Chapter 1 Inductive and Deductive Reasoning Pages/Teacher Pages/Margot Crewe...Chapter 1 Inductive...

1.1MakingConjectures(InductiveReasoning).notebook

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Chapter 1Inductive and Deductive

Reasoning


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Patterns are widely used in mathematics to reach logical conclusions.

This type of reasoning is called inductive reasoning.


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Predict the next number in these sequences:

1, 5, 25, 125, _____

5, 2, 4, 13, ______


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Inductive Reasoning - using specific data/information to draw general conclusions (patterns)

- characterized by drawing a general conclusion (making a conjecture) from repeated observations of specific examples. The conjecture may or may not be true.


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Conjecture- testable hypothesis based on available

evidence not yet proven.

- Conjectures can be tested and those that appear to be valid allow us to make conclusions.

• An educated guess (hypothesis)


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- Conjectures can turn out to be right or wrong.

- Additional evidence may support a conjecture but does not prove it.


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Examples:

The math class consists of 20 boys and 10 girls. Can a conjecture be made about the composition of the school?

Conjecture:1. There are more boys than girls at this school.

2. There are twice as many boys as girls at this school.


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Example: Draw two parallel lines and a transversal.

a) Measure the set of opposite angles as shown. What conjecture can you draw?

b) How could you strengthen your conjecture?


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Example 1: Study the following diagrams. Determine a possible relationship between the figure number and the number of small triangles present. (See textbook page. 6)


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Figure Number 1 2 3 4 5# of Small Triangles


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Georgia makes a conjecture that there will be 100 triangles in the tenth figure. Is she correct? To answer, organize the information about the pattern in a table like the one below.

Figure Number 1 2 3 4 5# of Small Triangles

Make a conjecture about the numeric pattern in the table.


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Can you come up with a second conjecture?

What do you think about Georgia‛s conjecture of 100 triangles in the tenth figure?

Use your conjecture to predict how many small triangles would probably be present in figure 12.


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Study the following data about precipitation in Vancouver. (See textbook page 7)

Use inductive reasoning to make some conjectures about precipitation in Vancouver.

What mathematical calculations could you use to help support your conjecture?


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What conjecture can you make about the shape created by joining the midpoints of adjacent sides of a quadrilateral?


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Make a conjecture about the number of squares in each figure.

Figure Number 1 2 3 4 5

# of Squares


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Use your conjecture to predict how many toothpicks would be used in Figure 10.


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What conjecture can you make about the product of two odd integers?

3 X 5 = 15

-5 X 7 = -35

-9 X -3 = 27

7 X -9 = -63

Conjecture:The product of two odd integers is an odd integer.


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Reflection:This conjecture is convincing because _________________________________________________________________________________________________________

This conjecture is not convincing because ________________________________________________________________________________________________

Conjecture:The product of two odd integers is an odd integer.


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What conjecture can you make about the difference between consecutive squares?

02 = 012 = 122 = 432 = 942 = 1652 = 25

1 - 0 = 14 - 1 = 39 - 4 = 516 - 9 = 725 - 16 = 9


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Investigation 1: (Handout)Using the Sept 2011 calendar, consider any two – by – two square around four of the dates.

Create a conjecture about what you notice. Test your conjecture with different two – by – two squares. Create as many conjectures as you can!


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Note: To make conjectures that are valid, based on a pattern of evidence, you need to have _a variety of sample cases__________.

Since any pattern requires multiple cases to support it, more than one or two specific cases are needed to formulate a conjecture. The more cases that fit the conjecture, the stronger the validity of the conjecture becomes.

The strength of a conjecture, however, does not substitute for proof. Proofcomes only when _ALL_ cases have been considered.


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SummaryKey Idea:• Inductive reasoning involves looking at specific examples.By observing patterns and identifying properties in these examples, you may be able to make a general conclusion,which you can state as a conjecture.Need to Know• A conjecture is based on evidence you have gathered.• More support for a conjecture strengthens the conjecture, but does not prove it.


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Examples: Study the pattern and predict the missing values.

1. 9 X 9 + 7 = 8898 X 9 + 6 = 888987 X 9 + 5 =9876X 9 + 4 =98765 X 9 + 3 =


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2. 92 = 81992 = 98019992 = 99800199992 =999992 =


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Examples: Study the pattern and predict the next two terms.

a) 2, 3, 5, 8, 13, ___, ___ (21, 34 add previous)

b) 20, 25, 31, 38, 46, ___, ___ (55, 65 add 5, 6, 7,..)

c) 10, 7, 12, 9, 14, ___, ___ (11, 16 add 2 every 2nd #)

d) 3, 6, 11, 18, 27, 38, ___, ___ (51, 64 add 3, 5, 7, ...)

e) 2, 6, 15, 31, 56, ___, ___ (92, 141 add squares)

f) 2, 6, 12, 20, 30, ___, ___ (42, 56 add 4, 6, 8, ...)

g) 15, 19, 25, 33, 43, ___, ___ (55, 69, add 4, 6, 8, ...)

h) 1, 2, 5, 14, 41, ___, ___ (122, 365, add 1, 3, 9, 27)

i) 3, 5, 11, 29, 83, ___, ___ (245,731, add 2, 6, 18, 54)

j) 59, 52, 55, 48, 51, 44, 47, ___, ___(40, 43,subtract 4 every 2nd one)


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Determine the number of matchsticks used in the 100th pattern.

a)

b)

c)


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A conjecture is an educated guess based upon repeated observations of a particular process or pattern.The method of reasoning is called inductive reasoning.


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1.1 Assignment: Nelson Foundations of Mathematics 11, Sec 1.1, pg. 12‐14Questions; 1 ‐3, 6 ‐ 9, 11, 16

Chapter 1 Inductive and Deductive Reasoning Pages/Teacher Pages/Margot Crewe...Chapter 1 Inductive...

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