Host your own version of Desmos for testing purposes

Even though it is a math app, as a physics teacher, I love Desmos!   I encourage my students to use it as their primary calculation tool.  If you are new-ish to Desmos, you might want to read a previous post about using Desmos in the physics classroom (here).


One feature I like is that Desmos can solve linear and quadratic equations, even in non-standard form.  Below is the solution to the question "A ball is thrown up at 25 m/s from the edge of a 30 m cliff.  When is it 15 m above the base of the cliff?". 

Before Desmos, I would see many students set up the correct equations from the equations of motion, but then foul up when transforming that equation into standard form, therefore getting erroneous answers.  Desmos removes that chance of screwing up.

However, the above feature is not available in the Desmos Test mode.  While there is a work-around (see the post-script below), it adds extra steps and and extra chances for student mistakes.  Allowing students to use the full-featured Desmos on tests and quizzes is problematic.  If students create a Desmos account, they can save past work, which could then be used as a "cheat-sheet".

Here is a way to allow students to use the full Desmos but in a secure way.  To do so, you need to be able to lock your students' electronic devices to a single website.  Since my school is an ipad school and we use Apple Classroom, doing so is easy for me.  You just need to create a website using Desmos' API. 

To create a revenue stream, the Desmos team wants commercial companies to embed the Desmos product into their website.  As such, Desmos has developed an API (application program interface).  As of this time, Desmos allows non-commercial sites to use the API for free.  You can check out what they offer at desmos.com/api .

To create my site, I used Google Sites and created a basic site.  I then went to the Desmos API site and opened up the "default state" example.  I then opened up the source code (exact steps dependent on OS and browser), copied that code, and then pasted it into my website.  I did do some modifications to make degree mode standard, but that was optional.  And now my students have access to all the computational functions of Desmos, and I have the knowledge that they cannot create cheat-sheets.  You can view this site at http://bit.ly/2m3C9Ei

Post-scripts
If you must use Desmos Test mode with your students, you will have to get them to transform their equation to standard form (...=0) and then replace 0 with y.  Unfortunately, those 2 steps can flummox many students which is why I did the above.  See the example below.

If you do decide to host your own version of Desmos, you should ask permission just to keep things legal.  There should not be a problem.  Here is what Desmos sent me when I asked such.

Hi Michael,

We're happy to have you integrating the Desmos API! So long as this is for non-commercial purposes, please go ahead and use this API Key that is listed in the Desmos API 0.6 documentation: xxxxxxxxxxxxxxxxxx. I would also encourage you to check out Learn Desmos to learn more about the functionality.

If you would like to embed screenshots as well, just follow these guidelines:

• Screenshot in offline content should include Desmos logo
• Screenshot in online content should include Desmos logo plus a Desmos hyperlink 
• A video without audio should include Desmos logo, plus Desmos hyperlink if online (e.g., YouTube)
• A video with audio should has same as above plus the speaker mentioning the graph is "powered by Desmos" (see Mathalicious example, Q3 -- bottom right)
• Branding and icons

If this is in fact for a business, just let me know and we can make a plan!

Best,

Kristin

Using Desmos in a Physics Classroom

Why you should use Desmos in your science classroom

You have probably come across the graphing calculator app Desmos sometime in the recent past.  It is a simple to use program that does a lot more than graph. Students can use it to get solutions to algebraic equations, do linear regression analysis, look at probability/statistics scenario, and just play with math.  You can get really creative if you want.

At teacher.desmos.com, Desmos has taken this great tool and given teachers a way to effectively guide and monitor students in their learning.  The team at Desmos has also created a community willing to share their creations.  This forum is not adequate to give even a cursory primer in making and using the activities, so I have links to activities I have made as well as ones written by others I have found useful.  You can learn a lot just by exploring these and by looking at the many examples at teacher.desmos.com.

Hints on using in a typical physics classroom

This is an activity that gives an overview of the tools available to students using the main Desmos app. 
Link to page

One person walk

This is an activity I use when students are still learning how to do linear regression analysis on data and how to use that equation to make predictions.  It also uses some advanced tools that allow data entered in one screen to be used in another screen.
Link to page

Two person walk

This activity expands on the "One person walk".  I use it to emphasize the inferiority of "distance" as a measurement compared to "position".  
Link to page

Feed the shark

This activity I use to allow students to demonstrate mastery of projectile motion analysis.
Link to page

Virtual Forces in Equilibrium Lab

Link to page 

Linearization of data using Desmos

Link to page 

 

Using Trigonometry with Lasers and Mirrors

Link to page

Some Mathematically oriented activities

Link to page

How Loud if Many Cats Meowed?

Cats can scratch your eyes out, but can they meow your ears out?

Recently, I came across a post looking at the question "If all the cats in the world "meowed" at the same time, how loud would the sound be?". The video is below.

Because I was proctoring tests, I had some time, so I decided to go a little further. I started with a basic assumption that cats like to be about 1 meter away from other cats. Starting here, I came across some neat mathematics.

How big?

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Let us assume that the approximately 600 million cats in the world want to be 1 meter apart. If we set them up on circles with radii of 1 meter, 2 meters, 3 meters, and so on, we can put 6 cats on the first circle, 12 on the second, 18 on the third, etc. How many circles do we need? 

\(6 + 12 + 18 + .............n = 6 \times (1 + 2 + 3 + .......m) = 600,000,000\)

and using the Gaussian sum formula for the first m integers (the legend about how Gauss derived this can be found here)

\(1 + 2 + 3 + ...... + m = \frac{{m\left( {m + 1} \right)}}{2} = {10^8}\)

 which gets around m = 14,000 circles. 

How loud?

Obviously, a cat further away will sound quieter, but there are more of them further away. How do we calculate that balance? As the video illustrates, we can add sound intensity levels, not decibel levels. Let us use the sound intensity level for a cat meow from 1 meter away as \({{I_{{\rm{cat}}}}}\). Remember that sound intensity diminishes as the inverse of the distance squared. So the total sound intensity of the 6 cats in the first circle will be \(6 \times \frac{{{I_{{\rm{cat}}}}}}{{{1^2}}}\), the intensity of the 12 cats in the second circle will be \(12 \times \frac{{{I_{{\rm{cat}}}}}}{{{2^2}}}\). The total intensity for all 14,000 circles will be

\({I_{total}} = 6 \times \frac{{{I_{{\rm{cat}}}}}}{{{1^2}}} + 12 \times \frac{{{I_{{\rm{cat}}}}}}{{{2^2}}} + 18 \times \frac{{{I_{{\rm{cat}}}}}}{{{3^2}}} + ......14,000 \times 6 \times \frac{{{I_{{\rm{cat}}}}}}{{14,{{000}^2}}}\). 

A rearrangement gives

\({I_{total}} = 6 \times 1 \times \frac{{{I_{{\rm{cat}}}}}}{{{1^2}}} + 6 \times 2 \times \frac{{{I_{{\rm{cat}}}}}}{{{2^2}}} + 6 \times 3 \times \frac{{{I_{{\rm{cat}}}}}}{{{3^2}}} + ......6 \times 14,000 \times \frac{{{I_{{\rm{cat}}}}}}{{14,{{000}^2}}}\).

Which gives

\({I_{total}} = 6 \times {I_{{\rm{cat}}}} \times \left( {\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + ....... + \frac{1}{{14,000}}} \right)\).

Notice the harmonic series in the parentheses. Using the definition of the natural logarithm, 

\(\int_1^n {\frac{1}{x}} dx \equiv \ln \left( n \right)\)  ¹,

we can approximate the sum of this harmonic series as

\({I_{total}} = 6 \times {I_{{\rm{cat}}}} \times \left( {\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + ....... + \frac{1}{{14,000}}} \right) \approx 6 \times {I_{{\rm{cat}}}} \times \ln \left( {14,000} \right) = 6 \times \ln \left( {14,000} \right) \times {I_{{\rm{cat}}}}\)

But what we hear is best measured not by the intensity level, but by the decibel level. How do we go from one the other? 

Most people are familiar with the decibel scale. The decibel level of a sound with an intensity of I watts per square meter is

\({\rm{dB = 10}} \times {\rm{ }}\log \left( {\frac{I}{{{I_0}}}} \right)\) .

Using this conversion, we get

\[\begin{array}{l}
{\rm{dB = 10}} \times {\rm{log}}\left( {\frac{{6 \times \ln \left( {14,000} \right) \times {I_{{\rm{cat}}}}}}{{{I_0}}}} \right) = 10 \times \left( {\log \left( {6 \times \ln \left( {14,000} \right)} \right) + \log \left( {\frac{{{I_{{\rm{cat}}}}}}{{{I_0}}}} \right)} \right) = \\
10 \times 1.8 + 10 \times \log \left( {\frac{{{I_{{\rm{cat}}}}}}{{{I_0}}}} \right) = 18 + 10 \times \log \left( {\frac{{{I_{{\rm{cat}}}}}}{{{I_0}}}} \right)
\end{array}\].

Recognize that \[10 \times \log \left( {\frac{{{I_{{\rm{cat}}}}}}{{{I_0}}}} \right)\] is just the decibel level of one cat from 1 meter away which the video says is 45 dB. All the cats in the above arrangement will give a decibel reading of (18+45) dB=63 dB.

Remember that an increase of 3 dB is approximately a doubling of loudness, 18 dB is 6 doublings, so all the cats in the world meowing at once will sound like \({2^6} = 64\) cats 1 meter away.

I will leave it as an exercise for the reader to show that if you reduce the distance between cats to 0.5 meters, the dB level will 69 dB and will sound like 256 cats 1 meter away.

¹ I realize that \(\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} +  \cdots  + \frac{1}{{14000}} \approx \ln \left( {14000} \right) + \frac{1}{2}\) is a better approximation to the sum of the harmonic series (look at the geometry of the area under 1/x and the areas from the harmonic series), but that extra term of 0.5 doesn't change the final result appreciably. 

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.

Hints on "Justify and Explain" questions on the AP 1 and AP 2 tests

Below is a conversation on the AP Community website. I reprint it here because I think it needs wider exposure. The response is from a long-time AP Reader who is held in high regard by many of the other Readers.

Justify and Explain questions... I need advice, or resources, that will help students answer ‘justify’ or ‘explain’ questions accurately and concisely (our school’s English department has tried to help, but they don’t seem to grasp the problem), and how to assess their answers accurately and efficiently. Next year, I’ll teach five periods of 40 students  (four of P1 and one of C (M,E+M)), which has always created lots of grading… but now, with the advent of the new exam, there’ll be more, especially of the ‘justify’ and ‘explain’ type. What do you do to improve your student’s answers and your assessment of them, while saving time?

RE: Justify and Explain questions... 5/20/14 9:30 AM as a reply to Martin R Kirby. Kirby, I sat next to Rebecca last year and scored that pilot test for Physics 1 and Physics 2. Here are some do's and don't tips for everyone. (1) Don't rewrite the question in an introductory paragraph. You are wasting precious space in the test booklet that gets you no points. Do use a label to shortcut your introduction but don't expect that label to earn you a single point. Rebecca said, "There are no more skinny points." In the old days, merely mentioning things like right-hand rule, Lenz's Law, Kirchoff's Junction Rule, Newton's Shell Theorem and so forth would get you the justification point. No more. (2) The justifcation point now comes from application of the label. Deborah says, "If you can't accomplish this in three sentences or less then you are probably not going to get the point." More importantly, the student is also losing time that could be spent getting more points else where. An example here would be something like,  "induced current will flow in a direction to replace flux loss (or cancel flux gain). In order to accomplish this, the right hand curl rule indicates that current flow must be clockwise." I would suggest that you have your students list maximum of four points in bullet form to begin on test day. The first bullet should be (i) label point that applies to this justification (ii) connect the label to the cause (iii) connect the label to the effect (iv) connect the label to the conclusion through physical application. Many students miss this last point and merely wound up repeating their conclusion. Award points for every bullet that they mention. Take off points for restating the question. I know extinguisment doesn't work but.. I have also considered giving them one essay question the night before just because our kids seem to write physics explanations like a five paragraph essay. Hope this is not redundant from past threads. I would have added a little more but the post was too long. Here is the rest. You can take some of the old AP Physics B FRQ's and use them to make practice problems. However, the problem with the old AP Physics B problems is that the changes always come at the end, rather than the beginning of the problem. Use the old images and move the change to the beginning of the problem. Consider the 2013 AP B FRQ where you had the catapult type problem and the mass was changed at the end of the problem and students had to explain the change in the range of the projectile. Practice that problem specifically. Tell the students to use "inertia" in their explanation. Do this when you are finishing up the Newton's Laws Chapter. After they do that, return to it again after the Work-Energy chapter. Have them repeat their justification using the Work-Energy Theorem instead of Newton's Laws. Then have a third go using momentum and impulse after you finish the chapter on momentum. Although the pulley makes the third approach more challenging it should help the students to synthesize three chapters of mechanics. Two more thoughts- (1) Look up T.I.P.E.R. questions and try to modify old AP Physics B FRQ's into those suggested formats. But move the change to the beginning and demand physical explanation without using mathematical logic or symbols. Your qualitative/quantitative practice can come from those old images but look at the pilot exam next month before you try this. (2) Start the year with bullets for the rubric and then try to grow your students into more complete and fuller explanations as the year passes. You can try  tycphysics.org/tipers.htm I don't know how to make this an active link yet.

What Physics Teachers Talk About Out Of Class

Physics Teachers being Naughty

One of the nice things about being a member of the Western New York Physics Teachers Alliance is the email list. Here we can share ideas with colleagues  we may see only once every month or two at our regular meetings. The following is an example of one such exchange (names have been deleted to protect the guilty).



I've got a PVC cannon that can fire ping-pong balls/dog toys, paper tubes/rockets, and dry erase markers.  It's electrically triggered, but I'm still trying to figure out what the max pressure it can take and still electrically fire.  Individual components are good up to 90 psi, but around 70, the sprinkler doesn't actuate.  40 psi will send a dog toy at ludicrous speeds.  I also went out and got an air compressor at Harbor freight, so we can basically fire the thing every minute or two (depending).  Air compressor is loud as dickens, tho.  Actual muzzle velocity calculations/high speed forthcoming.


It can fire dog toys; can it fire toy-dogs (not the plushy kind)?


If you can get them to fit in the barrel.  And not wiggle around too much.


We could get a Doppler-shift demo out it, too. 
 

"Yip-yip-yip-yip-yap-yap-yop-yop-yop-yop"