Math = Love: February 2019

Monday, February 18, 2019

Trig Identity Group Challenge - Reciting the Digits of Pi Puzzle

This past week, my pre-calculus students have been tackling verifying trig identities. I came down with a cold Sunday night, took Monday off at the insistence of my husband, and suffered through the rest of the week. Introducing trig identities while sick and lacking energy/motivation on Tuesday was not my best and brightest idea. So, I spent the rest of the week trying to make up for it.


Friday, I did a group activity with my students that I'm super proud of. So, I'm excited to share it here on the blog today.


 This activity started with a simple google search for a trig identity puzzle.

The first result was an answer key to a joke worksheet. I found several other copies of the same worksheet online from different sources, so I'm not who to give credit to for the original worksheet. If you know who credit is due to, please let me know!

Image Source: https://www.cabarrus.k12.nc.us/cms/lib09/NC01910456/Centricity/Domain/4422/Basic%20Trig%20Identities%20Practice%20KEY.pdf
Immediately, I was mesmerized. Why don't my answer keys ever look this nice?!? I was also amused by the fun fact that would be discovered by solving the problems. I'd just given my students a trig identity worksheet on Thursday, so I didn't want to do the same thing again. Plus, I was reminded the last time we did a joke worksheet a few weeks ago that teenagers cannot resist shouting the answer at the top of their lungs when they figure out the answer!

The questions were great, so I decided a little activity make-over was in order. Here's how I went about turning this joke worksheet into an activity that kept most of my groups engaged for almost the entire 45 minute period. A couple of groups did finish early, and a few groups did not finish the entire activity. I've had a good number of students out of class for various reasons, so many students were spending a good chunk of the hour helping their classmates understand how to verify trig identities because they had missed several days this week. So, I'm totally okay with the fact that these groups did not finish.


With a little bit of time, colored paper, and a laminator, I was able to turn this worksheet into a group activity that my students ended up comparing to Wheel of Fortune. 


The twelve problems from the worksheet were each printed on a separate sheet of paper with a challenge number at the top that corresponded to the problem number on the worksheet.



I ran out of time to laminate these challenge pages, so I just slipped them into sheet protectors. This ended up working remarkably well. I printed two sets of the challenge cards so that two groups could work on the same problem at the same time if they needed to. This is a lesson I've learned the hard way from designing other activities this year.


Because I kept the numbering from the worksheet, I was able to just print off the answer key I found online and use it as my own personal answer key for checking students' work during the group activity.

Next, I typed up the question that students would be attempting to answer throughout the activity. I laminated a copy for each group, but this was probably unnecessary. You could also just project the question for students to read/discuss before beginning the activity.


I had my students make predictions about what the correct answer might be before beginning the activity. Some of my students thought that figuring this out by performing calculations and estimations was the activity. I truly enjoyed listening to their attempts at answering this question.


Each group received a bag of cards to serve as their "answer bank." There are 10 possible answers and 12 challenges, so some answers may be used more than once. The previous day, we had verified trig identities where I told them exactly what the answer was. Students had to show the process to get from one side of the identity to the other side. This activity was more complex in that students didn't know exactly what they were working towards. Instead, they were simplifying the given identity until it matched one of the answers in their bank.


The last element of the activity that I had to create was the letters that students would earn with each completed challenge. I put them in piles on my desk by letter to make handing them out easier.


Each completed challenge would earn them a specific letter (as specified on the original joke worksheet). One more thing that I wish I would have created would be a tracking sheet for groups to keep track of what challenges they had and hadn't completed. The thought had crossed my mind while creating this activity, but I got side tracked and it ended up not happening. 


I had one student in each group write the numbers 1-12 on their dry erase board so they could erase/mark out each challenge as they completed it. This worked well enough, but a dedicated tracking sheet would have been nice. It would also be a good way to take a grade for this activity if you wanted to. 


After we discussed their guesses to the question at hand (On October 4, 2006, Akira Haraguchi broke his own record by reciting the number pi to 100,000 decimal places. Approximately how long did it take him to complete the task?), I handed each group a different challenge card to begin. Each group tackled their problem on their individual dry erase boards.


They continued rewriting and simplifying their given expression until it matched one of the 10 expressions in their answer bank.


When a group thought that they had successfully completed a challenge, they would bring the challenge card and the corresponding card from the answer bank to my desk. If they were correct, I would give them a letter to help them answer the question. If they were incorrect, they returned to their desk to rework the problem with their group. 


I would step in and help groups as necessary. Sometimes I would examine their work. Other times, I would give them a helpful hint in how they might approach a certain problem.


The most amusing part of this activity was getting to watch them guess what the answer to the puzzle might be as they started gathering letters. 


Some groups immediately tried to arrange their letters to form the answer. And, they revised their guess with each new letter. Other groups, tossed the letters in a pile and focused solely on completing as many of the challenges as quickly as possible. 


I do believe that I got more engagement from my students with this activity than I would have gotten if I had just handed out the worksheet as it was written. I definitely got a better sense of what my students understood and what they were still struggling with than I would have from a pile of papers to grade. I look forward to creating more activities like this in the future! 

Want the files to give this activity a shot in your own classroom? I have uploaded them here.

Wednesday, February 6, 2019

Hidden Animal Puzzlers by Frank Tapson

This year, I am teaching 4 sections of Algebra 2 and 2 sections of Pre-Calculus. Having four sections of one subject is always interesting because I find that keeping four different classes at approximately the same spot in the curriculum to be very challenging. One class is always finishing earlier than the others, requiring creative and quick thinking on my part.

A great resource I have found for engaging early finishers or engaging students on those days when so many students are gone that moving forward with the curriculum is impossible is Frank Tapson's Teacher Resources on Line website. Today's resource I'm sharing comes from his collection of Some Other Lessons.

Image Source: http://www.cleavebooks.co.uk/trol/trolqd.pdf
If you recognize Frank's name on my blog, it's for good reason. He is responsible for creating How Far Can You Climb?, Manifest Game, Skittles Game, Cover Up Game, and the Horizontal Number Line Poster I use in my classroom. 

Recently, the hidden animals puzzle from this file caught my eye. Perhaps because it's directly after How Far Can You Climb? which is always a student favorite. There's absolutely zero math involved in these tiny puzzlers, but they do make the perfect time filler! You could even post one a day as part of a bell ringer or opening activity.

Image Source: http://www.cleavebooks.co.uk/trol/trolqd.pdf (Page 35)
I didn't really want to print off the entire page of 30 puzzles and give them to my students as a formal assignment, so I decided to put them into a more useful format for just tossing up a few of the puzzles at a time as time permitted. I finally settled on creating a Google Slide presentation.


Can you find the hidden animals in the sentences below?



Instead of keeping the file to myself, I thought I should share it here on my blog. All credit goes to Frank Tapson for his original awesome resource. All I have done is turned the worksheet into a Google Slide Presentation/PDF for easier presenting.

My students have really enjoyed this the few times I've pulled it out with some extra time at the end of class. The hardest thing to balance is giving students enough time to find the hidden animal and moving on at a decent pace so the students who have already found it don't get bored.

You can find the Google Slides here and the PDF here.

Now, I just want to find a way to do this with hidden math terms!

Tuesday, February 5, 2019

Dividing Polynomials Puzzle Using the Box Method

After taking a 2 year hiatus from teaching Algebra 2 to dabble in teaching physical science and chemistry, I'm back at it this year. One of the things I really missed about teaching Algebra 2 was getting the opportunity to introduce my students to the super versatile box method for working with both multiplying and dividing polynomials.

This year, some students have been quick to embrace the box method for multiplying polynomials. Others have been resistant and insist on FOILing everything. This is okay. My goal is not to push my students toward a single method. Instead, my goal should be to expose students to different methods and let them choose which method works best for them. Exposure to the box method for multiplying polynomials is ample to allow them to use the box method for division.

Let's stop for a second and talk about other methods for polynomial division.

{\displaystyle {\begin{array}{r}x^{2}+{\color {White}1}x+3\\x-3{\overline {)x^{3}-2x^{2}+0x-4}}\\{\underline {x^{3}-3x^{2}{\color {White}{}+0x-4}}}\\+x^{2}+0x{\color {White}{}-4}\\{\underline {+x^{2}-3x{\color {White}{}-4}}}\\+3x-4\\{\underline {+3x-9}}\\+5\end{array}}}
Image Source: https://en.wikipedia.org/wiki/Polynomial_long_division

Long Division. Sigh. This is how I taught my Algebra 2 students to divide polynomials as a first year teacher. This was how I learned to divide polynomials when I was an Algebra 2 student myself. I had to spend precious class time reminding my students how to do long division of just numbers before we could ever delve into the world of long division of polynomials. It was very algorithmic. Then, there were those pesky sign changes that students ALWAYS forgot to perform. It was incredibly difficult to examine a student's work quickly and find their mistakes.

\begin{array}{cc}
    \begin{array}{c} \\ 3 \\ \\ \end{array}
    &
    \begin{array}{|rrrr}  
        1 & -12 &   0 & -42 \\
          &   3 & -27 & -81 \\
        \hline 
        1 & -9 & -27 & -123 
    \end{array}
\end{array}
Image Source: https://en.wikipedia.org/wiki/Synthetic_division
Synthetic Division. Confession Time: I can never remember how synthetic division works. I've tutored many a pre-calculus student who has come to me with a question about synthetic division. The first thing I always have to do is look at an example in the book and remind myself how it works. There's something about this algorithm that keeps me from retaining it over a long period of time. My other complaint with this method is that it's application is limited. It ONLY works if you are dividing a polynomial by a binomial in the form (x - a) or (x + a).

So, it is my disenchantment with these methods that has led me to embrace the box method, grid method, area method, or whatever else you want to call it for polynomial division. I love that this method does not feel as algorithmic as the other two methods.

After tweaking my approach to introducing dividing polynomials over the past few years, I have finally arrived at something I'm pretty happy with.

This year, we started with some basic notes. The only real guidance I gave to my students to start was that we were dividing instead of multiplying which means we have to set up our box slightly differently. When we multiply polynomials, we set up our box with the polynomials we are multiplying on the outside and work toward finding our answer on the inside of the box. With dividing, we know the final answer (inside of the box) and one of the factors (side of the box). Our goal is to find the missing side of the box.

With this limited guidance of where to put what, my students are usually able to figure out the rest of the process themselves. It's truly a beautiful thing to watch unfold.



I often get asked how the box method works, so I want to walk you step by step through a solution using this method. I'll be using a set of paper manipulatives that I created for my students to give them some much needed scaffolding between the problems we did together as notes and the problems they have to do independently on their own.

Students were given a dry erase template, a dividing polynomials problem, and a set of all of the necessary cards to complete the division process. I like to think of these as mini dividing polynomial puzzles for students to solve.


Since we are dividing by (x-4), we are considering that (x - 4) is a factor of the original polynomial. If it is, we will end up with remainder 0. If it was not actually a factor, we will end up with a remainder. To show that (x - 4) is a factor, we place it on the side of our box.


The polynomial that is being divided by (x - 4) represents the area of the box. The first term of this polynomial (when written in standard form, of course) will always go in the top left box. This is usually the point where I stop explicitly guiding students in how to set up a division problem and let them take over with their suggestions.


x times what equals x^3? x^2. Now, we have the first part of our solution. I can also multiply the x^2 I just found by the -4 to get -4x^2.


At this point, I always ask my students if we want -4x^2. Yes we do! So, I need a like term to add to -4x^2 that will not change its value. The only thing I can add to -4x^2 and not change its value is 0x^2.


x times what equals 0x^2? 0x.


Now that 0x is in our solution, I can multiply it by -4 to arrive at 0x.


Do I actually want 0x? No. My original problem tells me that I want -2x. What can I combine with 0x to arrive at -2x? -2x.


x times what equals -2x? -2, of course.


Now, what is -2 times -4? Positive 8.


Do I actually want 8? No. I want 3. At this point, I can't add any more terms to my solution, so I must resort to creating a remainder to get my 3. 8 - 5 is 3, so my remainder must be -5.


We can write this solution two different ways - using the R symbol for remainder or writing the remainder as a fraction. I normally let my students write the solution any way they wish, but I tell them that they need to recognize both ways of writing the solution and be able to switch between them depending on the context.


I ended up creating six of these polynomial division problems for my students to work through in groups. Each problem and its pieces were printed on a different color. These colors serve several purposes. First, if a piece ends up on the floor, it's super easy to reunite it with its set. Also, the colors allowed my students to keep track of which problems they had finished and which still remained to be done.


Running this activity with my Algebra 2 students four times over the course of one day allowed me to make some tweaks to my instructions/set-up AND think critically about changes to implement in the future.


I kinda threw this activity together at the last minute (thank you first hour planning period), so I didn't have time to make more than one copy of each problem. I had six problems and six groups. This meant that if one group finished a problem before another group finished, they had to sit and wait. Students sitting idle is something you definitely want to avoid in the classroom if at all possible. Next time I do this, I will definitely print 2 or 3 sets of each problem. Other times, another group did finish at the same time, but they had a color that the other group had already completed. If one group spent a long time on a single problem, it became an issue because all the other groups still needed that problem.


For my first class of the day, I asked them to simply work in groups to complete the division puzzle. I quickly noticed that one or two students tended to do most of the work. For my subsequent classes, I had students choose one person to handle the cards for each problem. The other students were allowed to help as much as they wanted, but only the one designated student per problem should be manipulating the pieces.


This made a HUGE difference. As I circulated, my students didn't exactly follow my instructions of only one student touching the cards each problem, but I did find that all of my students tended to be more engaged in the activity since they knew their turn was coming soon.


There was a lot more focus on HOW to solve the problem instead of rushing to finish.


Even with the colors, I did find that after my students had finished 4 or 5 of the problems, they started to forget which colors they had not yet solved. So in the future, I think I will make a stamp sheet of sorts for groups to keep track of which colors of problems they have solved and which they haven't.


Because I was running short on time and because I still was afraid that my activity might have a few typos, I didn't laminate these cards. They were a bit worse for the wear by the end of the day, so I will definitely be laminating any sets of these in the future for durability.


I would also like to expand the activity to feature division by a trinomial, but that will have to go on this summer's to do list since we have finished and moved on from dividing polynomials.


Another possible modification for this activity would be to introduce some decoy cards that are not part of the solution process. As the activity is currently designed, every single card will be used for every single problem.


A slightly different twist that I have also considered would be to take several cards out of each bag that students must supply themselves via dry erase marker on the template. I actually found that some of my groups were writing in what the terms should be in dry erase marker before they found the corresponding card to place on the template. To me, this seemed like they were just making more work for themselves, but I appreciated that they were thinking critically throughout the process. 


I have also had thoughts of expanding this activity so every single student is working on their own. I think I could have done this successfully at my previous school, but I think with my current class sizes of 30 Algebra 2 students that I might go mad trying to check that many students' work at once! 


For this activity, I printed my division templates on 11 x 17 cardstock (affiliate link). This fits perfectly in my 11 x 17 dry erase pockets (affiliate link) that are PERFECT for group activities. The colored cards are printed on regular letter sized cardstock. If you don't have access to 11 x 17 paper or have the ability to print on it, just print both PDFs at approximately 63% scale. This will allow you to print on regular, letter sized paper. Of course, the activity will be much more appropriately sized then for individual work than group work. 


I accidentally printed some of the work mat templates on letter sized paper, and my students ended up placing them in letter sized dry erase pockets (affiliate link) and using them as templates to complete their Delta Math problems. Students were very sad when they had to draw their own boxes to complete the division problems that involved dividing by a trinomial!


Want a little flashback to the past? Here's my attempt at teaching the same lesson back in 2015. It was the same basic idea, but it involved handwritten index cards and a lot of time spent making boxes out of painters tape on our desks.


Before I share the files, I do need to give some credit where it is due.

The problems that students were solving in this activity were not created by me but by my husband, Shaun Carter. They are featured in his work-in-progress Algebra 2 Practice Book.


So, thanks, dear husband, for making my teaching life easier as I made this activity to use with my students and share with other teachers.

Without any further ado, the files for this activity are posted here. Have ideas for making this activity even better? I'd LOVE to hear them in the comments!