Gabe Kramer's Math Blog – The Personal Website of Gabe Kramer

Best Practices in Teaching And Learning 2024 – Desmos Throughout the Calculus Sequence

Looking for any of the slides demonstrated during our BPTLM 2024 presentation? Look no further than this post!

Introduction to Desmos (Points, Graphs, and Sliders)

Limits

The Derivative

Secant Lines

Secant Lines Segments to the Derivative

Interactive Slope Sketcher

Area and Riemann Sums

Solids of Revolution

https://www.desmos.com/3d/fvmpfcxgjx

Sequences and Series

Polar and Parametric Curves

3D Points, Surfaces, Contour Lines, Parametric Curves, and Vector Valued Functions

https://www.desmos.com/3d/dcy7cty3b0

Double Integrals in Rectangular Coordinates
https://www.desmos.com/3d/kucmijoker

Standards-Based Grading Presentation

On June 16, 2022 I had the opportunity to give a talk titled A No-Nonsense Introduction to Standards Based Grading at Penn State Behrend’s Best Practices in Teaching and Learning Mathematics conference.

Here is a link to the PowerPoint:

Best Practices (Final) – POWERPOINT

As well as a few other documents:

Math 140 – Standards (Fall 2019) – STANDARDS DOCUMENT

Math 140 – Syllabus (Fall 2019) – SYLLABUS

Math 140 – Quiz 18 (Fall 2019) – SAMPLE QUIZ

Implementing SBG for the first time can be intimidating; it doesn’t have to be. If you have any questions or concerns, please email me: GDK5028@psu.edu.

Starting off Calculus for High School Students

After spending a week grading AP exams, a weekend at Desmos HQ, and a few days at PCTM I’ve had a lot of time this summer to think about how I teach calculus, so when gnylbe asked for my thoughts/ideas on teaching calculus to high school students, I couldn’t resist sharing a few (literal) shower thoughts.

Emphasize that calculus is the study of the infinitesimal and the infinite and ask students to be on the lookout for how both arise naturally throughout your time with them. A fun prompt for students is to ask them to start at zero and to find the next number. At this point, I’ve had some success talking with students about the countability of the rationals / uncountability of the reals with students who have only seen up through HS Algebra II. I’ve even asked how long are the rationals at this point and worked through showing that the rationals have measure zero with the aforementioned students.
Emphasize the difference between the static nature of algebra and the dynamic nature of calculus. In algebra students may be prompted to solve problems such as

A car is traveling down the highway at 67 miles per hour. How far will the car travel in 2.5 hours?

Whereas in Calculus, students can be asked very different questions. Such as

The velocity of a car is given by $v(t)=2\sin(\pi t) + 2.5$ how far does the car travel between $t=2$ and $t=4$ hours. Assume $v$ is in miles / hour and $t$ is in hours.

If you are feeling fancy, you can even tie these questions into Desmos.

3. Where I like to start Calculus content is by taking everything back to slope. I have a conversation like this with my students

Remember your first time taking Algebra when you learned about lines? You learned about points, and plotting points, and connecting them using lines, then BAM you were hit with slope: Rise over run. Change in $y$ over change in $x$ . Average velocity. Rates. Dollars per item. So.Many.Applications. Point-Slope form. Slope-Intercept form. Then what? You multiplied your line by $x$ and got quadratics, then cubics, then polynomials, then rational functions, radical functions, exponential functions, logarithmic functions, the trig functions, the inverse trig functions…. But what never comes back up? You guessed it, SLOPE! Well don’t functions have slope? You can’t tell me the graph of $y=\sin(x)$ doesn’t have slope! Sure it might not have the same slope everywhere, but I would argue that it does have slope.

From here I would run through parts of an activity like this one. And immediately after spend some time reviewing slope of lines and ask why the slope of a line won’t help us (yet)? (Hint: We don’t have two points to pick.)

Here we segue into secant lines, difference quotients, limits, and finally the definition of the derivative.

I’m not sure how much help this will be, but if I had to summarize my thoughts on all of calculus, they would go something like this:

Given enough time, I would be sure to teach my students that Calculus is much more than the mechanics that come from it. For instance, the limit shortcuts, and derivative rules, and integral properties are great and all, but they are not calculus, understanding the behavior of certain phenomena and describing it by using the language of calculus is calculus, the mechanics are just tools that allow us to do calculus. Much like saws, screws, and drills are not woodworking, but tools that help one to do woodworking.

The Heat Equation I

Imagine an insulated steel rod of length $L$ whose ends are held at a constant $0^\circ$ F. Let $u(x,t)$ be the temperature at each point along the rod at time $t$ , if the initial heat distribution along the rod is given by $u(x,0)=f(x)$ , find the temperature along the rod as a function of the position ( $x$ ) and time ( $t$ ).

Let the initial heat distribution be given by the graph below.

Since no heat is being added or lost along the length of the rod, the rod will aim to achieve equilibrium at $0 ^\circ$ F. Thinking about it, the temperature will be changing most quickly at the peaks and valleys. In particular, the rate at which the temperature changes is proportional to the concavity at each point along the graph, thus the Heat Equation:

$u_{t}(x,t)=\alpha^2 u_{xx}(x,t)$

Where $\alpha^2$ is the thermal conductivity.

The solution to the heat equation subject to the conditions above is given can be seen by clicking here.

Now how do we get there?

That will be answered in my next post.

Spring Mass Systems

This was a fun one.

A few weeks back we started second order differential equations in my DE class. After working through homogenous DE’s with constant coefficients, Cauchy-Euler DE’s, Undetermined Coefficients, and Variation of Parameters, we finally started looking at applications of second order DE’s. The big one that comes to mind are Spring-Mass Systems.

If the set up escapes you, it goes a little something like this..

Imagine you have a mass of m kg, the force a spring applies to that mass is proportional to the displacement of that mass from equilibrium, x m, in the direction opposite the displacement. This is known as Hooke’s Law, and is written as.

F=ma=mx”=-kx

Hooke’s law makes the assumption that the mass is sliding across a frictionless surface, however if you think about how friction will impact the sliding of the mass, the force due to friction will act on the mass proportional to the velocity of the mass, in the opposite direction, thus the sum of forces acting on the mass is given by

F=ma=mx”=-kx-bx’

mx”+bx’+kx=0

A second order, linear, homogeneous differential equation.

Now, remembering how these work we have three cases for solutions.

b^2-4mk>0 – In this case, the system is said to be over damped. The mass will slowly return to its equilibrium position.
b^2-4mk<0 – In this case, the system is said to be under damped. In this case the mass will oscillate, and if b is not equal to zero, then the mass will oscillate to its equilibrium position.
b^2-4mk=0 – In this case, the system is said to be critically damped. This is a sensitive case in the sense that decreasing friction / increasing mass or spring constant will cause the system to become under damped and increasing friction / decreasing mass or spring constant will cause the system to become over damped. The other way to think about the behavior of a critically damped system is that the mass will return to equilibrium as fast as possible without oscillating.

You can experiment with each case by playing with the values of m, b, and k in the applet here.

A little more about Slope

This is a continuation of my previous post on slope. Before going on any further, I am going to encourage you to explore the image below.

Some things to try/ask yourself…

Try dragging the blue point around, what do you notice happening to the secant line? What do you notice happening to h and the purple point?
Try dragging the purple point around, what do you notice happening to the blue point? What do you notice happening to h?
If the x-coordinate of the blue point is called x, what is the x coordinate of the purple point?
What is the slope of the line passing through the two points?
What happens when you put the two points on top of one another? (Other than them not being able to be separated again)
Double click on the graph and begin changing the function, do your answers to each of the previous questions still hold true?

Next I am going to present to you a series of graphs with a secant line passing through two points on the graph, (x,f(x)) and (x+h,f(x+h)). As you are looking through each of these, ask yourself: What is the slope of the line passing through those two points?

Notice that the slope of the line passing through each set of points is given by

$m=\frac{f(x+h)-f(x)}h$

This little guy is what we are going to refer to as the difference quotient.

Let’s look at an example.

Consider the function $f(x)=x^2+3x$ , find and simplify the difference quotient $\frac{f(x+h)-f(x)}{h}$ .

Since we know that $f(x)=x^2+3x$ we also have $f(x+h)=(x+h)^2+3(x+h)$ which we can simplify

$\begin{align*} f(x+h)&=(x+h)^2+3(x+h) \\ &= x^2+2xh+h^2+3x+3h \end{align}$

Substituting back into the difference quotient give

$\begin{align*} \frac{f(x+h)-f(x)}{h}&=\frac{(x+h)^2+3(x+h)-(x^2+3x)}{h} \\ &=\frac{x^2+2xh+h^2+3x+3h-x^2-3x}{h}\\ &=\frac{h(2x+h+3)}{h}\\ &=2x+h+3 \hspace{.25in} (h\neq 0) \end{align}$

Notice that this is the slope of the secant line passing through the two points $(x,f(x))$ and $(x+h,f(x+h))$ on the graph of $f(x)=x^2+3x$ . Now the question you should be asking yourself is how do I get from here to the slope of the tangent line passing through the single point $(x,f(x))$ ? That is going to require us to make $h$ close to zero, but how close is too close?

A little something about Slope

As mentioned in my previous post, I am going to be starting a series of posts with a focus on calculus preparation. Before we begin, let’s start by discussing the first major problem in calculus….

The Tangent Line Problem

The tangent line problem goes a little something like this…

Think back to your very first algebra class, back when you were first learning words such as variable and equation. Shortly after learning these words you began graphing lines on the Cartesian Plane, and if there is one concept from your first study of lines that was drilled into your head until you could recite it in your sleep, it was probably the slope of a line. The nice little formula..

m=Rise / Run

But as time went on, you started studying polynomials, radical functions, rational functions, exponential and logarithmic functions, then the trigonometric functions, maybe even functions in polar coordinates and conic sections. But what ever happened to our friend slope? Do these functions not have a slope?

To make a long story short, these functions do have a slope, just not a constant slope. For instance, if you look the graph below you will notice that the slope is the same depending on which two points you choose. (Feel free to drag the purple and red points or double click the graph to explore further.)

However, if we consider a different function, say y=sin(x),that the slope of the function changes depending on where the two points are.

In other words, explore what happens to the blue line as you move the red and purple points around. In calculus and algebra, this blue line has a special name, we call a line which passes through two points on a graph is called a secant line. And the slope of the secant line passing through two points on the graph of y=f(x) is given by

m=(f(x_2)-f(x_1))/(x_2-x_1)

However, in calculus our goal is not to compute the slope of the secant line (as this is relatively simple given a function and x_1 and x_2), our goal is to find the slope of the tangent line.

In the graph above the blue line is the the tangent line. Now, compare and contrast the similarities and differences between the secant line and the tangent line and ask yourself the following questions…

How difficult is it to find the slope of a secant line if I know f(x), x_1, and x_2?
How difficult is it to find the slope of a tangent line if all I know is f(x) and x_1?
What do I have to do to go from a secant line to a tangent line? In particular what do I need to do to the two points on the secant line to get to the tangent line?
(Bonus Question) According to the rules of algebra, why can’t I just put the two points right on top of one another?

These questions will be answered in my next post.

Let’s Prepare for Calculus!

Every so often, I see students asking /r/learnmath or /r/cheatatmathhomework what they need to do be better prepare themselves for calculus and although many of the answers are good, they are rarely comprehensive. For instance the majority of the suggestions fall into on of the following categories…

Get your hands on Calculus by Spivak and work through it before taking calculus. – Though an excellent text, I would hardly recommend this text to students as their first exposure to calculus as this book falls somewhere between a Mechanics of Calculus text (Stewart, Larson, etc.) and a Real Analysis textbook.
Make sure you have a very strong algebra background. – This is a good suggestion for any math course, however it does very little to help students prepare for their first semester in calculus. Throughout the next few posts, I plan on emphasizing the concepts you must have mastered in order to fully understand calculus the first time.
Watch Professor Leonard or Kahn Academy on YouTube. – Again, a great suggestion, however, I suggest using these resources to supplement a formal course in calculus as they lack the interactivity of a traditional classroom.

Throughout the next few posts, I will be briefly introducing several of the main concepts covered in first semester calculus with a focus on the algebraic techniques needed to master the concepts.

Part 1 – A little something about slope.

Fibonacci Numbers and Matrices

Fibonacci Numbers

On Wednesday, November 7, 2018. I attended a talk by Dr. James Sellers of Penn State University Park. It was a pretty interesting talk with a focus on using matrices to prove identities of Fibonacci Numbers. The bulk of the talk was based around the matrix

$Q= \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}$

Where

$Q^n= \begin{bmatrix} F_{n+1} & F_n \\ F_n & F_{n-1} \end{bmatrix}$

where $F_n$ is the $n$ th Fibonacci number and $F_0=0$ , $F_1=1$ , $F_2=1$ , and the remainder are based off the recurrence relation $F_{n+1}=F_n+F_{n-1}$ . This identity can be proven using a simple proof by induction (proof left to the reader). However, a proof that I found fascinating was the proof that

$F_{n+1}F_{n-1}-F_n^2=(-1)^n$

Before I give the proof, take a second and think through how you would prove it.

Okay.

Let’s go.

Consider $F_{n+1}F_{n-1}-F_n^2=\text{det}\left[Q^n\right]=\left[\text{det}(Q)\right]^n=[-1]^n$ .
QED.

Until next time.

Area, Area, Area!

Welp, Desmos saved the day again with its ability to approximate areas under curves using lower and upper sums and clearly demonstrate that we get closer and closer to the true area the more rectangles we choose to use.

I’ve created an applet which is pretty user friendly with some pretty neat features. You can find it here. A couple neat things to explore is that you can hide/show the upper sum, lower sum, and the actual area by clicking on the folder icons on the left. If you are interested in seeing the upper sum, lower sum, or actual area, you can find it by clicking the triangles next to “Upper Sum”, “Lower Sum”, and “Area”.