Category Archives: geometry

Teaching Past Grief and Cancer

Well, I had my surgery and I’m back to school. I’m fine. I have a few stitches and a small band-aid on my nose. This picture is with my husband, Doug, right after the surgery. We went sunscreen shopping.

I let kids knwp-1462574706701.jpgow the truth if they asked about it. It was almost always greeted with, “My mom had that!” or someone else they know. Then they told a little story about it. I smiled and said, “Yeah.” Then we got back to the math.

After blogging about things going on with me outside the classroom and how they subtly impacted the classroom in Teaching Through Grief and Cancer, so many people said wonderful things and shared their experiences. Most comments reminded me of the resilience and acceptance of my students. They were right on.

What prompted me to write about it, well, in fact, what stared me in the face making me very aware that I had not managed to make my absences seamless for my students, was when I gave a test and almost no one finished. That really surprised me. I gave the test last Thursday. Only a couple kids finished. At the end of the period, not sure what to do, I told the kids to go study, come in for extra help if they needed it, and continue with their test on Monday. Many of them still didn’t finish on Monday. So, I let them finish today. Turned out, I was giving a three day test over a week long period. That had never happened before. 

One of the unplanned benefits was that they were very motivated to study and knew what to study. Plus, the test wasn’t easy. It was on circles, secants, tangents… with lots of formulas and relationships and complicated diagrams. I’ll probably still end up curving the grades.

I also learned that I didn’t need to hold back on testing because of my absences. I had delayed their test because I had been absent so much. For my Geometry class, I had created a flipped unit, which began the day my dad passed away. I wrote about it here Flipped Circles Unit in Geometry and thought I had done a great thing by flipping the unit, making my absence a non-issue. Never mind that it ended right before a week break. When the kids came back from break, they’d probably forgotten much of what they’d done. Plus, some said the videos weren’t helpful. But, I was also hearing that from just a few kids.

So, we reviewed for a couple of weeks – three more absences for me, then we finally took the test. As described above, it didn’t go great at first. A week later, everyone has finally taken it. And, I missed two more days. That’s okay, though, I gave a final exam review packet. Actually, I gave lots of misc review materials when I was out this semester.

I wonder if the sporadic review packets also had the students going in too many directions. But, isn’t that also spiraling? Hmm… where’s the sweet spot on that? I certainly didn’t get there. That’s a great topic to research: disjointed and sporadic versus ‘spiraling.’ 

Anyway, I’m not blaming myself. Many of the absences were beyond my control. And, I think giving the test over several days ended up being a great differentiation strategy that I would possibly use again. They didn’t memorize the problems and all share the answers, as one might fear. They went away and studied. They came back and asked great questions. So, I think it all is turning out really well.

The last month of school looks good. There will be continuity and plenty of time to review. Everything seems like it’s getting back to normal. I’m so grateful for the ability to blog and reflect and interact with others about this. My spirits have totally lifted from the experience.

Thanks to Dan and Erin and Julie for your kind words on your comments. Thanks to all the friends and colleagues who stopped by today or asked how I’m doing. And thanks to all those friends and family who sent kind words on Facebook or through messages. Putting personal stuff out there is generally outside my comfort zone. I really meant to talk about how it was impacting my students. I didn’t think I would get such a personal response from others. The benefits were really beyond what I expected and I didn’t imagine that I would hear from so many people with similar experiences. I heard from old dear friends and felt the support of strangers. That is really wonderful.

Feeling grateful….

How 2 questions took 1 hour to solve

I was responding to a post by Dan Meyer about his awesome recent talk at NCTM. Here’s what I wrote in my comment:

I am working hard to keep engagement high. This week, we actually started with two blank triangles, one right and one obtuse. I asked kids to solve the triangle, and ask me questions. There were no measurements on the triangles. The triangles were to scale. They could ask me for one value.

From there, they had to measure another part of the triangle and let the solving begin. They could only measure one other part, no more. So, depending on what they chose, they would use different tools to solve the rest of the triangles. They used rulers or protractors, ratios, triangle sum rule, right triangle trigonometry and Laws of Sines/Cosines. They worked in groups to try to figure out what information they needed, they worked together to try different strategies. In the end, the only way they knew the answers were correct was by rationalizing whether or not they made sense.

We spent 1 hour on two problems. Engagement was high, completion rates were nearly 100%, participation was 100%. It was a really great day for me and I was able to coach them. They did the lifting. It was great. Great conversations.

So much of Dan’s work inspires mine. I love his 3-act-math ideas, though I don’t use them much. More so, I respond to the idea of opening up questions. I love the idea of putting out a skeleton and asking kids what they need. They actually answer, they actually engage in the problem and start to think and ask questions. Once they are invested like that, they don’t like to give up. Giving up has been like an epidemic in mid level high school math classes like geometry and Algebra 2.

Today’s geometry warm up looked like this:2016-04-22-13.44.41.jpg.jpg

We didn’t even get to the third problem. Yes, it was hand drawn. Like, free hand. That actually made the problem kind of interesting because I don’t really have a right triangle, do I? Just almost one. So, depending on what side or angle students measured (by the way I told them the base side of the triangles in both problems one and two was 15 units) the answers they got might be a bit different.

It was also a great talking point about how real life problems are rarely perfect and we rarely get to check the back of the book to see if we did it right. We have to trust our methods and our team.

Great conversations. Such a great day. It was hard for some of them to keep going to finish the second problem. They were a bit tapped out. We have block schedules, which means 90 minute periods. This took way longer than I expected. But, totally worth it. We’ll do the third problem next week. They actually asked for a book assignment when we were done. I guess their brains were tired. But, they did it! And, they did really well! Go team!

More about problem three in another post. Maybe.

Flipped Circles Unit in Geometry

I made a run at a flipped unit for my Geometry classes. I’ve done flipped lessons and some flipped units before, with pretty good results, so I felt good about putting this unit together. To see the plan, click HERE and click on the link for the unit plan at the top of my page. It will download a word doc which is editable and has links to the tutorial videos.

Why I did it:

  1. I knew I’d be taking a bunch of sub days, for various reasons. And, I wasn’t totally sure when they would all be. Some were known, others were a bit up in the air.
  2. There have been and would continue to be many student absences. We have a bunch of field trips scheduled during March and April, then special testing in May. Plus, it seems it’s just been a bad flu year, too.
  3. I want to expose students to lesson resources beyond just their teacher (me) because I think it will good for them to have more places to find information. Especially in light of our limited time together, thanks to reasons 1 and 2.
  4. When students have seen some of the concepts before class, they are more prepared when they are first discussed in class. They’ve had time to digest some of the information.
  5. When they have more class time to work problems and do activities, there is more of a chance that they will be able to ask questions with peers and with me. And, there is more chance that more students will complete most assignments.
  6. With more time for activities and interesting problems, our classroom climate is stronger. We all become learners and can ease some pressure and stress.

FYI, reasons 1-3 were my immediate concerns, but 4-6 are the reasons I’ve used flipped teaching in the past.

What happens:

Ideally, students watch a video lesson before coming to class. Then, the key points are summarized in class and students have more time in class to complete problem sets or other activities. It can be as traditional or project or problem based as the instructor determines.

I purposefully excluded Mondays as due dates. These are built in buffer days. They can be used for getting caught up after absences, explorations, team activities, short assessments, etc….

How I did it:

Well, I pretty much followed the topics in the order of the text, except for the first couple of sections. I merged and re-ordered those a bit. Some might be critical of following the text, but I think for the circumstances and reasons for the unit, it was a good approach. This way the students and substitute teachers have something to go by.

I would seek out videos on Khanacademy or YouTube, with YouTube really being my go to resource for video lessons. I would review them and go with the ones that I felt were most similar to the content and terminology of the text and my own vocabulary. I would try to find different instructors, so that kids could see the assortment out there.

When kids come to class I know they haven’t all watched the video. I’m not worried about that, because it’s meant to be an introduction and I equate with the fact that not all kids will get all their homework done, no matter what it is.

So, I ask the students what they remember from the video. We talk about vocabulary, and keep track of formulas. Then I supplement with more information and we do a couple of examples together.

Students spend the rest of the class working on an assignment. They have more time to work on it, because there’s been less time spent on whole class instruction. And, I have more time to get to students, group or partner them and have a better knowledge of what they understand well, and where they still need guidance.

Having Mondays left with no due dates was an unexpected gem. I think it’s a great take away in terms of leaving some breathing room in the plan. On a regular week at our school, every class meets on Mondays. Then, you see the students two more times for a 90 minute period. Absences are a real killer. Mine and theirs. For our school, flipped teaching (especially for math) makes a lot of sense.

What the plan doesn’t show:

Engaging openers and open questions. It’s so important to include these during class. There’s a book called 501 Geometry Questions. There are some great questions in there. We focus on HOW to solve problems. They are great to use for creating open questions. You can omit some information, or they may spark creativity to generate open question.

Assessment schedule: yeah, no planned quizzes in there. I really wasn’t sure when I’d be there. So, I left that a bit open. The kids don’t seem to mind. Because of the break next week, I may just do a project type assessment after the break. I plan to figure that out during the break. Sometimes, I have them take partner quizzes. They like those, because I tell them if something isn’t quite right. They go back and try again. Usually with some serious debate and excitement.

How I’ll make it better:

Before class: Someday, I’d like to have my own videos. But, that would defeat the value of reason 3 above (expose them to other resources).

During class: I’d also like them to have a more organized place for a running list of formulas, etc. And/or, hand out a practice test or something, so they can “document” their learning at certain points.

Also, I’d like to modify the assignment instructions so they have a choice over which problems they are working. I’d like to assign 20 and tell them to complete 15, of their choice, and let them know which ones are more straight forward, and which ones are more challenging.

After class: Move closer to a 2-4-2 homework model with the flipped lesson.

I’d also like to find a better way to regularly assess than the the big unit test on a certain date. I do that with quizzes, of course (not for this unit, granted), but I still would like an assessment that’s more interesting than that. At least sometimes. So, for this unit, I’ll be doing that over the break, and can add it to the plan for next year.

Open question: How do you find the hypotenuse?

In Geometry, we had a super-awesome-open-question to start the day. I just made it up on the spot, and had no idea it was going to turn out the way it did.

I was planning to put this problem up…


As I started to draw it (with a different orientation because it was from their about to be assigned assignment), I stopped here….


I changed the question from a sine question to an open question. I asked,

“What additional information do you need to find the measure of the hypotenuse?”

We had been learning about trigonometry and they knew about the Pythagorean theorem, so, I thought I’d get a range of suggestions.  And, I did. I asked everyone to take a minute to think about what they needed. Then I asked them to take share their idea with a partner. Then, I asked for ideas to write on the board. As students suggested their ideas, I replied, with, “Ah, okay. And how would you use  _____ to solve for the hypotenuse?” And, they would explain.  Then, on to the next idea.

Building their anticipation…. I didn’t tell them if they were RIGHT or wrong. It was really making them re-think as more ideas went on the board and were explained.Of course, there were many RIGHT ideas. When all of the ideas were out, I asked another question.

“Okay, there are several ideas out there. Let me ask you another question. Is there a way to solve for the hypotenuse with only ONE more piece of information?” There were many blurted responses. I really couldn’t understand what any one person was saying. So, I raised my hand as a reminder that I need a hand to go up. I called on a student and asked if they thought they could solve it with one other measure. They said YES, another angle measure.

So, I called on another student to give me a number between one and 89. They said 12, so I labeled one of the angles 12 degrees. We solved for the other angle measure, 78 degrees. We determined we still couldn’t solve it because we now had an AAA triangle, that could produce more than one possible triangle.

Okay, erase the angle measures. Ask another student for a side length. “Five.”

Okay, label the short leg 5. Can’t solve it. So, we decided we needed 2 pieces of information. But which two? Back to our list.

At this point in the hour, we had talked about many ideas, many possibilities and drew conclusions based on things we had learned. What I noticed (and knew I needed to write about) was how the level of engagement was so much higher than usual. Students who were normally a bit “checked out” were generating ideas and were “hooked” into knowing which method was going to work. Were they RIGHT?

So, I grabbed a ruler and measured the hypotenuse. It was 13 inches long. No one had suggested a ruler. So, that was just for fun. What I loved, was that no one suggested using a ruler, like they had at the beginning of the year. Back then, they would ask, can’t we just use a ruler or protractor to measure the sides and angles? Of course not, this is about relationships and we find the answers other ways. You cant’t assume things are drawn to scale, right?

We then went through each idea and tried it to see which ones worked and which ones needed more or different information. They all worked. I was pretty happy. Kids felt successful and it was a fun way to start the double block review period before the test.

I really saw how far we had come since last August. My students are amazing. God bless you, third period Geometry. I can’t wait to do this with my 7th period class nest week.

The beauty of the open question is that more kids can enter the conversation, the lesson is geared to the student’s knowledge and it’s exciting for the teacher, because you don’t know what they are going to say and you get to learn about your students. And, it keeps you very engaged, just like them. 🙂