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### Inductive Reasoning

- Conjecture: An educated guess
- Inductive Reasoning: Looking at several specific situations to arrive at a conjecture
- Sometimes, the conjecture is not true. To prove this, we find a counterexample. A counterexample is the opposite of what you are trying to prove.
- Remember, you cannot prove something true just by giving an example, because there could be a counterexample that you haven’t thought of yet.

### Inductive Reasoning

A(0, − 1), B(2, 3), C(3, 5)

Square root of any number is smaller than the original number.

If x can be divided exactly by 2, it can also be divided exactly by 4.

If ∠1 = ∠2 = 90

^{o}, and they share a vertex, then they are linear angles.

∠1 = ∠2 = 90

^{o}, and they share a vertex, but they are not linear angles.

If ∠3 = ∠4, and they share a vertex, then they are vertical angles.

∠3 = ∠4, and they share a vertex, but they are not vertical angles.

If point A is on plane P, and line m passes through point A, then Plane P contains line m.

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

### Inductive Reasoning

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro 0:00
- Inductive Reasoning 0:05
- Conjecture
- Inductive Reasoning
- Examples 0:55
- Example: Sequence
- More Example: Sequence
- Using Inductive Reasoning 2:50
- Example: Conjecture
- More Example: Conjecture
- Counterexamples 4:56
- Counterexample
- Extra Example 1: Conjecture 6:59
- Extra Example 2: Sequence and Pattern 10:20
- Extra Example 3: Inductive Reasoning 12:46
- Extra Example 4: Conjecture and Counterexample 15:17

### Geometry Online Course

### Transcription: Inductive Reasoning

*Welcome back to Educator.com.*0000

*This next lesson is on inductive reasoning.*0001

*For inductive reasoning, we deal with what is called conjectures; a conjecture is an educated guess.*0008

*When you look at several different situations, or maybe previous experiences, to come up with a final conclusion, then that would be inductive reasoning.*0017

*When you have repeated observations, or you look at patterns, those things would be considered inductive reasoning.*0036

*Basically, you are just looking at past experiences--anything that will lead you to some sort of conclusion is inductive reasoning.*0042

*Looking at patterns: if I have 4, 8, 16, 32, and I need to use inductive reasoning*0057

*to find the next several terms in the sequence, well, I can just see how these numbers came about,*0067

*and then I can just apply the same rule to find the next few numbers.*0073

*So, for this, 4, 8, 16, 32...how did you get from 4 to 8, 8 to 16, and 16 to 32?*0079

*Well, it looks like this was multiplied by 2; that means I would have to multiply by 2 to get my next answer.*0091

*So, if I multiply this number by 2, then I am going to get 64; if I multiply this by 2, then I am going to get 128; and so on.*0103

*Now, for this next one, I have to see...well, here is a triangle, a square, triangle, triangle, square, square...what will be my next shape?*0121

*Well, it went from 1 triangle, 1 square, to 2 triangles, 2 squares; so my conjecture will be that it will be 3 triangles, and then 3 squares.*0134

*Again, we are looking at patterns, or looking at some kind of repeated behavior, to come up with a conclusion (or what will happen next).*0155

*Now, a few more problems: if we make a conjecture about this, if AB = CD and CD = EF, what can I conclude?*0169

*Well, if AB is equal to CD, if I draw AB, here is AB; and here is CD;*0184

*so, if I see that this and this are the same, CD = EF, so this equals this; isn't it true that, if this equals this and this equals this...*0198

*doesn't that mean that AB will equal EF?--so that will be my conjecture: AB = EF.*0217

*Make a conjecture, given points A, B, and C: let me just draw out a coordinate plane.*0229

*A is (-1,0); it is right here; B is (0,2); it is right there; C is (1,4), which is right there.*0256

*If I look at this, A, B, and C line up; so my conjecture would be that points A, B, and C are collinear, because they are on the same line.*0272

*Counter-examples: just because you come up with a conjecture, you come up with some kind of conclusion,*0299

*based on what you see in your observations, based on the patterns, and so on, doesn't mean that it is going to be true.*0306

*Just because something happens 3, 4, 5, or 6 times in a row doesn't mean that it is going to happen again the next time.*0313

*Conjectures are not always true; and to prove that it is not always true, you have to provide a counter-example.*0322

*And a counter-example is the opposite of what you are trying to prove.*0331

*If you are trying to prove that something is true--let's say you saw something a few times,*0336

*and so you conclude--you make a conjecture--that the next time, it is going to happen again;*0343

*you can't prove that it is going to happen again just by showing that it happened.*0349

*You cannot prove something just by giving an example, because it might not happen the following time.*0354

*You might not be able to find a counter-example in order to prove that that is not true.*0364

*A counter-example is the opposite of what you are trying to prove.*0373

*Let's say, for example, that the first five cars you see today are black; does that mean that all cars are black?*0376

*That would be a conjecture; the conjecture would be that, since you saw five cars that are black...*0385

*my conjecture would be that the next car that I see will be black; and that might not be true.*0392

*In order to prove that whatever you concluded, your conjecture, is not true, you are going to provide a counter-example.*0403

*A counter-example would be to show an example of it not being true.*0413

*Now, let's go over a few examples of this: the first one: Any three points will form a triangle.*0418

*If I have three points like this, I know that it is going to form a triangle.*0427

*Is this conjecture true? It could be true, but just because I gave an example of it being true does not make this conjecture true,*0440

*because I know that three points will not always form a triangle.*0451

*And so, what I can do to prove that this is not true--to prove that it is false: I can give an example of when this is not true.*0458

*And that would be a counter-example: so three points that do not form a triangle...there are three points; they don't form a triangle.*0466

*This is my counter-example: by giving an example of when this is not true, I am proving my conjecture false.*0481

*So, this conjecture...sometimes it could be true, but it is not always true.*0493

*By showing an example of when it is not true, a counter-example--that is when you are proving the conjecture false.*0502

*The square of any number is greater than the original number: well, that could be true, but it is not always true.*0512

*To show that it is not always true, I need to provide a counter-example.*0519

*Let's say I have some numbers: let's say 2--if I square it...I am saying the square of any number,*0523

*so if I take a number, and I square it, then it is going to be greater than this original number.*0531

*If I square this, then it is going to be 4; well, this is greater than this number, the original number.*0536

*What if I have 0? If I square it, what do I get? 0.*0546

*If I have, let's say, 1/2, and I square this, I get 1/4.*0553

*Well, is 1/4 greater than the original number, 1/2? No, 1/4 is smaller than 1/2.*0564

*So, for this example, this is true; but this one is not true; this is false, and this is false, for this being greater than the original number.*0571

*0 is not greater than 0, and 1/4 is not greater than 1/2.*0582

*Here is a counter-example, and here is a counter-example, because these two examples show that,*0594

*if you square the number, then the answer is not going to be greater than the original number.*0602

*Counter-example, counter-example: by showing the counter-examples, I am proving that this conjecture is false.*0610

*Find the pattern and the next two terms in the sequence: 15 to 12, 9 to 6...the pattern here that I see is subtracting 3.*0623

*So, from here to here, I subtract 3; subtract 3; subtract 3; if you subtract 3 again, then you get 3; you get 0, -3, and so on.*0637

*This one: 1 to 2, 2 to 6, 6 to 24, and so on--this one seems a little tricky, but you just have to look at it.*0655

*Here, if I...let's see...1 to 2: I can either add 1; I can multiply by 2; here I can add 4, or I can multiply by 3;*0667

*here I can add something bigger...for this one, it doesn't seem like that would be the pattern;*0683

*so here it is multiplied by 4; here, multiply by 5; so then, for the next number, I can multiply by 6.*0691

*If I multiply this by 6, then I will get...let's see: 120 times 6 is 0, 12, 720.*0706

*And the next one...you can just multiply by 7.*0721

*This one right here: I have a square; then I have this shape in there; and then I have another one.*0728

*The next pattern will be...so then, here is the next step; that is up to there.*0736

*And then, my next one will be to draw a square within that square, like that.*0750

*And the next one would be to do the same thing; and you are going to draw another square inside.*0758

*OK, so you are given a statement, and you need to come up with a conjecture.*0769

*The given statement is that angle 1 and angle 2 are adjacent.*0776

*"Adjacent" means that the angles share a side and a vertex.*0781

*So, if I draw angles 1 and 2, they are adjacent; then what can I conclude?*0787

*Well, I conclude that angles 1 and 2 (since I know that "adjacent" means that they are next to each other) share a side;*0803

*so, angles 1 and 2 share a side and a vertex, because they are adjacent.*0817

*The next one: the given statement: line m (here is line m, and this is a line) is an angle bisector of angle ABC.*0842

*Here is an angle, ABC; and line m is an angle bisector--"bisector" means that it cuts in half.*0855

*So, this line cuts this angle ABC in half; that means that these two parts right here are the same--they are congruent.*0867

*If I label this angle 1 and this angle 2, what can I conclude?*0881

*If this line is an angle bisector, then these two parts right here, angle 1 and angle 2, I can say are congruent.*0887

*Or I could say that the measure of angle 1 equals the measure of angle 2.*0895

*Or I can say that angle 1 is congruent to angle 2.*0902

*That would be my conjector, since line m is an angle bisector of angle ABC.*0909

*The next example: Decide if each conjecture is true or false; if true, then explain why; and if it is false, then we have to give a counter-example.*0918

*Given that WX = XY, the conjecture is that W, X, and Y are collinear points.*0932

*The conjecture is saying that if I have WX...here is point W, point X, and point Y...W, X, and Y are collinear.*0947

*WX = XY; the conjecture is that these are collinear; is that always true?*0960

*Can you think of an example of when it is not true?*0968

*Well, what if you have WX, XY...WX is equal to XY; this still applies here, but it doesn't prove that the conjecture is true.*0971

*This is a counter-example, because this is an example of when my conjecture is false.*0997

*This one I know...this conjecture is false.*1005

*The next one: Given that x is an integer, the conjecture is that -x is negative.*1012

*So, if x is an integer (some integers are 2, 0, and -2; so these are x), the conjecture is that -x is negative.*1019

*If I make this negative, -2 is -x--is it negative?--these would be -x.*1033

*-2: this is true; this is an example of the conjecture; what about 0?*1045

*If I make it negative x, then that will be -0, which is just 0; and that is not a negative, so this is false.*1055

*And then here, -2: if I make it -x, then it is -(-2); well, that is a positive 2.*1067

*So, does that show that -x is negative? No, because this is positive 2.*1076

*So, in this case, this one is false; if x is an integer (these numbers right here), and you make those integers negative,*1083

*then the answer is going to be negative: in this case, this works.*1097

*If you make this number negative, it is not a negative; if you take the negative of this number, it becomes positive.*1103

*So, -x is positive in this case; so this one is also false, and then here is my counter-example, right here.*1112

*This one and this one are both counter-examples, because they show that this is not true; it is false.*1122

*Well, that is it for this lesson; thank you for watching Educator.com--see you next time!*1135

0 answers

Post by Melissa Wang on July 1 at 02:41:41 AM

soso

0 answers

Post by patrick guerin on July 3, 2014

Thanks for the lecture

0 answers

Post by Jinee Lee on October 19, 2013

this video is easy to understand

0 answers

Post by Manfred Berger on May 28, 2013

In exaple 1 subsection 2 I guess you could hust have set the cojecture up as x^2>x, solve for x and find out that it holds for all x>1, right?

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Post by Ding Ye on May 31, 2012

Great video!

0 answers

Post by Milo Barrera on February 15, 2012

nice and simple

0 answers

Post by amin khalif on September 6, 2011

the video was good