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## Animations Using Parametric Curves

This past semester I added a fresh coat of paint to a unit on parametric equations by challenging students to develop their own animations on Desmos. For many years I have used a Desmos activity to introduce the idea of time as a parameter within equations – https://teacher.desmos.com/activitybuilder/custom/576ed1058e03e695283c88b8 – and Desmos invites students to think about the role of time by inserting the idea into ordered pairs easily. Here’s how I introduced the idea. If you get lost or want to look under the hood, here is the final Desmos graph I share with students as a resource: https://www.desmos.com/calculator/7zuubmunhl

## STEP 1: Develop Your Vision

Before we dive into Desmos, think about what you want to animate and the path you want it to take. For this demo I have a vision of a ball starting in quadrant 3 and following 3 linear paths until the end in quadrant 1. Sketching out the vision is helpful:

The animation here has 3 linear phases – see the points below, and you can certainly allow for non-linear paths as well. There are a few things for students to attend to here: how the coefficients of t allow for movement, and notice how (t-7) and (t-10) are used in the second and third points in order to “trigger” the animations at the right time, matching the domain of t given for each of the points. Take time to build this with students. Define the second point to begin when the path of the first ends, then include the parameter t.

Allow students to interact with the points and alter them to their liking. Or have students develop their own paths.

## STEP 2: Introduce an Image

Next, find an image you would like to animate. I used a soccer ball here, and added a cute sun in the sky later. Upload the image to Desmos, and drag the corners to adjust the size of the image.

Now, a little heavy lifting with Desmos. Students will need to attend to precision and symbols here – encourage students to work together to follow the steps and syntax.

The goal is to replace the center of the image with a conditional statement using the 3 paths we defined earlier. The center I used with my soccer ball is shown below, and note the structure: for each path, start by stating a period for t, followed by the parametric point, separated by a colon. Then, a comma will separate each of the 3 stages.

It’s helpful to share the graph with students so they can dissect the command and make sure the syntax they are using is working.

Defining the center in this manner will then invite us to create a slider for t, which we will do here. Click the endpoints of the slider to define the start and end of the time period you would like. Then play and let the oohs and aahs wash over the room.

## STEP 3: Explore the Space

Now it’s time for students to build their own creations. As students build, they may become inspired to investigate new ideas. In my class, some things which came up are:

• Non-linear paths: these can be defined within the points
• Rotations: t can be used to define the angle of an image
• Image dilations and appearances: the slider for t can also be used to define the height and width of an image, as well as the opactity
• Backgrounds: students can find a general image to serve as a background. I encouraged students to lower the opactity of and background image so that the animation pops on the screen.

## STEP 4: Gallery Walks

Allow students to share their creations with each other half-way through the project and ask questions about procedures. In tech-based lessons, students are often their best resource, and inspiration for a new idea can come from each other. Here are a few student creations from this first project attempt.

Categories

## How Do We Assess Efficiency? Or Do We?

A problem on a recent assessment I gave to my 9th graders caused me to reflect upon the role of efficiency in mathematical problem solving. In particular, how much value is there in asking students to be efficient with their approaches, if all paths lead to a similar solution?  And should / could we assess efficiency?

The scene: this particular 9th grade class took algebra 1 in 7th grade, then geometry in 8th.  As such, I find I need to embed some algebra refreshing through the semester to dust off cobwebs and set expectations for honors high school work. For this assessment, we reviewed linear functions from soup to nuts. My observation is that these students often have had slope-intercept form burned into their memory, but that the link between this and standard form is weak or non-existant.  Eventually, the link between standard form and slope ( -A/B ) is developed in class, and we extend this to understanding to think about parallel and perpendicular lines.  It’s often refreshing to see the class see something new in the standard form structure which they hadn’t considered before.

The problem: on the unit quiz, I gave a problem which asked students to find the equation of a line parallel to a given line, passing through a point.  Both problem and solution are given in standard form.  Here is an example of student work (actually, it’s my re-creation of their work)….

So, what’s wrong with the solution?  Nothing, nothing at all.

Everything here answers the problem as stated, and there are no errors in the work. But am I worried that a student took 5 minutes to complete a problem which takes 30 seconds if standard dorm structure is understood?…just a little bit.  Sharing this work with the class, many agreed that the only required “work” here is the answer…maybe just a “plug in the point” line.

My twitter friends provided some awesome feedback….

Yep, we would all prefer efficiency (maybe except Jason). Thinking that I am headed towards an important math practice here:

#### CCSS.MATH.PRACTICE.MP8 Look for and express regularity in repeated reasoning.

It may be unreasonable for me to expect absolute efficiency after one assessment, but let’s see what happens if I ask a similar question down the road.

Confession, I really had no idea what #CThenC was before this tweet.  Some digging found the “Contemplate then Calculate” framework from Amy Lucenta and Grace Kelemanik, which at first glance seems perfect for encoruaging the appreciation for structure I was looking for here.  Thanks for the share Andrew!

Yes, yes!  Love this idea.  The beauty of sticking to standard form in the originial problem is that it avoids all of the fraction messiness of finding the y-intercept, which is really not germane to the problem anyway. Enjoy having students share out their methods and make them their own.

What do you do to encrouage efficiency in mathematical reasoning?  Share your ideas or war stories.

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## Put(t)ing Rational Numbers in Order

Many of my friends and followers have caught onto one of my guilty pleasures: my wierd fascination with The Price is Right (read about Price is Right and counting principles in this old post).  Here’s how a pricing game made for a fun review activity, and also made my life flash before my eyes (read to the end for that).

Here in Pennsylvania, we use the PA Core Standards.  For Algebra 1, here is a standard under “Anchor 1”:

A1.1.1.1.1 Compare and/or order any real numbers.  Note: Rational and irrational may be mixed.

Seems innocent enough.  Here is a sample “open-ended” task used to assess understanding on our state’s Keystone Algebra 1 exam:

Exciting….now let me go over here and watch the paint dry….

But during the NCTM conference, a lightning bolt hit. I was checking out a putting game at one of the booths, and I suppose rational numbers were on my brain….Hey – Golf + ordering rationals = feels like Hole in One to me!

In the Price is Right Hole in One game, contestants place groceries in order from least to greatest by price.  The number of items they can order until they are incorrect determines where they putt from. After a quick trip to the sporting goods store to find a putting cup, and some time with a Google Doc, we’re all set!

To start, I created a Google Slides presentation with 6 games.  Each game has 6 numbers for students to put in order:

During the game, all students in the class had about 2 minutes to place the numbers in order.  They, we randomly drew our “contestant”, who came to the board to fill in the 6 boxes on the board.

Next, we went through the numbers from left to right, and determined how far the contestant had gone in successful ordering.

On the floor, 6 lines were taped.  Line 1 was on the other side of the room, and the lines were closer and closer to the hole. If a student had 4 numbers correctly ordered, they were allowed to putt from line 4.  Two students were able to order all of the numbers and tried their putt from about 2 feet away.

Those who made their putts earned candy to share with their group.  In about 20 minutes, we got through 4 games – not bad for ending a Friday on a fun note.

But be careful! My last “contestant” – one of my less cooperative students and a sometimes hot-head – was able to putt from line 6 with the help of his group.  After missing the first putt, I reminded him that the game is really Hole in One – OR TWO, and had a second chance. Lining up the putt…he took it easy…and missed again.  This is when he raised the putter up and, for a brief second, it looked like the putter could end up flying in my direction.

“Sean, just pick up the ball and put it in the hole….here’s some candy…”