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!".

#flipclass FlashBlog On Parents, Support, and Homework

I try to keep the parents of my students informed about what is going on in my classroom. I could always do a better job.

I have one main "parent goal" for this coming year. I would like to get more feedback from parents about what is/isn't working for their child in my class. I collected email addresses at the beginning of the year and plan to form groups so I can send reflection forms to parents similar to what I have students complete throughout the year.

The biggest problem is finding the time to get all of that set up. Since I'm still in the process of finding my groove, email groups have been low on the priority list. But I don't want to forget about them.

Support from parents and administration for my flipped classroom has been amazing. My admin has been behind me from the beginning, and as positive comments started coming in from parents and students, support strengthened.

The only question I've been asked by a parent was during the Open House at the beginning of my first flipped year: "Are YOU making these videos?" OK...there were two questions. The follow-up question was "How are you going to have time to do this?"

Parent feedback is always positive. I hear from parents from time to time; I hear from their children what they are saying at home. I've even heard positive parent comments from tutors who work with my students.

I assign my videos as homework. There. I said it.

My videos are short (10-15 minutes), and I keep them to about 3 videos a week (sometimes more, sometimes less).

Parents and students appreciate the manageable, light homework load. Parents appreciate that they no longer have to try to figure out how to help their students with their math homework. They appreciate that they get to know me in a way as they listen to videos their child is watching.

Moving direct instruction outside the classroom allows me to do things in the classroom I would otherwise not have time for.

I am grateful for supportive parents and a supportive administration. I like the format of my flipped classroom and what it does for my students. I look forward to continuing to improve all its facets.

Trying to Find my Groove

I just finished Week 2 with students, and I have to say it's been a rough two weeks. My students are fine. I have one challenging class, but we are approaching the place where we understand each other. :)

I have just felt perpetually behind the past two weeks. I feel like I'm barely staying one step ahead of where I need to be. I don't have my sea legs, yet.

There are probably good reasons. We moved my son to his freshman dorm last weekend. I had to miss a day this week for a committee I'm on. There have been after-school appointments, which completely cramp my stay-at-school-until-I'm-done style.

I have been reminded of when I returned to teaching after 5 years at home, with two small kids. It was Christmas before I felt I was in a routine.

I really hope I'm not going to feel this way until Christmas.

The point of this blog, however, is not to whine about how hard everything feels right now (at least, not too much).

I want to reflect on what has happened in the classroom in two weeks.

I've pretty much followed what I did last year in Pre-Algebra and Algebra 1, but - as usual - I tweaked a few things.

I ditched last year's growth mindset lesson. I believe growth mindset is important and work on it all year, but I was bored to tears with the lesson I did last year. And if I'm bored, I know my students are.

This year I did this activity by Sara VanDerWerf to introduce cooperative learning and looking for patterns. I loved it! Students did, too! And it seems to have been beneficial; when we work together, students show many of the characteristics we talked about with this activity.

I did part of the lesson I blogged about here. It was a fine lesson; we had some good discussions. It highlighted some misconceptions. But it still didn't accomplish quite what I was looking for in my review of subtraction. I'll try another iteration - which I've already outlined - next year.

The tech start to the year has gone very smoothly. I've used Google Forms - including on a sub day! - and Flubaroo to quickly and efficiently assess students and adjust the next day's instruction. The newest updates to Google Classroom have me a very happy teacher.

The new annotation feature in Google Classroom is awesome!

I started my flipped lessons with Algebra 1 a little differently this year.

Typically, we watch an intro lesson to square roots together, then the remainder of the week's videos are completed at home.

This year, I went with an explore-flip-apply approach. I wanted student to build the conceptual foundation for square roots. So I started with this lesson from Illuminations. The extra time it took to do the discoveries and lay the foundation was worth it. I was so thrilled yesterday to hear a student - several days after the lesson and well into the next topic - define "square root" as the "side length that gives you the area of a square." SCORE!

We watched all the week's videos in class. Square roots, cube roots, sets of real numbers, classifying real numbers. My thought was students and I might benefit from me being with them as they watched their first videos. I could give tips for how to take notes. I could check their notes as soon as they finished a video. And all of that happened, but I think students were more distracted watching the videos in class. It also messed up my class time. My timing was completely off for class activities (mostly too short and having too much leftover time at the end of class). Another iteration will come next year.

One change I made in a particular video was very positive and much needed. I made my real number video two years ago, changed it a bit last year, and changed it again this year. It started entirely too long (about 15 minutes), and last year I got it down to 11. But I was still not happy. It was a lot of listening/writing.

So I decided to split it into 2 videos. The first video defined the sets of real numbers and gave examples, the second video looked at the Venn diagram organization of the system and how to classify numbers.

I'm happy, now. :)

Yes, it's two days instead of one, and if these videos are watched at home next year - which I think they will be - I'll have to find a few more meaningful activities to fill class time, but I'm happy with the two videos in place of one.

I continue to focus on shorter videos.

I didn't take a lot of pictures this week, but I did catch what is still one of my favorite activities: a search & order where students have to properly classify numbers and order them least to greatest. I heard lots of great discussions and answered lots of good questions. It was made even better with the new Classroom annotate feature.

So...I'm confident I'll find my groove. Before Christmas. I already feel better after completing the "big" activities currently on my calendar and staying at school way late yesterday to get a handle on things.

Do you know how quickly a school building gets dark and quiet on a Friday afternoon?

Goal for the Year: Keep Taking Risks

I didn't intend to blog today, but I had a "moment" yesterday, and I need to share it, mostly for my own benefit.

As a teacher, I do not tend to avoid risks. I flipped my classroom. I implemented a broad retake and redo policy. In the last few years I have used methods and strategies and lessons completely outside my comfort zone. I play around with new technologies.

Does everything always work smoothly? No. But the rewards of all those things have been greater than the hiccups.

So...it's a new school year (students start tomorrow) with new opportunities for growth. I have pushed myself and my students HARD the last couple of years, but there are still improvements to be made.

One area of focus is developing students conceptual understanding of math. WHY does math work the way it does? An understanding of the math is infinitely preferable to blindly using a rule and not having a clue why it works.

One topic where I find students' conceptual understanding to be particularly lacking is subtracting integers. Students really have no idea what subtraction means. They've been given rules to follow (this is improving!), but most students will miss subtraction most of the time.

Integer operations are not an 8th grade "thing." We use integers, but the foundation for integer operations happens in the 7th grade.

But I start the year in Pre-Algebra with a review of integer operations. It is familiar to students, it boosts their confidence as we begin the year, and it is very important to the work we do the rest of the year.

In years past, this has meant a review of "the rules." Singing songs, writing rules in foldables, practice problems.

This year's 8th graders didn't learn "rules" in the 7th grade. They learned how integers worked. So there will be no singing this year. We're going to work with number lines (I started this some last year).

I was looking around over the summer for a good, conceptual integer lesson, and I found one using temperature as a context that focuses primarily on subtraction. Perfect! I thought it would be great to have a nice collaborative lesson at the beginning of the year to show students that THIS is how this class is going to work.

Last night I printed the lesson out to look through it and get a better feel for it.

And I began to panic.

"This is confusing."

"This is too hard."

"This is too deep."

"This will take too long."

"They'll never get this."

"This will be a disaster."

"We're going to have to scrap this lesson. Now."

"We'll just teach the rules and do some practice."

And then I stopped myself.

Do I want my students to develop better conceptual understanding?

Do I want my students to "make sense of problems and persevere in solving them"? (Mathematical Practice Standard #1)

Do I want my students to improve their modeling and justifying skills?

Yes, yes, and yes.

So there will be no scrapping of this lesson. It may need some scaffolding and students will need lots of support and cheerleading. But my students can do this. I can do this.

If I want to continue to move my students where they need to be, I must continue to push myself. Teach the risky lessons. Don't revert to what's comfortable just because it's easier and less of a headache.

After I had this little conversation with myself, the inspirational people I follow on Twitter unknowingly confirmed my resolve.

"What's worse than failure is not trying in the first place." - @teachergoals

"Doubt kills more dreams than failure ever will" - Karim Sedikki, posted by @teachergoals

"Don't be afraid to scrape the paint off and do it again. This is the way you learn, trial and error, over and over, repetition. It pays you great dividends, great, great dividends." - Bob Ross, posted by @NFLaFave

The goal for the year (well, one of many areas of focus)? Keep taking risks. My students deserve it.