FOM 11 T16 – INDUCTIVE REASONING 1

FOM 11 T16 – INDUCTIVE REASONING 1

© Rob Ashby & Richard Obayashi 2015. Duplication by permission only.

MATH SPEAK - TO BE UNDERSTOOD AND MEMORIZED 1) INDUCTIVE REASONING = a process of using experience, experimentation and observation to gather

evidence which is used to make generalized conclusions called conjectures about patterns one notices.

2) CONJECTURE = a conclusion that is based on the evidence gathered through inductive reasoning. In science, a conjecture is called an hypothesis.

3) VALID = true.

4) COUNTEREXAMPLE = an example that proves the conjecture is not true for every possibility of a pattern.

5) EVEN NUMBER = a number that when divided by 2 produces an integer (a positive or negative entire number). e.g.

−56, −32, −20, −4, 2, 4, 6, 8

6) ODD NUMBER = a number that when divided by 2 produces a decimal number. e.g.

−55,−33,−27,−3, 5, 19, 61, 189

7) SUM = addition, or the result of adding numbers and or variables together.

8) DIFFERENCE = subtraction, or the result of subtracting numbers and or variables.

9) PRODUCT = multiply, or the result created when numbers and or variables are multiplied together. e.g. −7×5 = −35 −35 is the product of −7×5

10) MULTIPLE = a number created by multiplying a number by another number. e.g. 8×2 = 24 24 is a multiple of both 8 and 2 .

INDUCTIVE REASONING I) INDUCTIVE REASONING is the process of using one’s experience, observation and experimentation to gather

evidence which is used to make generalized conclusions called conjectures about patterns one notices. A CONJECUTURE is a conclusion based on the evidence gathered through inductive reasoning. In science, a conjecture is called an hypothesis. Conjectures are used to predict the results of similar experiences, observations and experiments. A) Humans have been using inductive reasoning for thousands of years to make predictions about the world they

live in and the people around them. People often used their long-term observations of the behavioral patterns of animals, plants, and their environment to make useful conjectures in the form of predictions. 1) First Nations people used their long-term observations of animal behavior to predict the arrival of the

seasons and changes in the weather. a) The return of small birds indicates that spring is almost here. b) The observation of crows returning to northern latitudes is an indication that winter is almost over. c) Seagulls tend to stop flying and take refuge on the coast when a storm is coming. d) Turtles tend to move to higher ground when large amounts of rain are one or two days away. e) The observation of cows lying down is an indication that rain is coming.

2) People have studied plant behavior to make predictions about their environment. a) Pine cone scales remain closed in high humidity and open in dry air. b) Oak leaves tend to curl in high humidity. c) The changing colour of the leaves of many trees is an indication of the start of fall and that winter is

approaching. 3) Sailors are able to predict the following day’s weather with a relatively high degree of accuracy by

observing the sunset. a) A red sunset very often predicts calm weather on the following day. [Ref. Picture on page 14 of your

text.] b) Thick dark grey clouds at sunset often predict rain or unsettled weather on the following day.



B) Inductive reasoning is extremely useful but limited. Each of the conjectures described above is not always true. A beautiful red sunset is sometimes followed by a rainy or stormy day. Not all turtles will move to higher ground when rain is coming.

II) USING INDUCTIVE REASONING TO MAKE MATH CONJECTURES A) Inductive reasoning can be used to predict the next picture in a series of pictures. The prediction you make is

also called the conjecture. 1) SAMPLE PROBLEMS 1: Study these examples carefully. Be sure you understand and memorize the

process used to complete them. 1) Study the series of pictures given below. Use inductive reasoning to make a conjecture about the next

picture in the pattern.

Analysis: The pattern contains three identical circles divided into quarters. The star is rotating through consecutive quarters in a clockwise pattern.

Ans: Thus, a reasonable conjecture is that the star will appear in the bottom left quarter as the picture below illustrates.

2) Study the series of pictures given below. Use inductive reasoning to make a conjecture about the value of x in the last picture in the pattern.

Analysis: The pattern consists of squares with numbers in each corner. 1) The product of the top numbers multiplied by the product of the bottom numbers creates the center number. OR 2) The product of the left numbers multiplied by the product of the right numbers creates the center number. OR 3) The result of multiplying the results of multiplying the numbers in opposite corners creates the center umber. Each option is illustrated below.

1) 2) 3)

Top Pair: 3 • 4 = 12Bottom Pair: 4 • 3 = 12Product of each result: 12 •12 = 144

OR

Left Pair: 3 • 4 = 12Right Pair: 4 • 3 = 12Product of each result: 12 •12 = 144

OR

Top left × Bottom right: 3 • 3 = 9Top right × Bottom left: 4 • 4 = 16Product of each result: 9 •16 = 144

Ans: Thus, a reasonable conjecture is that x will be 30 as illustrated these three options below.

Top Pair: 1• 2 = 2Bottom Pair: 5 • 3 = 15Product of each result: 2 •15 = 30

OR

Left Pair: 1• 5 = 5Right Pair: 2 • 3 = 6Product of each result: 5 • 6 = 30

OR

Top left × Bottom right: 1• 3 = 3Top right × Bottom left: 2 • 5 = 10Product of each result: 3 •10 = 30

3 5

64

43

360

5 64

4

23

240

4

4

3

144

3 1 2

5

3

x



2) REQUIRED PROBLEMS 1: Study these examples carefully. Be sure you understand and memorize the process used to complete them. {Answers for questions 1 & 2 are on page 8 of these notes} 1) Study the pattern below. Use inductive reasoning to make a conjecture and then draw the next figure.

2) Study the series of pictures given below. Use inductive reasoning to make a conjecture about the next picture in the pattern

3) Page 17: Question 1. {Answers are on page 527 of the text.}

B) IMPORTANT INFORMATION TO KNOW. BE SURE YOU UNDERSTAND AND MEMORIZE THE FOLLOWING.

1) Consecutive Whole numbers: e.g. 0, 1, 2, 3, 4, ... or ... ,23, 24, 25, 26, 27, ...

2) Consecutive Integers: e.g. ... ,−3,−2,−1, 0, 1, 2, 3, ...

3) Consecutive even numbers: e.g. ... ,−6,−4,−2, 0, 2, 4, 6, ...

4) Consecutive odd numbers: e.g. ... ,−5,−3,−1, 1, 3, 5, ...

5) Consecutive prime numbers: e.g. 2, 3, 5, 7, 11, 13, ...

6) Sum of two consecutive perfect squares: e.g. 4 + 9 or 49 + 64

7) Difference of three consecutive perfect squares: e.g. 9− 4−1 or 121−100−81

8) Product of two odd whole numbers: e.g. 1×5 or 23×49

9) Product of two integers: e.g. −8×5 or 0×27 or 14×91

10) Product of an even integer and an odd integer: e.g. −18×3 or 24×131

B) SAMPLE PROBLEMS 2: Study these examples carefully. Be sure you understand and memorize the process used to complete them. 1) Make three conjectures about this pattern of sums. Provide support for each conjecture.

1+ 3 = 43 + 5 = 85 + 7 = 12

STEP 1: Analyze the problem. This problem is asking you to make a conjecture (hypothesis) about the sum of two numbers. The numbers being added are odd numbers. The numbers being added are consecutive odd numbers. The numbers being added are consecutive odd natural numbers The results are even The results are divisible (can be divided) by 4.

Continued on the next page.

3 5

7



STEP 2: Write a conjecture(s). The sum of two odd numbers is an even number.

or The sum of two consecutive odd numbers is an even number.

or The sum of two consecutive odd natural numbers is an even number.

or The sum of two consecutive odd natural numbers is an even number that is divisible by 4.

STEP 3: Support your conjecture with at least two examples. The sum of two odd numbers is an even number. 7 + 9 = 16 ; 21+ 23 = 22

or The sum of two consecutive odd numbers is an even number. 17 +19 = 36 ; 55 + 57 = 112

or The sum of two consecutive odd natural numbers is an even number.

29 + 31 = 40 ; 55 + 57 = 112 or

The sum of two consecutive odd natural numbers is an even number that is divisible by 4.

29 + 31 = 40 where 40 ÷ 4 = 10 ; 55 + 57 = 112 where 112 ÷ 4 = 28

2) Make two conjectures about this pattern of differences. Provide support for each conjecture.

5−3 = 27−5 = 211−7 = 4

STEP 1: Analyze the problem. You are asked to make a conjecture (hypothesis) about the difference of two numbers. The numbers involved in the subtraction are odd natural numbers. The numbers involved in the subtraction are prime numbers. The results are even.

STEP 3: Write a conjecture. The difference of two consecutive odd natural numbers is an even natural number.

or The difference of two consecutive prime numbers is an even natural numbers.

STEP 3: Support your conjecture with at least two examples.

13−11 = 217−13 = 4

3) Make a conjecture about this pattern of products. Provide support for each conjecture.

8(9) = 7288(9) = 792888(9) = 7992

STEP 1: Analyze the problem. You are asked to make a conjecture (hypothesis) about the product of 9 and a number composed of only 8’s. The products (results) always begin with 7 and end with 2. The numbers 9’s between the 7 and 2 is one less than the number of 8’s.

Continued on the next page.



STEP 2: Write a solution to the problem. The product of 9 and a number composed of only 8’s always begins with 7 and ends with 2, where the number of 9’s between them is one less than the number of 8’s.

STEP 3: Support your conjecture with at least two examples.

8888(9) = 7999288888(9) = 799992888888(9) = 7999992

4) Make a conjecture about the product of two odd integers. EXAMPLE 2: page 8 of your text. STEP 1: Analyze the problem.

You are asked to make a conjecture (hypothesis) about the product of two odd integers. Solving this problem requires you to complete an investigation.

STEP 2: Complete an investigation. Your investigation must include at least three examples (more is better), each consisting of the multiplication of two integers. REMEMBER: Integers are positive and negative entire numbers. Be sure to include a variety of odd integers in your examples.

1×3 = 3 ; −7×5 = −35 ; −11×−9 = 99 Notice that the answer (product) in each example is an odd integer.

STEP 3: Write a conjecture. The product of two odd integers is an odd number.

or The product of two odd integers is an odd integer.

5) Make a conjecture about the difference of two consecutive perfect squares. EXAMPLE 3: page 9 of your text. STEP 1: Analyze the problem.

You are asked to make a conjecture (hypothesis) about the difference of two consecutive perfect squares. Solving this problem requires you to complete an investigation.

STEP 2: Complete an investigation. Your investigation must include at least three examples (more is better), each consisting of the subtraction of one perfect square from another.

4−1 = 3 ; 25−16 = 9 ; 64− 49 = 15 ; 121−100 = 21

Notice that the answer (difference) in each example is an odd integer.

STEP 3: Write a conjecture. The difference of two consecutive perfect squares is an odd integer.

C) REQUIRED PRACTICE 2: Pages 12 - 13: Questions 1, 2, 3, 7, 8, 9, 10 & 11. {Answers are on page 527 of the text.}

III) LIMITATIONS OF INDUCTIVE REASONING A) As useful as inductive reasoning is, it has significant limitations. Each of the conjectures described on page 1

of these notes is not always true A beautiful red sunset is sometimes followed by a rainy or stormy day; Not –all turtles will move to higher ground when rain is coming.



1) Because inductive reasoning is based upon experience, observation and experimentation, it is impossible to explore every possibility for a given pattern. This means it is possible to find an example that is not true thus limiting the validity of the conjecture. A COUNTEREXAMPLE is an example that proves the conjecture is not valid for every possibility of a given pattern.

B) SAMPLE PROBLEMS 3: Study these examples carefully. Be sure you understand and memorize the process used to complete them. 1) Make a conjecture about this pattern of sums. Test your conjecture.

3 + 5 = 85 + 7 = 127 +11 = 18

STEP 1: Analyze the pattern. You are asked to make a conjecture (hypothesis) about the sum of two numbers. The numbers being added are consecutive odd numbers. The numbers are consecutive prime numbers. The results are even.

STEP 2: Write a conjecture(s). The sum of two consecutive prime numbers is an even number.

STEP 3: Test your conjecture with at least four examples. 11+13 = 24 ; 13 +17 = 30 ; 17 +19 = 36 ; 2 +3 = 5

Notice that the last example produces an odd result. This is a counterexample. This means the conjecture as written is not valid and must be adjusted.

STEP 4: Adjust your conjecture. The sum of two consecutive ODD prime numbers is an even number.

2) Make conjecture about this pattern of differences. Test your conjecture.

−5−−3−−1 = −1−7−−5−−3 = 15−3−1 = 1

7−5−3 = −1

STEP 1: Analyze the pattern. You are asked to make a conjecture (hypothesis) about the difference of three consecutive odd integers. The results is either 1 or −1 .

STEP 2: Write a conjecture. The difference of three consecutive odd integers is either 1 or −1 .

STEP 3: Test your conjecture with at least four examples.

−11−−9−−7 = −5−3−−1−1 = −3−1−1−3 = −5

9−7−5 = −3

Notice that the above four examples produces an odd result that is not 1 or −1 . These are counterexamples. This means the conjecture as written is not valid and must be adjusted.

STEP 4: Adjust your conjecture. The difference of three consecutive odd integers is an odd integer.



3) Mary made this conjecture: The product of any two prime numbers is always an odd number. Is her conjecture valid? Justify your answer. STEP 1: Complete an investigation.

Your investigation must include at least three examples (more is better), each consisting of the multiplication of two prime numbers.

5×3 = 15 ; 13×5 = 65 ; 19×7 = 133 These three examples support Mary’s conjecture, which could lead one to conclude her conjecture is valid. However, 2 is a prime number and any number multiplied by 2 produces an even number as these counterexamples illustrate.

2×3 = 6 ; 2×11 = 22 ; 2×19 = 38

STEP 3: Adjust Mary’s conjecture. The product of any two odd prime numbers is always an odd number.

4) Hannah used the pattern given below to make this conjecture: The result of a difference of two consecutive perfect squares is always divisible by 3. Is Hannah’s conjecture valid? Justify your answer.

4−1 = 3 ; 25−16 = 9 ; 64− 49 = 15 ; 121−100 = 21 STEP 1: Analyze the pattern.

In each of the four examples given above the result can be divided by 3. Thus Hannah’s conjecture appears to be valid.


9− 4 = 5 ; 36−25 = 11 ; 49−36 = 13 The above are three counterexamples that invalidate Hannah’s conjecture.

STEP 3: Adjust Hannah’s conjecture. The difference of two consecutive perfect squares is an odd number.

5) Maxx used the pattern given below to make this conjecture: The result of a difference of any two perfect squares is an odd number. Is Maxx’s conjecture valid? Justify your answer.

4−1 = 3 ; 25−36 = −11 ; 16− 49 = −33 ; 121−64 = 57

STEP 1: Analyze the pattern. In each of the four examples given above the result is an odd number. Thus Maxx’s conjecture appears to be valid.


1− 4 = −3 ; 36−16 = 10 ; 25−81 = −56 ; 144−100 = 44

There are three counterexamples that invalidate Maxx’s conjecture.

STEP 3: Adjust Maxx’s conjecture. The difference between an odd perfect squares and an even perfect square is an odd number.

C) REQUIRED PRACTICE 3: Pages 22 - 24: Questions 1, 3, 4, 5, 8, 10, 14, 15 & 17. {Answers are on pages 527 & 528 of the text.}



ANSWERS TO THE REQUIRED PRACTICE

Required Practice 1 from page 2 1) Each row contains one more circle than the previous row. Thus, a reasonable conjecture is that the next row will

contain 4 circles as the picture below illustrates.

2) Each circle is divided into four quarters. A pattern of consecutive odd numbers moves top left to bottom right to bottom left. Thus the next picture in the pattern is illustrated below.

9



LAST then FIRST Name T16 – INDUCTIVE REASONING Block:

Answer these questions on your own paper. Show the process required to complete each problem to avoid receiving a zero grade for the question. Neatness Counts!!! (Marks are indicated in the italicized brackets.)

REMEMBER TO USE GRID PAPER FOR ALL ASSIGNMENTS!!! ∞ Copy the sentences numbered 1 - 8 then match it with the correct term listed below. (5)

Inductive Reasoning Conjecture Prime Numbers Natural Numbers

Whole Numbers Integers Perfect Square Difference

Sum Product Multiple Variable

1) A letter having a number you don’t know.

2) The set of positive entire numbers.

3) A statement you believe to be true that is based on observation or experimentation.

4) A number created by multiplying a number by itself.

5) The set of all possible entire numbers.

6) The math term that means multiply.

7) The math term that means addition.

8) Numbers that can only be divided by 1 and themselves to produce natural numbers.

∞ Answer these questions. 9) Study the diagram given below. Choose the letter of the picture that should replace the picture with the

question mark. Justify your conjecture. (2)

10) The triangle given below is an arrangement of numbers called Pascal’s Triangle. Study it carefully then predict the next row of numbers. Explain how you created the row of numbers. (2)

1 1 1

1 2 1 1 3 3 1

1 4 6 4 1 1 5 10 10 5 1

1 6 15 20 15 6 1

11) Anthony made this conjecture after his trip to the interior of British Columbia: As a person travels east, their altitude above sea-level increases. Is this conjecture valid? Justify. (2)

12) Martha makes this conjecture: The square of an even integer is an even natural number. Is her conjecture valid? Justify your answer. (3)

13) Billy made this conjecture: The sum of two consecutive prime numbers is an even number? Is his conjecture valid? Justify your answer. (2) Continued on the next page.



14) Study this pattern of sums, then answer the questions below: 22 +1= 5 ; 42 +1=17 ; 62 +1= 37

a) The results are all what kind of numbers (odd is not the answer)? (1) b) Use the above pattern to make a conjecture to predict the results of similar sums. (1) c) Continue the pattern for 2 more examples. (2) d) Is your conjecture valid? If not provide a counterexample. (2)

15) Study this pattern of sums, then answer the questions below:

−4+ −3+ −2 = −9 1+ 2+ 3= 6−3+ −2+ −1= −6 2+ 3+ 4 = 9−1+ 0+1= 0 3+ 4+5=12

a) What kind of numbers are being added together? (1)

b) What do the results have in common? (1)

c) Make a conjecture to predict the results of similar sums. Justify. (2)

16) Investigate then write a conjecture about the product of three consecutive natural numbers. (3)

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FOM 11 T16 – INDUCTIVE REASONING 1

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