Download Sample chapter from the Foundation Proof chapter

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Proof
This chapter will show you how to:
✔ tell the difference between 'verify' and 'proof'
✔ prove results using simple, step-by-step chains of
reasoning
✔ use a counter example to disprove a statement
✔ prove simple geometrical statements
22.1 Proof questions
Proof questions test the ‘Using and Applying’ part of
what you learn in Mathematics.
You need to:
1 Find examples that match a general statement, or give
counter examples which disprove a statement.
2 Show how one statement follows from another.
3 Explain your reasoning or criticise a piece of faulty
reasoning.
4 Know the difference between a practical
demonstration of something and a proof.
Many proof questions are based on the properties of
numbers. You need to know about:
● odd and even numbers
● prime numbers
● factors and multiples
● square and cube numbers.
Certain words occur in proof questions:
● consecutive (one after the other)
● integer (whole number)
● product (multiply).
You need to know the rules for addition and multiplication
of whole numbers:
Addition
Multiplication
odd odd even
odd even odd
even odd odd
even even even
odd odd odd
odd even even
even odd even
even even even
For the general statement
‘Dogs have 2 ears’. You can
find examples of dogs with
2 ears to match the
statement.
For ‘All dogs are black’ a
brown dog is a counter
example that disproves the
statement.
Chapter 14 covers most of
these. Look back if you
need help.
4
3
9
a2b
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22.2 Proof v. verify
Key words:
verify
proof
You need to know the difference between ‘proof’ and ‘verify’.
Verify means to check something is true by
substituting numbers into an expression or formula.
Proof means to show something is true using
logical reasoning.
Example 1 illustrates the difference.
Example 1
n is any integer.
Explain why 2n 1 must be an odd number.
Answer 1
Try n 1
integer whole number
2n 1 2 1 1 2 1 3
Try n 2 2n 1 2 2 1 4 1 5
Try n 3 2n 1 2 3 1 6 1 7
The answer is always an odd number.
This is not a proof because it does not explain why 2n 1
must be an odd number.
This answer verifies that 2n 1 is an odd number when
n 1, 2 and 3.
Will 2n 1 be an odd number for n 4, 5, 6, 7, …?
You cannot test it for all numbers – it would take forever!
Look at Answer 2.
n is any integer.
Explain why 2n 1 must be an odd number.
Answer 2
2n 2 n
Multiplying any whole number by 2 gives an even
number, so 2n is an even number.
So 2n 1 even number 1 odd number
2n 1 must always be an odd number.
Is it always?
The answer only shows it is
for n 1, 2, 3.
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Answer 2 is a proof . It shows whatever number you
start with (n could stand for any whole number), 2n 1
will always be an odd number.
This example shows that there is a huge difference
between verifying a result (simply checking with
numbers) and proving a result.
Verify is a practical demonstration of the result.
Proof shows how one statement follows from
another using simple chains of reasoning.
Example 2
Explain why the sum of any three consecutive integers is
always a multiple of 3.
‘Consecutive’ means one
after the other.
Verify
Try 1, 2 and 3
123623
Try 5, 6 and 7
5 6 7 18 6 3
Try 10, 11 and 12
10 11 12 33 11 3
The result works for these examples.
Multiple of 3 … a number
in the 3 times table.
Proof
If x is one of the numbers then the others are (x 1) and (x 2)
x (x 1) (x 2) 3x 3 3(x 1)
3(x 1) means 3 (x 1)
So the answer is always a multiple of 3.
For the consecutive
numbers you could choose
(x 1), x and (x 1).
Proof questions often have the word ‘explain’ in
the question.
You only get full marks if you prove the result.
You may get some marks for verifying (giving numerical
examples), but answers which just check the result using
numbers often score no marks at all!
Proof usually involves some algebra skills.
In example 1 you needed to know that 2n means 2 n, a
very basic algebra fact.
In example 2 you had to ‘collect like terms’ and ‘take out a
common factor’.
For help with these, look
back at Chapter 4.
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Exercise 22A
1 p is an odd number and q is an even number.
(a) Explain why p q 1 is always an even number,
(b) Explain why pq 1 is always an odd number.
2 n is a positive integer.
Explain why n(n 1) must be an even number.
3 x is an odd number.
Explain why x2 1 is always an even number.
4 If b is an even number, prove that (b 1)(b 1) is an
odd number.
5 x is an odd number and y is an even number.
Explain why (x y)(x y) is an odd number.
6 Explain why the sum of 4 consecutive numbers is
always an even number.
22.3 Proof by counter example
Key words:
counter example
Proof by counter example asks you to show that
a statement is incorrect by finding one example
where the stated result does not work.
You can substitute numbers into an expression or formula
until you find a case where the result is not true.
Example 3
Tony says that when n is an even number, 12n 3 is always
even. Give an example to show that he is wrong.
Even numbers are 2, 4, 6, …
Try n 2
Try n = 4
1n
2
1n
2
3 21(2) 3 1 3 4 … even
3
1(4)
2
3 2 3 5 … odd
When n 4 the result is not true, so Tony is wrong.
Sometimes you need to try several values before you find
a counter example.
The case where n 4 is a
counter example.
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Example 4
Simon says that when you square a number, the answer is
always bigger than the original number.
Give a counter example to show that Simon is wrong.
Try 1
12 1 1 1 … the same
Try 2
22 2 2 4 … bigger
Try 3 32 3 3 9 … bigger
Trying 4, 5, … will be even bigger answers than these.
Try 2 (2)2 (2) (2) 4 … bigger
Try a decimal number smaller than 1, such as 0.5
(0.5) 0.5 0.5 0.25 … which is smaller
2
(0.5)2 gives a smaller number, so Simon is wrong.
Exercise 22B
1 Sam says that when k is an even number,
k2 12k is always odd.
Give an example to show that he is wrong.
2 Heather says that m3 2 is never a multiple of 3.
Give a counter example to show that she is wrong.
3 p is an odd number and q is an even number.
Andrew says that p q 1 cannot be a prime number.
Explain why he is wrong.
4 a and b are both prime numbers. Give an example to
show that a b is not always an even number.
5 Ian says that n2 3n 1 is a prime number for all
values of n.
Give a counter example to show that he is wrong.
6 Give a counter example to each of these statements:
(a) the square root of any number is always smaller
than the original number
(b) the cube of any number is always bigger than the
square of the same number.
Work systematically, trying
different values.
Squaring numbers bigger
than 3 will give bigger
numbers.
negative negative
positive
(0.5)2 is a counter example.
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542 Proof
22.3 Proof in geometry
To prove a result in geometry you use step-by-step
reasoning, showing clearly how one statement
follows from another.
For geometry proofs you need to know:
● angle properties of parallel lines
● angle properties at a point and on straight lines.
Don’t just verify by finding
examples that work.
Never use a protractor to
check angles. The diagrams
are not usually accurately
drawn.
Example 5
Prove that the sum of the angles of a triangle is 180°
B
This result was proved in
Chapter 5.
D
b
Start by drawing a diagram.
A
a
y
c
x
C
E
Draw a triangle ABC.
Extend side AC to E.
Draw line CD parallel to AB.
Label the angles as shown on the diagram.
x a (corresponding angles)
y b (alternate angles)
x y c 180° (sum of angles on a straight line at point C)
So
a b c 180°.
The sum of the angles of ABC is 180°.
To set out a geometry proof:
● State each step clearly.
● give a reason for each step.
Steps need to follow on from each other.
Look back at angle
properties and parallel line
properties in Chapter 5 to
remind yourself of these
facts.
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Exercise 22C
1 Prove that the exterior angle of a triangle is equal to
the sum of the two opposite interior angles.
2 Prove that the sum of the interior angles of a
quadrilateral is 360°.
3 Prove that the interior angles of a regular pentagon
are 108°.
Summary of key points
You need to know the difference between proof and
verify.
Verify means to check something is true by
substituting numbers into an expression or a formula.
It is a practical demonstration of a result.
Proof means to show something is true using logical
reasoning.
Proof questions often have the word 'explain' in the
question.
Proof by counter example asks you to show that a
statement is incorrect by finding one example where
the stated result does not work. You can substitute
numbers into an expression or formula until you find
a case where the result is not true.
To prove a result in Geometry you use step-by-step
reasoning, showing clearly how one statement follows
from another.
Questions on proof can be set at various grade levels
depending on the content of the material. They will
typically be at grades E, D or C.
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Examination Questions
1 TO FOLLOW!!!