Did you recently get your ACT student score report back and are wondering what modeling is? I know this is a question that one of my own students asked this week because he got his results back and one of the areas that he missed the most questions on was modeling. In fact, it seems that modeling takes up a lot of the math section, and yet it does not fall under something that is easy to parse, like geometry or algebra. So in this blog, I’m going to talk to you guys about what modeling means, and I’m going to give you a few tips that, if you’re missing modeling problems, might help you perform a little bit better on the ACT on these questions.
My name is Brooke, and I’ve been coaching the ACT for over two decades. I’ve written two books on the ACT math section, and I’m going to use some of the problems from those books in this video today to explain to you what modeling is. So, it has nothing to do with fashion, as you probably realized. Modeling is essentially any question that has real-world scenarios attached. Word problems, problems that apply to real-world situations, or anything like that gets thrown into modeling.
So, improving on modeling questions can be kind of tough to address because it’s a little bit of almost every discipline within ACT math. It’s part of geometry. It’s part of algebra. It’s part of all of these things. But let’s talk about some really quick tips that can help you steer clear of common mistakes on these problems.
Make it Real
My first tip for you guys, if you want to get more modeling problems correct, is to make it real. One of the things that I like to do with word problems is to help me set them up, and I know some of my students say they’re not good at word problems. It’s such a lie. It’s not that you’re not good at word problems; you just need to dive into them a little bit more and figure out some more strategies so that you can tackle them.
Let’s take a look at an example:
A fabric store sells cotton and velvet fabrics. Myra pays $50 for three yards of cotton and two yards of velvet. Jamie pays $22 for one yard of cotton and one yard of velvet. How much does one yard of velvet cost?
Some of you can set this up really quickly, really easily, but if it doesn’t click right away, one of the things that I like to do is just make up numbers as if I’m shopping in the store. If I’m shopping in the store and I’m going to pay some money for these fabrics, what’s going to happen? If I’m going to buy three yards of cotton, that cotton is going to cost me something per yard. It’s going to be, maybe, $10 a yard. And I’m going to buy three yards, and then I’m going to do 3×10, and I’m going to get $30 for the cotton. What I’m doing here is I’m making numbers up and I’m looking at what I am doing. I’m multiplying the 3 by the 10. That’s the strategy. So, then I could solve this problem: 50 = 3C + 2V.
But the thing is to figure out that this is how I set it up. If I don’t see it right away, I pretend like I’m shopping in the store. I pretend like I’m going to pick up a yard of cotton. It says $10 a yard, so I’m going to multiply my three yards by the $10, and that’s going to give me $30, and then my velvet is $10 a yard. So I’m going to do two times the $10. That’s not actually the answer because it won’t work when I look at the other one, but it at least helps me understand, okay, this is a variable that means the cost per yard of cotton. And then this is going to be the cost per yard of velvet. And that helps me line it up.
So, make it real. That’s going to help you line it up. Then you can set it up with flying colors, and then you can really easily solve it. For the problem, you get 44 = 2C + 2V, and after subtracting down, you’ll get that cotton equals $6, but I don’t need cotton. I need velvet, so I need to be really careful to not just put 6. I have to reread the question and be very careful with the details. I need to know what one yard of velvet costs so I can look at the original equation of 22 = C + V. I’m going to plug in 6 for C now plus velvet, and I get velvet equals 16.
But again, make up numbers and throw them around. Pretend like you’re shopping in the store. If you have a real-world problem, the beauty of it is that you can put yourself in the real-world situation in your brain, and then you’re going to understand how to set up the algebra better.
Be Careful About the Details
My second tip is to always be careful about the details. I would say half the time when people miss modeling questions, they don’t miss it because of their math. They miss it because they miss the detail. That detail could be rereading the question at the end. That’s really important, like the previous example with the one yard of velvet. If you solved for the cotton because that was easier to solve for, you have to plug it back in to find the velvet.
Here’s another problem that we’re going to talk about today:
The pep club is organizing a bake sale and is planning to purchase baking supplies.
Each dozen cookies requires 1.5 cups of flour and 7/8 cup of sugar. Originally, the club planned to purchase enough flour and sugar to make exactly 200 cookies, adjusting the recipe scale as necessary, but found that ingredients were cheaper when purchased in specific sized quantities: 14-cup bags of flour and 8-cup bags of sugar. How many extra whole cookies was the pep club able to bake assuming it purchased at least enough flour and sugar to bake 200 cookies?
You can see this big long word problem, and it says each dozen cookies requires this. The biggest mistake my students make is that this word, “dozen,” flies under the radar because it’s not a number. They think each cookie requires this. They don’t realize it’s a dozen. They think it’s one cookie. And then this is 200 cookies, not 200 dozen cookies. Paying attention to these kinds of details and logging them and understanding what they mean and not screwing those up is huge.
Take It One Step at a Time
I had a student I was working with the other day, and he missed a geometry question. He said he didn’t solve for a particular length because he needed another length. And I told him that you have to solve for that first length because if you find that length, then you can find the other length, and then if you find that length, then you can find the length you need. It’s like this domino effect. And so remember that oftentimes modeling problems are multi-step problems. Real-world problems, they’re not just like, do this, find this; it’s not a single input-output. They could be a little bit complex. It can be hard to see the forest for the trees.
And so a great tip for these is to just take it one step at a time. Do not expect to get to the final answer. Don’t try to be so strategic that you say, “This is what I need, so that’s the only thing I’m going to solve for. I’m not going to solve for anything else.” A lot of times you just have to take what you know and what you need, put them together, and move a step forward and see what you can get next. And then move another step forward and then move another step forward and just start walking through the forest until you get a little bit closer to where you want to be. So this is a bit of an involved word problem. The other thing I’m going to say about these involved word problems, if you are missing modeling problems, is that they can be very time-consuming. And one of my tips as you get toward the end of the test: I often recommend that my students skip these questions because they can take a lot of time. And if you’re missing a lot of them anyway, if you skip these questions or work on the questions that maybe are a little bit more straightforward and don’t involve so many steps, you can potentially get a few more points because it’s not like questions on the ACT are worth different amounts of points. They’re generally worth about the same amount of points per question except for the ones that don’t count at all, but we don’t know which ones those are because they’re experimental. But otherwise, everything’s worth the same. So a long-winded word problem could be sucking lots of your time up.
But in any case, I’m going to show you how I get into a problem like this (the same one from before):
The pep club is organizing a bake sale and is planning to purchase baking supplies.
Each dozen cookies requires 1.5 cups of flour and 7/8 cup of sugar. Originally, the club planned to purchase enough flour and sugar to make exactly 200 cookies, adjusting the recipe scale as necessary, but found that ingredients were cheaper when purchased in specific sized quantities: 14-cup bags of flour and 8-cup bags of sugar. How many extra whole cookies was the pep club able to bake assuming it purchased at least enough flour and sugar to bake 200 cookies?
So, this is a little bit complicated, but I’m not going to get into how many extra cookies I can bake. The first thing that I’m going to figure out is how many cookies I was going to bake. I need to make at least 200 cookies. Let’s see how much flour and sugar I bought and how many cookies I baked. The first thing I’m going to do is I’m going to figure out how many dozen cookies I have, which is 200 divided by 12. And I get 16.66666666666, and I can round it. And you’ll get that in the calculator. You can just put that in your calculator. And 12 is the number of cookies in a dozen; you should probably know that. So this is how many dozens I want. And so now what I’m going to do is I’m going to figure out how much flour I need to make this required number of cookies. I’m going to multiply this by 1.5. And when I do that, I get 25. And then I’m going to take 7/8 of a cup of sugar, and I’m going to multiply that by 16.6666666, repeating. And we’re going to see what we get there: 14.583, repeating. So now what I’m going to do is I’m going to go down to my purchases. I need a 14-cup bag of flour. So if I need 25 cups of flour, that means I’m going to buy 28. And I can just kind of do that math in my head; I can see the 14 and realize I need to buy two of those. Then I’m going to figure out how many of these sugar bags I’m going to buy. Which is two, because 16 is greater than 14.58. So, I’m going to have 16 cups of sugar.
And now what I need to do is subtract these amounts, which are going to make the 200 cookies that I originally was planning to make. So first I’m going to subtract 25 from this 28, and I get three cups of flour. And I’m going to subtract 14.583 repeating from 16. I’m actually going to go the other way. It’s going to be negative, but I don’t have to reenter it then. And I get 1.416 repeating cups of sugar. And so now it may not be clear to me which one of these is the limiting factor, if that makes sense. I know with three cups of flour—because each dozen requires one and a half cups of flour—I can make two dozen cookies. Now I’m going to do the same thing for the sugar. I’m going to take 1.416, repeating, and I’m going to divide it by 7/8, which is the amount per dozen, and that’s how I’m going to figure out how many dozen I can make from that. That gives me 1.619 dozen.
So how do we deal with this? Sugar is going to be my limiting factor. I can make two dozen with the flour, but I have a limiting factor of sugar. Now, I know I need 1.619 dozen. So I need to figure out 0.619 times 12, right, because that’s going to tell me how many of the dozen I have. I’ll get 7.428, and I can’t have 0.428, so that’s seven more cookies. So I have one dozen plus seven cookies, which is a total of 19 cookies. So they get to make a bonus of 19 cookies because they bought these larger bags.
That was a lot, guys. Do you see why I’m telling you to skip these? Because they take too much time, and they may not be worth the points. So this is my first tip: skip them. But if you are going to do them, just dive in. Start to figure out what you can calculate, and by the end of the question, you’ll be able to figure out what the answer is.
When in Doubt, Draw it Out
Okay, my last quick tip: when in doubt, draw it out. You might get a problem like this one:
An arborist (a scientist who studies trees) comes across a tree that juts out of the ground at 12 degrees. When the sun is directly overhead, so that its rays are perpendicular to the ground, the tree’s shadow is 8 meters long. If it can be determined, what is the length of the tree?
So what do I do? I draw a tree jutting out of the ground at 12 degrees. Then, I draw a sun overhead. So there are rays perpendicular to the ground, and you can see that it’s a triangle. There’s a tree’s shadow that’s eight meters long, which is what’s under the sun and the tree. And so then I have a picture, and then I can do my SOH-CAH-TOA. So this is CAH, which is adjacent over the hypotenuse, and I can find that hypotenuse at the height of the tree. So, cosine(12) equals 8 over the hypotenuse, h. And then I rearrange this and get that h equals 8 over cosine(12).
So, draw it out. When in doubt, draw it out. You guys should always be drawing pictures. If you’re not drawing a picture, you’re being too lazy and you’re missing points, and with modeling, you really need to draw. So do not be lazy; get out that pencil. The more you draw, the better you are.
I hope you guys liked this blog and that it was helpful! If you want more practice on the ACT math section or if you need more modeling help, our math books have a ton of it, and you can buy them on Amazon. We are revising it right now for the new and updated ACT, but spoiler alert, the math section is not much different. It just is missing the answer choice E. So, what we were doing in our revision is we’re going through and we’re paying attention to the changes made, but we haven’t found any significant content changes. We also have an online course for the ACT, so if you’re looking for video lessons or if you like learning by video, it is the mothership of video learning, where I give you all my tips and strategies for every section of the ACT, and you can check that out at supertutortv.com.
