Category Archives: open questions

Unit 1 Planning: Get them engaged on Day One!

(8/21) I will have my first day with my Algebra 2 students on Thursday (8/24). Here I sit, thinking about what to start with. Last year, we learned how to write numbers in different bases. Kids enjoyed it and asked to do more with it. But, we didn’t have a lot of time for it and it did not show up on the first test. It’s not part of the Algebra 2 curriculum. However, I remember learning about different bases somewhere in my high school math classes. It’s disappeared from the curriculum. Or, maybe it’s reappearing somewhere else.

Having kids learn to write the number 25 in base 5  (looks like 100), 7 in base 7 (looks like 10), 12 in base 4 (looks like 30), really is interesting for them. Their little neurons start firing. They say things like “Whoa!” and “That’s so cool!”

We also spent time issuing texts, getting kids signed up on Remind , and doing getting to know you activities. We went over some expectations, etc. But, I knew I had won them over. Then I started teaching.

This year, I want to do that again. And get them on Desmos quickly – download the app, etc. Play Marbleslides: Lines, etc.

(8/23) I make a plan often based on all of the great ideas and inspiration I get from Twitter math teachers (#mtbos #iteachmath), Jo Boaler, Dan Meyer and many others. I recently read a great article about not grading students in the first month (or some time frame like that) and I thought, “Wow, what a great way to build culture and address equity issues.”

So, I’m thinking about that right now.

I feel that the most important things I can do as a teacher is invite students to be curious, let them know they are an important part of the class, and teach them mathematical concepts. After meeting with my excellent colleagues, I’ve come up the following plan:

We are going to see if those iPads work. If they do (fingers crossed) I want to get students to do a card sort activity, and hopefully play some marbleslides and then get into some vocab around linear functions and translations. We will get signed up on Remind and learn about a few classroom expectations. Seriously kids, no phones and limit your bathroom breaks.  Be nice. That’s really it. However, when working with teens, there is always a need to discuss these things, come to an agreement and then move forward. They will test these agreements. We will need community building. I look forward to that.

I’m really excited to get going. I’ll be working some great activities and mathematical ideas into my lessons. We’ll explore some history, look at different bases, play games on Desmos, be creative, and have another great year!

Excellent Summer PD: Desmos Summer Institute

This summer I went to a two day Desmos training in sunny San Diego that was completely dedicated to improving student learning through activity builder.   Dan Meyer, their CEO, kicked off the training with his typical charismatic, humorous and collaborative way of learning about us and generating the goals and plans for the next two days. There were about 25 teachers and a group of Desmos staff and Desmos Teacher Fellows. So, we were surrounded with amazing support, creativity and collaboration.

The Desmos staff said we can share the materials from the workshop. I love that attitude of sharing. In that spirit, I’d like to share with you one of the great shares of Day 1, the Desmos scavenger hunt. The Hunt provides tasks and solutions so you can test your skills and learn new ones. We had great fun with that.

Using the activities or making your own requires you go to I recommend you get started by watching the one minute video and then create an account. You can login and start searching existing activities.

I recommend starting with searching the existing activities and activity bundles. There are individual activities or bundles of activities by topic (there is a list on the sidebar to the left of the screen). For instance, in Algebra 2, there is a bundle for exponential functions. There are currently seven activities. By doing a quick preview of each activity, you can decide which ones you want to use and when to use them during your unit. As you preview the activity, you’ll see a green pop-up that gives teacher tips. They are really helpful.

To make your own activities, I recommend using the materials provided by the Desmos team to help you build a great activity. To build your own, use this link to get started Or, sign in to, and click ‘Custom’ under ‘Your Activities’ on the left side bar. You can click ‘New Activity’ on the right and then click the ‘Get Started Here!’ link to take you to, to see helpful videos and examples. I also really recommend that you use the Teacher Guide when creating your activity. Each activity has a printable guide to help you build your activity and lesson plan. [FYI, the link to the Teacher Guide is an example, you will get one that corresponds to your activity]

I love the activities in Desmos and have had great success with them. Students work and I am freed up to circulate and help as needed. At a glance, I can see where every student is from the teacher dashboard. I can use the teacher tools to anonymize student names and project individual student work or entire class screen overlays so students can see multiple ways of solving problems. Students get instant feedback on their work. Pacing is individualized and many activities get more difficult as you progress, which challenges every student. Students naturally start to ask each other questions. I can partner students as needed. I can even pause an activity – which always leads to groans and the question, “Why did you stop it?!?” You can read about my marbleslides experience What’s great about marbleslides, if you want more details on a specific activity.

My takeaways:

  1. There are amazing educators out there. Find them and stick with them. Then, find more. Share your ideas. Share your lessons. Build better lessons together.
  2. There are great activities for Algebra 2 already built in Desmos. I’m not sure I want to build any myself. It’s not easy to do and more are being added all the time. There are teachers all over the country adding to the bank of activities.
  3. The activities are varied and can be used in many ways – introducing a topic, vocabulary builders, practice, and formative assessment. So far, the ones I’ve used have all provided differentiation for students.
  4. I will build some activities for my AP Economics classes, which I will be teaching for the first time and am very excited about. I’m picturing supply and demand shocks as a starting place. I built my story board using post-its, as recommended, and am ready to start building!

Having the time to really delve in and learn the tools and process for building activities was a gift. There is a second training August 10-11 in San Mateo, CA. Sign up by July 21

Teaching Through Grief and Cancer

Teaching math is always a tough job. Kids often need more from you than you can give. The bell rings and kids are still not understanding the material.

Throw in some personal problems for the teacher. My dad died about six weeks ago. I got a cancer diagnosis about five months ago, that I basically ignored, because my dad was really sick, on chemo, and shrinking. So, I didn’t want to be laid up or worrying anybody with my own cancer.

Granted, my cancer wasn’t that big of a deal on the surface. That’s kind of funny because it was skin cancer. Get it … surface? Ha ha. It was just basal cell carcinoma. On my face, my nose. So, I’ll be going to school with an obvious wound. However, it was my third cancer. The others were a bit more serious. So, I’m a bit upset about it. I’m only forty-eight. Okay, truthfully, I’ll be forty-nine this month.

So, today was my surgery. It went really well. Mohs surgery. I only needed one round. That’s rare. Most people need two or more. So, I’m feeling pretty good.

But, my students have suffered. My daughter is having to deal with a pretty solid amount of turmoil, grief and upset feelings during her last semester of high school. She and I are at the same school. So, she may need to field questions about my stitches and bandages on my face. I’m worried about all of this and I’m still grieving.

So, how are my students doing? Well, I’m not sure. I haven’t been there much. I’ve been out of the classroom a lot this semester because I also served on a hiring committee (two days) proctored the SBAC (one day) and serve on a countywide committee for Algebra 2 alignment (two days).  My daughter went to State championship for wrestling (two days) my surgery was today and I’m taking tomorrow (2 days) plus out for bereavement (three days).

So, my plans for my classes are all blown to shit.

And, even when I have been there, I’ve been less than I’d like to be. Even though I do forget about my troubles when I’m at school. I love teaching. I have good relationships with my students. But, they need more than that from me.

So, this blog outlines my problems and hopefully sets the stage for my next blog, which is going to be about how my students are doing in light of it all. And, about how I’m planning to compensate for it in the last month of school.

Many of us have dealt with or will deal with these kinds of serious issues while teaching. If you have, post a response below. How did you handle it? What do you wish you had done differently? What would you keep the same? What advice would you give a teacher going through these types of things?


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. 🙂


Attitude Matters, Everyday…

Attitude – sometimes mine’s not as good as I’d like it to be. What’s my attitude towards my students, towards teaching math, towards my colleagues, my administration, etc? My attitude may change throughout my day, week, year, and over the life of my teaching career. Or, maybe I just lose touch with why I got into teaching and why I decided to teach math, thanks to all the hurdles that seem to toss themselves in front of you.

I actually really like my students and I think my job, teaching math, is really important. But, things get in the way sometimes and my attitude can suffer.

Luckily, attitude isn’t a fixed frame of mind. It’s a changing and evolving beast and while it can get bent out of shape by other things, such as fatigue, illness, tough teaching assignments, difficult colleagues, etc. it can also be straightened back out. We really have control over our attitude. I’ve decided to focus on it for the rest of the month, and see what happens. For a better breakdown of the impact of life’s events on our attitudes, check out this article by Micheal Graham Richard, Growth vs. Fixed Mindset. It asks which one are you, but my theory is that will very likely evolve and shift.

Staying in touch with your good attitude towards your students and teaching is probably the most important thing to do everyday. In fact, I’ll argue the most important thing to prepare each day, before lessons and tests, homework, etc., is our attitude about our students. It will define our approach to problems that arise during the day. It will allow for open mindedness and acceptance when our lesson doesn’t go as planned (especially when it is way below expectations).

My plan is to think of my students as the multi-faceted creatures that they are. They have interests and math may or may not be one of them. My goal is to try to inspire students to enjoy math, feel challenged, but not overwhelmed. Sometimes, this attitude really drives the activities and sometimes I lose touch and get caught up in the ‘listen, take notes, here’s your 20-25 problem homework assignment.’ this is usually when I get concerned about how much I’m supposed to teach them in a year and how little time I feel that I have to do it. I also realize that there’s nothing wrong with notes, lecture and lots of problems. However, that can be a drag for a lot of kids, so I don’t do it everyday, and when I do I try to give lots of class time to work the problems together. Even better, I love it when I can engage them in open questions. I think that’s one of the best times I have with them.

For the rest of March, I’m challenging myself to adjust and refocus my attitude each day, before the kids arrive, and to have some good open questions ready. My hope is, by giving attention to my attitude and this one teaching tool for the next few weeks, these two things will become second nature. I’m hoping my attitude will not only be positive, but will evolve and become better as I open up to outcomes with my students. And, as I open up my questioning. I’m hoping to open my mind enough that I transform my classroom and kids experience for the better.

I think our attitudes are really important and I know I don’t give mine enough attention. I want that to change. I think attitude may be like a muscle and for it to be strong, it needs attention and proper feeding. Otherwise, it feels like I’m sometimes being swayed by the event in front of me, or the person in front of me. I want my attitude to be really strong and drive my response to that event, or that person. Teaching math isn’t easy. So, I need to tend to my thoughts, my views, my attitude towards my students, towards teaching, towards my school, my colleagues, etc.