# Connect the Dots Like a Numerical Analyst

I have to say, teaching Numerical Analysis is one of the highlights of my job. Granted, my primary responsibility at Wayland is the Virtual Campus Director, and I will never teach Numerical Analysis online. Nevertheless, I LOVE it. In fact, the course banner that I use in Blackboard reinforces that fact to my students every time they log in:

Just as a for instance, I was able to get them to “solve” the age-old Connect-the-Dot problem. What is that, you ask? Well, simple: We all know, from the time we are toddlers, how to complete a Connect the Dot worksheet:

BUT, what is the mathematical solution? After all, math majors should look at the connect-the-dot worksheet and wonder, “What’s the equation of the solution?”

So today, as an introduction to using splines for interpolation, we derived the simple formulas for a piecewise linear interpolant:

Given a set of $n+1$ points with coordinates ${(x_j, y_j)}_{j=0}^n$, we can uniquely describe the piecewise linear function $S(x)$ where $S(x_i)=y_i$ for all $i=0, 1, ldots, n$, as follows:

$S(x)=Big{ S_j(x), xin[x_j,x_{j+1}]$, for $j=0, 1, ldots, n-1$

where $S_j(x)=a_j x+b_j$,

$a_j = displaystyle frac{y_{j+1}-y_j}{x_{j+1}-x_j}$,

And $b_j = y_j - a_j x_j$ for $j=0, 1, ldots, n-1$

At least, that’s the solution I told them in class today.  The truth is that’s not correct.  In fact, this will only “solve” the limited case where you always move left to right and never go back the other way.  What we really need is a parametric approach.  Given the initial data set above, we assign a parameter $t in mathbb{R}$ to each point, say $t=j$ for the point $(x_j, y_j)$.  Then we have the following solution to the Connect-the-Dot problem:

Given a set of $n+1$ points with coordinates ${(x_j, y_j)}_{j=0}^n$, we can uniquely describe the piecewise linear parametric function $bar{S}(t)$ where $bar{S}(j)=(x_j,y_j)$ for all $j=0, 1, ldots, n-1$, as follows:

$bar{S}(t) = Big{ biglangle S_{j,x}(t), S_{j,y}(t) bigrangle$, for $j=0, 1, ldots, n-1$

where $S_{j,x}(t)=a_{j,x} t+b_{j,x}$ and $S_{j,y}(t)=a_{j,y} t+b_{j,y}$

$a_{j,x} = x_{j+1}-x_j$ and $a_{j,y} = y_{j+1}-y_j$

$b_{j,x} = x_j - a_{j,x} t$ and $b_{j,y} = y_j - a_{j,y} t$for $j=0, 1, ldots, n-1$

That’s better, don’t you think?  From there we launched into a derivation of linear system approach to interpolation by natural cubic splines.  Then I ran out of time before finishing the derivation, which lead to the instagram post below…

That moment when class is over but you haven’t finished the proof… #mathteacherproblems

A photo posted by Scott Franklin (@splineguy) on

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# The Importance of Your Worldview

This week, I have the privilege and honor to lead the discussion in the Faith and Science course at Wayland.  The topic of discussion will be the importance of your worldview.  We start with a discussion on the 19th century masterpiece, “Flatland: A Romance of Many Dimensions” by Edwin A. Abbot.

Then we’ll discuss a couple of readings:

Are Scientists Biased by Their Worldview

The Importance of Worldview

Slides for guided discussion:

# 18 Basic Tips and Tricks for the Teacher’s iPad

The best thing you can do to familiarize yourself with the iPad is just to play with it.  You’ve got to be willing to explore by tapping, pinching, and swiping away.  One of the core design principles at Apple has been that their systems should be intuitive.  As you learn some of the basic interactions, you simply need to explore these common icons and gestures in different apps. Below are some the most basic tips and tricks that help teachers (and most general users, as well) to navigate their iPad.

## 1. Launching and closing apps

When you are on the Home Screen, you can simply tap on an app’s icon to launch the app on the device.  Once an app is launched, all you need to do to exit the app is click the home button at the bottom of your device: Apps don’t completely close down when you move to the home screen.  They also don’t “run” in the background unless you have Background App Refresh enabled for the app.  An app will save its state and you can return to the app later. To completely close an app, double tap the home button and then swipe across to find the app you want to close.  To close the app, swipe the app up and away.

# Teaching in the One iPad Classroom (Updated)

This Friday, May 16, 2014, I will be conducting an all-day iPad workshop at the Education Service Center in Lubbock.  I’ve updated the handout that I use for these workshops so I’m posting it here for the participants to access.

Everyone else is welcome to take and use the handout, as well.

Teaching in the One iPad Classroom – Handout

# Review Activity Ideas

During the Teacher Quality Grant class today, we began preparing for the Post-Test that will take place in two weeks. Class participants shared a variety of ways that they conduct reviews in their junior high and high school classrooms. This group of teachers shares some very interesting games that they have used to engage students and get them more excited about math. Here are the ideas they shared:

1. Jeopardy®: This is a classic game where students compete for points by selecting a category and point value. The question (or answer) is revealed and they are given the opportunity to buzz in and answer. Some of the teacher use PowerPoint presentations designed but this can even be done low-tech with a board with numbers and the questions read off of a handout.
2. Around the Room: Papers with an answer at the top half and an answer on the bottom half are hung around the room. On each page, the answer does not match the question. Students are sent to a starting point (each can have a different starting point). Once they figure out the answer to the question, they find the page that has that answer at the top. Then, they answer the question at the bottom of that page and hunt for its answer. This trek takes them Around the Room.
3. Clicker Questions: If the technology is present, teachers and create clicker questions and use a personal response system to have students answer the questions as they go.
4. Bingo: Just like regular bingo, except the cards have the answers to questions instead of just numbers. The questions are read one at a time and the students get to mark their square if they have the answer on their card. One great idea to create the cards was to have a long list of answers on the board and let the students create their own cards at the beginning of the review. This way you don’t have to figure out a way to generate random bingo cards.
5. Easter Egg Hunt or Scavenger Hunt: Hide the questions in Easter eggs around the room. The hunt is fun enough by itself, but you can have them accumulate points by answering the questions.
6. Relay: Place the questions at one end of the room, have teams run down and get the question. Once they get the right answer, they can run down and get the next question.
7. Tic Tac Toe (very low prep): Divide up into two teams, each team takes its turn by getting a question. If they get it right they can take their turn on the tic tac toe board. If they get it wrong, the other team can have an attempt at the question. Keep going until someone wins the game.
8. Football, Baseball or Basketball: Prepare a list of questions that are worth different values. For example, in football, questions could be worth 5 yards, 10 yards and 15 yards. In basketball, they could be a free throw, 2 pts or 3 pts. In baseball, they would be a single, double, triple or homerun. In each game, you can follow the usual rule so of the sport but the way they make plays is by answering the question.
9. Trash Can Basketball: Questions are given a value equal to some distance from the trash can. The easier the question, the farther away from the can. If the team gets a question right, they get to shoot a wadded up piece of paper at the trash can. If they make it they score a point.

# How to Insert Equation Numbers in Word 2010

How to Insert Equation Numbers in Word 2010.  In most cases, I’m using LaTEX to typeset my math docs but when I am in a hurry or I’m having my students write up reports, I need to use Word.  Here’s a quick demo for adding equation numbers in word that auto-number and can be referenced in the text.

# Graphs of Sine and Cosine Functions

I just posted a new video to the Trigonometry Lecture Series.  This is the 11th in the series.

In this video, I cover how to identify properties of sine and cosine graphs, determine the amplitude and period of sinusoidal functions, graph the sinusoidal functions using key points, and find an equation for a sinusoidal graph.