How amazing would it be to turn off your brain in the middle of the ACT math exam and actually get a lot of questions right in the process? In this blog, I’m going to talk about how you can do exactly that by utilizing the TI-84 calculator. There are so many powerful things this calculator can do, whether you have a Plus CE or an Evo. And if you have the silver edition, you can watch my full-on review of TI-84 calculators to learn why you should upgrade. But I’m going to get into tips, tricks, shortcuts, and programs all the rage on the TI-84. And yes, it’s not Desmos, but man, can it do a lot of powerful things in the same vein as what Desmos does on the SAT. So if you’re taking the ACT and you want your math score to bump, this is the blog for you. My name is Brooke, and I’ve been teaching the ACT for over 20 years. I’ve also written two books on the ACT math section, which you can check out at Amazon.com. We’re in the process right now of revising those for the enhanced ACT, but the old books are still awesome. And if you buy them now and you want some bonus material that is some of the new questions and chapter information that I’m putting in the upgraded version, we will send you a PDF of that if you go to the book owner’s area.
So first, one thing every student should know is how do you clear the screen if you want to get out of wherever you are. Often clear will work when I’m just on the regular calculations page, but if that’s not enough, you can click the two buttons up in the corner, second quit, or a combination of second quit and clear. If you just keep clicking second quit and clear for a couple of passes, you are guaranteed to get out of whatever mess you’ve gotten yourself into on the calculator and get it cleared out so you can start fresh.
Number two: how to store values. Storing values on the calculator is useful if you have to plug something in, for example, and you need to put it over and over into the calculator. You can do that by simply using the STO button. So you push a number like eight, you push STO, and then you can store it in any number of the variables that are available. You can see on the calculator there are a bunch of green letters. On the Evo, they’re on the keys themselves. And you just pick a letter to be your variable, one of those green letters. And what you’re going to do, you’re going to press store after you input the number you want to store, and then you push alpha, the green button, and then you push the key that corresponds to the letter you want. I did P, and then you click enter, and now P has stored the value of eight. So if you have a complex equation, where you have to plug into a function and P is the variable in the function, you can store eight in P, and then it will essentially plug it in for you as you type out the whole equation. If I have to evaluate an expression, it will evaluate it all in one step. So, that’s a cool thing. And then finally, the answer: if I want to take 89 and I want to divide it by three, you can just click divided by three, or if I want to do three divided by that, right? I can do three divided by, and then click second and ANS, and that’s going to pull up the answer like that. So you can see how we use ANS there. Those are the basics of the three functions you need to know.
Now let’s get into some of the ways that we can use this calculator. The first thing I’m going to talk about is decimals to fractions and fractions to decimals. This is a tool that I personally use. I took the June ACT, yes, as an adult. Luckily, I took it in a room of adults before the National Test Prep Association annual conference, which was cool because I got to take it with people my age in a hotel room as opposed to a bunch of teenagers and official administration that they coordinated with ACT. But what I will say is from my experience on this test, I used this probably six times in the math section. There are only 45 questions. That’s a pretty good number of questions that I’m using this tool on. I use the Evo on the test, and on the Evo, it’s super easy. The fraction to decimal, decimal to fraction button even tells you if you click it and you don’t know what’s going on. It says, “Toggle fraction to decimal and view syntax.” I don’t know what that means, but it toggles fraction to decimal. So, let’s take a look at that in action. I can do five divided by six, and that’s going to give me a decimal. And let’s say I did a bunch of calculations and I got a decimal, but I actually need a fraction for my answer because of the multiple-choice answers. I click that button, and it changes it back to a fraction, which is super cool. It can go the other way too. I can click it back and forth, and boom, it’s going to toggle between fractions and decimals. And I loved that button.
If you have a TI-84 Plus CE, we’re going to use a slightly different method, which is I’m going to do my five divided by six, but how do I turn that decimal back to a fraction? I’m going to hit math. It’s going to first highlight frac, which is number one. I click enter, and then boom, I get five-sixths. So, same process, but a couple of extra keystrokes in there. It takes a tiny fraction longer. It’s also less right where you’re calculating, because on the Evo, it is right down by your enter key. It’s in a really convenient spot to just click really fast as opposed to the TI-84 Plus CE. So, that is how you go from decimals to fractions.
We’re going to take a look at that in action now. There is a table of data, and you can see that a high school surveys a group of students. They ask whether they play an instrument and whether they take honors math, and we have this two-way table. Take a look at the video to see the complete table and fully understand this example. When I have a two-way table like this, I have to see what I need. I need the probability that the selected student plays an instrument if I randomly select any student. So what I need is the total at the bottom, and then I need that they play an instrument. Once I have those numbers, I’m going to take them and divide them by each other, but then I want to turn it back into fraction form because I’m pretty sure this probability in the answer choices is in fraction form. And obviously, you need to look down at the answer choices. I don’t have any here because I just made up this question because I’m trying to show you guys how to use this, so I didn’t make up all the answer choices. But if all the answer choices are in fraction form, I’m going to use this to then reduce the fraction, and the calculator is going to help me do that. So, I can fill out the table that way. How do we fill out this table? Well, if I have the total, what I do is 216 minus 182 to get 34. And then I get 88 plus 34 to get 122. And that’s one of the numbers I need. And then I can add two of the numbers to get 154, and then I can add 154 plus 216 to get 370. I didn’t really need to solve for the 248; that’s optional. But you can see I can just then take 122 divided by 370. So, that’s what I’m going to do, and now I have the decimal. But if all my answer choices are in fraction form, I’m going to click math. It’s going to select frac, and I’m going to click enter, and then it’s going to give it to me in lowest terms. On the Evo, I could even have put it in a fraction. I could use my fraction button. Another thing I love about the Evo is it has a fraction button. It’s so fun. There are two ways to do it on the Evo. You can either use that fraction button or you can do how I did it on this calculator and then click the toggle button. But either way, you can see how both of these calculators make it really easy to manipulate between fractions and decimals and decimals and fractions, and you 110% should be using that on your tests because it just makes life so much easier.
Now we’re going to talk about fraction operations. Fraction operations is the idea that you’re adding and subtracting fractions and doing computation with them. Again, the calculator makes it easier to do that. My favorite way to do it on the TI-84 Plus CE is just to use parentheses and division, and then at the end toggle back to the fractions. So, I’m going to show you how I do that, and make sure to check out the video to see the full problem.
We’ve got two houses here. One is four and a half miles north. This other one is three and seven-ninths miles south. So what I essentially need to do is figure out how far away these are. Those are all relative to the center of town. So if I drew a picture really quickly, G is four and a half miles north of the center. H is three and seven-ninths south. And I need to essentially add these together to find the distance from G to H. So what I’m going to do is 4.5 plus 3 plus—and I like to just separate out the fraction—7 divided by 9. I click enter, and then go to math. I enter the frac and then click enter again, I get 149 over 18. That’s going to give me my answer choice. If for some reason I needed the decimal, I could see that too, but usually it’s in fraction form.
One other note is for if you need a mixed fraction. You can also do that in the calculator. You go to math, and then you go over to the frac menu, and then you can see the mixed fraction option. That actually lets you write mixed fractions, but we want to go down to the “turn your improper fraction into a mixed fraction”, so you can go back and forth. By the way, the toggle that is on the Evo is also in this menu. So, if you want that toggle button to go back and forth, it’s here, but it’s kind of buried in a menu, so it’s not as fast as just using the frac. These also let you use fractions in the calculator and type them in as fractions, just like the fraction button on the Evo does. So, if you’re jealous of the Evo and you really want to do things that look like fractions, you can use that. I find it slower again than using parentheses and division, so I don’t use it. And the Evo has that same menu as well. If you want to do it in the Evo, you can kind of figure it out from that menu. And in the Evo, I would do the same thing, but instead of using that division, I would just use this fraction button. And again, if it’s not already in an improper fraction form, I just use the fraction button for the portion of the number that’s a fraction, and I add it to the whole number, and I find that that’s just faster than me messing with trying to make it into an improper fraction.
The third thing we’re going to talk about is “i”. So, “i” is that little letter that is indicative of imaginary numbers. And you can do “i” questions completely brain off, calculator on, on the ACT really easily by just using the built-in “i”. And whichever calculator you’re using, it is on the period button, and you just have to click the second and then the period button, and you’re going to get to the “i”.
I’m just going to solve a problem really fast, but I will give you a quick warning: if your calculator is not a Plus CE edition, there are varying levels of sophistication to the “i” key on older versions of the calculator. Several generations ago, the TI-84, “i” could only go to the fourth power. Going back to the problem, first, I’m going to do negative seven, and then I’m going to do times, we’ll click second “i”, times four, second “i”, times second “i”. And I click enter, and it gives me the answer “28i”. Brain off, calculator on.
We’re moving on to absolute value. Did you know that you can type in absolute values right in your calculator? So, again, brain off, calculator on. First, I’m going to go into abs. In here, I’m going to go to math, and then I’m going to go to num. Then, I’m going to go to one, which is abs. So you just go over to num, enter abs, and then it creates absolute value signs. Now, really important when you use absolute value signs: when you are putting negative numbers, you have to use the negative sign on this calculator. You cannot use the subtraction sign to stand in for a negative number. For the problem, I have to do negative seven, and then I’m going to arrow over, and then I’m going to do minus, and then I’m going to do absolute value again. I’m going back into that math menu over to num, enter absolute value, and then I’m going to do 12 minus 31. 12 minus 31. Enter. Boom, and it gives me the answer, negative 12. If you put this in your calculator and you get an error, what you probably did is you put a subtraction sign in front of the seven instead of the negative sign. That’s really important. This here on your calculator is the negative sign, and you have to use that. You can’t use minus, you have to use the negative sign. So, beware.
Okay, next thing I’m going to talk about is trig functions. Now, here’s the deal. Using the stored values of trig functions is not always the best idea in the world. And oftentimes, trig questions are best done without a calculator at all, I’m going to be honest. But if you don’t know how to do something, if you forget an identity or if you forget a principle, your calculator has every trigonometric value stored in the world. And what I would advise you to do is guess and then come back to it. And if you have time at the end of the test, you can pummel it with your calculator.
I’m going to give you an example. Let’s say you have a problem that says cosine squared x equals one-third. What is sine squared x? And maybe you don’t remember that sine squared x plus cosine squared x equals one. Because if you did, you would really easily know the answer is two-thirds. And that’s how you’re supposed to do this question. But if you totally forgot that, the way that you could solve this problem and any problem that involves trig is that you could derive what x is. So if cosine squared x is one-third, we can take one divided by three, enter, and now take the square root of that, by clicking second square root of answer. I click enter, and now that gives me cosine x. Now that I have cosine x, I can do arc cosine, which is cosine negative one to get back to the angle that gave me that cosine. So I’m going to click second cosine, which gives me cosine negative one, and then I’m going to click answer, which is second ANS, and I’m going to click enter, and so now this is the angle. Now what I’m going to do is I’m going to go back out the other direction. I’m going to click sine, and then click answer. As you can see, this is way slower than my method of memorizing the Pythagorean identity. And now what I need to do is just square that because that’s sine of x, and I need sine squared x. And if I do math, frac, enter, I get two-thirds. Obviously, knowing the identity is the much better way to solve this problem. But if you are stuck and you have time at the end, remember your calculator knows every trig value in the world. You can go inside and outside and jump around and just pummel at it with the fact that your calculator has that knowledge.
Now we’re going to talk about averages and data distribution. Some of my favorite tools on the calculator are in this section for me. In general, it might just not be super time-consuming to just do averages questions by hand. But when I’ve got these one, two, or three part questions where I have to solve for a standard deviation and a mean, or I’ve got two data sets, sometimes it’s nice to use the built-in tools on the calculator. And the way that this works is that we can actually enter the data set into the calculator, push one button, and then we get all the statistical data on a single screen.
So, the way that we’re going to do this is we’re first going to click stat. And then I’m going to click one, edit, and it’s going to give me these lists. Now, if there’s stuff in these lists, all you have to do is arrow up to the name of it and then click clear, and then it will clear the whole list at once. Now, I’m going to just put my list of all of the data in here. I’ve got 103, 107, 165, 189, 240. Now, for list two, I’m going to do list two, and I’m going to type the other set.
The first thing I’m going to say is we’re being asked the standard deviation. I’m going to show you how to calculate the standard deviation, but on the ACT, you generally never have to calculate the standard deviation. If you have just a standard deviation question, you probably don’t need to get out of the tables. It’s overkill unless there’s something that’s so close together or weirdly hard that you’re not 100% sure of yourself. In general, standard deviation is how far away from the mean you have to go to encompass about two-thirds of the data in a normally distributed data set. Now, not every data set is normally distributed, and so it gets a little sticky in there, but that’s a good kind of ballpark understanding of how it works. And the question that I ask between two data sets is how far spread apart these data points are. If they’re spread apart really far, my standard deviation is going to grow because to get two-thirds of the data set from the mean means that I have to go farther out, so that’s going to have a larger standard deviation. If all of the numbers are really tightly packed, we’re going to have a smaller standard deviation. This one is really tightly packed, so it’s going to have a very small standard deviation.
I already know list one’s standard deviation is much bigger than list two’s, so I can do that right away, even before I do any of this math, the standard deviation of set A is a lot bigger than that of set B. So I know that that’s true, and I’m going to get rid of all these other standard deviation things, but I also need to compare their means because I don’t know if that’s true or not. How am I going to compare their means? Well, once I get these lists in, I’m then going to go to stat again, but instead of edit, I’m going to go to calc, and I’m going to go to one variable stats. I’m going to go in and then it’s going to ask me which list. First, I’m going to find the statistics of list one. I’m first going to look at list A, and then I’m going to hit calculate, which is that button in the middle. You have to click F3. So, we know the mean here is 160.8. Now I’m going to go back and do the same thing, but when I do calculate, I’m going to do one variable stats again, but instead of L1, I’m going to do L2. I click L2, you see it overwrites, and then I need to hit calculate, which is actually F3 down here. And I got 160.8 again. So, I got the same mean for both. There is no differential between them, and the answer is A.
Like I said though, it also solves for the standard deviation. The Sx is actually the standard deviation. We can compare these two really quickly. You can see that Sx is 0.444. I told you set B is going to have a really small standard deviation because all those numbers are super tight together. Everything’s 161 except for one number, so all those are super close together. The other one, on the other hand, when I click stat, and then I’m going to go back to calc, and then I’m going to go back to L1 instead of L2, and then click zoom. You can see that standard deviation is 57.7, way more ginormous.
Going through some of the other things in these stats: “n” is the number of items in the list. The minimum value of x is the bottom of the range. And then if you scroll down, there’s actually some really cool data in here. If you arrow down, you can see the rest of the data, and the n is five, and the minimum value is 103. The maximum value is 240, and it gives you the first quartile, the median, and the third quartile. These are the significant points on a box and whisker plot. So if you ever have a data set and you need a box and whisker plot, you can find the exact values calculated for you with the calculator on, brain off, by using this tool. You can also do a stat plot. So if you are looking at pictures of those, and you need to figure out the numbers, you can trace them. You hit trace and you can get to all of the quartiles. But it can do the whole box and whisker plot for you, too.
Next up, we’re going to talk about frequency tables. Sometimes mean and median data before you find the mean and median is actually displayed in something we call a frequency table. This is the way a frequency table works. It basically is usually doing a survey or something like that, and multiple people are saying the same answers. So, for the example in the video, if I were to write out a proper data set, I would write zero 18 times, and then I would write one 42 times. And that’s what the frequency table stands for. But we can actually put this into our table, our stat table, and calculate the mean a little bit easier here. The way that I’m going to do that is to go back to edit, and hit enter. I’m going to go up and clear it out.
Once I’ve cleared those out, now I’m going to set this stuff up. I’m going to put the numbers on the left and the frequency on the right. So, we’re going to do zero, one, two, three, and on the right is the number of exercise classes they take. We have 18 people take zero, 42 people take one, 51 people take two, and 39 people take three. So I have all that data in there, and then I’m going to go back to stat, and calculate, and then one variable stats. But now you see this thing that says frequency list that I never filled in before. I’m actually going to put the frequency list as L2. So again, I’m going to go second and L2. And then I’m going to hit calculate, and it’s going to give me all the same stuff. It’s going to give me the mean, which is 1.74. It’s going to give me the standard deviation, which is 0.97, and it’s going to give me those quartiles. So if I need a box and whisker plot, I know what it should look like. And it did the whole question for me.
Again, if you have something like this, you guys know what to do. Now you enter all your data points, you get them in here, and you’re going to go to second plot. You’re going to make sure you have a plot on. You’re going to select box and whisker in the plot, and then you’re going to hit zoom and zoom stat, which is nine, and then it will literally graph it for you.
Up next, we’re going to talk about built-in apps and equation solvers. These are some of those powerful tools that are on your calculator. What I love about the TI-84 Evo, if you haven’t bought a calculator yet, is that it loads all of these right on the main page on the graphical menu here. So I’m going to show you both how to access these on the Evo and on the TI-84 Plus CE.
The first thing we’re going to talk about is the numeric solver. A few caveats on the numeric solver. Whenever you press enter, make sure you press enter twice because sometimes it stores values from the last time that you used it, and it can be confusing, and you can think that that’s the answer and put the wrong answer. So be very careful with numeric solver. But I’m going to show you where it is and how to use it. So on the Evo, you go to the home menu, and go down where it says numeric solver, and then you press enter. We’ll start with the equation solver: what I do is I’m going to set equation one equal to equation two, but it’s expression one equal to expression two, actually. So I can type something like x plus seven on the one side equals, and then we’re going to go down. Remember, the screen should be moved above the keyboard, and so that graph button is the next. But here’s the really important thing. This doesn’t mean the answer is 103. I have to push enter to solve it, or I have to push solve, which is this button again. That’s because it stored a value from the last time I did this. You have to be really careful using this tool that you always make sure you hit the solve button or enter to make sure that it actually runs the calculation. And then it will solve this for you.
Where do you find it on the other calculator? On your TI-84 calculator, you’re going to go into math, and then you’re going to click upward and it’s going to send you to the bottom of the list, which is C, which says numeric solver. When you click enter on that C, what’s going to happen is it will pull up exactly what you guys saw over here on the Evo simulator, and it works the same way, and that’s how you find it.
Now let’s look at a couple of questions we could use this numeric solver on. So, I have an isosceles triangle that has equal bases A and B. If A is 40 degrees and B is 3x minus 8, what is the value of x? Brain off, calculator on. I’m going to go to my graphical menu and to my equation solver, and I have 3x minus 8 equals 40. So then I can just do 3x, but instead of minus 8 on the one side, 40 is the next one. So I just arrow down and clear, and then 40. I take that, and then I push enter, and then I push enter again or solve, and again, I press it three times because I’m always worried it’s holding old values, and I get 1.5, and I’m done.
The next question: the ordered pair (B, 9) is a solution for this equation. What is the value of B? So if it’s (B, 9), that is (x, y) with y equaling nine. And then I get 2x plus 9 equals 36. So again, brain off, calculator on, to re-enter this. Okay, going back into the equation solver. I’m going to do 2x plus 9 equals 36. Again, super easy. Make sure to hit solve multiple times. At first it said 1.5, which was wrong. You have to keep hitting the button until something changes. And then always double-check when you use the numeric solver. Because like I said, there is this issue with you having to push return and it having to register. Sometimes it’s a bit buggy, so just be careful with it. x equals 9, 3 times 9 is 27, plus 9 is 36. We got the answer.
You can even use it with things like crazy exponents. So 3 to the x minus 3 equals 81. I could do that. Back to my numeric solver. So I’m going to click second quit and clear just to try to clear it out because it seems to be storing old stuff. What do I need? I need three to the x minus three. So I’m going to do three, and then I have the power button to the parentheses for x minus three and put that as equals to 81. Then I’m going to click next, and then I have to click solve and get x equals seven. Then, I can test it. Three to the fourth is 81. That sounds right. Brain off. Calculator on, boom. But just make sure you press solve, or you could be in trouble.
The next tool that I’m going to show you is System Solver. System Solver solves a system of equations, but these are linear equations. So, if you have nonlinear equations, we have to use something else. But for systems of linear equations, this is what we’re going to use next. And you can see how it sets up my system of equations if you take a look at the video. This is two by two, which is the standard. Now the real way to do this really fast is actually to stack and add. I actually do this without the calculator, but I like to show you the calculator trick just in case. And some students are weaker in math than others. And if you’re trying to up your math score and you got an 18 on the thing, just learning these calculator tricks might get you five points in three days. If you’re already getting a 34, you can probably do this without your calculator faster. The way that I like to do this is stack and add. So I just do 4x, and the y’s cancel, which equals 14 plus 18, which is 32. And then I just divide, and I get 8 is x. And I don’t have to do much work, and it’s super fast. So I like to do this stack and add because those cancel really fast, and it’s a really easy elimination problem. But again, if you’re shaky with your algebra, you just use a system of equation solver. And when things get more complex, this is actually really helpful. Because not everything is this easy. So then, we have y equals 14. Going back to my other equation. I have 2x minus y equals 18 to x. And then I’m going to hit the graph button, and then again. And I get x, which would be 8.
Let’s talk about the TI-84 Plus CE. Where do you find this? You find this under apps, and then I only have one app here because this is like a simulator that doesn’t have the rights for all of these applications. But on your TI-84 Plus CE to get to this exact same thing, you’re going to go to an app called PlySmlt2. For me, it’s number nine in my apps list. So, I click apps on my calculator, and then I go down to number nine, and then after I hit nine, it’s the simultaneous equation solver, which is the system solver. They’ve changed the name of it. They used to call it Simultaneous Equation Solver, which is less clear to me than System Solver. System Solver sounds like a system of equations solver. They changed the name, and that makes it easier. But you just go here, and then it’s the exact same tool. You click next, and you get the exact same window.
All right, next up in that PlySmlt2, you might have recognized something else called polynomial root finder, also something that’s super valuable. On the Evo, it has moved to our front graphical menu. So, you can click enter, and then it gives you the setup, and you can say the order. The order means the greatest exponent that exists. So if x to the third power is the greatest exponent that exists, then that order is going to be three. If it’s an x to the fifth, you’re going to make the order five. Then you’re going to say whether it’s real or whether it’s imaginary; usually it’s just real. Dec, normal, float, radian, or degree doesn’t really matter, but you can do whatever you want, and then click next. Then it’s going to pull up a dialogue, and you can fill that out. Again, to get to this on the TI-84 Plus CE, you’re going to go to applications, and I, again, go to number nine. My on-screen one doesn’t have number nine, but you’re going to go to PlySmlt2, and then within PlySmlt2, you’re going to key to the polynomial root finder and press enter. And then everything from there forward is going to look just like my TI-84 Evo.
All right, what are we doing? We have a trinomial x squared plus 13x plus 36. I’m going to go to my polynomial root finder, and I have an order of two because it’s an x squared; it’s a quadratic. So, I just do an order of two. The next one again is a graph key. Then, I fill out the rest of the information and hit enter. So, now here’s what’s going on. We’re actually factoring, but the way we can factor is if we find the zeros. If I have a factor of x minus four, that indicates a zero with the opposite symbol. So, it’s positive four. The signs of the zeros are the opposite of the signs of the factors. I can infer all these factors based on what my zeros are. The first thing I’m going to do is find the zeros using the polynomial root finder. So, I enter all the information in, and then I just hit solve. And I get negative nine and negative four, but negative four is the solution. The factor is the opposite sign, so that’s x plus four. And that is an answer choice, so I’m done. Now, negative nine was also an answer, but that would have given me x plus nine. That’s not available, so I know it’s not right.
The next thing that I’m going to show you guys is how to do linear regression. Now I know a lot of you may know how to do linear regression in Desmos, which is really fun and easy and quick, but you can also do it in a TI-84. It’s a little less intuitive, but stick with me and I’m going to show you how it’s done. First, we have a funky function. And it asks which of the following values is closest to the average rate of change of the function from x equals negative eight to x equals 12. Really quickly, all that we’re doing here is we’re going from x equals negative eight to x equals 12. And what we essentially want to know is the average rate of change. Guess what the average rate of change is? It’s just the slope of the line that connects those two points. That’s it. So knowing that I have these two points, I’m going to write them out. The first one is (-8, 15), and the one on the right is (12, 5). I have this pair of points, and what I’m going to do is a linear regression on this pair of points using the stats functionality in the calculator. I’m going to go to stat, then click edit and enter. If there’s anything in here, again, all you have to do is arrow up here and then clear it out. Then, just put in this set of ordered pairs. It’s really important, again, to make sure I use the negative sign and not the minus sign, or things get really wonky.
To do the linear regression, a couple of things. When I have to go into mode, I’m going to turn the diagnostics on. On a CE edition, you just go to where it says stat diagnostics, and that’s what you’re going to turn on. In the Evo, it’s just diagnostics on. Now I’m going to go to calculate, and then I’m going to go to linear regression, which is number four. Notice you can also do quadratic regressions here. If you have a quadratic, that’s something that you could use. There’s also exponential power regression, and they give you the form here, which is nice on the Evo. On the older edition, I don’t think it gives you the form, which makes it more confusing. But at least if you have any equations like this, you could do a regression in here as well, but you can’t customize the regression. Maybe you can; I’m not really sure.
So, the X list is L1, and the Y list is L2. There is no frequency list, and I’m just going to go down to calculate, and in my linear regression, my slope is negative one-half, and my B is 11. So, that’s how you do an “mx plus b” linear regression in a TI-84. The other way that you can find the equation of a line is to use a program.
The next way I’m going to show you how to do that is with a program. Caveat on programs: ACT has some rules when it comes to programs. What they say is they’ve forbidden the use of programs that use what they call CAS, or computer algebra systems, functionality. It’s really hard to find or dig up evidence of what that means according to ACT, but they released an official document several years ago, and I happen to have copied all of the language on it into a blog on my website. And if you ask AI what that means, they’ll actually potentially point you to information from my site. In any case, the basic idea of avoiding that is one: we want simple programs; they have to be less than a certain number of lines of logical code. And two, we don’t want anything too complex; it has to be single functions. So, we don’t want it to do a ton of stuff. We just want it to do like sort of one simple thing. And the example that they give of something that’s okay is the quadratic equation, so we’re definitely going to throw the quadratic equation here into the calculator programs that we think pretty much adhere to these. We can’t guarantee that you won’t have a proctor that tells you not to use them or that they won’t get you in trouble or won’t, you know. I think when it comes to getting in trouble with ACT proctors, it has way more to do with what random proctor you have than what the rules actually are.
In our estimation, we think these are probably cool, but we’re not ACT, so we can’t guarantee that. And you are using these programs at your own risk. That being said, I’ve worked with hundreds, if not thousands, of students over the years, and I’ve only ever had one student have an issue with programs, and that issue was that the proctor made them all clear the memory and all of the programs that they had on their calculators before they began the exam, and that’s about it. So nothing bad has ever happened to my students except for not being able to use the programs. And like I said, I just did linear regression, so at least you know how to do that without a program. So, first of all, with programs, I have videos on how to download programs to your calculator, which you can go and check out. If you are trying to upload programs to your calculator and you have an Evo, I have a whole video on how to take old programs for the TI-84 and convert them into Evo programs, and you can go watch that calculator video as well. So, I recommend that you check out these other two videos if you’ve never transferred programs to your calculator. But after you’ve done that, you can actually get some programs up. We have a package of programs, and if you would like these totally for free, you can just go to SuperTutorTV.com/free and download our package of programs. I think we’re also going to be adding the Evo programs in there as well soon, so you don’t have to convert them. We’ve already converted them, so then you can just download the Evo package.
Here are some of these programs. The first one that I’m going to do is the slope one, which is just slope. I’m going to press enter, and this is the other alternative. You want to find the equation of a line in slope-intercept form, but you don’t want to go into the stat menu and list and all of that stuff because it looked confusing and you don’t want to have to check the mode, so you can just use this instead. My x1 is going to be -8, and y1 is 15. And then x2 is 12 and y2 is five. This is a little bit simpler than using the regression in the other way, and then it just gives you the equation, and we know that the m here is the slope, which is -0.5. So, a super simple and easy program that you guys can use and that I would recommend. And again, I usually do that more often than using the regression, but it’s good to know how to do both, just in case you get that nasty proctor. Once you get them on your calculator, you just go to program on the Evo and go into TI-BASIC. On the Plus CE, if you don’t have Python, you just go into it; it just pulls them all up automatically. And if you have the Python, which I have the Python CE, you also have to check into the BASIC because these are all BASIC programs.
Next up, we’re going to talk about the Pythagorean theorem, a super simple program. We have this right triangle, and we need to find x, but in order to do that, we first have to do something clever, which is figure out the area of the triangle. The best way to do that is the Pythagorean theorem, unless you happen to know this Pythagorean triple, which is kind of a doozy, and most of my students probably don’t know it. Most of the time on triangles, you should be using Pythagorean triples. You shouldn’t need the Pythagorean theorem. But if you do need it, I got it right here in programs. So I’m going to go down to the Pythagorean theorem, press enter, and they’ve got the exact one I want, which is I want a leg when I know the hypotenuse. I’m going to select that and put in the values, and it tells me that the other side is 20. And then what I’m going to do is find the area of this triangle by taking the two right legs, multiplying them together, and then multiplying by one half, which is going to be 210. One-half of 20 is 10 times 21 is 210. So, that is the area of the triangle. I also know that the area of the triangle has to be equal to one-half of 29x. Okay, so I can write that out, and then I’m just going to multiply both sides by two. I get 420 equals 29x, and then I’m just going to divide by 29. 29 is a really ugly number, and it’s going to give me a very ugly fraction. So, then I would look down at the answer choices, and it really just depends on if the answer choice is rounded or exact. And then I might want to divide these in my calculator and find the decimal answer, and I’m going to put the closest to that. Again, if this is an easier triangle, I want you using Pythagorean triples before those times when you need the Pythagorean theorem program.
The next programs that I want to talk to you about are for the law of sines and the law of cosines. A quick review of the law of sines: the law of sines is a formula that essentially creates a ratio between the sine of any angle and its opposite side length. And that ratio is equivalent to the sine of another angle in the same triangle over its opposite side length. So, there’s this special ratio, and that’s created by the law of sines, and it basically allows us to solve triangles where we know either two sides and an angle, that angle being opposite one of the sides we know, and we want to solve for an angle that’s opposite the other side, or we know two angles and one side, and we want to solve for one of those other sides. I know this is a little bit complicated. You really should be getting my ACT math books if you really want to deep dive into this, or you can Google it. But I’m going to show you a program that basically does the law of sines automatically.
We’re going to go to LawSine2. I just revamped this to make it more user-friendly and to work on the Evo, because our old one stopped working on the Evo, even when we converted it. If you have a newer version of the CE edition, this works better than the old SuperTutor one. So if you’d like to download this and the other programs that I’m using here, you can download them for free by going to SuperTutorTV.com/free. Let’s first find a side. I’m just going to go to number one choice and then enter, which is the angle opposite the unknown side. That’s going to be 100. Then I need the opposite side, angle B, and opposite side Y. Angle B is going to be 30, and side Y is going to be opposite that, which is 5. So I’m going to enter that, and then it tells us the missing side. It’s 9.848. If the numbers are not in that format, they might be 9 and then the root of something, or it might be different syntax on your paper.
You should also familiarize yourself with the law of sines because sometimes with law of sines questions, you don’t have to solve the triangle out completely. But anyhow, this is kind of a handy trick to have in case you do have to solve a triangle. The second one is to find an angle. It works pretty much the same way. I just put in my known angle, and enter the information from my board. We also have a law of cosines program.
I’m going to go second quit, clear, and we’re going to go back to programs, and I’m going to go to TI-BASIC, and then I have the law of cosines. The law of cosines is a little bit different because we use it to solve for a single angle when we have three sides of a triangle. So here, for example, I have 5, 7, and 11. So one option is to find the angle from three sides. The opposite side that we need is 11. The other side one is 5, and the order here doesn’t really matter, and then I have a 7, and it gives me the angle is 132.1774186. The other thing I can solve for is if I have 5, 7, and 132, I can also solve for that. So, I’m going to find the third side from SAS, which is what that says. And so I’m going to do choice one, and the first side is 5, the second side is 7, the angle between is 132, and you can see boom, I get 10.99, which is around 11, which is my other side. Those are helpful as well when you’re trying to solve triangles. Law of sines and law of cosines, remember, we use when we don’t have right triangles. When you have a right triangle, it’s usually faster to use the Pythagorean theorem or SOHCAHTOA to solve those. But when you don’t have a right triangle, these two tools are going to be helpful.
Okay, number of diagonals. We have a program that will tell you the number of diagonals in a polygon. This may only come up about once every 10 or 12 ACTs, but when it comes up, this makes it really easy. Here’s all we do: we count the number of sides. There are five in a pentagon, right? You can count them yourself if you want, and we want the number of distinct diagonals. So, how do we do that? Well, I come back over here to my Evo and go to my programs, then TI-BASIC, and I’m going to look for num diags. Number of sides: it’s a pentagon, and we get that the number of diagonals is five, and I’m done. And it’s that easy.
Next up, a couple of other programs. We have a midpoint program. If I have (5, 7) and (6, 8), it will find the midpoint of those. So, a really simple program. A lot of times on the ACT, it’s actually kind of how that Pythagorean theorem problem was, where they give you the midpoint, they give you one of the points, and then you have to find the other point. So that program is not as useful for those kinds of things, but we do have it, which sometimes is useful, but not as useful as a lot of things I’m going to cover.
Next up is the quadratic equation, also a pretty self-explanatory kind of program. We actually have two of them. I don’t know if both of them are in our package. One of them is the quadratic formula. And so here I can just do 1, 4, and 16 and get a non-real answer. It gives you a non-real answer when there’s a non-real answer. There’s also quad form, and you press enter, A, which is 1, and B, which is 4. And then when I’m going to do a negative, I’m going to put it in parentheses just to be extra safe and make sure you use the negative sign, not the minus sign. So, the answers are 2 and -6, and it just solves for the answer. The other one that I have, I don’t know if we distribute, but it’s a little bit more intuitive because it shows you the formula.
Okay, so getting into another problem. We’re going to use the distance formula on this. When is the distance formula useful? Well, occasionally on coordinate geometry questions, you might need a distance interstitially to solve a question. You can see here, this is a right triangle, and we want to find the area of it. And so this is a time when I could use this distance formula, but it’s going to be kind of ugly. So, in order to do that, what we’re going to do is we’re going to go into programs, TI-BASIC, and distance. And you just enter the points, and then it’s going to find the distance between the two points. Obviously, knowing the distance formula is good. It’s basically built off of the Pythagorean theorem, but it’s much easier to just do this. Enter in (2, 7) and (8, 11), and it’s just going to tell me the distance, which is 7.2111 or so. Then I do it again for the other points: (8, 11) and (1, 14), and I get 7.6157. And then I can just multiply these two by each other times one-half, and I’m going to get the area, which is going to be something like 27.46. So, there is a way that you could use the distance formula if you have a question like that on the SAT. Coordinate geometry is definitely a good place to do that.
Next up, I’m going to talk about graphing. Graphing is kind of a superpower of the TI-84, and there are some things that you should definitely know. First of all, you need to know how to graph conics. It might not be super intuitive because you actually have to use an app in order to do it. On the CE edition, you’re going to click apps, and then you’re going to go down to four, which is conics. It might not be four on your calculator, depending on what you have loaded on there. You can actually manually load things on there, and it’s possible to delete things. And once you’re here, you can graph a circle, an ellipse, a hyperbola, or a parabola. And you can graph a parabola that might be in the form that isn’t y equals, and that’s where this gets kind of tricky. You can see none of these have the y isolated, and that makes it really hard to graph in the typical y equals fashion. This one I actually recommend kind of thinking about before you start to plug things in. If you have to plug in four different equations and type them all out and graph them all out, that’s going to be really time-consuming. Use your brain first, see if you can sort of get it down to what you think it is, and then double-check the one that you think is right. You can see for this, if it were centered, it would be at (0,0), and there would be no shift of y or x. But you can see here it’s either -4y or 4y. We don’t quite know what it is. To graph conics on the Evo, you actually go to the lines and conic app. And I can go to add an equation, and then it’s going to give me a little menu, and I can choose what I want, and I want a circle. So I’m going to do a circle in this format because that’s what I have to work with. So what I have is x squared, and then I have y squared, and then plus 0x. And then let’s try -4y. And then I’m going to do plus -5, because it doesn’t let me adapt the zero. Then I hit return, and then I’m going to hit graph. When this is -4y, it’s y minus 2 and x minus h, and y minus k is the center. So if it’s negative, it moves up. If you want, you can then go on and graph all the rest of them as well. Up to you. And you can obviously trace if you want to trace; you can click trace, and then you can try to see what’s going on here.
In particular, I tend to use graphing as a tool to solve things on the ACT, particularly when I have nonlinear situations. Inequalities are really great too. If you don’t know how to graph inequalities, I recommend that you check that out. I’m not going to go too deep into the graphing, but I did want to show you the conic app. You can also use this to solve a system. Remember if we’re solving a system of equations, I could go back and add another equation. And I can add an equation that’s just a line, and I can add ax plus by equals c form, but if I want, I can add a line, and then we can find intersection points. And what’s cool about the Evo is it actually will show you things like y-intercepts. The other thing that’s really great to use graphing for is key features of graphs. If you need to know y-intercepts, x-intercepts, maximums, and minimums, again, you can totally trace those out. And like I said, when you are on the Evo, it will actually kind of jump to those points and label them for you.
Next up, we’re going to talk about probability. There are three probability functions in your TI-84 calculator that are good to know. One is nCr, which we use for combinations. Remember, combinations are when we take something so many at a time. You’re taking 10 items three at a time, and so the order doesn’t matter in a combination. A permutation is nPr. When we have a permutation, remember those are arrangements or order questions in which order matters. So what you pick in what order is important. And finally we have factorial, which is just that exclamation mark, which means multiply all the numbers from the number in front of the exclamation mark and count down by one all the way until you reach one. So if you have five factorial, that means five times four times three times two times one. That’s what it means, but the calculator is going to calculate it automatically for you, which is cool.
I’m going to show you a couple of the problems where you can use some of these tools. So here we have Panda Salad Emporium. There’s a salad trio dish, and it invites customers to choose one portion of three different salads from their six summer salad options. So, this is six choose three, and that’s a combination. We’re going to go into math, and then we can go to prob, and you can see right down here we have nCr. So I have six different salads, and I’m going to choose three of those six. And so that’s just six choose three, and it gives me the answer: 20. I could also teach you to do this using the fundamental counting principle. If you want to learn that, buy my math books or my online math course. I have an online video-based course for the ACT where I go through every chapter in my math books super in-depth. I also have really awesome grammar lessons and some tips for reading and science. You can check that out at SuperTutorTV.com. But this is such an easy way to solve a combination problem. The biggest thing is you have to understand that it’s a combination.
We’re going to do another one. Dwayne and three of his friends go on a carnival ride. They can sit four people front to back. They only go on the ride once, but Dwayne is curious how many different ways he and his friends could sit on the ride. How many different ways can they sit on the ride? So here, order matters. So, I have four permutation four, because order matters. So I’m going to go back into that menu, math, and prob, and then I’m going to go to nPr. And here I have four people, and I want to know how many ways they can sit on this log ride, which is 24 ways. Be really careful with these wordings. Dwayne and three of his friends—you wouldn’t believe how many of my students try to do like three permutation four, which does not work. You can’t choose four things from three items.
Okay, then we have factorials. For the problem in the video, we need to find x factorial. The first thing I want to figure out is what is going on here. So, the numerator is going to be two more than the denominator. That means it’s going to be 7 times 6, or two numbers in a row that equal 42 times 5 times 4 times 3 times 2 times 1. But I want to cut off 5 times 4 times 3 times 2 times 1. So, x is essentially six here. If the numerator is x plus 1 factorial and the denominator is x minus 1 factorial, it’s going to be some number that’s two more. So, x minus 1 plus 2 is x plus 1. And then I know that whatever’s left over, and doesn’t get canceled out. If you do this enough, you’ll figure it out. I know that’s kind of what the scenario is going to look like, and then I need two numbers that are next to each other chronologically that multiply together to be 42, and that’s going to be 7 and 6. So I know that x is going to be 6, but that’s not what they ask for. They ask for 6 factorial, and that’s when I’m going to get out my calculator. I’m going to go back to this menu. First, I’m going to click 6, and then I’m going to go into math, and then I’m going to go to prob, and then I’m going to go to number four, my exclamation mark, and hit enter, and it gives me 720. So, I don’t have to sit here and go 6 times 5 times 4 times 3 times 2 times 1. Too much work, and I don’t want to do that. I can just use the factorial button, and I’m good.
Exponents are up next. Now, if you have the Evo, these are a little bit easier to find. There is an nth root button now, and it’s on the mainframe of the calculator. It can be a little bit harder to find on the Plus CE, and that’s why I’m going to show you where it is really quickly. Where you want to go on the CE if you want to get into exponents is you go over here into math, and then the nth root key is right there. Cube root, cubed, and the x root are all in this math menu. So make sure that you’re familiar with that math menu and you know where it is and how to use it, because that can come in really handy on the ACT. It’s something everybody should know. And again, on the Evo, it’s right on the keys. You shouldn’t miss it; you should know where it is too.
Another program we have is Simprad, which will simplify a radical. It’s not going to simplify it when you’ve got variables in it, but it will simplify a number to its lowest terms. So that’s another program to check out in our program package. Right now, though, let’s take a look at the problem in the video and how we might solve this. One thing I might do is I might take the fourth root of 16, and again, I can do that on this calculator; it’s just going to be second, and then I’m going to put a four up top, and then I just put 16 at the bottom. It’s the same thing on the CE; you just have to go into the math menu. And in fact, it’s probably still here. What’s cool about the calculator is they didn’t move stuff out of the math menu if it used to be there on the Evo; it’s still in the same place that it is on the CE. So, if your muscle memory is plugged into a CE and you switch to the Evo, it’s not that daunting. Then we have to do the b to the 12 to the fourth root, and then we just do fractions, so we just turn the root into a fraction. That’s 12 divided by 4, so it’s b to the third. What I essentially need is 2b to the third. Now, here’s something that’s kind of funky. When I look down, there are two things with b to the third, and that’s when I have to invoke my brain a little bit. When I take something to an even power, here’s what happens. If it happens to be negative, all that negative gets sucked away, and it becomes positive, and then I can take an even root of it, and it’s still positive. But this is for all real values of b. It doesn’t say that b is necessarily positive. And so this is actually safer when I think about it, and the answer is C. So does having the calculator and understanding where the root thing is help me? Yes, for sure, but it’s not going to do the whole problem for you. You still have to think.
Scientific notation is up next. EE is what these calculators use for scientific notation instead of 10 to the something. So if you see 5 EE 17, that’s 5 times 10 to the 17th. So I’ve got a couple of problems really quickly here.
The average diameter of a monocyte is 2 times 10 to the -5 meters. And it asks how many times this diameter is of the diameter of a red blood cell, which is 6.5 times 10 to the -6 meters. So, what I have is 6.5 times 10 to the -6 times n equals 2 times 10 to the -5. That’s how we write this in calculator language. I’m going to first enter 6.5 times 10 to the -6, and then I’m going to do 2 times 10 to the -5 divided by that. And we get 3.08. I would recommend playing around with this because it is a little bit funky. It doesn’t like parentheses; it’s really picky on syntax. So just be careful with it.
Finally, the last two things I’m going to teach you guys are LCM and GCF. Really quickly, LCM is like when we have these word problems where a flashing light flashes over and over again. Or here, we have pens sold in packs of six and bundles of eight. And if both sold the same number of pens, that means the number of pens sold has to be divisible by both 8 and by 6. And I want the smallest number that each could have sold. Now you could do this manually and find the answer that way. And that’s not that time-consuming. Let’s be honest, it’s not that bad. But it’s also totally built into your calculator, and you can do it in about two seconds. And maybe it’s slightly faster than that, especially on problems that are a little bit more challenging. So, you’re going to go over here to num, and then you can see here we’re going to do the least common multiple of six and eight. Click enter, and it’s going to give me 24. So there you go; there’s the built-in least common multiple. The greatest common factor, remember, is what is the biggest number that is a factor of both numbers. This obviously is going to be much faster if I use this tool. So I’m going to go again to num, but instead of LCM, I’m going to do GCD, which is greatest common denominator, the same as greatest common factor. I have 4620 and 1575, and it’s going to solve the problem for me, and the answer is 105. So, if you have the greatest common factor, the least common multiple, or a word problem that’s using that concept, you can use these tools to do them super fast.
I hope you guys like this blog, and I hope it helps you level up your score on the ACT. We have an online self-paced program. So, if you want more amazing tips via video and practice drills, in particular for the English and the math sections, that includes access to all of the items that are also in my ACT math books, which you can also purchase at Amazon.com and walkthroughs of every single chapter. If you want a math tutor walking you through every single thing in my math books, which are super crazy comprehensive, you can get it online on our online course: check it out at SuperTutorTV.com.
Be sure to check out our FREE ACT® Calculator Programs and ACT® Math Formula Cheat Sheet here!
Good luck on your ACT!
