3D Plots on a TI-nspire

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With the end of exams, it’s time to think about fun things we can do with math. Today’s post comes from a emailed question: “Can I produce a 3D plot on a TI-nspire?”

The answer is yes!
First, go to the home page. you want to select the button with the parabola on it (second from the left). It will say “Add graphs to new document.” 


Now press the menu button and select View and then 3: 3D graphing.

You will now see a 3D cube and a place to enter a 3 dimensional function.

Ensure that your equations are in the form z as a function of x and y. Enter each one and it will be graphed on the 3D cube with each equation drawn as a different colored surface. my equations here are z=ex-ey in grey and z=sin(x)+cos(y) in purple.

You can use the arrow buttons to rotate your graph. The Range/Zoom button has a lot of options for changing the graph as well.

All I Want to do is Zoom! (Position, Velocity, Acceleration) Part 1

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One of the most extraordinary things about calculus is how it gives us a new way to connect concepts and one of the most beautiful is position, velocity and acceleration. We know we can start at a point, jog forward at some speed, and end up in a second spot. If we make the journey again, now running at a higher speed, we can travel farther in the same amount of time (change in distance, holding time constant) or arrive at our destination more quickly (change in time, holding distance constant). We can also alter our speed by speeding up or slowing down during the run itself, changing our acceleration.

These two changes, position influenced by velocity and velocity influenced by acceleration, can be modeled using derivatives. We start with an equation for position, in this case

f(x)=-(x-2)2+4.

This results in a lovely parabolic curve that starts at a height of zero at time zero and then goes up reaching a height of four at time two and landing back on the ground at time (height of zero) at time four. Try to describe the velocity in words. Remember that velocity can be negative.

We go up from time 0 to 2 and down from time 2 to 4. Our forward velocity is then positive from 0 to 2 and negative from 2 to 4. We calculate the velocity and graph it.

The velocity is the purple line. Desmos will graph derivatives for you: you can define your position with a function like F(x) then go to the next line and type

G(x) = d/dx F(x)

to see what it looks like (we will need the G(x) when we look at acceleration. Notice when the purple graph is positive (time 0 to 2) and negative (time 2 to 4) for the same intervals we said the object was traveling up and down. The very highest position is where the velocity crosses y=0.  This is going to be super important later: at the top of a peak or bottom of a valley the first derivative will be zero because that is the point where you change direction. And what is the first derivative?

G(x)=-2(x-2)(1)=-2x+4.

Note that this is the equation of a straight line with a slope of -2 and a y-intercept of 4.

Now let’s look at the acceleration by taking the derivative of the velocity, G(x). The derivative of a derivative is called the second derivative of the initial function and it is also going to be very important (we can keep taking derivatives until we get zero, but most of our work as undergraduates will be with the first and second derivatives).

The acceleration is now green … and it’s a negative constant! This makes sense mathematically as

d/dx ( -2x+4) = -2.

The slope of the velocity is the acceleration. What does constant acceleration look like? You’ll probably find out in your physics one lab but if you can’t wait, find out if your school has a programable model car your classmates and you can borrow. Find a long flat quiet space and set the car loose at a very slow acceleration. See how long it takes before it is zooming at top speed!

So your teacher may ask you, when is the object speeding up and when it is slowing down? In this case, we can see from the graph that the object slows down as it gets higher until the peak, at which point it speeds up as it get closer to the ground. The point where the velocity is zero? That’s the critical point on our graph. Finding where velocity is zero is key to determining when a graph is slowing down or speeding up … but as you can guess, acceleration plays a part too. Next time we will look at what happens when acceleration isn’t constant!

Impermanence

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I feel it is very important to take yourself seriously as a math student, even if you have no intention of taking more than the class or two required for your major. This means honoring your math practice. One of the most important lessons I’ve learned is to embrace uncertainty.

Every mathematician I know makes mistakes and is proud of them. Mistakes often lead to discoveries. It is difficult to do math if you don’t feel free to explore, so give yourself permission to experiment and that means lessening the pressure.

  • Keep your class notes separate. The notes you get from your teacher are a map to get you where you need to go in your class. Yes, they can be difficult to understand at times but this is why you must keep them together in a form that is easy to use. Sometimes it takes another lecture before today’s notes make sense. Therefore they need their own place. You can add observations about your teacher’s notes (and I recommend you do) and even the occasional explanatory doodle but don’t mix anything else in with your class notes.
  • Scratch paper. If we reserve our notebook for notes only we need a place to work. No one sits down to do math and produces perfect work the first time. If they do, they aren’t really doing math (more about this later). Math is about challenge and flexible thinking and you need space and freedom to explore. I do my work on loose leaf paper in pencil or on portable white boards (you can buy a pack meant for children for very little). I never plan to turn in my first draft, ever. This way I can try the problem in different ways, make observations, make mistakes, and make discoveries.
  • Pencil. I recommend buying a mechanical pencil. Find one that feels good in your hand and reflects who you are. If you feel you must use a regular pencil, make sure you keep it sharp. Pen is great for making a bold statement but it makes a mess of your work when you make a mistake. If scratching out your work is going to get confusing, don’t use a pen. Just as an artist makes a sketch before a finished drawing, a mathematician treads lightly during the exploration phase.
  • Eraser. You will wear through your pencil’s eraser in no time so it’s best to have an eraser. I use a little one for little mistakes and a giant one for huge mistakes.
  • Play with Graphs and Figures. We are fortunately to have a variety of tools close at hand to help us understand our math. Often your instructor is tasking you with developing a gut feel for the math and that means play. Using your calculator, Desmos, or Excel to explore equations can really deepen your understanding. If you are taking higher math you might want to learn a language like Matlab or Python (these are great to have on your resume): ask your professor what they recommend.
  • Rewrite your work. A big part of doing math is how you present your work. You want your ideas to shine as clearly as you would in an English paper. So after all the craziness on your scratch paper, condense your math into neat, legible final form.
  • Show your work. So you’ve done all your scratch work and you are ready to present the final product. Your instructor is not as worried about the correct answer as you might think, they want to see your thought process. You don’t need to include every step, but show enough that the instructor can follow along. If your final answer is wrong, showing your work can get result in partial credit and a comment on where you made your mistake. I know instructors that won’t give credit at all unless they see the work!
  • Write for a non-professional audience. Often when we do math, we are thinking about our professor who seems to know everything. This is not the best practice because we can end up taking shortcuts that she won’t like. Instead, I recommend writing for a friend or family member. Make your work clear enough that someone that doesn’t know the assignment can follow along. If you can describe what you did to someone out of the class then you really understand what you are doing!

Exploring the Unfamiliar

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I read a fascinating essay from Professor Barbara Oakley today. Professor Oakley has spent a lot of time thinking about learning, especially learning about mathematics. I highly recommend her book, “A Mind For Numbers,” it taught me a lot about how I learn and how to help others that might learn differently than me.

Illustration from Barbara Oakley’s Essay in the New York Times

She has a new book, “Learning How to Learn” written for students K through 12 but applicable to anyone at any age hoping to establish good study habits. On her website she has some wonderful worksheets to guide you on the best ways to study.

One of the things she suggests is to study in different places. I feel it is very important to set up a nice study space in your home, a place you customize to be perfect for you. And it always feels good to have a special seat in your class. Having a study routine is a good idea. At the same time, you must remain flexible. Sometimes your professor will rearrange the class for exams or even use a different room and you must be sure that you can perform your best under any condition. Other times you get your favorite spot but there’s more noise than usual coming from the hall. Using different spots when you study is a great way to ensure that your concentration won’t be blown if something unexpected happens.

So have a dedicated place to study at home but also at least one place to study on campus and try different seats. I have three places on campus: the study rooms in the math department of my school, the community room outside the tutoring center, and the library. Also have some places to go that are neither home or school. Tech workers call these places “third spaces” … they aren’t where you live or where you go to work but they are places that feel good for studying, writing, and solving problems. These can be any place that won’t mind you sitting at a table for a few hours. Coffeeshops and public libraries tend to be great for this, but some restaurants, bookstores, and museums can be good too … so don’t feel like you have to learn to drink coffee to have a third space!

Why Math?

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Everyone has reasons why they study what they do. I want to share why I study math.

I went to an arts high school in Virginia where I studied painting and sculpture. One day we had a very interesting project: make a cardboard chair. This project is based on Frank Gehry’s famous Wiggle Chair seen below.

 

What my classmates and I did was take a bunch of cardboard and slap it together willy nilly then adding more and more for strength and stability. The results were fine, the chairs could support our weight and they looked cool but we used a ton of cardboard because we weren’t sure how much we needed. My chair was not as elegant as Gehry’s chair.

Later, in collage I took some math and engineering classes and had an epiphany. I could have researched the properties of cardboard and designed the chair using math as Gehry (a trained architect) had done. I could have saved on material and made something really groundbreaking!

To me that’s the power of math, it makes life better and beautiful. Without math, we spend a lot of time guessing and that can be expensive, time consuming, and inefficient.

Beginning Desmos

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I often need to help students with their graphing calculators. It can be hard at first to learn to use a graphing calculator, there are quite a few steps to entering your equation and getting the window set up just right. There are also many different kinds of calculators, and while they are similar, it can be frustrating when the instructor shows something on his calculator and it’s not the same as yours! It’s even worse if he doesn’t know how to use your calculator and you are stuck trying to figure it out on your own (luckily Google is great for finding instructions). They are also expensive. If you are going to become a scientist or engineer you want to get the best graphing calculator you can, but what if you only need it for one class?

All these reasons are why more and more students (and professors) are using Desmos. Desmos is a free online calculator that also comes as an app for your tablet or phone. Press the start graphing button and you can create your first graph in seconds!

This is the Desmos window. When it comes up it will have the cursor in the top left corner ready for your equation. You can type it in using the same shortcuts you use for your electronic homework, or press the button on the bottom left (it looks like a keyboard) to bring up a list of basic function. From there, click on ABC to bring up letters or functions to bring up more advanced functions.

Adjusting the graph is a breeze. Grab the graph with your mouse and drag it in any direction. Use the + and – buttons to zoom in and out. If your graph is tall and narrow or short and wide you can change only one axis at a time by clicking the wrench button at the top right and type in the bounds of your graph.

From the phone you can just use your fingers to pinch in and out. You can also carefully pinch to change only one axis at a time.

Finally, you can edit your graph at any time and immediately see the results. If you want to compare two graphs, just hit return from the end of your first equation, or click right under it (where the 2 is). This will start a second equation which Desmos will graph along with the first.  In the picture below, I am comparing x2 to x3.

Now your professor probably went over these steps in class so this is just a review. I will be posting more on using Desmos soon … and we will definitely learn some fancy tricks!

Introduction

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Welcome to my new blog! As a mathematics graduate student and tutor I am always learning new things that are worth sharing. My background is in art so I come at math from a slightly different angle. If you think math is just about calculators, equations, and proofs I hope to prove you wrong! My favorite part of math is drawing and building.

You can also visit my art blog at www.catsgrin.com. I’m not doing much creation right now as school keeps me busy but you can see what I do when I have free time!