18 October 2015

Linearizing Data on an iPad

Some of you are forced to use an iPad at your school. While it has many "whiz-bang" features that impress administrators and non-science people, you realize that the iPad is very limited. You would like your students to have a device that does what you want to do, not just what Apple thinks you should be able to do.

One of the basic concepts taught in most physics classes is the linearization of data. This concept usually comes up when looking at data gathered from an acceleration experiment, be it the distance traveled on a ramp versus time or the speed at the bottom of a ramp vs distance on that ramp. Graphing that data raw gives a curve. Just looking at the curve gives no indication whether is is a a quadratic (x^2). cubic (x^3). exponential (n^x), or other function. It is only by looking at the data many different ways that the true relationship shows.

The standard approach to this task is with a spreadsheet (I always recommend Gnumeric which is free and cross-platform). But full-featured spreadsheets on the iPad cost a lot of money and free options are limited. So, for years, I used one free app to graph data and another free app to manipulate that data. While they got the job done, the procedure was cumbersome for students.

Recently, the people at Desmos added some features that allow us to do everything in that one free app. Let me sketch what I show my students to do.

While I am concentrating on using Desmos on the ipad, all instructions are applicable for the web-based application.

Step 1 add the data

After students install the Desmos Graphing Calculator app and start it, they have the option of creating an account. I don't require it, so each student makes that decision. They then add the data via the Table option as seen below. Have students experiment with two-finger squeezing and stretching to better see the structure of the data.

Step 2 manipulate the data

Now come techniques specific to Desmos. You now need to generate a new set of points based on your chosen manipulation. In this example, we want to square the independent variable but keep the dependent variable unchanged. 

Start a new box on the data side. Type in what you see below (if you just type "x1". Desmos is smart enough to make the "1" a subscript). As soon as you complete the manipulation (seen below), you see the new points on the graph to the right. With some two-finger stretching and spreading on the graph, you can see the points clearly.

Now you need to bring those points from the graph to tabular form. Click on the octagram just above the data and you will see something similar to below.

Step  3 display the new points

Click on the Table icon and you will see the manipulated data.

Step  4 get the line equation

To get the linear regression equation for the above data, start a new box on the data side. Type the equation as you see below (the tilde "~" is important). Now you have the needed equation of the line of best fit.

PS Desmos can also solve quadratic equations 

Another nice feature is that Desmos can solve the typical equations encountered in motion analysis. Below is the equation to determine the time that an object is 25 meters above the ground when is thrown upward from a 15 meter platform at 20 m/s. Notice that the equation does not have to be put into the standard form as per most other web-based equation solvers.

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