Category: Algebra

Categories
Algebra Middle School Technology

Desmos Online Graphing Calculator

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:

desmos1

Inequalities can also be graphed easily and nicely:

desmos2

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:

desmos3

Points can be traced.  Check out how the minimum here is communicated with appropriate symbols:

Photobucket

Yep, we can do piece-wise functions too:

desmos5

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:

Horsham-20120502-00199, Uploaded by Photobucket Mobile for BlackBerry

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!

Categories
Algebra Geometry

NCTM – Thursday

Very exciting day….quite a buzz as I arrived today.  Looking forward to some interesting sessions in the next few days.  The hard part is deciding which sessions to choose.

Teaching Proof: Lessons From an Action Research Study.  Pete Johnson, Eastern Connecticut State University

Yesterday, I attended a research session on proof,where’s a 5-person panel each explained research they had done on encouraging justification and proof in both middle and high school math courses.  From that session, there were two takeaways for me:

  • When we provide a proof, for whom is the justification intended?  Who is the audience for the communication? Let’s focus on the end user.
  • An interesting activity is to provide a number of givens, then having students work to  support the “strongest claim”.  We often tailor arguments to fit our pre-determined conclusions.  But what are the possibilities, given provided information?

My first session today continues my focus on proof.

“writing proofs” is not a topic or a bit of content, it is a a process, a way of thinking that evolves over time

Much of the discussion in this session focused on the following challenge:

Prove that if n is an odd positive integer, then n squared is an odd positive integer.

A few approaches emerged.  Let n = 2k + 1, then simplify n-squared.  Are we guaranteed that the result is odd?

Another attendee suggested letting n = m + 1, but then how do we know that m-squared is even?  What is the assumed toolkit of knowns and agreed upon principle in this problem?  What does it mean for a number to be odd?

Also, the group tended to focus on the oddness, but have we proven that the result is positive, or  an integer?

Findings: Teaching proof as a “separate topic” does not work.  Also, instruction in formal logic does not seem to transfer well to mathematical proof.

Engaging Activites for Your Classroom: technology in Middle School Mathematics

The main event of this session featured activities utilizing the TI Nspire CX Navigator system.  A few years ago, I acquired the Navigator system for the TI84 calcs, and had used it in some of my classes.  But, over time, I found the system cumbersome, and that the classroom payoff was not often worth the set up required.  I was eager to try this updated system, as it is now wireless, and integrates with the new color CX calculators.

My first impression is that sending files has become more intuitive, and the entire interface is cleaner and less clunky than the 84 software.  The examples demonstrated today were pulled from the TI Activity Exchange, and could easily be edited for use with TI Publish View, which mentioned in an earlier blog post.

Philadelphia-20120426-00191, Uploaded by Photobucket Mobile for BlackBerry

Looking forward to more great math discussions tomorrow!

Categories
Algebra Geometry Middle School

101 Questions

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 few items 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.
101qs

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.
66-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:

The Ticket Roll Problem on Prezi

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!