Desmos App for iPhone: Almost Better than Gold!

Last Friday, I found the following announcement in my inbox:

If you could have seen me jumping around, you would have thought I just won a bunch of cash.  But for math teachers, like me, Desmos is better than cash - it's a type of math gold.  (Or at the very least, it can help my students feel like a million bucks, as they discover math ideas for themselves!)  Giddy with excitement, I couldn't get the app on my iPhone fast enough!!  I had tried to show Desmos to some collegues back in June of 2013, but could not really show off its power on my iPhone.  Now I can show off Desmos to those colleagues at the holiday party in December!

So...what does Desmos look like on my iPhone??  Click on the image below or the phrase "Demos for iphone" under the image to watch the screencast and see for yourself. 

Desmos for iPhone

The iPhone version is great, but there are two benefits to the online version of Desmos that are not part of the iPhone version - being able to save your work and access to the "file cabinet" of Desmos demonstration files.  This past summer I had students explore demonstration files on equations of circles, because they wanted to add circles and semicircles to their projects.  In less than one minute, I opened the Desmos demonstration file on circles and the students figured out how to create circles and semicircles from equations with no verbal input from me.  Yes - I would say that the Desmos app for iPhone is almost better than gold!

(Note: This blog would have been posted sooner, but it took me a while to figure out how to create a screencast of my iPhone screen without jailbreaking my phone.  I ended up using an iPhone with ios 7 mirrored to my MacBook Air.  Mirroring was done by using Reflector and screencasting was done by using Screencast-o-matic.)

Looking Back & Looking Forward

Looking Back...

One of my main goals this past school year was to have my Geometry students see math from multiple perspectives.  This goal was a direct result of Jo Boaler's "How to Learn Math" MOOC.  The fact that very simple problems can be seen from multiple perspectives is one aspect of mathematics that I truly love and my hope was to instill that love, or at least an appreciation, with my students during the last school year. My second goal was to have my students see the value of mistakes - that mistakes are part of learning.  Not all student answers are perfect, but we can learn from all student answers.

So...how did I do?  I asked students at the beginning of the year to use 2 words to describe math and I had them do that again at the end of the year.  The first Wordle was based on the words students used from the beginning of the year and the second Wordle was based on the words students used from the end of the year. The larger the words, the more the students used that word. (Note: There are some interesting words that only a few students used.  I recommend zooming in on the pic to see them.)

Beginning of School Year	What are the "Top 5" words students used to describe math prior to the school year starting? 1) interesting 2) challenging 3) complex 4) fun 5) ubiquitous
End of School Year	What are the "Top 5" words students used at the end of the school year? 1) useful 2) challenging 3) interesting 4) fun 5) ubiquitous

It is interesting to note that not many words in the "Top 5" list changed.  Although some students still found math "confusing" or "intimidating" at the end of the school year, there are many new words in this list had including, rewarding, elegant, stimulating and exhilirating.  Not all of my students loved math like I did, but there was more of an appreciation of its power at the end of the school year.

But what about my orignial goals: math from multiple perspectives and students valuing mistakes for what can be learned from the mistakes.  I asked about this in a google form and although some students said simply "Yes" or "Sure", others gave more detailed responses.

One student said the following: I think that mistakes were also seen as ways to learn in the class, but not personally because I hate making mistakes. I do realize that I should learn from them more now, from this class.  Embracing mistakes is something that I still personally find challenging and I can relate to the struggles of this student.

Another student said: Viewing math from different perspectives not only showed us the many dimensions of mathematics, but also helped those who may learn differently than others. By looking at math in different ways we grow to appreciate the different ways of thinking of our classmates. And here I wanted my students to see math from multiple perspectives to help them understand the beauty and connections within mathematics.  But this student gets that different perspectives are important because (drumroll, please) not all students learn the same way!  I know this, but didn't think of this as a reason for pursuing the goal of multiple perspectives.

Looking Forward...

In the fall, I will continue to teach Honors level Geometry.  But I will also have a section of College Prep Geometry.  How do I get this group of students to undertand that there is value for them in seeing math from multiple perspectives?  How will I get them to embrace mistakes as a form of learning?  Of course I can do some of the same things I did last year, but these students haven't had as much success with math and are less enamored with the subject.

What will I try? We need something that is accessible to all students, but can challenge those students who are ready for higher order thinking. Enter Low Floor High Ceiling tasks, some of which can be found at youcubed.org and others at nrich.maths.org.  In the next week or two, I have the opportunity to try out a problem or two in a summer transitions program. Two tasks I am considering using are What's the Secret Code? and Beelines. Beelines seems quite challening, but I like the visual aspect of this problem and the ability to link it to GeoGebra or keep it simple with paper and pencil. Here is a video about the Beelines problem.  Looking forward to hearing the Buzzzzzing of voices as students tackle this one.

Around the World: Surprise! (Part 2)

Before you read part 2 of this blog, I suggest you read part 1.  The problem I gave my students was as follows:

Imagine that the Earth is a perfect sphere and that a metal wire is snugly wrapped around its equator. Now imagine that we cut this wire in one spot and splice in an additional 100 meters of wire.  We take up the slack by using posts to raise the wire an equal distance all the way around the Earth.  How high above the surface of the earth will the wire be?

Students worked together to find an answer of about 15.9 meters.  Groups had to explain their solution to me before they could continue on to the next phase - finding the solution for a second spherical object.  

On the front board I had written the following:

1 - baseball

2 - moon

3 - Jupiter

4 - beach ball

5 - basketball

Groups selected an index card at random and were assigned the object based on what they selected.  They needed to research the dimensions for their particular object and do the calculations again.  They submitted their results through a google form and when I displayed the results to the entire class, this is what they saw. (Note: These results are for all 3 of my classes combined.)

WHAT????  

No matter how big the object, the wire would always be 15.9 meters above its surface. 

Wait...WHAT????  But, why???

That was the reaction I was hoping for.  But these are the reactions that I got.

Period A - Huh. Interseting, I guess. (I think they were still on spring break.  It was Monday and 8:20 AM.)

Period B - That's cool.  Why does that work?

Period F - We are not surprised at all.  You do stuff like that with us all the time.  

So...now you may be asking Wait...WHAT Why does this work?  The derivation I did with one of the classes appears below.  The reason it works is because there is a direct linear relationship between the radius of an object and its circumference.  (Students know this direct linear relationship as C = 2*pi*r.) When the circumference increases by x, the radius increases by (x/2) divided by pi.

I love problems like this that are a SURPRISE to students and go against what student think should actually happen.  I would love to have more problems like this to share with my students.  If you have any problems like this, please send them my way.

Around the World: Surprise! (Part 1)

Back in the late 90's I worked with The Math Forum Problems of the Week. (POWs) Specifically I worked on the Trig/Calculus POW
and the Discrete Math POW.  Submissions came in from around the globe and I enjoyed seeing the variety of methods students used to solve the problems. There was even one time when a student used calculus to prove she found the minimum number of moves needed to solve the discrete math problem for that week!

This past week I was on spring break and I pulled out a wonderful GeoPOW problem from my files.  I plan to use it on Monday, because we often miss students on the first day back from spring break due to families extending their spring break.  Rather than having several students miss new material, I thought it would be fun to look at this surprising problem.  In the GeoPOW files, this problem was called "All Around the World".

Imagine that the Earth is a perfect sphere*, and that a metal wire is snugly wrapped around its equator. Now imagine that we cut this wire in one spot and splice in an additional 100 meters of wire.  We take up the slack by using posts to raise the wire an equal distance all the way around the Earth.  How high above the surface of the earth will the wire be?

*many students are surprised to learn that the earth is not a perfect sphere*

Collaboration is a big part of learning math in my classroom; students will work on this problem in groups.  After each group gives me an explanation that convinces me that they have the correct answer (Note: The answer is about 15.9 meters.), the group will reach into a hat and pull out a slip of paper that will have one of the following items written on it: basketball, moon, Jupiter, beach ball or Mars.  The groups will then repeat the problem that they just did with the Earth, but with their new spherical object.  After students have finished the problem, they will go to the tinyurl written at the front of the room to enter data into the following form:

After students click the Submit button, they will see the message "We will review the results from the class shortly."  The surprise will come when I turn off the AV Mute button on my projector and we see the results at the same time!  Come back for part 2 of this blog when I share how the lesson went and my students' reactions.  But before then, do the math and predict why my students will be surprised.

Teaching Trig Ratios in Geometry

Although I have probably taught trig ratios (sin, cos and tangent) 2 or 3 times each year for about 15 years, I was never really satisfied with how it turned out.  Yes, students could parrot the definitions and find sides and angles until the cows came home, but did they really understand why the sine ratio was always a certain number for a specific angle?  I was doubtful that the big picture was being understood...until today!  OK...I am still not satified with the entire lesson, but I think it was better.

As much as possible, I want students to construct their own knowledge/understanding of concepts with my guidance at the side.  In the past, I would have them draw right triangles that were similar, measure segments and calculate various ratios. (See handout below.)  But guess what the problem was.  Yes, that is right.  Inaccurate measurements led to the ratios not being equal and me saying things like "Well, the ratio 0.75 is pretty close to 0.76.  So, close that if you had measured accurately, I bet they would be equal."

Bleh!!!

The world would be a utopia if the students could measure accurately with 25-cent rulers and 25-cent protractors.  But they haven't and can't.  How could I get more accurate measurements?

ANSWER = Geogebra  

I was going to create a worksheet for them to do, but didn't quite have enough time.  (Both enough time in class and enough time to create the worksheet.) So, then I decided to create a screencast; it is uploaded to my youtube channel and I was planning on showing the 6 minute screencast in class. But since I really wanted students to experience the discovery for themselves, I led them through the activity as a whole class with each student working on his or her own laptop.

As long as students did exactly as I did and clicked in the exact same order, we were ok.  If they clicked in a different order, then we had different outcomes; their segment d was the hypotenuse and my segment d was the leg.  Luckily, only one or two students clicked in a different order and they figured out how to make adjustments for what they had on their own computer.

Students in my one class were initially surprised that we all got the same values, and when I asked them why this made sense, they understood!  Students chimed in that all the ratios must me equal,  because the triangles we had on our screens were all similar to each other by Angle-Angle similarity!

I think some students didn't find this too amazing, because as a class we decided what angle we would use and we all used the same angle.  Their feeling was "So, what?" Next year I still plan to lead the students through this activity rather than have them follow a set of typed directions.  However, I would have the data collected by table. I would have each table group choose what acute angles their right triangle would have.  This means there would be 5 sets of similar right triangles and they would see that it is not just one specific set of triangles where the sin ratio is the same, but the sin ratio is the same for any one given angle.

Fun With Grow Objects

I would like to dedicate this blog to today's snow day.  Without you, this posting would not be possible. (Or definitely not at likely.) This is the story of a polar bear, fruitcake, snowflake, Christmas tree, candy cane, penguin, snowman, Santa, gingerbread man and a Geometry class.

At the beginning of December, I went into Michael's to purchase what I call "grow objects".  (Very cheap - only $1 each!) You place them in water and they expand in size.  The claim on the package is they grow up to 600%.  Perfect!  We were studying similar figures in Geometry and students easily agreed that the enlarged objects should be similar to the original objects.  This mini project was perfect for sooooo much mathematics.  Prior to putting the objects in water, we considered the following questions:

1. What does it mean to grow 600%?  What is the scale factor?
2. Is the growth rate linear, exponential or something else? What equation could be used to model the growth?
3. Do all the objects grow at the same rate? When does each object grow the fastest?

Here you can see the objects at the end of 10 days of growth.

Due to snow days, midterm exams and Christmas break, my Geometry students have only investigated the answer to the first question.  What does 600% growth mean?  Does it mean all dimensions will be 6 times larger?  To answer this question, we considered a cube with a side of length 1 and then a cube with a side of length 6.  Students quickly saw that if all dimensions were 6 times bigger the volume was 216 times bigger!  

So, what would the side of the cube be if the volume was 6 times larger than the original?  With a little discussion, we decided it would be the cube root of 6 or about 1.8 units.

Students then reasoned that this would mean that the objects would grow to a little less than double for each of the dimensions.  So, how did the objects do?

All objects more than doubled in each of their dimensions!  In fact, for most of the objects the actual growth was between 900% and 1100%.  Maybe the manufacturers should relabel the packaging.  After all, 900% and 1100% sounds much more impressive than only 600%.

Unfortunately, our unit on coordinate geometry and modeling doesn't happen until April and questions 2 and 3 don't lend themselves to the study of the Pythagorean theorem, our current unit.  However, in researching for this blog, I found out that I could buy some brains, lions, spiders, geeks, castles, money and unicorns for growing at onlinesciencemall. The students had so much fun with this that we will definitely be doing this again.

However, you are probably wondering the answers to questions 2 and 3 and I was, too. So, I graphed two sets of data from my students (Christmas tree and polar bear) with desmos (a wonderful FREE online graphing calculator) and found some equations to fit the data. It should make sense that a cube root function works best since the growth is impacting the volume of the object and students were only plotting days against one of the three dimensions of the object. So that answers question 2. (Click here to see/download this desmos file that I placed on google drive.)

But what about question 3?  With Desmos we can easily plot the derivative of each function.  Notice that they are very similar to each other. This makes sense since f(x) and g(x) have the same basic parent function, the cube root function.  Also, from the derivative we can see that the fastest growth occurs between days 0 and 2, where the derivative has the largest positive value.  The asymptotic nature of the derivative also makes sense since the growth of the objects slowed down considerably by day 5.

Who knew that polar bears, fruitcakes, snowflakes, Christmas trees, candy canes, penguins, snowmen, Santas, and gingerbread men could help students learn so much about similarity, modeling and rates of change?