# 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:

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.

# The origin of a project…

It wasn’t until they actually took a bite of the hot dog that they had an authentic experience. That bite transformed what was kind of a silly project meant to be fun into a life skill for cooking food when they are hungry and have no other resources. Like camping. Or, maybe lunch.

Two of my enthusiastic students were so happy with their results, they asked to have a couple more hot dogs, skewers and buns so they could cook more during lunch.

Yes, we made solar hot dog cookers.

This year, I said, let’s make them. I didn’t really have a rubric and didn’t want to give them instructions. After all, this stuff is all over the internet (just Google it). I also didn’t want to make this so complicated that I would feel overwhelmed. And, I didn’t want to take away from the much more involved project one of my collegagues does with his Engineering student (we have a couple students overlapping our classes).

So, to me this is a great evolution of a project. Think about it, try it, formalize some things for next year. The kids said I should always do this, so I have to take that feedback at point value.

This year, I made it optional. Next year, everyone must do it. This year, I planned it for this week, which is right before spring break. I thought this was a fun thing to do during this week where lots of students are absent due to trips or have big tests or papers due in other classes. Next year, I’ll do the same. This was a good week for this, luckily.

This year, vague rubric written on board:

For a C, it must be parabolic and made from inexpensive materials, with the hot dog at about the right place.

For a B, document your process: how did you decide your shape, take some pictures while building, record problems and your solutions to the problems. Present in a power point, a paper, a movie or a poster board (or whatever other great idea you have).

For an A, all of the above and a calculation showing how you determined where your hot dog should go. And, maybe present it to the class. Actually, I have one student who wants to present, so he is.

Next year, I’ll type that up. I actually think it’s pretty good and the kids didn’t balk, complain or ask for clarification. Well, maybe some clarification. But, next year, I’ll have photos and example to show!! Yay!

# 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. 🙂

# I didn’t teach today…

… I facilitated. It’s a review day. The kids will take a test next time I see them. My Algebra 2 kids are ready for the test. Even though I didn’t teach, I was able to check in with everyone as they worked on problems, got caught up and helped each other out. They asked good questions. I feel like they are ready.

I told them so. They seemed to like that feedback.

So, I ‘taught’ but it was on individual levels, not the group as a whole. I think I had a great day and was relaxed in part because I had spent time this morning writing about and thinking about my attitude. I brought my respect for my students to the forefront of my mind.

It was also because they are prepared, and know what they will be tested on. They felt confident, too.

Here’s a copy of last week’s quiz, a model for this week’s test…

# 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.

# Teaching Conics in Algebra 2

I really like opening the day with an open question. They’ve been kind of easy to think of so far. But, what about conics? What’s a good group of open questions that can be used with conic section lessons? Before I could think of that, I really had to look at the new standards for Conics. In doing that, I realized I really hadn’t examined exactly what the kids are supposed to learn. So, I had to research and think about that for a while first.

I did a little research about conic section topics and standards that need to be covered in Algebra 2. I checked this publication on the California Dept. of Education website: California Common Core State Standards for Mathematics. You’ve probably all seen it, if you’ve been working in California.

Here’s the conics standard for California – yes, there’s only one, but it’s loaded (p. 83):

3.1 Given a quadratic equation of the form  ax² + bx + cy² + dy + e = 0, use the method for completing the square to put the equation into standard form; identify whether the graph of the equation is a circle, ellipse, parabola, or hyperbola and graph the equation. [In Algebra II, this standard addresses only circles and parabolas.] CA

Um… okay. Let me think about that. first of all, is this the same or different for what we’ve been doing at my school for teaching conics. Do we need to address the directrix and focus or foci? Because, I talk about those whenever I talk about conics. Even when introducing them in Geometry.

Here are the Geometry standards (p. 74):

Translate between the geometric description and the equation for a conic section.

1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

2. Derive the equation of a parabola given a focus and directrix.

In the past, I taught about the foci of an ellipse and parabola and hyperbola. Last year, we didn’t test on graphing hyperbolas. According to the Algebra 2 standard above, it seems like it’s all four, except for that part in the brackets. Is that only for other States? I needed to find out. After all, I’m serving on a County wide committee to discuss teaching this class and we should really know what those apparently contradictory statements mean. The bold is supposed to be California.

So…. here’s what I found out… Go to this website IXL, scroll down to whatever standard in which you are interested, hold your cursor over the standard and a sample questions will pop up. Wow. Great stuff. No foci/directrix stuff until Pre-Calculus. Okay – I guess they’ll handle that in Pre-Calc. Looks like just graphing parabolas, those that open horizontally or vertically, and graphing circles. For circles, be able to find the center. I think they still need to be able to tell whether the conic is a circle, ellipse, parabola or hyperbola from the equation, though.  Please make a comment below if you understand this all to be different than what I’m writing here. This seems much less than what I’ve taught in the past. So, maybe that’s a good thing. 🙂

Next, what will the lessons be? Then, I checked the NCTM website, Teachers Pay Teachers the NRICH websites for ideas related to those standards. And, of course, Desmos. Well, on the day I had to start the topic, I didn’t have a good ‘open question’ opener. I just asked kids about the equation of a parabola, in vertex form. I asked about the equation of a circle. I put it on the board. I asked if they’d seen that before. I asked them to notice that there’s a x-squared and a y-squared term. I asked if that meant it’s a function. So, it was a weak start compared to what I would have liked. But, it was the day before break and my goal was to introduce conic sections. I had them watch this Conics video from YouTube This was pretty much a vocabulary lesson with a graphic that was pretty good for getting them to understand the basic concept of what conic sections are.

Then, the fun began…  as they started to use the Desmos activity, Polygraphs: Conics, found here. Now, I’m figuring out my unit plan. I have the week off. I plan to find open questions, interesting activities and relevant homework for them. Something that spirals old stuff, too. I plan to write more about it, too….  Ideas?

# Kids really bombed the quiz…

Dear Mathies,

So, teaching logs, and missing two days last week, I gave a quiz to see where the kids are. I’m feeling pressured, because of the February break, to get a test in next week. It’s also the end of the grading period that Friday before the break. So, I’m feeling like a quiz is needed to let me and the kids know where we are this week.

Well, it seems like we’re overall pretty far from where we’d like to be. 🙂

So, do I labor through the grading, or do I hand them back without grading them, and allow them to work through them again? Do I worry about cheating, grade inflation, lack of “data” for issuing the grade for the reporting period? Yeah, I do worry about all that. I also worry about their stress level, because I want them to like math and math class. And, logs are not one of the easier Algebra 2 topics. They will learn more if they enjoy themselves and aren’t experiencing ‘stress-brain.’ But, I could feel the tension in the room as they we reviewed for the quiz and as they took the quiz.

So, I want to get creative. I want them to get the feedback they need, without the low score. So, I’m thinking that I will give them the quiz back today and let them correct them with a pen. Then, if they want to turn it in or retake it, they can decide. So, there’s still some accountability and they have a bit less anxiety and a bit more control.

My questions for you are:

1. What do you think of this idea?
2. Do you have a better idea?
3. How have you handled these conflicting pressures around assessment?
4. What else should I be thinking about? What am I missing here?

If you can give me feedback on any of these things, I’d appreciate it!

Here’s what happened: I did the above, and many want to retake, but many kept their score. They got some feedback, for sure. However, I didn’t let them keep the quizzes. So, some asked, “How will I study for the re-take?” Great question…

I’m going to send a couple of tutorial links (via Remind and put links on my website) and have them review their assignments to ‘study’. I hope this will help them learn how to study, too. Maybe we’ll focus on that some more, too. Regularly.

Thanks,

Laurie

# Asking Open Questions, why it matters for math

I’ve been asking open questions and I think you should too. Maybe you already are. If so, please reply to this blog with some of yours so we can share! 🙂

The other day, I started my Algebra 2 class by asking students to think of 10 numbers. I know I’m not a mind reader, but if what they say is true and 90% of communication is non-verbal, then I was 90% reading their minds…

I knew they had easily thought of their numbers as I continued to give instructions, “draw a number line and try to think of numbers that would represent the number line… .” Their faces fell as they had to rethink this. They were once again happy and satisfied when I demonstrated and started to draw my number line and scaled it -5 to 5 and started to plot points above the number line at -3, 1, etc. I asked them to think of maybe some negative values, maybe a decimal or a fraction, maybe a mixed number… I asked them to try to think of numbers that represent numbers they’ve learned about over the years.

They all started to write down new numbers. So, at this point, one of the great things about open questions was that everyone was involved, not just the kids who could ‘figure it out’. Everyone could think of numbers. Everyone understood a number line and negatives and fractions and decimals – this was an algebra 2 class, after all. This was feeling easy for them. They great thing is that they could all have different numbers and all be ‘right.’ Satisfying.

So, even better, they were hooked. I asked if there were any other interesting values they could think of. One student said, “Pi!” That was perfect. I marked it on my number line on the board. Many kids marked theirs. I said what about 2π? or -π? or 1.5π? Any others? Wait…. nothing, that’s okay, I’d leave radicals for another conversation.

Then I asked, “What is my number line missing? What am I not including?” They had to think. I asked about my domain. They said it was very small. I agreed that I had only included small numbers. Then I asked, who picked the largest number? Hands went up. The largest ended up being only 100, the smallest -100. I asked if we should have larger numbers or infinity symbols somewhere.

Anyway that wasn’t the point, but it was fun. So, then I marked a new number on my number line, at about 2.7. I asked if anyone new what the value was. They shouted out guesses, finally I heard 2.7!. I said, yes, that’s very close! Then I wrote the expression lim as x→∞ of (1+1/x)^x on the board, over the point. From there, we talked about limits, what happens as x gets larger, we made a table of values and tried larger and larger numbers for x, only to see that y was changing less and less and moving towards, 2.718···. Then I said this is a special value, just like π, and it’s called the natural number, e

I’ve introduced and derived e with my students many times over the years. The difference this week, was that they were all hooked, all involved, they all had money on the table (intellectually speaking). Everyone was writing and thinking. You could see it from the expressions on their faces.

In Algebra 2, asking open questions feels so important to me. It’s a tough class to teach. There’s a lot of material and a short amount of time. There’s a wide range of students in the room – varied grades, varied backgrounds and varied attitudes towards learning math. Open questions allow for many entry points which generate stronger feelings of success and inclusion.

The new standards seem to want us to go deeper than we have in the past and students need to think more critically. Engagement needs to be high. Asking open questions really helps them engage and think and stay tuned to see what happens next. Even if what happens next is a bit of traditional lecture and on to some problem solving. They enter it with more curiosity and more confidence.

I’ve been doing more open questions this month and I’m seeing a change in the culture in the room. I have to thank the great educators and researchers who introduced these ideas to me at CMC-North this past December. Steve Leinwald in particular, Dan Meyers and Michael Fenton and the dream team at Desmos. My teaching has transformed (see It All starts to Gel…  http://wp.me/p73p86-2)

Resources for me have been:

• NRich website
• The book: More Good Questions, Great Ways to Differentiate Secondary Mathematics Instruction, by Marian Small and Amy Lin  link to amazon
• The book: Styles and Strategies for Teaching High School Mathematics, by Thomas, et al., 2010 link to amazon