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Writing an Equation to Describe Pictures - Problem 1
When asked to write an equation based on changing patterns in a picture, first identify what is the same between the pictures or the images. This will be the constant term in the equation. Next, identify what is changing between the pictures. This will be the variable being multiplied or divided by a number. Write the patterns using words, then replace the words with corresponding variables. Use this equation to determine how many units the nth picture will have by plugging in the value to the equation.
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.
Looking down here, we can see this 0 pattern has 2, that one has one jitter of 5, one jitter of 5, 6, 7 and 8, whatever that is. We're going to have to find a rule to describe how this is changing. In the next part, what we're going to have to do, is figure out how many boxes will the 8th picture have.
Let's go ahead and start looking at a couple of 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. Take a second and see if you can see which part stays exactly the same. It's these two right here. These 2 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 2 boxes that are always there.
Then we have to look at what part is changing. Here we go. I have this up and down part here has 3 boxes, then it has 6, here it has 9, 3 6 9. Hopefully that's a pattern that is starting to ring bells in your head.
What this is is this number down here times 3. I have 3 boxes there which is 3 times 1. Then I have 3 times 2; I have 6. 3 times 3. 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 then plus 2; because we have those two on the bottom that stay the same the whole time. Let me tell you how that that one more time. The number of boxes in each grouping is equal to 3 times the picture number. Because I have 3 going vertically and then you can see this is 3 times 1, 3 times 2, 3 times 3. 3 times the picture number and then plus those that stay constant.
If I were to write this only using mathematical symbols, it would look like this; y equals 3x plus 2. Number of boxes equals 3 times the picture number plus that 2.
Those of you who've already learned about slope and y intercept, might recognize that that kind of makes sense. The 0 grouping when your x number is 0, that's equal to the y intercept number. It's pretty cool I think it's neat when you can connect pictures to graphs and I think it's kind of neat.
Part b; how many boxes would the eighth picture have? If you want to, you can do this like the meat ball way where you go through and you draw like gazillion boxes. And it takes you all day and you're like trying to know how many they are the eighth. There's a short cut. Use your equation. We know the number of boxes is equal to 3 times the picture number plus 2.
To solve this part b, I would say, number of boxes equals 3 times the picture number plus 2. That's 24 plus 2 which equals 26 boxes. I like that short cut because, you didn't have to go through and draw 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.
Again guys, just want to summarize, the important thing to look for when you're given a series of pictures, is to look for patterns and 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.