Ugh, What Now?

     I presented this problem to my 9th grade honors geometry students, and we got stuck. While we were working through it there was a retired math teacher in the room as well who was also stumped. After class, I showed it to a few of my colleagues; also stumped. We figured out a few possible paths that might help, but we are trying to solve it strictly within the constraints I gave my students, using knowledge that they would be able to grasp.

     We've been working with all of the different types of angles, quadrilaterals, triangles, and all of their properties recently. We've stumbled upon regular figures and what we can do with them. The end goal is to be able to do this, but we've been going through similar problems as a class to lead up to that point.

Here's what I gave them today (please ignore my phone shadow):

     I gave my students a regular hexagon, drew in some lines and labeled some angles to be calculated. I love these problems because it takes all of the knowledge that they've gained over the past few weeks and puts it into one beautiful package. They have to know the types of angles, symmetry, properties of various figures, etc. to solve for all of these angles. It's awesome to watch their brains working and the excitement on their faces when they figure these out. Even students who ask, "Why do I need to know how to do this?" are still engaged and have a desire to figure it all out. But anyway...

     In this problem, the black numbers are the names of the angles and the blue numbers are their measures. If you look carefully, we've calculated all of the angles that I drew in, whether or not they're labeled. Except for angles 14, 15, 24, and 25 at the top.That one line that passes by all of them (connecting the upper-left vertex to the midpoint of the other side on the right) is really messing things up.

     To start the problem I gave the students no angle measures; they had to calculate all this by themselves by figuring out a possible starting point. It is a regular hexagon, and it is drawn to scale, but that's all the info they had. I did not give them any side lengths either since we were focused on just finding the angles. 

     Angles 14, 15, 24, and 25 really threw us off. We've tried extending some lines outside the hexagon, we've tried drawing extra lines in, and I even had a student line up a congruent hexagon along the side formed by angles 6 and 16 to see if that would help her. My next step is to give the sides a length and see if trig. will lead me to freedom, but I was hoping to get the answer using only angle measures. 

     Can you figure out the measures of angles 14, 15, 24, and 25 using only knowledge of angles, polygons, and symmetry? Or is more information required? If so, what else do you need? Any guidance would be greatly appreciated.

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**Note: I did figure this out, but it required me to work with the side lengths, using law of sines and cosines. Theoretically, my students could do that if they recognize the fact that the length of the sides doesn't matter. I've taught them law of sines/cosines, and I have some really bright students that could possibly make that connection. However, I am still curious to see if there is a way to do it without referencing the sides and using only angle measures. I'm leaning towards 'no, it can't be done' but still not fully convinced.

3 Acts = Desire To Learn

Last week we started working with 3D figures, surface area, volume, etc. We went through developing all of the necessary formulas, practiced using them, and the whole class was a set of all stars. So the natural thing to do was move apply these beautiful concepts. Initially I thought of starting this unit with application and developing everything from examples. Instead, since we just came off the area/perimeter unit, we used those skills to develop everything necessary for this. In my opinion, these two units are the easiest to find applications because everyone has to paint a room or put something together at some point in their lives. The natural progression was to throw some 3 Act problems at my students and watch what happened.

They started with Dan Meyer's world's largest coffee mug. The questions started flying, I provided information, and they were off to the races.

I like the image below. When I noticed the approach they were taking, it allowed me to generate some decent discussion among the students.

They took their mug and split it into 2D boxes, or so it seemed. In speaking with the group, they were thinking in terms of 3D cubes but they had not taken into account the fact that cubes would not completely fill in the mug. They actually were having trouble finding a way to make up for the curved walls of the mug. For some reason, thinking in terms of a cylinder did not occur to them. In our dialogue, they did have that 'Ah ha' moment and realized an easier take on the problem. I could tell that they were thinking in terms of volume, and they were keeping in the 3D realm, but breaking a round surface into cubes creates a few problems. The good news was that they recognized something was wrong and were problem solving in how to make up for the missed area, they were just having issues coming up with a clear solution. This kind of outside-the-box thinking makes me happy because even though there was a much easier solution, they didn't give up. They got stuck, but they kept the discussion going and tried new ideas. I believe if I wouldn't have come over, they would've continued working and come up with something that brought them closer to the answer, but not quite a fully correct one.
From this point we moved onto Dan's water tank. After showing Act 1, I actually got the question, "Do we have to watch this thing fill up? Its going to take forever!" I had never done this problem with my students, but thinking back to Dan TED talk, I got the exact reaction he described. Before Act 1 was even over, they were complaining about how long it would take and some actually starting focusing their attention elsewhere. When the video stopped, everyone erupted into, "Hey wait, isn't it going to fill up?" "Uh, Mr. Brandt, the video broke." "I wanted to see how much water their was!" The questions flowed naturally and they got to work. Below is a mashed up, kind of, gross, panoramic-type view of my board when all was over.

I had the students present their solutions because I noticed many different approaches. Throughout their work, I got the question of "How do you calculate the area of an octagon?" numerous times, to which I responded, "I don't know, but I can tell you its a gross formula." Some students wanted to look it up, which I allowed (most were not successful in finding it and those that did couldn't use it correctly). Others broke it into familiar shapes and went from there. Some went for 16 right triangles, some 8 isosceles triangles (both recognized the need for trig without prompting in these processes, which was sweet)., and some broke it into a rectangle and two isosceles trapezoids (which for some reason I never thought of). There was actually only one group that did this last thought process and it came from a group that I did not expect, which added to the awesomeness. Each group explained their thoughts and why they approached it how they did and everyone learned something. I could see on the students' faces surprised looks of enlightenment as other groups were presenting; it was a beautiful thing.
As I explained to my students after this problem, this is what pushed me towards math. I love the fact that every problem can be solved in numerous ways, some more creative than others, but they all take different aspects of math and combine them to form a solution. Its wonderful how all of these numbers and concepts fit together into a nice, complex package that is eerily consistent. This is exactly what I want to push onto my students. I want them to see where it all comes together and to see that any problem can be solved, even if you don't have the knowledge that you think is required. Through these 3 Act problems and allowing the students the run the curriculum through questioning, I believe I'm on my way. After my speech to them, I didn't have any students saying "Yeah, but you're a nerd. You're a math teacher, you have to care about this stuff." Instead, I saw heads nodding in understanding, as if to say, "Yeah, this is kinda cool."
I loved the way this day went. After these two problems, the students saw me project Dan Meyer's google doc list of 3 Acts, and they started requesting them. They were begging me for more problems! As a strict teacher that sticks to the curriculum, I said "Nope." NOT! We did some more! I teach geometry, but we were doing problems that weren't geometry related. Why? Because the desire to learn was present. No one in the room cared that we were moving on to topics outside of our current area of study because they wanted to be perplexed and solve some problems. If this isn't the goal of the teacher, I don't know what is.
All in all, it was a pretty sweet day.

Mistake?

I was grading some student work today and came upon this mistake.

In the class, we are currently studying problem solving strategies. This particular problem came from the 'Look For A Pattern' lesson. The problem asks the student to find the next four rows in Pascal's Triangle. Now, the majority of my students have not seen or worked with Pascal's Triangle before, so to them this is a seemingly random set of rows of numbers. I did not give them the background knowledge of the Triangle beforehand either because I wanted to see what they came up with as their answer. Usually students get it correct right away. This, however, was new to me. Because I only gave the first five rows, if you look at them as whole numbers rather than individual digits, they are all powers of 11. Pretty neat. This student went with that and continued. Unfortunately, Pascal's Triangle differs from this point on. She did find a pattern, thinking outside the box. Now, I did mention specifically in the problem that it is Pascal's Triangle, but she found a pattern.

Does she have an understanding of finding patterns? I believe so. Do I mark her wrong because her answer is different than what I've got on my answer key? If I'm grading on finding patterns (which I am), then no. If I'm grading on their understanding of Pascal's Triangle (which they may have never seen before), then yes. But then again, why would I grade on something they've never been exposed to?

I thought this solution was interesting; thought I would share.

Snow, man.

Checked out my 101qs submissions today to see the low perplexity scores. After checking up on Dan Meyer's blog I realized that my 'Snow, Man' pic had an honorable mention in his Top 5 for this week. I won't lie, I got excited. However, after reading his response to it, I realized that I had missed the mark slightly. My picture gives way to some basic questions, but could be better if planned out.

I would imagine if I showed this picture to my students I'd get questions like, "How much snow is that? What is the ratio between the three balls? How long until it melts?" and so on. These aren't bad, but are they really any better than a textbook problem? After all, the whole point of Act 1 is to get the students engaged and really interacting with the math behind the situation. I need something that will stimulate independent thought and make the students care; something that will allow them to envision themselves in it. Everyone builds snowmen, but who really cares about how much snow they've used to build it?

After reading the comments about how to redesign the problem, I've got some ideas to reproduce this in a few months. I liked the idea of comparing it to another snowman in a neighboring yard - introduce the competition element (after all, everything's better when there is a winner and loser). I could start by showing an aerial view of my yard along with a time-lapse video of me building the snowman in a methodical way, clearly using a certain amount of my yard. I could then pan out showing my yard again with the amount of snow that's been used already and the amount left. 

This could still produce the same questions as before, but now it could be expanded to "Who's snowman is bigger? What is the biggest possible snowman that could be built? How long would it take to build such a snowman? Could you combine the two to make a super snowman?' While these questions may not seen anymore advanced or intense, the redesign could get my students involved more. It wouldn't have as much of a 'ugghh, not this again' context, but would be more inviting. It would be more practical, more relevant, more realistic to their lives. Getting them engaged and thinking is the key, and this might just do it. I'll let you know in 9 months.

This has made me realize that I still have some work to do in creating these First Acts. While it can be done with almost any picture or video that inspires thought, they key is using the best possible option and format. Taking a basic picture like that above works, but does it work as well as I think? I need to think about these from a students' perspective and a teacher's perspective to get the best possible Act. Practice, practice, practice. Thank you 101qs and Mr. Meyer!

Order Matters at 5-Below

I was walking through 5-Below this afternoon looking for some fun stuff for my daughter and saw this hanging on the wall: a customizable iPod case with 'over 17,000 possible combinations!' Whenever I see advertisements like this I always wonder if they have a math guy that figured that out for them or if they just estimate it. I also wonder how many people will change the way their iPod looks 17,000 times.
If I showed this to my students while teaching permutations and combinations, which would they argue that it is? Is this an engaging enough picture to get them to think about it, or would they quickly move on?

Oh, the possibilities...

Any (mathematical) Questions?

Because It's The Cup

In the spirit of the playoffs beginning today...

What's the first question that comes to mind?

How Awesome.

The above image was a problem that I presented to my class a few weeks ago. We were concluding our unit on area and perimeter, so I decided to kick it up a notch and ask them to find the area of this figure. We had practiced irregular figures such as this one already (actually about 90% of what we did was irregular figures due to the fact that they are literally everywhere in the world), but I don't think anything this abstract or complex. I just made it up on the spot, purposely giving them little information. Below is what occurred when a student volunteered to solve it on their own in front of the class.

Naturally, I did not turn this brave individual down. I encouraged the class to copy down the problem and try it on their own while this brave young soul gave it a shot. The picture above does not due the entire situation justice. As a geometry teacher, I try to teach my students the curriculum (obviously) but more importantly I teach them how to think, a skill that many lack. The conversations that went on throughout this problem (I'd say it took them a good 20-25 minutes until they collaborated, asked questions, fixed mistakes, etc.) were absolutely amazing. At one point, I looked at the clock and realized how long they'd been working; I considered stopping them but I let them go. Why should I stop them from working and thinking original thoughts? I come back to a previous point I made: as a teacher its my job to make sure that they are thinking and problem solving; nothing should stand in the way of that, its in their best interest (but more on that later). Just look at the difference in pictures; I love when students write all over diagrams. It sheds light on their thought process and how they internalize information. As I said, I intentionally didn't draw this to scale and left certain parts unlabeled. I actually tried to give them as little information as possible so they would be forced to think and ask questions. Ultimately, I could care less about their answer, I want to know what's going on in their minds. They had to ask if the 'triangle' on top was isosceles or if the bottom was a rectangle or if both sectors were ninety degrees or if those two vertices were 'midpointish.' And most of the time, I replied with vague answers and forced them to come up with ways to verify their own questions (in the beginning of the year, they hated this, but now they're used to it and rise to the challenge). The majority of the students came to an answer, and most were correct. Those that weren't understood where they went wrong when we discussed it as a class. Its amazing what happens when you just let your students experiment even if it means taking extra time. 

I try to do stuff like this all the time. Some of my colleagues have argued with me because its not real-world, or because it takes 25 minutes for one problem, or because its not entirely written 100% into the curriculum and there is no time. All of these are bogus (while relevancy is an important aspect of teaching, especially math, it is not the only aspect that should be focused on). In the end, if its good for the students, I'm going to do it. I didn't have one student complain about this problem and I had 100% participation without fighting with them. In talking with them, I could tell that many of them learned from it. Students love a challenge when they know they have a chance at success, and that's what I gave them.

I could go on and on about this experience, but ultimately it comes back to a question I pose to my department members on a regular basis: If it's good for the students, why would I not do it?

Curiosity?

Now I'm curious. I wonder what this would like. Would you be able to predict the number of letters in any number (even though it would be easy to count)? What kind of relationship exists? I smell a project....

12 Inches of Math (informer!)

We built our first snowman with our daughter this weekend, and it was a tremendous amount of fun. She didn't do much work, but she gets excited everytime she sees it.

I've been trying to harness my inner-nerd this year; trying to find math in everything I see and do. I figure if I want my students to see math in the world, I need to be able to see it too. I imagine this is how Mystery Guitar Man approaches his videos: searching for music in his everyday surroundings and discovering how he can use those sounds to create something new and original that is both visually and audibly pleasing in order to send the message that music is everywhere (hence my previous post, worlds collide). I feel that I'm getting better at recognizing various situations and their mathematical values, but it's taking some time to develop this mindset.

After building our snowman, I realized that there really wasn't that much snow on the ground to begin with, which you can tell from the dirt and mud in the picture, it was just the perfect wet snow for building. I wondered, how much snow did we use in building that snow man? If we would've used all of the snow in our backyard, how big would our snowman be and would that beat the world record? How much snow actually fell (not just how many inches deep)?

I realize that if I'm following the mathematical storytelling model, you should come up with these questions, but they were just some thoughts. Essential info will be posted later (before the snow melts).

We Built This City



How tall is he? And how much does he weigh? How much does he have to eat in order to live a healthy lifestyle? How does he rock so hard?

Betelgeuse, Betelgeuse, Betelgeuse!!

I was talking with one of my colleagues about how we can improve problem solving skills within our students to encourage higher-order thinking skills. In the past, I've shared with him Dan Meyer's 3 Acts and he was as amazed as I was. Together, we've been working on creating problems for algebra1, 2, and geometry that are set up for this process, and he shared with me a video similar to this one that he showed his students. It's a simple video; there's nothing fancy about the way its made and there is no narration, but the possibilities are endless in the way its interpreted. He used it to illustrate scientific notation. My first thought was to ask the question, 'How many Earth's fit inside Betelguese? How many suns fit inside VY Canis Majoris?' You could relate it to similarity, 3-dimensional space, fractions, or a hundred other topics. Regardless of how I end up using it, I plan on removing the diameters from the video so that my students can do some research to find the information that they think they need to answer their questions.
This video further illustrates to me how many questions can be generated from any picture or video. The phrase 'a picture is worth a thousand words' has become more real. I now try to look at my everyday surroundings from a mathematical perspective. I always tell my students that if math didn't exist, nothing else would either, but they always blow me off. Now I'm beginning to gather solid, concrete examples that they can see. I think the way I'm going to approach it from now on is show my students these examples so we can explore them as a class, and then use them to illustrate that math truly is everywhere. If I can get them to believe this, them I'm one step closer to getting them to appreciate everything that happens around them at any given moment.

So, what are your thoughts? What questions come to mind when you see this video?

P(calculating the odds)

So, I was listening to the radio tonight and heard this story and I thought to myself, "Wow, what are the odds?"

But seriously... what are the odds?

Natural Geometry

Mathematically beautiful. I woke up one rainy morning a few months ago and saw this on a bush next to my back porch. It caught my eye for obvious reasons. So many questions came to mind. I've seen spider webs before, and while they're fascinating to look at and to think about, this one was different. I've never seen one in a nearly-perfect parabola/cone shape before. Its times like this that I wish I would've stopped to take some measurements or had all of my students standing next to me. This could've been (well, I guess it still could be) a great exploration of paraboloas and calculus. I'm just glad I stopped to take a picture. When I got home it was no more.
And how the heck does it appear to be levitating?