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…

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

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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.

 

 

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

 

What’s great about marbleslides

Marbleslides is a great teaching and learning game created at desmos. It’s been a great addition to my units in Algebra 2. So what’s great about it? Well, it’s tough to know…

…where to start the list…. hmmm…

How about instant feedback for students? They know immediately whether or not they ‘got it right.’ If not, they try again. Don’t you wish they would do that on their homework? Check the answer, if it’s wrong, try again. This is so interactive and quick, they are more likely to stick with it. ‘Stick-with-it-ness’ is a new term I’ve invented. You may have invented it, too, or some version of it. 🙂

The marbleslides activities allow kids to stick with it, even if they are not as far along as other people in the room. They get to work ‘where they are’ without getting behind the rest of the class. It’s a way to differentiate seamlessly, without it being obvious, because of the high engagement level students experience. You can very easily spend more time with the kids who need you. You can make suggestions, but never give away the answer. You can remind them to read the instructions if they missed them (which happens a lot). They can then ‘reset’ the problem and give it another go with much increased success. They are feeling challenged whether they are on slide 7 or slide 17.

For the teacher: Very little planning time is involved and you get to use that time to assess and reflect. It provides instant information to be used for formative assessment. And, kids can complete the activity later if desired by you or by them.

Pretty cool.

Here’s the play-by-play of how I used the Desmos marbleslides activities with my Algebra 2 students:

First, I had students review graphs of rational functions using the marbleslides activity here. We had already learned this and I thought it would be good to start them with something familiar before moving on to new functions. I had hoped this would also strengthen their understanding of the transformations of that parent function. It did. Yay!

So, they were able to learn to navigate desmos and how marbleslides works. Then, a week later, after introducing exponential functions, I had them do the marbleslides activity for that function. I heard comments like, “Whoa, I understand this now.” If you are interested, go to teacher.desmos.com and create an account. Use their pre-made activities and/or the learn.desmos.com tutorials. You will be up and running in no time. You can email their team or me if you want more info about what I did.

Will this always work to get every student super in love with the topic/lesson/learning goal? I’d love to say YES!, but it may be more realistic to say that probably not everyone will fall madly in love. However, this will certainly raise engagement and increase understanding. And, time flies when they’re doing the activity. This definitely deserves space as another tool in the toolkit to increase overall engagement and mix up your activities. What’s needed? Well… devices – iPads or computers. It won’t work on the phones… yet. So, you may need to schedule some lab time. Totally worth it. 

Should you do it everyday? No. Keep mixing it up so you capture or engage your audience. Some kids will be more engaged one way, some kids another. Some like partners, some like to work alone, some like lecture, some like ‘discovery.’ But, what’s great is, by ‘capturing’ them one day, they will most likely increase their general interest level and they are more likely to be willing to go with it another day with another method. Read Styles and Strategies for Teaching High School Mathematics by Thomas, Brunsting and Warrick for great ideas on differentiating your practices.

More on that book next time.