THE NATURE OF PROOF: Induction 1: A Helpful Tool
NOTES:
In the last activity, you learned that you cannot trust observation alone as a method of proof.
However, when you observe a series of events, you can conclude that the events will occur in a similar pattern.
This process of observing patterns from a series of repeated observations of experiments is called INDUCTIVE REASONING.
We call a conclusion reached by inductive reasoning a GENERALIZATION.
One example of inductive reasoning comes from this repeated observation:
"The sun rises in the morning."
And:
"It sets in the evening."
We conclude that the sun will rise tomorrow morning and set tomorrow evening.
Even though induction is useful in many situations, we cannot be absolutely certain that the conclusions we reach will always be true.
The sun might explode today... if it does, you won't have to worry about tomorrow's geometry assignment.
In mathematics, using inductions may also lead to false conclusions.
***
Suppose a man has a circular piece of property.
He gives each of his 6 children a peg to place somewhere on the boundary of the property.
He connects each of the pegs with every other peg to form as many pieces of land as possible.
He wants to be able to give a piece of land to each of his 32 grandchildren. How many pieces of land do you think he has marked off in this way? ANS:31
To find the number of regions that are formed, a geometric representation can be made.
The property can be a circle. The pegs can be points. The points can be connected by line segments.
One way to solve such a problem is to solve simpler problems and look for a pattern... consider a simpler problem.
How many regions are formed using 3 points? ANS:4
You may have wrongly used inductive reasoning to predict that 6 points would produce 32 regions.
But no matter how you "slice it," there will not be more than 31 regions.
***
The process of induction, where we look for patterns in simpler problems to help more complex problems, can be very useful in many situations.
This problem, howerver, shows us that induction is not infallible. We need to be cautious in using this strategy.
WORKSPACE:
ANS:
31 , 4 , 8 , 2 , 1 , 16 , 31
THE NATURE OF PROFF: Induction 2: A Good Start
NOTES:
Although it has its limitations, inductive reasoning has contributed greatly to science and mathematics.
The famous Italian astronomer Galileo used induction when he discovered that the time required for a pendulum to complete one swing through an arc is always the same whether the arc is small are large.
***
Consider applying the principle of induction the the following problem involving quadilaterals...
"The line segments joining the midpoints of the sides of any quadilateral form a parrelogram."
...even though this statement is true for every quadilateral we have tried, that does not guarantee that every quadilateral produces this same result.
Suppose we now look at how the area of a quadilateral compares to the area of a parallelogram formed by connecting the midpoints of the sides of the quadilateral.
Watch as the 4 small triangles move to the interior of the parallelogram...
The area of the parellelogram equals the sum of the areas of the 4 small triangles.
After seeing these results, we might be willing to make this statement:
"The line segments joining the midpoints of the sides of any quadilateral form a parallelogram whose area is one-half the area of the quadilateral."
...even though this statement is true of every quadilateral we have tried, that does not guarantee that every quadilateral produces this same result.
In order to use induction to prove this statement for all quadilaterals, we would have to draw all the quadilaterals possible and check each one.
Another approach would be to find a general proof that will apply to any quadilateral.
In later lessons we develop a method for finding general proofs of mathematical statements.
WORKSPACE:
ANS:
#4 , Parallelogram , #1 , Parallelogram , #2 , Parallelogram , #3 , Parallelogram , parallelogram , #1 , One-half , #2 , One-half , #3 , One-half , #4 , One-half , one-half
INDUCTION II: Test
This lesson will evaluate your knowledge of inductive proof and observation.
NOTES:
WORKSPACE:
ANS:
a , c , a , b , a , b , d , c , b , NOT:a,d
ROOTS AND RADICALS I: Pre-Test
This lesson will evaluate your skills in simplifying expressions that contain roots and radicals.
NOTES:
WORKSPACE:
ANS:
NOT:b , NOT:a , NOT:a , b , NOT:c , NOT:a , NOT:d , c , NOT:c , d ,
ROOTS AND RADICALS: Square Roots
- Find square roots of numbers
NOTES:
In this activity we will solve equations like:
2
x = 5280
By the end of this activity we will find values of x that make these kinds of equations true.
Let's start by solving some easy equations.
2
x = 4 so x = 2 or -2 because 4 = 2 * 2 and 4 = (-2) * (-2)
2
If x = 7 then x = ?
There is no whole number of fraction that makes this equation true. The symbol \/-- is used to represent "irrational" number that satsify such equations.
__
\/7 is read "the square root of seven."
Expressions like \/23 , \/14 and \/b are called radicals. These radicals are the square roots of numbers, so:
___ ___
\/23 * \/23 = 23
___ ___
\/14 * \/14 = 14
__ __
\/b * \/b = b
__
The symbol \/ is called the radical sign.
The expresiion under the radical sign is called the radicand.
***
Before pressing ENTER, desribe in your own words what \/a stands for.
The \/a is the square root of "a" and is the number, which when multiplied by itself is a.
__ __
\/a * \/a = a
***
The principle square root of 25 is written \/25 and equals 5.
The negative square root of 25 is written -\/25 and equals -5.
***
2
Notice that x = 25 has two answers, 5 and -5. However, \/25 has only one answer, that is 5.
In this course the \/a will always mean the principle square root of "a."
***
Before pressing ENTER, describe in your own words what the principal square root of a number is.
The principal square root of a number is the positive square root of the number.
For our purposes, in a radical expression like:
__
\/n
n MUST BE non-negative (n >= 0).
This is because there is no real number which, when squared, results in a negative number.
***
Let's use a factor tree to help find the square root of a number.
225
/\
5*45
/\
5*9
/\
3*3
225 = 5 * 5 * 3 * 3
____ ______________
So \/225 = \/5 * 5 * 3 * 3 = 5 * 3 = 15
Let's use another factor tree to help find the square root of a number.
4
254x
/\ 4
245*x
/\ /\
5*49 2 2
/\ x*x
7*7/\ /\
x*x x*x
____4 __
So \/245x = 7 * x * x * \/5
2 __
= 7x \/5
***
Here's a short-cut way to find the square roots without using a factor tree:
____ ________________
\/576 = \/2*2*2*2*2*2*3*3
= 2*2*2*3 = 24
____ ________ __
\/242 = \/2*11*11 = 11\/2
____2 ______________
\/588x = \/2*2*7*7*3*x*x
__
= 14x\/3
***
Before pressing ENTER, describe in your own words how we find the square root of a number.
One way is to use a factor tree (see above).
***
Recall that we can factor the equation:
2
y = x - 9
by noticing that it is a difference of squares.
y = (x + 3)(x - 3)
We can also say that the equation:
2
y = x - 7
is a difference of squares.
2 __ 2
y = x - (\/7 )
This is how we factor the equation:
__ __
y = (x + \/7 )(x - \/7 )
***
Compare the factored form of the following equations to their graphs... (impossible to do without graphics :-( )
***
Before pressing ENTER, describe in your own words the factors and the graph of this equation:
2
y = x - 3
The factors of the equation are (x + \/3)(x - \/3). The graph intersects the x-axis at -\/3 and \/3.
WORKSPACE:
ANS:
3 -3 , 4 -4 , \/6 -\/6 , \/29 -\/29 , 6 -6 , \/53 -\/53 , 7 , -7 , 4 , -4 , 2 , -2 , 4\/21 , 21x^3 , 14 , 15x\/2 , 9x\/3x , 8\/7 , 6y\/2y , 2\/19z , 12\/5 , 4m^2\/3 , 6 6 , 10 10 , 35 35 , 43 43 , a , a , b , b , a , 6 , 7 -7 , 4x\/2 , 10 , 9 -9 , \/19 -\/19 , -13 , 12r\/2r , 10\/5 , -10y^2\/6y
ROOTS AND RADICALS: Operations on Radicals
- Add, subtract, multiply, and divide radicals
- Rationalize denominators of fractions with radicals
NOTES:
In the previous activity we simplified expressions like:
__ __ __
\/25 -\/36 \/18
| | |
\ / \ / \ /
- - - _
5 -6 3\/2
In this activity we will simplify expressions like:
_ _ ____ __ __
\/3 * \/4 ; \/16/9 ; \/18 + \/32 ;
__ __
\/12 - \/27
Each of these expressions is called a Radical Expression because each contains a radical sign.
***
Notice how we multiply radicals:
_ __ ______
\/3 * \/75 = \/3 * 75
__________ _____ __
\/3 * 3 * 25 = \/3 * 3 * \/25
3 * 5 = 15
***
_ __ ______
\/6 * \/18 = \/6 * 18
_________ _____ _
\/6 * 6 * 3 = \/6 * 6 * \/3
_
6\/3
***
_ __ ______
-3\/8 * 5\/32 = (-3)(5)\/8 * 32
_________ _____ _
-15\/8 * 8 * 4 = -15\/8 * 8 * \/4
-15 * 8 * 2 = -240
***
WORKSPACE:
__ __
\/21 * \/42
*ARGH! HOW do I do THIS??! Blah.*
*Presses Help*
_______
\/21 * 42
*Ahh... I can't beleive I needed help! I missed the first part of my notes! ROFL!!*
_________________
\/2 * 3 * 3 * 7 * 7
_
21\/2
***
_ ___
3\/5 * \/125
_____________
3\/5 * 5 * 5 * 5
75
***
___ ___
-2\/15t * 3\/45t
_________________________
(-2)(3)\/3 * 3 * 3 * 5 * 5 * t * t
_
-90t\/3
***
__ __
\/28 * (-\/42)
_______
-\/28 * 42
_____________________
-\/2 * 2 * 2 * 3 * 7 * 7
_
-14\/6
***
ANS:
21\/2 , 75 , -90t\/3 , -14\/6