Laws of Sines Derivation and the Ambiguous Case

I really wanted to make these lessons more meaningful, so I started off with a 3 act math task from the website http://www.101qs.com. This one addressing finding how many pieces of sod cover the triangular garden shown below:

 This led to some good questions from the students that made them want to learn how to solve oblique, or non-right triangles.

First, students derived the Law of Sines

Then they practiced a bit. Then they derived the area formula for a triangle using the Law of Sines:

After practicing that a bit, they were able to solve the sod problem...though they got 18 pieces and the video got 20. Hmmmm.

Anyway, the next day we tackled the Ambiguous Case (SSA) for the Law of Sines. We talked about why there could be an ambiguous case for sine and not cosine, i.e., there are two angles less than 180, one that is acute and one that is obtuse for every sinx = a number between 0 and 1. For cosine, if cosx = negative number, the angle would be obtuse and if positive, the angle would be acute. 

I told them that there were three possibilities with SSA-two triangles, one, or none. They were reminded of the discriminant at that point! I had each case on a separate piece of paper with scissors, pink paper, rulers, and a glue stick. For each case, they had to draw line segment AB about halfway down the paper, knowing that they did not know it's length (this is also side "c".) They may need to draw it longer or erase from it. But they had to measure the given angle A and cut length "b" from the pink paper. Then they had to cut length "a" to try and have it meet side c (AB). Here's what they did for the first one.

They measured the sides and angles that were missing and then solved by Law of Sines and saw that the answers were very close...and that there was only one possibility for this triangle. When they solved for B, we talked about how B' = 180 - B was the other solution that gave the same sine value. However angle B' + angle A > 180, so there would only be one triangle.

We did this again for case 2, and they saw that no triangle exists, as shown below. 

We proved this mathematically, which was pretty cool. I should point out that my sides were a bit long, and next year I will make them smaller...however, students knew that the ratio of the sides just had to be in proportion, so they divided by two in some cases to get them to fit on their paper. 

We all had enough of the arts and crafts and for the last case, two solutions, we drew it by hand. It was very cool to see the students grasp the concept of side "a" swinging in to make an oblique angle, and that the two angles, I call them B and B', one acute and one obtuse, both added to 180 degrees. They also saw that angle B' + angle A was less than 180 degrees, so they had room for a third triangle, and therefore there were two possibilities algebraically. Or shall we say, trigonometrically.

They solved for the missing sides of both triangles, and practiced one of each type after that. I felt that they understood the ambiguous case much better.

Tomorrow, we will derive the Law of Cosines and after that, Heron's or Hero's formula. 

Here are my notes for Law of Sines and Area derivation and practice, Law of Sines Ambiguous Case.

Puzzle Day: Human Jumping Puzzle, Riddle of the Tiled Hearth, and 8 Queens

Today was a fun day in problem-solving. I have been wanting to do the frog jumping puzzle Shaun Carter blogged about since his first post last year. He also made this interactive puzzle shown below:

We did the human jumping puzzle by taking 7 chairs and putting them in a row and making two teams: 3 blue and 3 reds. With AP exams, we happened to have exactly 6 students in class, so it worked perfectly. I recommend that you have students wear a colorful tag because even though they knew who was on each team, I could not keep it straight. The rules are above, but in terms of humans,

• Blue team can only move to their left and red team can only move to their right.
• You can move forward one chair if it is empty and next to you. 
• You can move forward two chairs if you "jump" over someone
• The puzzle is solved when all of the blue team moves to where the red team was and vice versa (order does not matter.

The goal was for my students to do this in as few moves as possible. They did it about 3 times (they are a bright bunch) until they got it right: 15. I wish I took pictures!

We then took away chairs and did it again with 2 blue and 2 reds and also 1 blue and one red and found the pattern to the number of moves it takes, which you can also see on Shaun's post. My students recognized the pattern as perfect squares - 1, starting with (n+1), where n is the number of half the counters (so, only the blue counters, for example.) 

This reminded me of another puzzle, which I read about in this (parentheses), [brackets], and {braces} blog, called The Riddle of the Tiled Hearth. There is a great story that goes along, and we used unifix cubes and played with this for a while. In a nutshell, in a 4x4 grid, you have to place as many of four different tiles (or colors) in the grid without any two colors being in the same row, column, or diagonal...kind of like a type of Sudoku...using the least amount of blank spaces as possible. This took quite some time, and it was interesting how the kids really thought they could do it without having any blank tiles (you can't.) We talked about how tiles had to be "knight moves" away from each other, from chess, so that they were not in the same row, column, or diagonal.

From ppbmath.weebly.com

Then THIS reminded me of ANOTHER puzzle: The Puzzle of the Eight Queens. They loved this one, too...and I did not have an 8x8 sheet ready, but next year I will. The object is to place 8 queens on a chessboard so that none can attack one another...again, none can be in the same row, column, or diagonal. Students could adapt this from the last puzzle by using 8 cubes of one color. The website I linked is cool because students can check their work if you project it on the board and have them tell you their answers.

These were all really fun, and I joked with the kids that they could take the rest of the day off in terms of thinking in the rest of their classes...our heads hurt! Not all of them could solve every problem, but it certainly gave them background info to solve a similar problem in the future. 

   

How Having a Math Lab Can Benefit Your School

I have been asked this question a lot, so I thought I would blog about it. At my school, we are fortunate to have a Math Lab staffed by almost all of our math teachers every period of the day. We have seven periods in the day, so that means seven teachers staff it and four do not. Some years, we have two teachers staff it over lunch. It really depends on scheduling.

Here's how it works. A teacher sits in the middle of tables, shaped in a U, and students come in and leave as they please during their free period. They sign in on an iPad on a google doc that any teacher, grade chair, or administrator can access and sort. This is helpful, for example, if a student tells their parents they were in the Math Lab, and then we can check. It's also good when we are writing comments and we can quickly see if a student never goes, and then we can recommend that they do. Finally, it's good to see that the Math Lab is being utilized constantly and that the benefits outway the costs.

I did not have a picture of our Math Lab, but this is a general idea.
A teacher is in the middle and can move their chair around to assist students.

Students can just work on homework and ask questions as they work through it, or they can study and ask specific questions. If they were absent, they can come in and get a lesson taught to them that they missed. The teacher in the Math Lab needs to be well-versed in all subjects. Admittedly, it is difficult to help BC Calculus students if you have not taught that material recently. I bring my computer and often Google questions that I don't remember. Often, someone else has posed that question and there are several solutions on the internet you can work from, or you can use the solution manual, which we keep in the Lab.

We also use it for make-ups. If a student was absent, they often go to the Math Lab during their free period to make up an assessment. We have a locked filing cabinet with a folder for each teacher. In the folder, one side says "to take" and the other says "taken." Teachers write the students' names on the test, show whether they can use a calculator, and write the maximum amount of time students are allowed to use. We have a row of cubbies in the Math Lab for students who take those tests. We have silencing headphones we bought from Home Depot for students to use since it can get noisy in there. We also have a small dry erase board on top of the cubby, and we write the time the test ends so all we need to do is look up and check as we are assisting other students.

Benefits:

• Students can get help during their free period in addition to before and after school with their individual teacher.

• Students can get a second way to have the material explained to them. One note: It is important to make sure the teachers explain the problem the way the student's teacher does it so as not to confuse them. For example, when completing the square to find the vertex of a parabola, we would ask the student, does your teacher divide by a or factor it out? We may ask to look at a student's notes.

• Students can get caught up quickly when absent.

• Students have a quiet place to study and they really appreciate that.

• Teachers get exposed to all kinds of math in the Math Lab, therefore, they review all topics constantly. So though I have not taught Geometry in 25 years, I teach it every day in the Math Lab.

• Teachers get to know students and vice versa that they would not necessarily have had in the classroom. There is a great rapport with students in there, and they feel comfortable coming in to ask questions in a "safe" place.

Cons:

• It can be costly as I think it costs about the same as a teacher's salary to staff it. So at our school, teachers have five classes and two free periods. Math teachers in the Lab have four classes, a Math Lab, and two free periods.

• If a teacher who staffs the Math Lab is absent, there will often be a non-math sub in there who can't assist students that day. Sometimes we do have math subs, so this does not always happen. 

We truly believe that the benefits outway the cons, and I'm so happy our school can support this. We had it at my previous job in a public school 20 years ago, too. We did not always have it at our school. A few teachers, including myself who had worked in one, asked for it, and we did it on a trial basis a few periods a day. The response was overwhelming. Students and parents asked for it full-time. It took a few years to happen, but when it did, there was no looking back. It is often shown on the tours because it is a big selling point for our school--students get math help anytime they want. What parent wouldn't want that? We also have a Writing Lab set up in the same way. Questions? Let me know and I will be happy to answer :)

Can You Solve the Locker Riddle? A Ted-Ed Lesson and How to Write One

Two of my favorite problems have become Ted-Ed riddle videos. I wrote about the Ted-Ed Lesson "Einstein's Riddle" here. I worked with the creator, Dan Van der Vieren, the week before his lesson came out when he taught my class how to solve the Rubik's Cube. I was so impressed with his riddle that I asked him how he was able to create it. He told me about the nomination form, and I filled it out and wrote about the problem that I do every year in my classes, The Locker Problem.

The idea, of course, is not to write out all 1000 lockers, but to create a smaller, similar problem and look for a pattern. I use this to teach my students about tenacity, collaboration, and the value of using the solution to a small problem to solve a bigger one.

Within a few weeks, I heard back and was scheduled for a 30-minute interview with someone from Ted-Ed, and we talked about how to change the problem to have a story line, and having something in the lockers that remain open. I brought it up to my Problem-Solving class, and we ended up making a riddle that would involve having to solve a problem to earn an inheritance. Once that was done, I sat one night and wrote the riddle, very excited that the number of words in the sentence that I wanted in the lockers was ten, and that there were exactly ten lockers open. Often when I give this problem, I ask, "Which lockers are touched exactly twice?" So I threw that piece in their as well. As an author of a Ted-Ed riddle, you also need to come up with 5 multiple-choice questions, 3 open-ended questions, additional resources to explore, and a guided discussion question. 

The people at Ted-Ed are awesome! They are extremely professional and always got back to me within 24 hours, offering advice and helping me with any questions I had. And the animation is THE BEST!! Who cannot love that little girl??

Within a few months, it was up! The idea is that teachers can even customize a lesson of their own with the riddle. Here is the actual link to the entire Ted-Ed page that contains the riddle: http://ed.ted.com/lessons/can-you-solve-the-locker-riddle-lisa-winer.

My students LOVE these riddles, the first five of which can be found here. They beg for more. Do you have a good riddle? Consider nominating yourself and writing one for Ted-Ed. 

Hello my name is x+3--activities for teaching polynomial and rational functions, by A. Mosaad

This is a quick post to tell you all about an easy and fun activity to use for polynomial and rational functions. I learned of it by attending a workshop of the same name as the title at the Rutgers "Good Ideas in Teaching Pre-Calculus and..." workshop that I also blogged about here. Here is the abstract:

It began with the presenter, Amro Mosaad, giving everyone a nametag with a factor on it, like “x-1” and x+3” and have them pair up and give them one of the following worksheets. They have many tasks, such as find the zero and graph their product and then graph their quotient...this is so good after you’ve taught polynomial graphs and rational functions. They can switch papers with another group and have them check work on Desmos.

This can also be done with zeros and complex zeros. Amro said, "Find your twin," meaning, find your complex conjugate.

Amro also suggested the "mystery polynomial" activity: Get into groups of 2 or 3 with no one you have been in a group with yet. Give your group the polynomial you just came up with and your “name” and your new partner(s) that you never worked with before will need to figure out who your other partners were. Awesome!

Here are the links. Thanks to Amro for sharing them with me!

Rational Functions activity
Polynomial Functions activity
Complex Zero activity

I thought this was a great formative assessment activity.

One attendee I was working with suggested that students check work on www.mathway.com wow!! I have never seen this website before, but I will definitely be using it in the future!

Three Act Math Tasks

Who hasn't seen Dan Meyer's video on how long it will take to fill this tank?

I have, but I did not know anything about Three Act Math Tasks until I went to a workshop this past Friday called "Good Ideas in Teaching Pre-Calculus and..." which has been put on by my former math professor at Rutgers University for the last 30 years.

Growing up in NJ and now living in South Florida for the last 21 years, I have lost a LOT of my NJ accent. I no know longer say "warter," as in "how much warter fills the tank above?" but I still refuse to call the fruit of the same color "oar-ange" (I prefer "are-ange").

The woman who gave the "tawk" on Three Act Math Tasks had such a thick accent, I giggled at the sound of my former self saying "Three Ayct Mayth Taysk," which she said over and over again. But I digress.

She posed this question, which she said was from the Discovering Geometry book that I adore:

and said, instead of posing this question, she directed us to Dan's 100 questions website http://www.101qs.com/2352-meatballs, where we watched this video (click on the former link to get to it.)

All I have to say is: game changer. This is very cool, and I will definitely be using these tasks in the future. Rather than give students a problem to try as part of their homework, show them a video, ask them to ask the question, figure out what they need for the solution, and then show the solution.

Someone in the class also mentioned that some tasks can also be found at http://robertkaplinsky.com/. I looked at his site and remembered happening upon it before. Today, his blog happened to come up as a suggested one in an email by Bloglovin'. I went to follow him on twitter...I already had! 

To me, these tasks previously would have been a 5 act math task, because I would have had to think of the task, film myself doing it, and then apply the 3 act math task. But it's already done for you...and here is the spreadsheet link of all of the ones out there, and how they apply to the standards. It's amazing how much is out there! Look how many people are on it right now, randomly, at 1:54pm:

I hope to use these upon return from spring break. The presenter said she does them about twice a chapter. I'm sure Dan Meyer would say use them more. 

Dan Meyer and Robert Kaplinsky are gurus here that I plan to follow...is it weird that both pictures above show Italian food? Now I'm hungry.

Since this blog is also about eating and playing, I wonder if Dan will share his meatball recipe in the comments??

If you haven't seen Dan's Ted Talk, it's called Math Needs a Makeover and I highly recommend it. 

And here is a clip that is also shown over at robertkaplinsky.com - How many hot dogs and buns should he buy?

Again more food!

Finally, other sites with math tasks are mathalicious, which I blogged about here, and yummymath...why is everything sounding delicious? Get over to one or all of these four sites to get your kids thinking about some real-life problem solving with purpose. Your kids will buy in faster than you can say "You're from Jersey? What Exit?"

Fangirling over Jo Boaler's Growth Mindset...and Why You Should Be, Too

I love the fresh faces on the first day of school. No opinions formed yet, no knowledge of who will be my top student, who will struggle. I set the bar high, and I hope all will reach said bar. Of course, along the way, there can be some students who will not get there. But it won't be because I didn't try to get them there. Perhaps they gave up along the way due to circumstances out of their control. Maybe they still don't get that going to extra help really DOES help. Maybe they don't realize the amount of time the successful students are putting into their homework outside of class. Maybe their parents told them that they were not good at math as children, and, therefore, they are not good at math...here's where Jo Boaler, from Stanford University, comes in. She claims that all students can achieve a high level of mathematical success and that teaching with this in mind will dramatically improve student achievement. Her website, https://www.youcubed.org/ is well worth looking at, and her book Mathematical Mindsets is on my desk for summer reading. In it, she professes to "banish math anxiety and give all students a clear roadmap for success." Who wouldn't want that??

This video is well worth the 9 minutes, and next year I plan on showing it to my classes. But today, I got an email from Jo Boaler (as I am a youcubian--yes, that's what her followers are called, and I'm a big "fangirl" of Jo Boaler) in which she shared a short handout for parents. It's excellent, and I hope to give it to my students' parents next year.

Here is the actual link to the handout.

Here is another link to a poster on Positive Classroom Norms where she focuses on Growth Mindset. Below is the first page of it, but there is much more in the link. I will use these in my classroom next year. She has done a lot of research on it so we don't have to!

And finally, here is what she says about Growth Mindset: 

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As an aside, I am going to also add a little of the "play" that I lose from time to time in the name of my blog...something for health, perhaps. After writing my recent post on learning how to say no, I've been finding some things that have been coming my way, and this is one of them that I received in an email from Ted.com. I've only had time to watch the first one, but it was great, and it highlighted the importance of self care. Here is the playlist that came to my inbox: http://www.ted.com/playlists/299/the_importance_of_self_care?utm_campaign=social&utm_medium=referral&utm_source=facebook.com&utm_content=playlist&utm_term=social-science. 

And here is a link to the first video. Enjoy!