Chapter2InductiveReasoningB

In this video, we will look at a few examples that will let us practice inductive reasoning.

Now remember, inductive reasoning is any time you are making conclusions based on observations of patterns.

So inductive reasoning has a lot to do with patterns.

In Example A, it says a dot pattern is shown below.

The first question: How many dots would there be in the bottom row of the fourth figure?

So if we look, we see we have a first figure, second figure, third figure.

Now, when you are working on patterns, its often helpful to just start by extending the pattern yourself, so that you sort of get a feel for it.

So I notice in the first figure there is just one circle.

In the second figure there is one circle and then two circles under it.

And then in the third, there is one and then two and then three.

So I would guess in the fourth, there would be one, and then two under that, and then three under that,

and then another row with four under it.

So it looks like basically the number of circles in the bottom row always corresponds to the figure number.

So the third figure had three circles in the bottom row.

And the answer to this question, in the fourth figure there would be four dots in the bottom row.

Next question: What would the total number of dots be in the sixth figure?

So now we are trying to look at the total number of dots.

Well as we already talked about, if we look at a specific figure, so say the third figure,

the way that figure is created is there is one circle on the top,

two underneath that, and then three underneath that.

And it keeps going until you reach that term number.

So if you are in the fourth figure, you are at one, two, three, and then four all together.

So there is one circle, two more, three more, four more.

So by the time you get to the sixth figure, it's going to start with one circle on top, and then two under that, and three under that, etc., until you get to six.

So the total number of dots would be one plus two plus three plus four plus five plus six, which would be 21 dots.

Now lets go on to Example B, which says how many triangles would be in the tenth figure.

So lets again just sort of look at the pattern and try to get a feel for it.

And I want to start by actually counting how many triangles are in each one.

So I noticed in this first figure, there are four triangles; one, two, three, four.

In this second figure, there are one, two, three, four, five, six triangles.

In the third figure there is one, two, three, four, five, six, seven, eight triangles.

And just by looking at these I notice each time we are going up by two.

So one way to come up with the answer to this would just be to keep going and write out all the way to like the tenth figure.

And then just keep track of how many triangles there would have to be if you keep adding by twos.

So in the fourth figure there would have to be ten triangles.

In the fifth figure there would be twelve, then fourteen, then sixteen, then eighteen, then twenty, and then twenty-two.

So I know that the answer would have to be 22 triangles if it keeps going up by two triangles each time.

You could also try to come up with a rule that would help you figure that out more quickly, as opposed to having to count all the way up to ten figures.

Because if the question had been about figure 100, it would be annoying to have to go all the way out to figure 100.

So if you want to think about that, the fact that you are adding by two every time means that all of the numbers of triangles,

all of these numbers, are all multiples of two. And they're a specific multiple of two.

It is the original figure number, plus one, times two.

So one plus one is two; two times two is four.

Two plus one is three; three times two is six.

Three plus one is four; four times two is eight.

So basically, if you add one to your figure number, and then times by two, you will get the number of triangles.

So, just remember both of these examples were inductive reasoning where you are looking at some patterns,

trying to generalize, to come up with conclusions just based on those patterns.