CHAPTER 1

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1.1 – MAKING CONJECTURES

A conjecture is a testable expression that is based on available evidence, but is not yet proved.

Inductive reasoning is drawing a general conclusion by observing patterns and identifying properties in specific examples.

Deductive reasoning is drawing a specific conclusion through logical reasoning by starting with general assumptions that are known to be valid.

So, what’s the difference?

How did Georgia form this conjecture? What patterns do you see?Forming a table can often help to organize our thoughts!

Figures 1 2 3 4 5

Number of Triangles

1 4 9 16 25

• How many triangles are in Figure 3? • What pattern do you see?• How many triangles might be in Figure 4? Figure 5?• So, according to our pattern, is Georgia’s conjecture

that Figure 10 will have 100 triangles a reasonable conjecture?

• Is there a different conjecture that we could make about the pattern we see?

Make a conjecture about the product of two odd integers.

Questions:• What does product

mean?

• What is an odd integer?

• What different types of odd integers are there?

• What kind of examples can we make?

EXAMPLE

So, what is our conjecture?

EX: MAKE A CONJECTURE ABOUT THE DIFFERENCE BETWEEN CONSECUTIVE PERFECT SQUARES

G E O M E T R I C A L LY N U M E R I C A L LY

Test:

32 – 22 = 9 – 4 = 5

42 – 32 = 16 – 9 = 7

62 – 52 = 36 – 25 = 11

82 – 72 = 64 – 49 = 15

Test:

What is our conjecture?

Make a conjecture about the shape that is created by joining the midpoints of adjacent sides in any quadrilateral.

Questions:What is a quadrilateral? What does adjacent mean?What’s a midpoint?

TO TRY ON YOUR OWN (YOU’LL NEED A RULER!)

Everyone should draw at least three different quadrilaterals, and make a conjecture!

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pg. 12-15# 3, 5, 8, 9, 10, 14, 15, 16, 21.

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1.2 – EXPLORING THE VALIDITY OF CONJECTURES

How can we check the validity of our conjecture?

After we make a conjecture we need to test to check if it’s valid! That’s why we gather evidence. However, we cannot prove a conjecture based on inductive reasoning alone—the best we can do is say that the evidence either supports or denies it.

1.3 – USING REASONING TO FIND A COUNTEREXAMPLE TO A CONJECTURE

EXAMPLE 1

# of Points 2 3 4

# of Regions

2 4 8

Kerry believes that as the number of points increase by 1, the number of regions Increases by a factor of 2.

We need to test her conjecture. Here are the next two examples:

So, we have disproved Kerry’s conjecture. Sometimes,

you just need to look at more examples to check if

your conjecture was correct.

Matt found this pattern:

EXAMPLE 2

Do you think this pattern will continue?

What will happen once it gets to the tenthstep? How will we write the expression?

It might be written several ways:

None of these possibilities works within the pattern, so we know that

Matt’s pattern will not always work. We can revise our conjecture by

limiting it and saying that the pattern will hold from 1 – 9.

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pg. 17# 1, 2, 3

pg. 22# 1, 2, 4, 5, 10, 12, 14, 17

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ARITHMAGONS

Fill in the arithmagons on the handout, and then make some of your own andchallenge a partner to solve them.

1.4 – PROVING CONJECTURES: DEDUCTIVE REASONING

A proof is a mathematical argument showing that a statement is valid in all cases,or that no counterexample exists.

Can we prove it? How?We want to make a

generalization.

PROOF

• First, check that the conjecture is true in the examples that Jon showed.

• 5(3) = 15• 5(-13) = -65• 5(-1) = -5 It works!

• Try another example.• 210 + 211 + 212 + 213 + 214 = 1060• 5(212) = 1060 It works!

• Make a generalization, using variables.• Let x represent any integer.• Let S represent the sum of five consecutive integers, with median x.

• S = (x – 2 ) + (x – 1) + x + (x + 1) + (x + 2) = 5x + 3 – 3 = 5x

• So, since the sum of five consecutive integers is equal to 5 times the median for any integer, the conjecture is true for all integers.

We want to put the conjecture in general terms, using a variable that

could represent any number, to prove our original statement. This is an example of deductive reasoning.

EXAMPLE

All dogs are mammals. All mammals are vertebrates. Shaggy is a dog.

What can be deduced about Shaggy?

So, Shaggy is a mammal and a vertebrate!

EXAMPLE

Proof:

Let’s try the same technique to prove that <AED = <CEB.

EXAMPLE

In our first lesson, we made a conjecture that the difference between consecutivesquares is always an odd number. Now we will try to prove it.

Ex: 262 – 252 = 51

So, we will let x be any natural number. Let D be the difference between consecutive perfect squares.

D = (x + 1)2 – x2

D = x2 + 2x + 1 – x2

D = 2x +1

Since 2x will always be an even number, then 2x + 1 will always be odd. So, we cansay that the difference of consecutive perfect squares is odd for all natural numbers.

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pg. 31-33,# 1, 2, 4, 6, 8, 9, 10, 12, 14, 17, 19

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We will be having a quiz on sections 1.1 – 1.4.