Desmos and Functions

In Algebra, we just finished our unit on functions and a colleague of mine had been talking about Desmos, so I decided to explore it a bit and see if they had any resources on functions. I discovered Function Carnival and thought it would be a perfect mini project for applications of functions. The activity has students watch three videos simulating different carnival activities (a cannon man, bumper cars, and a Ferris wheel) and then asks students to draw the scenarios as functions. After each scenario, the students are also prompted to help find errors in a provided graph and explain how to fix it. Before creating their own scenarios, I wanted to be sure students fully understood the situations they had been practicing on Desmos. I thought it might take some students more tries than others to get their graphs correct (and it did), so I also added to the project by asking students to redraw their graphs on graph paper after they completed the Desmos portion. With their hand drawn graphs, they then had to correctly label the independent and dependent axis with names and units, and then find several key features of the graph such as domain, range, y-intercept, and extremas. That way, the concepts were reinforced and they were able to analyze correct graphs before creating their own. (Below are student examples of graphs).

cannon bumperferris

I am so happy with how this project turned out for several reasons:

  1. Every student was engaged in the Desmos activity. Giving students real world scenarios to physically see and digitally manipulate peaked students interest. The technology was easy for students to understand, but the scenarios were complex enough to capture their interest. Furthermore, students had the ability to self check their answers and when they weren’t exactly right, they wanted to go back and fix their mistakes to get the simulation to work precisely. It was tricky for some to understand why their path wasn’t matching, but I liked letting them have some time of productive struggle. They often called me over when they got their graphs perfect and wanted me to see their accuracy…I loved that!!
  2. Desmos incorporated a writing component for students to synthesize their learning. I really liked seeing how students interpreted the given scenarios and how they explained the errors that they were asked to correct. It gave students a chance to write in their own words about the mathematics of the situation. Telling students that I can also see this portion on my teacher view in Desmos perked them up to write in more complete sentences and will give me a chance to evaluate their grammar and mathematical vocabulary. Below are some examples:
    1. “The graph says that cannon man will have no suspension period in the air and that he will fall at the same speed going down without deploying his parachute. He should try adding a still period at the apex of cannons man jump and then making him fall fast. Afterwards he should slow him down before he hits the ground.”
    2. “The bumper car would end up going backwards because its going back towards the start. she needs to make it into a straight line.”
    3. “The graph says the bumper car goes back in both time and distance. To help her, I tell her to measure how far along the road the car´s traveled, NOT draw the car´s path.”
  3. The additional portion I added required students to create their own units for the graph and reflect on what was happening in the scenario. When students had to create their own axis, they had to logically think about what would make sense…some started to write that their cannon only shot up to 5 or 6 feet tall…that wouldn’t make for a very fun carnival event!! After understanding this, they fixed it and were able to justify more appropriate units. They also realized without units, they could not accurately state their domain and range.
  4. Vocabulary from our unit was reinforced at several points throughout the project. Several students quickly noticed when they drew extra lines in the Desmos window it would make more cannon men or bumper cars appear…which was quite fun :)…but beyond that, they explained to me that it was no longer a function because their graphs did not pass the vertical line test and there were too many outputs for one input. We also talked about the slope of the lines when the cannon man’s parachute deployed, when the bumper car crashed and stopped, and when the Ferris wheel ride had a constant speed. It brought meaning to the lines and why they were less steep, flat, or constant.
  5. Lastly, I got to use this super cool half graph and half lined notebook I requested for our math department!! 🙂 notebook

I’m looking forward to more Desmos projects! If you’re interested in the additional pieces I added to the project, Desmos Carnival Project.

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Conditional Statements in Geometry and Computer Programming

While planning for the logic unit in geometry a few weeks ago, I wanted to increase the rigor and apply a more meaningful experience to the topic than I had in previous years. In the past, I have had kids do a project in which they found ads in magazines that used (or could be rewritten to use) conditional if-then statements. Then they rewrote the ads to include the converse, inverse, and contrapositive statements. I think this is a fun activity and it is a really great way to have students recognize if-then statements in the real world, however, I began to think that this experience might not be project worthy. I found that asking students to rewrite the statements did not have a lot of meaning to them and some of the mathematical logic got lost when they simply repeated a statement about shampoo or men’s deodorant.

So this year instead, I planned to use the ads as just a warm up by showing students an Allstate commercial and asking them to identify the if-then statements. Then, I started thinking about what else in our world uses conditional statements and how I could make a more meaningful and rigorous project. I realized computer science uses if statements for a program to do something if a condition is met. Several years ago I worked for a company where I used coding, but I knew I needed to brush up on my understanding before I could teach it to kids. In my search to make this project of relating conditional statements in geometry to computer programming, I stumbled upon two helpful resources: Pearson and Khan Academy.

In our Pearson textbook, there are enrichment activities and coincidentally, the textbook had a similar idea to mine for the logic unit. They gave students some code and asked them to identify the hypothesis and conclusion in it. However, they used GOTO which is a bit outdated and not used a lot anymore. So, I held on to their idea about dissecting code and rewriting the hypothesis and conclusion, but searched around for a more relevant platform.

In my mind I wanted students to actually write code, manipulate it, and see it work with this project. I soon found Khan Academy’s tutorials on if statements and as I was working through them, I found that it was the perfect match to what I wanted. The tutorials teach students how to edit code with different scenarios and then students do a similar challenge to try to master the code. In the first challenge that I had students do, they were shown a ball that drops off the screen and they had to use if statements to make it go back the other way, displaying a bouncy ball effect on the screen.

For the project, I created a worksheet for students to use while they watched the tutorials, practiced the code, and performed challenges to see that their code actually worked. It asks the student to rewrite the if statements as a hypothesis and conclusion (similar to Pearson), decide if there is a biconditional phrase, and then also use the inverse code to manipulate it again. (The worksheet is below and a Googledoc is linked here). Then after completing their code, I planned to have them reflect on their work by writing a blog post with two key aspects in mind: Connections: The student can recognize, explain and use connections among mathematical ideas and Problem Solving: The student can apply mathematical algorithms (series of steps), tools, and/or representations to accurately solve problems.

prog 1prog 2
prog 3
prog 4

Finally, I showed my ideas to one of the deans at our school who is much more knowledgeable in computer programming than me to get some of his insights on the activity. He helped me create a short piece after students finish the challenges that helped them pre-write before their blog post. It also helped me create a rubric (below) to guide students as to how I would grade them including accuracy of conditional statements, problem solving throughout the activity, and the written expression of their blog post.

rubric pic

Overall this project went great…it has now become one of my new favorites!! Every student was engaged in the tutorials and the challenges; I saw them have a sense of ownership in their learning because they could self direct and work at their own pace. When students got the code, several of them got so excited and called me over showing off their successes. I also had a lot of really interesting conversations during and after the project…one student said they can now imagine how complex creating a video game would be with all the coding involved. Several students said they understood why their program needed both the if and the then components to make it work and really liked seeing their work actually do something. I am excited to read their blog posts soon (they’re due on Tuesday) and hear how they summarize the project and learning objectives!