Pythagorean Theorem

I love the wide variety of resources available for exploring the Pythagorean theorem!  I decided to go pretty low tech this year although I've seen some fun explorations using Geogebra and Sketchpad.  Yesterday I started with a set of squares and had students try to form a right triangle by using 3 squares as the sides of the triangle.  We recorded all of the trios of numbers they found, and then set that list aside.

Next, we explored a 3, 4, 5 right triangle, and built squares on the three sides and looked for patterns.  Students pretty quickly identified that the sum of the areas of the squares on the two shorter sides was equal to the area of the square on the longest side.  Using this information, we were able to generalize and "discover" the Pythagorean theorem.

Going back to our list of trios of numbers, students were able to check to see if their triangles were right triangles. We developed a list of Pythagorean Triples, and then just looked for other trios of numbers that would work.

Today I brought out Dan Meyer's Taco Cart Problem.  Students did a great job of asking questions and figuring out what they needed to know, and we brought back in a lot of the vocabulary we had used yesterday when exploring right triangles.  I supplied the information for Act Two and sent them on their way to work out the problem in groups of 3 or 4 on 2ft by 3ft whiteboards.  This was the first problem they had solved with the Pythagorean Theorem, and there were some great misconceptions for us to discuss!

When I started the discussion, I asked what was the first piece of information that we needed to agree on.  Students suggested the length of the hypotenuse, as all other calculations were based on that distance.  Some misconceptions: Students squared the legs and found the sum, but left that value for the hypotenuse instead of taking the square root.  Students added the legs and recorded that value as the hypotenuse.  Students squared the legs and found the sum, but then thought that dividing by two was equivalent to taking the square root.

The next part of our discussion was how to use the rates.  Also some misconceptions here, but I love that we were able to discuss if groups had a strategy that would have worked if their hypotenuse had been right.  Common misconceptions here:  multiplying the length by the rate of 2 or 5 ft per second instead of dividing.  Forgetting that one person had part of the walk on sand and another part on the sidewalk.

Monday we'll be looking at a few more problems, including a version of the television 3 act problem from Timon Piccini.  Also really liking revising the Wizard of Oz from Robert Kaplinsky.  Also looking forward to using Tilted Squares from NCTM as an enrichment activity- I think it's from an article that you have to be a member to see, but there is an illuminations activity that is pretty similar.

What are your favorite Pythagorean Theorem tasks or explorations?

Single Riders

I noticed this sign while visiting a local amusement park this week, and as we waited for the ride I noticed that there were often several open seats for each cycle of the ride.  Now I'm a bit of a Disney fan, and even though they sometimes have long lines, there are rarely empty seats on the big ticket rides.  They do this with single rider lines, and assigning people to specific rows so that they know how many single riders they need to fill the space.  Less wait=happier park visitors!

Then the questions start. How much would it cost to add single rider lines? How many open seats are there really in an hour?  Day?  Are people actually annoyed that there are empty seats- and how much time would adding a single rider line really take off of the average wait.  Certainly Disney went through this process when they decided to add the single rider lines, but how much more volume does Disney have per day over my local amusement park?

When I think about  SMP#3- Constructing Viable Arguments and Critiquing the Reasoning of Others- I want my students to really look at the world around them and use math as a tool to prove whether something does or doesn't make sense. To do this, students really need to observe and question what they see. They need to know how to collect data and organize it in a way that can be communicated with others.

How do we as teachers help to build this capacity in our students?  How can we use SMP #3 in our classrooms to help students model their understanding of a concept and identify misconceptions that are common with that concept?  How do we build a classroom culture where students are comfortable critiquing and having their understanding critiqued.

More to come....suggestions and thoughts are welcome!

Graphing Inequalties Exploration

This is an exploration lesson for students to discover the properties of linear inequalities in two variables. The link will download a copy of the files that I print out for students as well as a possible facilitation guide. The general idea behind the lesson is that students are taking a piece of a Cartesian plane and identifying whether points on the graph are or are not solutions to the inequality. If it is a solution, the point gets a "closed circle" sticker. If it's not, the point gets an "open circle" (a reinforcement sticker). Students work in small groups to fill up the plane with open and closed circles. I verify and ask questions as I walk around.

 When they are finished I pair up the groups and have them note things their graphs and inequalities have in common. There are 8 groups, but they are strategically paired. For example in the first rotation students are comparing inequalities that both have a boundary line with the same slope and y-intercept, but one is greater than or equal to, and one is less than or equal to. By the end of the discussion students should be able to identify solutions to inequalities in two variables, understand the concept of a boundary line, and connect what they know about slope and y intercept to the boundary line. 

The third part of the lesson is a set of cards for students to match up to inequalities. The purpose of this is to introduce the different boundary lines when the inequality is just less than instead of less than or equal to. For example, in the first pairing, all groups have inequalities with the same slope and y-intercept, but different inequality symbols.

Graphing Linear Inequalities Exploration Lesson