUATIONS ............................................................... 101

CHAPTER 5: STATISTICAL REASONING 5.1 5.2 5.3 5.4 5.5

5.6

EXPLORING DATA .................................................................................. 108 FREQ TABLES, HISTOGRAMS & FREQ POLYGONS ................................... 111 STANDARD DEVIATION .......................................................................... 115
STANDARD DEVIATION OF GROUPED DATA (ENRICHMENT) ................. 121 THE NORMAL DISTRIBUTION ................................................................. 122 A. Z-SCORES ...................................................................................... 128 B. USING Z-SCORE TABLES TO DETERMINE PERCENTAGES ................... 132 C. USING PERCENTAGES TO DETERMINE Z-SCORES .............................. 139 CONFIDENCE INTERVALS ...................................................................... 142

CHAPTER 6: QUADRATIC FUNCTIONS 6.1

6.2 6.3

6.4

6.5

A. EXPLORING QUADRATIC RELATIONS ................................................ B. VERTICAL STRETCH y = ax2 ........................................................... C. IDENTIFYING QUADRATIC FUNCTIONS .............................................. D. EXPLORING THE EFFECTS OF a, b AND c ......................................... STANDARD FORM OF A QUADRATIC FUNCTION ..................................... A. FACTORED FORM OF A QUADRATIC FUNCTION ................................. B. FACTORED FORM OF A QUADRATIC FROM ITS GRAPH (a=1) .............. C. FACTORED FORM OF A QUADRATIC FROM ITS GRAPH (a≠1) .............. D. FACTORED FORM OF A QUADRATIC (WORD PROBLEMS) ................... A. VERTEX FORM OF A QUADRATIC FUNCTION ...................................... B. VERTEX FORM OF A QUADRATIC FUNCTION FROM ITS GRAPH ........... C. DETERMINING THE NUMBER OF X-INTERCEPTS ................................ MAXIMUM/MINIMUM WORD PROBLEMS .............................................

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146 148 150 152 155 162 167 168 169 172 179 184 185

Mathematics 2201 Mathematics 1201

CHAPTER 7: QUADRATIC EQUATIONS 7.1 7.2

7.3 7.4

SOLVING QUADRATIC EQUATIONS BY GRAPHING .................................. SOLVING QUADRATIC EQUATIONS BY FACTORING ................................ A. FACTORING REVIEW ...................................................................... B. SOLVING QUADRATIC EQUATIONS BY FACTORING .......................... C. DETERMINING A QUADRATIC EQUATION FROM X-INTERCEPTS ........ A. THE QUADRATIC FORMULA .......................................................... B. THE NATURE OF ROOTS OF A QUADRATIC EQUATION .................... WORD PROBLEMS USING QUADRATIC EQUATIONS ..............................

193 202 202 208 210 211 216 221

CHAPTER 8: PROPORTIONAL REASONING 8.1 8.2 8.3 8.4 8.5 8.6

COMPARING AND INTERPRETING RATES ............................................. SOLVING PROBLEMS THAT INVOLVE RATES ........................................ SCALE DIAGRAMS .............................................................................. SCALE FACTORS AND AREAS OF 2-D SHAPES ..................................... SIMILAR OBJECTS: SCALE MODELS & DIAGRAMS ................................ SCALE FACTORS AND 3-D OBJECTS ....................................................

227 233 235 237 241 243

SOLUTIONS TO PRACTICE EXERCISES ..................... 248

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Chapter 1: Inductive and Deductive Reasoning

1

CHAPTER INDUCTIVE AND DEDUCTIVE REASONING Contents.............................................................. Contents

1.1

MAKING CONJECTURES: INDUCTIVE REASONING .................. 1

1.2

EXPLORING THE VALIDITY OF CONJECTURES ....................... 6

1.3

FINDING A COUNTEREXAMPLE TO A CONJECTURE ............ 7

1.4

PROVING CONJECTURES: DEDUCTIVE REASONING ............. 8

1.5

PROOFS THAT ARE NOT VALID ....................................... 12

1.6

REASONING TO SOLVE PROBLEMS AND PUZZLES ............ 14

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1.1 MAKING CONJECTURES: INDUCTIVE REASONING If the same result occurs over and over again, we may conclude that it will always occur. This kind of reasoning is called INDUCTIVE REASONING. REASONING. Inductive reasoning is reasoning based on detailed facts and general general principles, which are eventually used to reach a specific conclusion. It is a type of reasoning by which generalizations are drawn from general patterns in observed data. Mathematicians use inductive reasoning to guess at what might be true. For example, example, consider the sum of positive consecutive odd numbers starting at 1, you get the pattern: 1+3=4 1+3+5=9 1 + 3 + 5 + 7 = 16 All of the the sums appear to be perfect squares squares: 4 = 22, 9 = 32, 16 = 42. You might make a conjecture that the sum of any number of positive consecutive odd numbers, beginning with 1, is a perfect square. A CONJECTURE is a testable expression that is based based on gathered evidence but has has not yet been proven proven. More support strengthens a conjecture but does not prove it. We have have only demonstrated this pattern for a few examples, but but what if there is a later sum which is not a perfect square? Is this conjecture true for all?

PRACTICE 1.

Study the pattern and use your conjecture to determine the missing value. a. 1 x 1 = 1

2.

b. 1 x 9 + 2 = 11

c. 92 = 81

11 x 11 = 121

12 x 9 + 3 = 111

992 = 9801

111 x 111 = 12321

123 x 9 + 4 = 1111

9992 = 998 001

1111 x 1111 = _______

1234 x 9 + 5 = _______

99992 = ____________

Study the pattern and use your conjecture to determine the two missing values. a. 2, 4, 7, 14, 17, 34, 37, _____, _____

b. 3, 4, 6, 9, 13, 18, _____, _____

c. 2, 6, 15, 31, 56, _____, _____

d. 1, 2, 5, 14, 41, _____, _____

e. 3, 5, 11, 29, 83, _____, _____

f. 1, 1, 2, 3, 5, 8, 13, _____, _____

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3

Examine the matchstick pattern:

a. Complete the table below for the first 5 figures. figures. Number of triangles Number of matchsticks

1 3

2

3

4

5

6

7

b. What numeric numeric pattern do you see in the table? This would be your conjecture.

c. Extend the pattern for Figures 6 and 7 to determine the number of matchsticks. matchsticks. d. How did you use inductive reasoning to develop your conjecture?

4.

Examine the matchstick pattern:

a. Complete the table below for the first five figures. figures. Pattern number Number of matchsticks

1

2

3

4

5

...

200

b. State your conjecture for the pattern.

c. Use your conjecture to determine the number of matchsticks in the 200 200th figure. figure. d. How did you use inductive reasoning to develop your conjecture?

5.

Examine the dot pattern: a. Complete the table below for the first three figures. Pattern number Number of dots

1

2

3

...

100

b. State your conjecture for the pattern.

c. Use your conjecture to determine the number of dots in the 100th figure. figure. Sharpe Mathematics  2017

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6.

4

Examine the cube pattern: a. Complete the table below to determine the number of visible faces. Number of cubes Number Numb er of visible faces

1

2

3

4

5

...

200

b. State your conjecture for the pattern. c. Use your conjecture to determine the number of visible faces in the 200th figure. figure.

7.

Examine the cube pattern: ,

,

a. Complete the table below to determine the number of cubes. Pattern number Number of cubes

1

2

3

...

50

b. State your conjecture for the pattern. c. Use your conjecture to determine the number of cubes in the 50th figure. figure. 8.

A bathtub initially held 240 litres of water, water, but someone pulled the plug to let the water drain out. Water drained from the bathtub at a constant rate until completely drained, as shown in the following table. Time (in minutes) after plug pulled Volume (in litres) remaining in bathtub bathtub

0 240

3 180

6 120

9 60

12 0

Make a conjecture about the rate at which the water drained from the bathtub.

9.

The total cost of getting the internet installed in your home and then using it regularly is made up of a fixed installation fee plus a charge for each hour online. The relationship between cost and time is shown in the table below. Time online, online, in hours Total cost, cost, in dollars

10 39

20 54

50 99

80 144

200 324

Make a conjecture about the cost of internet use. Test your conjecture using different values. values.

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5

Krista’s cell phone company charges an initial base fee plus a set rate per minute. Several bills are shown based on cell phone usage. Time, Time, in minutes Total cost, cost, in dollars

0 25

10 26.20

20 27.40

30 28.60

40 29.80

a. Make a conjecture about about the cost of using the cell phone.

b. Test your conjecture by determining Krista’s bill if she used her phone for 100 minutes.

11.

Make a conjecture about the sum of two odd numbers. Test your conjecture with several examples.

12.

Make a conjecture about the product of an even number and an odd number. number. Test your conjecture with several examples.

13.

Nancy construct constructed onstructed several scalene triangles and measure measured easured each angle and each side. What conjecture could she make make about the relationship relationship between the smallest angle and the the measure of the smallest side?

14.

Dann construct constructed onstructed several pairs of intersecting lines and measure measured easured the pairs of vertically opposite angles. What conjecture could he have made?

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1.2 EXPLORING THE VALIDITY OF CONJECTURES A conjecture is based on evidence that is available at the time, but can be supported or even disproved with additional evidence. Optical illusions, for instance, can “trick” your eyes and can result in an invalid conjecture. conjecture. Let’s consider the sequence 5, 12 12, 19, 19, 26 26, … The next next term appears to be 33, since the given numbers are all increasing by 7. However, However, what if the sequence represents the dates on a calendar? If so, then then the next number could be 2 (if 31 days in a month), 3 (if a 3030-day month), month), 5 (if Feb with a 2828-day month) month) or even 4 (if Feb during a leap year with 29 days) days)! With more information made available, a conjecture can be revised or modified in light of new evidence. Mathematicians, athematicians, understandably, are not always satisfied with inductive reasoning as it can lead to guesses, or false conjectures. This will be explored in greater depth in the coming sections.

PRACTICE 1 5.

Make a conjecture regarding the length of both horizontal lines. Test the validity of your conjecture using a ruler.

C

16. 16.

Make a conjecture conjecture regarding the lengths of AB and CD. Test the validity of your conjecture using a ruler. A

17. 17.

B

Make a conjecture regarding the length of both horizontal lines. Test the validity of your conjecture using a ruler.

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D

Chapter 1: Inductive and Deductive Reasoning

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1.3 FINDING A COUNTEREXAMPLE TO A CONJECTURE As we have seen, sometimes your original conjecture may may be incorrect and require revision. One way of determining if a conjecture is wrong is by finding a counterexample. A COUNTEREXAMPLE is an example that does not support or contradicts your conjecture. A single counterexample counterexample is all that is needed to disprove a conjecture. If a counterexample is found, you would then be required to revise your conjecture.

5 cm

A square that is also a rectangle.

5 cm

5 cm

5 cm

For example, Emily conjectures that if all squares are rectangles, then it must also be true that all rectangles are squares. As you can see from the drawing below, this is not always true as the second diagram satisfies the conditions for a rectangle but is clearly not a square.

5 cm

A rectangle that is also a square.

8 cm

A rectangle that is NOT a square.

PRACTICE 1 8.

For each inductive reasoning example: example:

• Write two examples that confirm the conjecture. • Write one counterexample that disproves it.

a. In Newfoundland, we get more snow than the previous year during our winter season. b. The Toronto Blue Jays have improved every season since 2012. c. If a number is not positive, positive, then it must be negative. d. The square of a number is always greater than that number. e. The sum of of two prime numbers will always result in an even number. f. All natural numbers can be expressed as the sum of consecutive natural numbers. g. A number divided by another number will result in that number becoming smaller. h. If the diagonals of a quadrilateral quadrilateral bisect each other, then the quadrilateral is a rectangle. i. If the diagonals of a quadrilateral are perpendicular, then the quadrilateral is a square.

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1.4 PROVING CONJECTURES: DEDUCTIVE REASONING REASONING Mathematicians often use proof to verify that a conjecture is true for all instances, instances, not not just for a few specific examples examined. examined. Proof is also known as DEDUCTIVE REASONING, and is reasoning from proven facts using logically valid steps to arrive at a conclusion based on statements we believe to be true. Proofs often help justify why a certain certain pattern is true and helps to deepen your level of understanding. understanding. For a statement to be deemed a proof, it must be valid for ALL cases, and no counterexample exists to disprove it. INDUCTIVE INDUCTIVE REASONING vs. DEDUCTIVE REASONING As we have seen, inducti inductive nductive reasoning makes conclusions based on observations or patterns. patterns. Inductive reasoning, however, does not guarantee that a conjecture is valid for all cases, regardless of how many examples support the conjecture. As a result, inductive reasoning can not be used to prove a conjecture. Deductive reasoning makes conclusions based on statements we believe to be true. Deductive reasoning is a method of proving a conjecture is true for all instances. Furthermore, inductive inductive reasoning makes conclusions from a specific specific to a general case whereas deductive reasoning goes from general to specific specific. pecific. Let’s consider an example to demonstrate the difference between inductive and deductive reasoning. In the first table we will use specific numbers and work through each instruction. instruction. In the second table we will perform the same steps, but this time using a more general approach. Complete the table and make a conjecture using inductive reasoning.

Prove your conjecture using deductive reasoning.

Instructions

Case Case 1

Case 2

Case 3

Instructions

General case

Choose any natural number

5

11

16

Choose any natural number

n

Add 3

8

14

19

Add 3

n+3

Multiply by 2

16

28

38

Multiply by 2

2(n + 3) = 2n + 6

Subtract original number

11

17

22

Subtract 2

9

15

20

Subtract 2

(n + 6) – 2 = n + 4

Subtract original number

4

4

4

Subtract original number

(n + 4) – n = 4

Subtract original (2n + 6) – n = n + 6 number

Based on the second table, using deductive reasoning we have proved that regardless of the original number chosen, the final answer is always 4.

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Deductive reasoning can also be used to prove algebraic and geometric conjectures. EXAMPLE 1 Prove deductively that when two odd numbers are added, the result is an even number. Solution: Let 2a + 1 = first odd number and 2b + 1 = second odd number ∴

(2a (2 + 1) + (2b (2 + 1) =2 2a + 2 2b + 2 2(a + b + 1) = 2(

Since 2 is a factor of the sum, sum, the result must be an even number.

EXAMPLE 2

D

E

C

Given quadrilateral ABCD is a square and ΔABE ABE is an Prove deductively that when two odd numbers are added, the result is an even number. equilateral triangle. Use a twotwo-column proof to prove that ΔAED AED is isosceles. Solution: 2n + 1 = first odd number

A

B

2m + 1 = second odd number Solution: (2n + 1) + (2m + 1) = 2n + 2m + 2 = 2(n + m + 1) Statement Justification Since 2 is a factor, the result must be an even number. AB = AD Properities of a square AB = AE

Properties of an equalteral triangle

AD = AE

Transitive property

ΔADE ADE is Isosceles

Triangle with 2 equal sides

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PRACTICE 19. Complete the table and make a conjecture using inductive reasoning. Instructions

Case 1

Case 2

Case 3

Prove your conjecture using deductive deductive reasoning. Instructions

Choose any natural number

Choose any natural number

Square it

Square it

Add original number Divide by by original number

Add original number Divide by original number

Add 7

Add 7

Subtract original number

Subtract original number

General case

Conclusion:

20. Complete the table and make a conjecture using using inductive reasoning. Instructions

Case 1

Case 2

Case 3

Prove your conjecture using deductive reasoning. Instructions

Choose any natural number

Choose any natural number

Add 2

Add 2

Multiply by 2

Multiply by 2

Subtract by 2

Subtract by 2

Divide by 2

Divide by 2

Subtract original number

Subtract original number

General case

Conclusion:

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21 .

11

Gina wishes to prove that when an odd number is squared, the result is an odd number. Her partial proof is below but it is incomplete. incomplete. Fill Fill in the missing parts to make it true. Let 2n ∈N represent any odd number. 2 + 1, n∈

We can use the notation 2n to denote even numbers numbers, and 2n+1 +1 to denote odd numbers numbers, where n∈ ∈N

(2n + 1)2 = (2n + 1)(2n + 1) = = 2( Since

) + 1 is in the form

,

thus thus when we square an odd number we end up with an odd number number. er.

22. 22.

Use deductive reasoning to prove the following conjecture: “The sum of any three consecutive integers is a multiple of three”.

23. 23.

Use deductive reasoning to prove the following conjecture: “The sum of any two odd numbers is an even number”. number”.

24. 24.

Use deductive reasoning to prove the following conjecture: “The product of any two odd numbers is an odd number”.

25. 25.


 Given line BX bisects ∠ABC. Complete the twotwo-column proof to prove that ∠1 ≅ ∠3.

Statement Statement
 BX bisects ∠ABC

A

Justification

X 1

∠1 ≅ ∠2

3

B

2

C

∠2 ≅ ∠3 ∠1 ≅ ∠3

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1.5 PROOFS THAT ARE NOT VALID Sometimes when trying to provide a proof, an error may occur which can lead to an invalid conclusion. Some examples of common errors include: • • • • •

A false assumption, assumption, meaning what you are trying to prove has an error A mathematical error in reasoning, such as dividing by zero Circular reasoning, which is reasoning that follows from what you are trying to prove A calculation error Faulty logic logic

PRACTICE 26. 26.

One of the most famous invalid proofs is the attempt at proving 2 = 1. The proof that follows contains an error. Can you find it? STEP 1 : Let a = b a+a=a+b STEP 2 : 2a = a + b STEP 3 : STEP 4 : 2a − 2b = a + b − 2b STEP 5 : 2(a − b ) = a − b STEP 6 :

2(a − b ) a−b = ( a − b ) (a − b )

STEP 7 :

2( a − b ) a−b = (a−b) (a−b)

STEP 8 :

2=1

The error is located in STEP

Add "a" to both sides Simplify LHS Subtract "2b" from both sides Factor out "2" from LHS Divide both sides by (a − b ) Simplify

.

The error is

27.

.

Roger wishes to prove that the sum of an odd number and an even number will result in an odd number. His His proof is below but contains an error. Find and correct the error to make it true. Let 2n, ∈N represent any even number. 2 , n∈ Let 2n ∈N represent any odd number. 2 + 1, n∈ ∴ (2n) (2 ) + (2n (2 + 1) = 4n 4 +1 Since 4n is even then 4n+ 4 + 1 is odd and in the form 2n + 1. 1. Thus Thus when we add an odd number and an even number, the result will be an odd number. number.

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Chapter 1: Inductive and Deductive Reasoning 28. 28.

13

Here is another famous invalid proof attempting to prove 2 = 1. The proof that follows contains an error. Can you find it?

STEP 1 : STEP 2 :

Let a = b a×a = a×b

STEP 3 :

a 2 = ab

Multiply both sides by "a" Simplify

STEP 4 : a 2 − b 2 = ab − b 2 STEP 5 : (a − b )(a + b ) = b (a − b ) (a − b )(a + b ) b (a − b ) STEP 6 : = (a − b) (a − b) STEP 7 :

Divide both sides by (a − b )

( a − b )(a + b ) b ( a − b ) = (a−b) (a−b)

STEP 8 : STEP 9 : STEP 10 :

Simplify Simplify Replace "a" since a = b Simplify

a+b=b (b) + b = b 2b = b 2b b = b b 2=1

STEP 10 : STEP 11 : The error is located in STEP 29. 29.

Subtract "b 2 " from both sides Factor both sides

Divide both sides by "b"

. The error is

.

Identify the error made in the following proof to prove that 5 = –5 : Let’s start by assum 5 = –5 assuming ssuming that Squaring Squaring both sides we get: get: (5) 2 = (– (–5)2 Thus 25 = 25 25 = 25 Square root of both sides: This statement is true, 5 = –5 true, so

30. 30.

must be correct.

Ashley is trying to prove prove the number trick below. Every time she tries the trick, she gets 5. Her proof, however, however, does not give the same result. What was Ashley’s error? Instructions

Case 1

Case 2

Case 3

General case

Choose any natural number

5

8

12

n

Add 3

8

11

15

n+3

Multiply by 2

16

22

30

2n + 6

Add 4

20

26

34

2n + 10

Divide by 2

10

13

17

2n + 5

Subtract original number

5

5

5

n+5

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1.6 USING REASONING TO SOLVE PROBLEMS & PUZZLES differencess between inductive and Based on what we have examined so far, there are clear difference deductive reasoning. Use what you have learned to help with making a decision for each of the practice exercises that follow.

PRACTICE 31. 31.

Classify each of the following as using either deductive or inductive reasoning. a.

I heard heard a lot of barking this morning. morning. The neighbor’s dog must’ve been alarmed by something, since he rarely barks.

b.

All cats purr. Mittens is a cat, so she purrs.

c.

The majority rules in the House of Commons in Parliament. There are are an odd number of seats in the House, thus there is no way there will be a tie when voting on issues.

d.

Based on a random survey of 30 graduating students at Highlander High School, 76.3% indicated that they preferred a “Las Vegas” Vegas” theme for their grad. Therefore, approximately 76% of all graduates in the school will vote for the “Las Vegas” theme.

e.

All mollusks are invertebrates. Snails are mollusks, so snails must be invertebrates.

f.

Paul is taller than Kelly. Kelly is taller than Linda. Therefore, Paul is taller than Linda.

g.

I will not rest until I get the signatures of every player in the roster for the 201720172018 Toronto Maple Leafs. There are a total of 16 players. Thus, I will need to get 16 signatures.

h.

Dogs almost always chase after cats. There is a dog, so it will most likely chase cats.

i.

No cod can survive in fresh water. Windsor Lake is a fresh water lake, so there are no cod present in it.

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32. 32.

15

Choose the image that completes the pattern:

Which shape below completes the sequence?

A

B

C

D

E

33. 33.

A snail is at the bottom of a well and wants to get out. He manages to crawl up the wall 3 feet each day, but at night night slips back down 2 feet when he rests. If the well is 30 feet deep, how long will it take the snail to get out?

34. 34.

Two girls, girls, Lisa and Kim, and two boys, boys, Bob and Jack, are all on school varsity teams. One is on the basketball team, one plays volleyball, volleyball, another plays softball, and another is on the hockey team. During a sports banquet, the four of them sat around a square table with one person at each side.

• • • •

The basketball player sits on Lisa’s left The softball player sits across from Bob Kim and Jack Jack sit next to each other A girl sits on the left of the volleyball player

Based on this information, who who is the hockey player?

35. 35.

Adam, Billy Billy, illy, Chris and Debra all love ice cream.

• • • • •

Each person likes just one flavor of ice cream The flavors they like are vanilla, chocolate, strawberry and orange Billy likes orange The tallest person likes chocolate, and the shortest likes strawberry Chris is taller than Debra, but shorter than Adam and Billy

Based on this information, what flavor of ice cream does each person person like? like?

Sharpe Mathematics  2017

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Mathematics 2201

Chapter 1: Inductive and Deductive Reasoning

36. 36.

What number should appear in the centre of Figure 4?

3

5

4

2

2

8

18

21 3

6

3

4

1

4

6

50

5

Figure 1

37. 37.

16

2

2

Figure 2

Figure 3

Figure 4

The following pattern appeared on an IQ test.

Which shape below continues the sequence?

A

B

C

38. 38. a. Place the numbers 1 through 9 in the circles so that the sum of the numbers on each side of the triangle is 17.

D

E

b. Complete the Sudoku puzzle so that each row, each column and each of the nine 3×3 grids contain one instance of each of the numbers 1 through 9.

3 7 5

1 3 6 9 8 4 3 3 8 9 1 2 6 4 5 1 6 7 5 2 9 6 8 9 4 Sharpe Mathematics  2017

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6 4 8

7 4

1 5 3

Mathematics 2201

Chapter 2: Properties of Angles and Triangles

17

CHAPTER PROPERTIES OF ANGLES AND TRIANGLES Contents.............................................................. Contents

2.0 REVIEW .............................................................................. 18 2.1

EXPLORING PARALLEL LINES .............................................. 20

2.2 ANGLES FORMED BY PARALLEL LINES ................................ 24 2.3 ANGLE PROPERTIES IN TRIANGLES ................................. 25 2.4 ANGLE PROPERTIES IN POLYGONS .................................. 30 2.5 EXPLORING CONGRUENT TRIANGLES .............................. 36 2.6 PROVING CONGRUENT TRIANGLES .................................. 40

Sharpe Mathematics  2017

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Mathematics 2201

Chapter 2: Properties of Angles and Triangles

18

2.0 REVIEW In this section our focus will be mainly on geometry as it relates to angles and triangles. There are, however, a number of terms and concepts that you should know from junior high. The following is presented for review purposes only, as you will need to have them available and at your disposal for many of the questions in this chapter.

ACUTE ANGLE an angle measuring less than 90° 90°. That is, 0° < x° x° < 90° 90°

RIGHT ANGLE an angle measuring exactly 90° 90°. That is, x° = 90° 90°

OBTUSE ANGLE an angle measuring measuring between 90° 90° and 180° 180°. That is, 90° 90°