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Writing an Equation to Describe Pictures - Problem 3
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To help understand the patterns between pictures that show various units, start by making a table of values listing the picture number of one column and the number of units on the other column. Next, identify what is changing. In other words, try to identify what is the pattern by which the pictures are changing. Write an equation in words describing the pattern of the pictures. Lastly, replace the words with corresponding variables to complete the equation. Use this equation to determine how many units the nth picture will have by plugging in the value to the equation.

This is a problem where, I'm given a series of pictures, and I'm asked to write an equation to describe how many small triangles there would be. Then, the second part, what I have to do, is figure out how many small triangles will the 9th picture have. I'll tell you guys I just had to draw these. They're not easy to draw. I do not want to draw out to the 9th picture. Hopefully, we can get this using a method that doesn't require drawing all these triangles.

First thing I want to do, is try to organize my information. I'm going to try to set it up in a way that's a little different from this, because it's hard for me to look at this, and be able to tell right away how to approach this problem.

I'm going to set up picture number and then number of triangles in a table. Then my picture numbers are 0 1 2 3 4. Then I have to count how many triangles there are. 0 has 0. 1 has 1 that's not so bad. 2 has 4. 9 and then 16.What I want you guys to start thinking about is what patterns start to like pop up in your brain.

Hopefully, when you see this group of numbers, bells and whistles go off in your brain. This is something that's really important in math. 0 1 4 9 16 those are what we call perfect square numbers. I'm going to write that up here, perfect squares. In that what do you mean squares those are triangles? I know what you mean.

When I'm talking about perfect squares, what that means in math, is you're taking your picture number, timesing it by itself, to get your number of triangles. Like 3 times 3 gives you 9. That's called squaring and you write it like this, 3² equals 9. These groups of numbers 0 1 4 9 16 those are called perfect squares because each one has the square root that's the picture number.

Let's go back and answer the question; write an equation to describe the number of small triangles in the pattern. Well what I did was, I took my picture number and I times it by itself. Picture number we're going to write it as squared. That's what that little tiny 2 means. That's equal to the number of triangles.

If you wanted to write it using only math, you will write y equals x². Which means x times x. That's your equation that describes how many triangles there are in each picture.

Now for part b, we're not going to have to draw the triangles thank goodness. We're going to say how many will the ninth picture have. I have to do my picture number squared or times itself, that's going to give me 81. That's the cool thing about math you guys. It's all about short cuts. I could have drawn out all these gazillion 81 triangles, it would have taken me all day or I can just use this rule. Finding the pattern helps me find the rule so I can just jump to the answer.

One thing that's really helpful in math, is finding these short cuts. And if you guys can start recognizing patterns like these perfect square numbers, you're going to be well on your way.

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