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Writing an Equation to Describe a Table - Problem 1
When asked to write a function, remember that this means the equation will be written as "f(x)=". Writing a function requires understanding what the pattern or relationship is between each input, or x, value and each output, or f(x), value.
In this problem we're given a series of pictures and we're asked to write an equation to describe the number of boxes that show up. So looking down here we can see this zero pattern has two, that one has 1, 2, 3, 4, 5, 1,2,3,4,5,6,7,8, 1,2,3,4,5,6,7,8,9, whatever that is we're going to have to find a rule to describe how this is changing and in the next part we're going to have to do is figure out how many boxes would the eighth picture have? So let's go ahead and start looking at a couple of the strategies for how to approach this problem.
The first thing is to look for what part of the picture is staying the same in each one of these groupings. So take a second and see if you can see which part stays exactly the same. It's these two right here, these two boxes on the bottom they're not changing at all. There's always like plus 2, there's some 2 that shows up somewhere in my equation because there's two boxes that are always there.
Then we have to look at what part is changing so here we go. I have this up part, up and down part here has three boxes then it has six, here it has nine, three, six, nine hopefully that's a pattern that's starting to ring bells in your head. What this is is this number down here times three, right? I have three boxes there, which is 3 times 1, then I have 3 times 2, I have 6, 3 times 3.
So first I'm going to write this equation using words and then we can write it using mathematical symbols. The way I would write it in words would be to say number of boxes is equal to 3 times the picture number, 3 times picture number and then plus 2 because we have those two on the bottom that stay the same the whole time.
Let me tell how to write that one more time. The number of boxes in each grouping is equal to 3 times the picture number because I have three going vertically and then you can see this is 3 times 1, 3 times 2, 3 times 3, three times the picture number and then plus those two that stay constant.
So if I were to write this only using mathematical symbols, it would look like this, y equals 3x plus 2. Number of boxes equals three times the picture number plus that two. Those of you who have already learned about slope and y intercept might recognize that oh that kind of makes sense. The zero grouping when your x number is zero that's equal to the y intercept number pretty cool, I think it's neat when you can connect pictures to graphs. I don't know I think it's kind of neat.
Okay part B, how many boxes would the eighth picture have? So if you want to you can do this like the meatball way where you go through and you draw like a zillion boxes and it takes you all day and you like to find out how many there are in the eighth and count them up. There's a shortcut use your equation.
We know the number of boxes is equal to 3 times the picture number plus 2. So to solve this part B I would say number of boxes equals 3 times the picture number plus 2. So that's 24 plus 2 which equals 26 boxes. I like that shortcut because you didn't have to go through and draw tons and tons and tons of patterns and you might have made a mistake or whatever, you can just go right to the answer when you know the equation.
So again guys I just want to summarize, the important thing to look for when you're given a series of pictures is to look for patterns in what stays the same, what's the constant as opposed to what's changing. This is the part that's changing in this particular example.