Recently, I have been noodling around with the Desmos Online Graphing Calculator. I have used Texas Instruments products exclusively and extensively in my classes for years, but am always on the lookout for tools that are easy to use, functional, and (most importantly) cheap! The Desmos calculator aligns quite nicely with my personal motto, “If It’s For Free, Then It’s For me!”
The calculator has an interface which is intuitive, and it’s easy to dive right in and start graphing:
Inequalities can also be graphed easily and nicely:
Trig functions are formatted in readable form as you type them, and you can choose to have the x-axis “count by pi”, which is a pretty cool feature:
Points can be traced. Check out how the minimum here is communicated with appropriate symbols:
Yep, we can do piece-wise functions too:
What appeals to me about this calculator is that it is web-based, fires up quickly, and is ready to use. This site should be on everyone’s links for students, shared on Edmodo, or whatever resources page you use. While I love my TI software, it often takes too long to load, and you need to be a real Nspire user to navigate around. The site is also usable with Ipads:
But keep in mind that this is a stripped down calculator. It graphs stuff, and that’s about it. You’ll still need your TI’s to perform calculations, analyze data, or do more in-depth analysis, like intercepts or integrals.
Sometimes, less bells and whistles are better.
EDIT: check out the comments for news from Eli, founder of Desmos, who gives some more great information about this tool!
One of the more intriguing math-related websites I have been following this year is 101qs.com by Dan Meyer. The site has a simple concept: you are presented with a picture or short video clip, and are asked to contribute the first question that comes to your mind. I have contributed a fewitems to the site, and reading some of the questions posed often leads me in directions I hadn’t initially considered. How neat! You can also view questions which others have contributed for each item. The pictures and videos are meant to serve as “first acts“, mathematical conversation-starters which lead to problem-solving discussions.
What I like most about this site is that there are no answers. Rather, our focus shifts to posing interesting questions, facilitating meaningful discussions of problem solving methods, and working towards plausible solutions.
As the site became populated with more “first acts”, I recruited volunteers in my district to find a way to use this site with their classes. I found two high school teachers, who were eager to share their Academic (our most basic) Geometry classes. It’s a shame that we often reserve interesting, open-ended tasks for our highest achieving kids, so I was interested to see how these groups would take to the project. And while my high school colleagues were enthusiastic about using the site to develop a task for their students, there were some natural questions about managing the task: How will kids react to having such an open-ended task? Will they persist in completing the task? How will we assess their work?
Note: one teacher I am working with attempted to utilize the site, after we had some discussions of a project, but found that her students were blocked from the site at school, due to its YouTube links. I have since taken care of this snag, but you may need some coaxing with your higher-ups.
We settled upon a structure to help kids step through the task. In day 1, partnerships of students will:
Select an item to explore.
State your question.
Develop a plan of attack and list measurements you will need to consider.
List the math (formulas or concepts) you will need.
The partnerships will then meet with the teacher to discuss their ideas and revise, if necessary. The task then moves on to day 2:
Complete the plan of attack.
Answer the question.
Reflect upon your process and state any changes or improvements.
To complete the task, students will create a presentation which steps through their question. In order to help students understand the task and our expectations, I visited the classes, and modeled the process for one of the Top 10 pictures from the site: the Ticket Roll.
The class discussions were rich, and allowed many students to provide ideas: How many tickets are there? How long is the roll? How will we find the thickness of a ticket? How precise do we need to be? Why are we doing this? In both classes I visited, we discussed the dis-comfort we feel when we have a question without a known answer, and how rare it is to have this happen in math class. To complete the ticket roll problem, I shared a Prezi I made to model our expectations:
As students complete this task, look for an update here and I will share some of the presentations. Would love to hear all of your ideas for how to utilize this rich resource!
This year, as I have moved from a role as classroom teacher to a math coach, I find myself sharing some of favorite lessons and approaches with my colleagues. Taking a cue from Oprah, this is the first of a series of my “Favorite Things”, hopefully without the rampant screaming or random things hidden under your seat.
The Monty Hall Problem on NLVM
The famous “stick or switch” problem is one which will generate tremendous discuss in a classroom. The premise is simple: you are offered the choice of one of three doors, behind one of which there is a great prize. After the host reveals a non-winning door, you are offered the chance to switch your guess. Should you stick or switch? The problem rose to a next level of fame with fierce debate in the “Ask Marilyn” article by Marilyn Vos Savant in Parade Magazine. After having included the problem in my prob/stat course for any years, the discussions rose to a crest as the problem was featured in the movie “21”, with even students I did not know stopping me in the hallways to ask me about its logic (note, this video has a number of ads on it):
As an emerging stats teacher, I would play the game using cardboard doors and hand-drawn pictures of cars and goats. In later years, I used an applet at the National Library of Virtual Manipulatives to play the game with my class. Look under Data Analysis and Probability for “Stick or Switch”. What I really like about this applet is that it will play the game 100 times quickly and display the results.
There are many ways to explain this game to your students. One straight-forward approach is to have students attempt to write out the sample space (all possible outcomes) for this game. For example:
Car behind 1. Pick 1. Shown 2. Stay. WIN
Car behind 1. Pick 1. Shown 2. Switch. LOSE
Car behind 1. Pick 1. Shown 3. Stay. WIN
Car behind 1. Pick 1. Shown 3. Switch. LOSE
Patterns in Pascal’s Triangle
In one of my favorite lessons each year, the many, many patterns in Pascal’s Triangle were discussed. This would come after an investigation of combinations, and students would often recognize that each entry in the triangle is actually a combination, along with other patterns like the counting numbers or perhaps the triangular numbers. But how about the Fibonacci sequence and the “square” numbers? What’s the “hockey stick” theorem?
Over the years, I have used a number of sites to display the triangle and invite students to share their ideas. One nice resource which explains patterns in the triangle can be found here.
The cherry on the Pascal sundae would come when I would invite students to approach the board and circle all of the even numbers. Often, only 15 rows or so would be visible, so the chance to make a generalization is rich and there to discuss. How would this look if we colored 100 rows, 200 rows, 10,000 rows? And do we only get patterns for even numbers? How about multiples of 3, or 5, or 24? Check out this site from Jill Britton, which includes an applet that will color multiples in Pascal’s Triangle. Watch the expressions from your students when you reveal multiples of 5 in the first 128 rows of the triangle:
Try different multiples and observe the great symmetries. Prime numbers like 17 and 23 provide the most surprising results. Digging deeper, start a discussion of the modulus function with your class (it’s just the remainder, no big deal….). What happens when we not only color multiples, but also color the remainders similarly. From the Centre for Experimental and Constructive Mathematics, we get this great applet, which will color remainders based on specifications you provide:
Electric Teaching
While I have been using the Monty Hall problem and Pascal’s Triangle in my classrooms for many years, this last resource is fairly new to me. Electric Teaching by David Johns is an excellent site with an effective interface which is ideal for SMART Boards. The site challenges students to match-up equations with data tables and their graphs.
The site has problems for linear, quadratic and trigonometric functions, along with conics and even problems which include derivatives. What a great way to not only have students participate, but verbalize their thoughts as they identify the matches. The YouTube channel has additional resources from Dave, including a tutorial of the Electric Teaching site.
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