Math = Love

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!

Monday, February 4, 2019

Monday Must Reads: Volume 53

I can't believe it is already Monday once again. The weekends seem to be going by faster and faster lately. I have several blog posts in the works that I need to finish and share. Hopefully I will be able to knock a few out before the next Monday Must Reads rolls around. If you're new to the blog, Monday Must Reads is my weekly-ish attempt at capturing the amazingness of other teachers (mostly math) on twitter.



Mark Trushkowsky shares a fun twist on a classic puzzle.

Image Source: https://twitter.com/mtrushkowsky/status/1091745013279715330
I am so impressed by the geomARTry created by Anna Vance's students.

Image Source: https://twitter.com/TypeAMathLand/status/1092116209594781696
I'm also saving this sorting activity that Anna created for next year's Algebra 2 class!

Image Source: https://twitter.com/TypeAMathLand/status/1091817206592225280
Lindsey Sharp's bulletin board could make a great introduction to systems of equations!

Image Source: https://twitter.com/LPSharpD300/status/907747070077014016
Michelle Ervin introduces polynomials with a Latin 101 lesson. LOVE it!

Image Source: https://twitter.com/FindingX_Ervin/status/1091506206156488705
Jennifer Fairbanks shares some great prompts to elicit student creativity.

Image Source: https://twitter.com/HHSmath/status/1091111767764287488

Image Source: https://twitter.com/HHSmath/status/1090233364454486017

I love how Cassandra Valenti incorporates literature into her lesson on exponential growth and asks for student predictions before finishing the story.

Image Source: https://twitter.com/MathWithMrsV/status/1091393201578168326
 Mary Bourassa shares a great puzzle from one of her textbooks.

Image Source: https://twitter.com/MaryBourassa/status/1091386042668797953
Kristen San Filippo inspires with this approach to a sub day pop quiz.

Image Source: https://twitter.com/LV_sanfilippo/status/1091083275148369920
Looking to incorporate reflections on student data? Check out this idea from Valerie Muller.

Image Source: https://twitter.com/muller_va/status/984581704936456192
Miss Dulowski shares a new-to-me math joke. Click here for the answer.

Image Source: https://twitter.com/missdulowski/status/1042414710908497923
Robbyn Glinsmann shares a great visual prompt for counting triangles.

Image Source: https://twitter.com/GlinsmannMath/status/1089160484874522624
Fawn Nguyen shares a vintage mental math prompt.

Image Source: https://twitter.com/fawnpnguyen/status/1090667977731596288
James Robinson creates a brilliant twist on flash cards for trig functions. What is missing from each card?

Image Source: https://twitter.com/Mr_Robinson_23/status/1090648780385517570
These roller coaster projects from Bridget Herrman look fun!

Image Source: https://twitter.com/BHerrman/status/1090410726798299136
Mong Kon Mo shares a 4 in a row activity for solving basic equations.

Image Source: https://twitter.com/MkmoTeacher/status/1090467811946721280
Mrs Stromberger combines play doh and right triangles for a fun activity that gets students moving around the classroom and solving problems created by other students.

Image Source: https://twitter.com/Mrs_Stromberger/status/994323689918394369
Another interesting task from Mrs. Stromberger involves a giant Twix and a system of equations.

Image Source: https://twitter.com/Mrs_Stromberger/status/1090371244074196998
Kristen Clingenpeel shares a fun method for setting up a comprehensive end-of-year or end-of-semester review.

Image Source: https://twitter.com/MsClingenpeel/status/1000112252060610560
I love how Elissa Miller uses transparencies to reinforce matching congruent figures.

Image Source: https://twitter.com/misscalcul8/status/1090437114766995456
Dan Walker shares a lovely little volume problem.

Image Source: https://twitter.com/360maths/status/1090351550634094592
Mrs. Tangel shows how to use factor trees in a festive manner. Keep this in mind for next Christmas!

Image Source: https://twitter.com/Mrs_Tangel/status/803317799154237446
Mary Platt shares a great water-color based pi day lesson.

Image Source: https://twitter.com/Mary4Math/status/974005563074936832
Kristina Clark shares a way to turn masking tape into a data collection activity for scatterplots.

Image Source: https://twitter.com/KClarkGMS/status/998998813665787906
Need a way to make substituting values more concrete and less abstract? Check out this idea from Ms Blackshear.

Image Source: https://twitter.com/MsBlackshear17/status/1085733901073555456
Teaching geometric constructions? Check out this compass petting zoo idea from Mark Kaercher!

Image Source: https://twitter.com/shskaercher/status/1090257368045228032
John Rowe shares some triangle based area puzzles inspired by Naoki Inaba.

Image Source: https://twitter.com/MrJohnRowe/status/1059190995298877440
Kanchan Saxena shares the brilliant idea of physically modelling 3D trig problems.

Image Source: https://twitter.com/kanchan24saxena/status/976332651463303171
John P lichaa shares a great idea of introducing congruence and similarity with marshmallows.

Image Source: https://twitter.com/JPlichaa/status/895230145916583936
All I have to say about these trig projects created by John's students is Wow. Just Wow.

Image Source: https://twitter.com/JPlichaa/status/1020450278733111297
Katherin Cartwright shares a great problem based off of a giant banana landmark.

Image Source: https://twitter.com/kath_cartwright/status/1083583095486349312
Emily Pratt shares a photo of Eddie Woo's 1 through 6 challenge. This sounds really fun!

Image Source: https://twitter.com/EmilyPr79582111/status/1040753728670715904
I also love this new use for Prime Climb in simplifying fractions shared by Emily!

Image Source: https://twitter.com/EmilyPr79582111/status/1040592825052622848
Miss Aldridge shares a great way to measure student understanding before beginning a topic.

Image Source: https://twitter.com/SJBMissAldridge/status/1088496246162100226
LOVE this transformations card sort from Tara Maynard!

Image Source: https://twitter.com/tmaynard5/status/1088100401239547904
Jocelle Skov shares a favo(u)rite lesson, and it's a definite winner!

Image Source: https://twitter.com/mrs_skov/status/1087827307484463105
Helen Scott shares some great questions for working with algebraic perimeters!

Image Source: https://twitter.com/HelenScott88/status/1085519366253355008
Could your calculus students find the area of a traced body? Mrs. Tomalesky's can!

Image Source: https://twitter.com/MASTmathlady/status/837324212658454528
Check out this awesome bulletin board from Maher Calculus. It's all created with piecewise functions!

Image Source: https://twitter.com/arrows4deb/status/1049717380019052544
Jennifer Fairbanks shares a great domino based puzzler.

Image Source: https://twitter.com/HHSmath/status/1087085044559040512
Until next time, keep sharing your awesome ideas and resources!