# Polynomials

Oh Polynomials. My least favourite unit by far in the Foundations of Math and Pre-Calculus 10 course I am teaching. Find the greatest common factor, least common multiple, factor these trinomials, collect and simplify like terms, the swimming pool has a width of 5x + 1 and a length of x + 2… YAWN.

## The Challenge

How can I frame a boring, completely algorithmic and skill-based unit into something that’s relevant and meaningful for my students? I am borrowing Dan Meyer’s definition for relevance here.

## It Begins with a Question… ​

A colleague asked me today, “How much time do you have for homework at the end of class?” This was a surprising question to me, and as I thought back over the 10 day unit, my answer was almost none. The question sparked a great dialogue between us about our approach to teaching the same content in our respective classrooms. It really made me think. I realized that while I still dreaded teaching polynomials, I had found a way to improve the way I taught it from first semester that required less rote work and more thinking.

One thing that has not changed, however, is that I avoid teaching FOIL method like the plague. It only  works for expanding binomials and does not apply for polynomials with more than two terms. After I read this article I was convinced I would never need FOIL in my classroom:
http://www.makesenseofmath.com/2016/11/why-i-will-never-teach-foil.html
For a good laugh:
https://saravanderwerf.com/2017/04/01/why-ive-started-teaching-the-foil-method-again/

Some things that came up in our discussion:

1. Algebra tiles – benefits and fall backs
2. Picture talks
3. Factoring method (criss cross or sum-product?)
4. WODB
5. Progress checks
6. Taboo
7. Human Bingo

## 1. Algebra Tiles ​

Definitely a hate-hate relationship. As a math teacher, I am obligated to entertain this idea and I do admit it has its benefits, especially in lower level math classes when students are initially being exposed to distributive property and the like. The problem, however, was that my students were already armed with the skills and knowledge of multiplying and factoring polynomials. Moreover, the limitations of using tiles far exceeded the benefits, in my opinion. Algebra tiles do NOT work for: polynomials higher than degree two, multiplying more than two polynomials, and multiplying polynomials with more than three terms. This meant that it took more effort for students to understand how and why it works.

Nevertheless, we spent a few classes examining algebra tiles and their usefulness. Rather than approach it from the typical standpoint of using algebra tiles as a manipulative, I wanted students to see the link between the algebraic and pictorial representations of polynomials. This took work and was not as straightforward as it seemed. A big takeaway for me was that students gained much more out of the experience when they were able to physically manipulate the tiles and arrange them into their “factored forms.” Last semester, I “taught” algebra tiles by merely showing them examples and drawing them on the board.  It took a bit more prep, but this semester I printed eight sets of tiles (positive and negative) in my classroom and had students manipulate them instead.

If we were to spend any more time on the unit, or if this was a lower grade level, as an enrichment activity I would have students discuss the limitations of algebra tiles and look for ways to address them.

## 2. Picture Talks

I like to use Sarah VanDerWerf’s Stand and Talks as a format for students to discuss picture prompts. I find that the buy in for engagement is much higher when the prompt is linked to physical movement. My favourite questions for photo prompts are: “What do you notice?” and “What do you wonder?”

What are my photo prompts, you ask?

That’s right. Algebra tiles.

Goals for students:

• Make observations and ask questions
• Use math vocabulary
• Share ideas with their peers

​I like this activity because it is easy to differentiate and works well as a “minds on” for any topic. Asking students a general question like what they notice/wonder means that lower ability students can comment on ANY aspect of the photo (e.g. “there are blue and green rectangles”) while higher ability students can be pushed towards making observations based on any mathematical patterns or relationships they observe (e.g. “the green tiles represent positive polynomials and red tiles are negative”).

## 3. Factoring Method – Criss Cross or Sum Product?

I’ve had the great fortune of only having one prep and a spare block this semester (for friends and readers who don’t teach, that’s teacher jargon for FREE TIME, kinda. The details are not important). Anyway, I’ve been making drop-in’s to my fellow colleagues classrooms with my new-found “free time” and one thing I picked up was the importance of proper SEQUENCING. For instance, a natural progression for factoring trinomials might look as follows:

• Common factors
• ​Ex:    12xy + 3x
• Trinomials with leading coefficient of 1
• ​Ex:     x^2 + 4x + 4
• Factoring when the GCF is a binomial
• Ex:      x(x + 1) – 2(x + 1)
• Trinomials with leading coefficient not equal to 1
• ​Ex:     3x^2 + 8x + 4

That, together with a quick exercise on sum/products, helped me push students towards seeing the relationship between the factored form of  a trinomial, and the sum/product method.

I prefer this method over the traditional “criss cross” method for a few reasons

• Focus is on noticing patterns
• Less trial and error work
• Allows them to answer questions like this:

## Which One Doesn’t Belong? (WODB)

Fantastic activity for building up thinking skills and vocabulary. Each student picks one of the expressions and must argue why that one doesn’t belong.

Benefits:

• Everyone speaks (I always encourage full sentences and proper use of math vocabulary)
• More than one argument may arise for each expression

“27x^2 doesn’t belong because it is the only expression that has a coefficient with a perfect cube”
“45x^2 doesn’t belong because it is the only expression that has a coefficient with 5 as one of its prime factors”

More WODB prompts can be found here.

## Taboo ​

How it works: One student is chosen to stand/sit at the front of class facing the audience, they are in the “hot seat”. Behind them, a vocabulary term is shown for the rest of class to see. Students in the audience must help the student in the hot seat guess the vocabulary word by miming, explaining the definition, or giving examples. They may not use any part of the word in their explanation.

Modifications: Differentiate by giving students the option of bringing a “cheat sheet” of vocabulary terms with them. Prepare students for the activity by giving them cross word or fill in the blank exercise reviewing the vocabulary words for the unit. An “expert round” can include vocabulary not on the cheat sheet. “Challenge round” can be facing a peer or the teacher. Can play in teams or as a class.

## Human Bingo ​

 chapter_5_human_bingo.docx

This activity was shared by a good colleague of mine. To get BINGO, students must find one “expert” in the classroom to answer each question on the bingo card until all the questions have been answered. The student who answers the question must sign their name. A student may not be asked more than once to answer the same Bingo card.

Benefits:

• Gets students moving
• Students can pick the question they want to answer
• Fun review activity for a test or quiz!

# Reflections on My First Semester Teaching in China

Semester one of my first year living and working in China is officially over! Since my last post about the first day of school, I realized haven’t blogged at all this entire semester. I am a little disappointed that I had skipped through all the middle bits, but regardless, here we are.

This past semester I taught Math 10 and 11 of the British Columbia curriculum at an international school in Suzhou, China. With the exception of a handful of students, all of them are English Language Learners. Some might argue that this does not pose a big problem in mathematics, since the language of mathematics can be viewed as a combination of abstract signs and symbols separate from the English language.  The problem is, it is one thing to understand mathematical ideas and concepts, but another to be able to communicate them. Someone who is well versed in a mathematics should theoretically be able to describe the same concept in more ways than one – numerically, algebraically, graphically, and verbally. Mathematicians strive for precision in expressing ideas, and this is not always simple. Aside from students having to approach mathematics from an ELL standpoint, the issue is compounded when you consider all the ways in which ambiguity arises in the English Language. Take for instance the word “and”; conjunction in mathematics is commutative (A^B is the same as B^A), but you can see from the example below that “and” in everyday English is not commutative.

The sentence, “John took the free kick, and the ball went into the net,”  would have a very different meaning if the conjuncts were reversed (Devlin, Introduction to Mathematical Thinking).

For my most challenging students, the issue wasn’t so much as getting them to communicate their mathematical ideas well, but getting them to communicate at all. For students with extremely low level English ability, being afraid to speak or ask questions in class was a huge roadblock in developing  a good grasp on the mathematics we aim to study. The most frustrating times were when students didn’t even bother to try. Perhaps this has something to do with being in a culture where “saving face” is important, but students were sometimes so afraid of being wrong that they left entire test pages blank, multiple choice even! (Yes, I know, I was stunned!) You’ve probably heard this a million times but I’ll say it again, mathematics is not a spectator sport! You have to do it to get it, like riding a bicycle. (Am I preaching to the choir here?)

My biggest goal this semester is to get students talking more. About mathematics. In English. A large part of my success will depend on how well I set up a classroom culture of trust and acceptance. This is huge. If I have any hope of getting students to share their original thoughts and ideas they need to know they are safe doing so. Luckily, I’ve got some ideas to help me get started, but the rest will be trial and error (as is most of my teaching anyway). I also plan on working in a slower progression at the beginning of the year to first get students acquainted with some of the language used to describe mathematical expressions before we dive into what exactly mathematics is. With any luck, every student will be able to describe, in English, what we are learning in any given unit.

Things That Went Well in Semester 1
1) I finally found a groove! Lesson planning no longer takes up hours and hours each day (#win), and I also have a nice support network of experienced teachers to draw ideas from and borrow resources from. Establishing daily routines early on in my classroom (and enforcing them!) also worked wonders.

2) Brain breaks. I was a little hesitant about these at the start since they seemed silly and unnecessary if the lesson is well-chunked. I learned early on though, not all lessons are made equally and some days really are a drag, especially when are teaching 80 minute blocks. Taking a short 5-10 minute break to stretch/play a game/go on your cell phone provides both myself and the students some much needed refuge from a long period of work.

3) First week activities. As I mentioned earlier, setting up a warm and inviting classroom culture is key to being able to get students to talk more math, and learn more in general. I spent about a week doing activities and playing games related to math with my students last semester before I started diving into teaching any curricular content. I plan on spending about the same amount of time, if not more, this coming semester settling in with my new classes.

# First Week Activities

Another year, another country, and another school! Phew, all this moving around is getting tiring, and I’ve been teaching new courses every year. I’m so grateful to the MTBoS community (Math-Twitter Blogosphere) for sharing resources and teaching tips and tricks, it makes me so happy to be teaching math! I can’t praise it enough! Sarah Carter from Math = Love, Sara VanDerWerf, and Dan Meyer have been my go-to’s for classroom activities and lesson ideas.

I’m teaching high school math (grades 10 and 11) this year. My school runs on 80 minute blocks. Here’s what I did.

Algebra Seat Finders and Visibly Random Groups – Rather than making a seating plan or having students choose their own seats I greet students at the door and hand them each a card as they walk in. On the card are algebra problems involving one or two step equations and order of operations that are easily solvable via mental math. The answer to the question will tell them which table to sit at. I’ve arranged my tables into groups of four and have signs taped to the side of the desks so they can easily find the group number. (If you would like to download copy of the seat finder cards I used, they are available at the bottom of my post).

​I do the same thing each day, so that every day students will sit in different groups. I like this activity because students are doing math as SOON as they enter the classroom.  Some students will cheat and trade cards with other people so they can sit with their friends, but you will come to notice this quickly. I tell students that in this class we are a community and that they will always be working with different people so they get to experience different perspectives and meet everyone in class. Even if certain students don’t get along, it’s low stakes because the seating changes every day. On Fridays I give them a break and tell them to sit anywhere they like. It was interesting for me to notice that given the choice, students tend to sit with classmates with similar level. Peter Liljedahl has done some cool research on visibly random grouping, check out his free webinar here

All these cards solve for x = 1, as my class is arranged in groups of four.
Day 1

• Bell Work – Who I Am
• Start the class with low key student profile sheet from Dan Meyer as I take attendance. Gives students a chance to tell me about themselves. My favourite questions on this sheet are the “Self Portrait” and “Qualities of a good math teacher.
• Numbers Quiz
• Adapted from Sarah Carter. I beef this up a bit and use this as an opportunity to talk about test/quiz expectations (no talking, no asking a neighbor to borrow an eraser or calculator…etc.), and the consequences for cheating. I tell them that this is a difficult quiz and so far no one has been able to obtain a perfect score. All I ask is for them to try their best, and if they don’t know an answer, guess. When I tell them to flip their papers over I usually hear a few chuckles or giggles. Again, I enforce that the room should be silent and let them know I mean business.
Next, I tell them a bit about myself and we grade the quizzes. #2 and #6 (distance questions) are a good chance to incorporate number sense and reasoning as most students have no idea how far it is from China to Canada or how long it takes to run a 21 km race.

• Student Quizzes
• Next, I give them a chance to write ME a quiz about themselves. I take their quizzes and return it to them to be marked. Most students asked basic questions like “What is my favorite subject?” or “What is my favourite food?” Others were more creative and decided to have a bit of fun with the activity…
• Personality Coordinates (Dan Meyer)
• Originally planned to complete this activity the first day, but I was over-ambitious with my planning so ended up introducing it and coming back to it later. First I showed students this diagram:
I asked them to silently think of things they notice/wonder about the diagram. Then I did my first ever Stand and Talk and went around listening to conversations which gave me a chance to check in on students’ English ability. I teach EL (English Language) learners so I found it helpful to model how a conversation might go the second time round:

Student B: I notice there are two perpendicular lines. What do you notice?
Student A: I notice the four dots are arranged in a square. What do you wonder?
Student B: I wonder what the teacher will ask us to do with this diagram. What do you wonder?
Student A: I wonder if this is a function.
..etc.

We discuss and review parts of the coordinate plan. I ask them a few questions about the dots. (Which two dots share the same x-value? Which dot has the lowest x and lowest y value? etc.)

The next time we revisit this activity I start with an example:

• Name Tents (Sarah VanDerWerf)
• ​At the end of each class on the first week I asked my students to choose ONE question and answer it in their name tents:

• ​1. One thing you enjoyed about today’s class?
• 2. One question you have.
• 3. A suggestion for class.
• I write back to them every day. This is a big commitment but worth the time in my opinion.
• Some positive feedback I’ve gotten: Fun, engaging class, students enjoy group work and team activities
• Some things I need to work on: talking slower, writing bigger on the board
• Some questions I’ve been asked: When do we get the textbook? When do we have our first quiz? Is math difficult?
​Day 2

• Syllabus Quiz
• Rather than giving a long speech about course expectations, school and class policies, I wrote a quiz. Even though I assign syllabus reading for homework most students will not do this. The quiz is open book and is graded (can be done in pairs), and I count it towards their “English proficiency” grade for the course.
• Talking Points
• This one MUST be modeled to students. It’s a little complex, especially for EL Learners so it’s important to explain clearly and minimize the amount of instructions given. The main point is to get ALL students talking and sharing their opinions.  To model the activity, I pick three random students to do a “practice round” with me. This was less effective with my grade 10 students as they are new to the immersion program. Next semester I might film a teacher example of this activity to show students instead.
• What is Math?
• Share our ideas of what math is, give a common definition of mathematics that we will use for the course.
• Expectations for the Year
• Go over things like: cell phone policy, asking to go to the bathroom, materials needed for class, binder expectations, course evaluation…etc.
• Name Tents
• Again, end the day with student writing me some feedback.

Day 3 – Day 5
Teach some content and continue reviewing and practicing start of class and dismissal routines.

 Algebra Seat Finders – Groups

# Mindset

Picture a circle on the center of a blank page. Along the circumference of the circle are six spokes, evenly spaced. If you were to write down one word for each of the spokes that defined who you are, what would you write?

For me, these words are: female, older sister, Chinese, Canadian, teacher, learner… These are important parts of my identity, they fundamentally shape who I am and how others view me, however, if I am not careful, they can also label me and lock me in. We all have assumptions about ourselves that can hinder us from reaching our true potential. To be more specific, I recently had a conversation with a good friend of mine who told me about an article she read that said the reason why many females are overqualified for their jobs are because women tend not to apply for a position if they feel they do not fulfill all the requirements, whereas males will if they feel they fit most of the criteria. I wondered how many opportunities I missed because I told myself I wasn’t good enough to try.

I recently interviewed for a position that required teaching AP physics. With my measly, almost-two years of full time teaching, and zero experience with physics (or AP for that matter), I definitely did not think I had all the requirements for the job. But I thought about what my friend told me, and I said- to no one in particular- “Heck, what do I have to lose?” Lo and behold… I was stunned when I landed an interview… and even more amazed when they called me back for a second one.

If such a small shift in my thinking could have led to such a significant outcome, no doubt this can apply to all areas of life and learning as well. I am currently reading Mindset by Professor Carol Dweck. I wonder a lot about how I can help my students uncover the hidden assumptions they have about themselves in order to develop a growth mindset. We talked about what it means to have a fixed versus growth mindset at the beginning of the year and what that looked like for different people. We explored the nature of science and how important it is to acknowledge failure in science. We discussed our ideas about how success is like an iceberg; magnificent and grand on the outside, when in fact much of it is submerged and hidden below the surface. I try to make it real for my students and have them connect it to their own lives, but most of all I’m trying to build a classroom culture that enables them to feel safe taking risks, making mistakes, and to fearlessly embrace new challenges. I struggle with this every day. Sometimes I feel like I am making good headway, and other times I feel like I’m picking my students up by the feet and trudging  them through the mud, shouting, “Come with me! There is a light at the end of the tunnel!!! Just keep moving!”

And with that last bit of imagery, I shall kindly remind myself that learning is a process, and that we each move on our own time.

When I think about Carol Dweck’s research on mindset I am reminded of my grandfather, who, throughout all the years I have known him, has shown me in his own way that it is NEVER too late to learn a new skill or to grow your mind. When I was eight or nine, I remember grandpa practicing to get his truck driver’s license. He had only been in Canada for a few years at that point, had never driven a truck before, did not have access to one, and was unable to take lessons, but that did not stop him. He took us out to Canadian Tire and bought a toy truck with remote controls. I remember watching him maneuver it around the carpet in his bedroom, studying it from different angles, gathering information about the spacing, and so on. He practiced like this diligently for days before his driving exam. Even I tired of watching the little truck move around in endless loops, turns, and parking maneuvers, but grandpa always aimed for perfection. This was the type of man my grandfather was.

I used to hate going to Chinese school on the weekends, but grandpa insisted that I persevere because he was afraid that I would lose my heritage and that my future children would forget their ancestry. This thought frightens me also. I never used to think learning Chinese was very important. I just knew how going to Chinese school made me feel – stupid and inadequate. It was like being sent to a correctional facility for not being born to the right circumstances. To hide my feelings of inadequacy I worked even harder to get good grades. I memorized difficult words, I practiced spelling them out over and over, and people told me how smart I was.

It wasn’t until one day my grandpa said something to me that I finally was able to breathe. I didn’t even know it then, but I was suffocating. I had been trapped by the need to prove how good I was, that I too could read and write, two things that seemed to come so effortlessly to others. I used to cry myself to sleep because it seemed that no matter how hard I tried or how much I worked at it, I would never be fluent in Chinese like my family. So, when grandpa said those words to me I knew the facade was up. I didn’t have to pretend anymore. He said, “Even if you are not very smart or talented at something, with effort and practice we can make up for the things we lack. This is me, your grandfather.” And then he said, “You and I, we are both hard workers, no?” I will never know what prompted grandpa to say those words to me, but I just know that when he did, at that very moment, I felt true clarity and a huge sense of relief. It didn’t matter that I wasn’t great at something, what mattered was that I tried.

# Dan Meyer, I Wish I Met You Sooner

“I teach high school math. I sell a product to a market that doesn’t want it but is forced by law to buy it.” — Dan Meyer