# Mental Math Monday

I must first dedicate this post to my mom. She first introduced me to mental math when I was in elementary school. As I tell my students, I used to have to do a series of mental math before I was allowed to eat dinner every night…I’m not sure that’s a true story, but it’s what I like to remember (and I like to tell my kids so they get a glimpse into my math loving family). So, in celebration of my mom, Monday warm ups in my class are dedicated to mental math. In fact, last year, my mom and I had a competition between my students and hers…unfortunately Skype didn’t work the day we planned for, so we did the competition separately…I think we should try again this year, what do you say, mom?! Rematch, 2016??

Anyways, I love doing mental math as a warm up and my students actually do, too. Ironically, when first introducing the idea to my students, there was a whole lot of whining…”Awww no calculators…no, I can’t do math in my head…but, it’s Monday.” So, after a little convincing that everything is going to be okay, my kids soon realized that they loved mental math more than they ever thought! As it creates a little friendly competition amongst each other and themselves, it also reinforces concepts like square roots and any other operations students have been studying, helps students grow their quick thinking algebraic skills, and reinforces their listening skills.

Here’s how it works…ask students to clear off their desks…it’s all done in their heads and the less distractions the better. Tell students not to talk, not to say things like “wait, wait” or “ahh I lost it” (they will do that), because it’s all about listening and processing. Tell students when you are done, and only then, to raise their hands if they got the answer (mine tend to just blurt it out at the end which is okay sometimes, too, if you can’t control the excitement!) Call out a series of operations starting with a single number. I just make it up as I go and calculate in my head as I say it, but you can create the series if you want before and read it off.

Example: take the number 25, add 5, divide by 3, double it, divide by 4, add 2, square it, add 3, subtract 10, divide by 2, subtract 1.

You can get more or less complicated…throw in fractions, negatives, triple digits, ect. Today, I had my kids make up their own problems, ensuring they had operations that could be calculated in their heads and nothing ended  with something like 157 divided by 17. Then, I read a few of them out, and in some classes, several students volunteered to try calling the ones they created out to the class.

If you try it, have fun with it…kids get really into it as they race to get them all correct!

# 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).

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!! 🙂

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