Pythagorean Stackin' #flipblogs

Twas the day after Labor Day. As has been the case the past several years, we were on an activity schedule for a pep-rally. Shortened classes. Excited students. Brains a little rusty after the three-day weekend.

On Friday, Algebra 1 looked at how to use the Pythagorean Theorem to find unknown sides of right triangles.

Today was our practice day, using one of my favorite activities: Pythagorean Stacks.

I got the activity from Jan at Equation Freak (who has since retired). It's a great activity to not only practice using the Pythagorean Theorem but to also hone those diagram-interpreting skills. I've blogged about the activity before.

Students did well with the activity. Once they saw how to use the diagram to find the information they needed, most of them took off with it. (In the rush of each class period, it never even crossed my mind to take any pictures of kids working!)

Due to the shortened classes, we will finish the activity tomorrow. Then we'll move into the converse of the Pythagorean Theorem before looking at applications of the theorem.

(By the way, I've used Jan's Pythagorean Short Stacks with my Pre-Algebra kids. The diagram is a little more straightforward and accessible to my inclusion students.)

I Notice...I Wonder...The Pythagorean Theorem in 3D

I am pretty tired and close to brain-dead, but I wanted to post while today's activities were fresh in my mind.

I had a brainstorm Saturday morning. Algebra 1 was finishing up the Pythagorean Theorem this week, and I wanted to take a look at the Pythagorean Theorem in 3D. I tried it last year with not a lot of luck. A few saw it, many did not.

My "A-Ha" came at the strangest time, but I went with it and began planning. What if I approach it with an "I Notice...I Wonder..." angle? What if I can get kids to see where we're headed without telling them?

I decided to show kids a diagram of a rectangular prism with the interior diagonal in blue and ask what they noticed and wondered. Then I showed them the same diagram with the dimensions of the prism and the question, "What is the length of the blue line segment?"

I split kids into groups of three, gave them a copy of the diagram they could write on, and I gave each group a box (various sizes) with string as the interior diagonal so they could have something to touch and see.

And then I turned them loose.

I was scared. What if they all stared at their papers and said, "We don't know what to do!"? What if I had to walk everybody through step by step by step? What if it was a total flop of a lesson?

My fears were unfounded.

Students noticed and wondered some good things. They were advised to give serious, math-related observations before they began. :)

As I distributed diagrams and boxes, everyone got to work. Almost everyone tried SOMEthing. Maybe two groups all day said, "We don't even know where to start."

I tried to not give too much away as I helped groups. "You're on the right track." "Tell me why you approached it that way." "Does what you found have anything to do with the length of the string?" I'd ask a question or two, smile, and walk away.

I would say by the end of the activity, most students saw the two triangles they needed to see to find the length of the string/blue line segment.

It was interesting to see their incorrect approaches and misconceptions. Many wanted to start with area of the rectangular faces or the volume of the box. Some would find the diagonals of the rectangular faces and add them. A few had visualization difficulties (interestingly, some of my strongest math-y students had the most trouble with the visualization of the string in the box).

A few went right for what is the formula/shortcut for finding the length of a box's diagonal, extending what they knew about the Pythagorean Theorem, but since they couldn't explain to me why it worked, they had to find another way (the "8th grade way") to get the correct answer. Those groups got mad at me. :)

The cardboard boxes were a MUST. I didn't start the day with enough boxes for every group, but after some searching by a dear friend, I ended up with enough, and I'm so glad I did. Being able to hold, touch, and outline what they were finding on a real box helped make the lesson a success. One kid even said, "I'm glad we had this box; I can't tell anything on the picture!"

I'm wondering if I needed some triangles of the proper sizes cut out so that students could SEE the triangles they were trying to see. I would need the same size boxes for everybody to make that process easier. It's something I'll think about before I do this activity with my Pre-Algebra kids.

I wish I had added one thing to the notice/wonder start to the lesson. After their initial observations and questions, I wish I had said, "If I told you we were dealing with the Pythagorean Theorem in 3 dimensions, does that add anything to what you notice or wonder?" I wanted to tell the students as little as possible, but it took a few of them a little bit to realize this activity was tied to what we've been doing in class for the last week.

The next part of this lesson I need to tweak is the wrap up. All the groups in my 3rd Period class finished, saw both triangles they were supposed to see, and found the length of the string/segment. But I'm pretty sure a couple in my 5th Period and a few more in my 7th Period never got there, even with prompting and hints. I hate feeling like some students leave a lesson thinking, "That was nice. What on earth were we doing?!?"

So...how to get them all "there" and tie up the lesson with a nice little bow without just giving it all away?

I'll keep thinking on it...sleeping on it...and waiting for the next Saturday morning "A-Ha!".

Spending a Week with Pythagoras

Standards dealing with the Pythagorean Theorem fall in our eighth grade course of study. If I don't cover those standards with my eighth grade Algebra 1 students, they will miss learning and using the Pythagorean Theorem. And high school teachers will come after me. :)

I always cover the Pythagorean Theorem with my Algebra 1 students, but this year I decided to follow the lead of a blogger friend who also teaches Algebra 1 to eighth graders and start the year with it (after discussing square roots and real numbers). I covered the material in fewer days than I will with my Pre-Algebra students, because, after all, we still have a lot of Algebra 1 standards to get to.

We started the unit with a visual proof of the theorem. It's a proof out of a book I've used since teaching high school geometry another lifetime ago, Visualized Geometry (I just looked it up; the book was published in 1990!). A right triangle has squares drawn off each side, the squares from the legs are cut and arranged to cover the square off the hypotenuse. After some productive struggle with getting the pieces to "fit," students readily see that "a^2 + b^2 = c^2." (I didn't get any good pictures of the final product. And as we were disposing of all the pieces, a student in the last period of the day said, "We should have glued that in our INBs!" Brilliant! Why didn't I think of that? I made a note of it for next time.)

I have to cover the Pythagorean Theorem, its converse, applications of the Pythagorean Theorem, and using the Pythagorean Theorem to find the distance between points in the coordinate plane. Students had 3 videos dealing with these topics.

In class I tried several new activities this year. I was very pleased with the work students were able to do with the content.

To practice using the Pythagorean Theorem, I used this activity from Equation Freak. It was SO much better than a page of "find the missing side" problems. Students got great practice with visualization and finding lengths of sides in a diagram that will help them in future math classes. And they had one, final goal. We did not get completely finished with the activity in class, but I gave students the option to finish it at home. A good number of students brought it in completed the next day!

"Pythagorean Stack" by Equation Freak

We practiced applications of the Pythagorean Theorem using task cards I've had for a few years from Live Love Math. I added QR codes to the back so students could check their work as they went.

Pythagorean Applications

To find distance on the coordinate plane, I used Rockstar Math Teacher's "Pythagorean Zoo" activity. I have not done a lot with Socrative, but the activity went smoothly, and the kids enjoyed it. The funniest comment all day was, "Mrs. Gibbs! Are you ruining the zoo with MATH?!?"

"Pythagorean Zoo" by Rockstar Math Teacher

I was confident my students understood and could use the Pythagorean Theorem, so the day before the test I was able to do a few activities I had not had time for earlier in the week. We did a couple of problems with the converse of the theorem. In hindsight, these problems should have been done earlier in the week, after they had seen the converse in a video. Many students did not remember seeing anything like that, and I had to point them to their notes. I should have reinforced the topic right after they had watched the video.

We looked at the Pythagorean Theorem in 3D. This problem led to some great class discussions and was also good for their visualization skills. Having taught Geometry in that former life, I know how important those skills are!

We finished the unit with another lesson I snagged from Equation Freak. We watched a clip from The Wizard of Oz and "corrected the scarecrow." Like Jan, I had a few students who had never seen the movie. The activity was a great exercise in using precise mathematical vocabulary.

"Correct the Scarecrow"

I received another funny comment with this activity: "Now you're ruining The Wizard of Oz with math!" But I'm hoping students will begin to see we can find math EVERYwhere.

I have more pictures than normal because I'm participating in #teach180 on Twitter. I'm trying to tweet a picture of something going on in my classroom every day. Some are blogging every day, but I know better than to try a daily blog post. :) The highlight of my day for the last couple of weeks has been scrolling through the pictures posted with the hashtag. I've already gotten (and used!) a couple of ideas from others' posts.

I'm also trying to focus my blog this year on what we do in class. I make the videos, my students watch the videos, but the real magic happens in the 51 minutes I have with my students.

As I've said before, none of these meaningful, engaging, beneficial activities would have been possible in my traditional classroom. Or the unit would have taken twice as long as it did. I have much more fun in my flipped classroom, and I know my students do, too.