Team Teaching: Factoring Quadratics

THE GOOD:

My peers felt that the use of algebra tiles as manipulatives worked very well in explaining what the factored and expanded form of a quadratic looks like. It kept them very engaged and it was sort of a fun puzzle solving activity because finding the right configuration to get a rectangle was fun. It also helped students realized that there are some configurations that don't really work. As well, they found the lesson to be clear, concise and got the point across well enough.

I felt very much the same about what my peers felt. When Enrique, Nathan, and I were discussing how we would introduce factoring quadratics as such and got to discussing the algebra tiles, I saw that the activity would really engage my peers. I think this would totally work in a classroom one day. In fact, during my practicum, if I'm asked to do an introduction to factoring quadratics like this, I might adapt this lesson plan.

THE NOT-SO-GOOD:

My peers thought that time management was an issue. They also saw that there were some issues with classroom management as some students were chatting and not paying attention after the group activity had ended and some students felt that the group activity should have deserved a bit more time.

I saw the issue with time management as well. We tried our best to get our point across as quickly as possible but I felt that this lesson could have gone a full hour given the activities and the discussions that it could have garnered. I wanted to let my peers figure out the pattern between the two ways of calculating area of the rectangles they formed as this would show them, not only FOIL, but how to factor a quadratic polynomial. But as such, time was an issue and I felt that the point of the exercise was lost due to time constraints.

With regards to classroom management, I think I could have handled myself a little better. There were a couple of times where I "snapped" and shot the infamous "look" but I recovered quite nicely once I got the students' attention again. That might be something I want to look out for in my future practice.

Factoring Quadratic Polynomials: A BOOPPPS Lesson Plan

Bridge: Introduce quadratic polynomials by using algebra tiles and relating them to areas of rectangles.

Learning Objectives: Students will gain a basic, entry level idea of how to factor basic quadratic polynomials of the form ax² + bx + c.

Teaching Objectives: Teachers will be able to use group work to teach students the patterns of factoring quadratic polynomials using manipulative such as algebra tiles.

Pretest: Ask students what kind of simple geometric shapes can be made with the algebra tiles. Ask, in what ways, can one find the area of the rectangle. (Should be two distinct ways. More if the class is creative.)

Participation: Put students into groups of 3 with a set of algebra tiles. Have them form various rectangles and note, using the different ways, what the area of the rectangle is. Make note that these are the expanded and factored forms of the quadratic polynomial.

Post Test: Ask students if they found a pattern to the two ways that the area was calculated.

Summary: From here on, discuss the various other forms of quadratic polynomials as the group work. Discuss factorable and non-factorable quadratics in that manner. Talk about the quadratic formula and what it can do to factor any quadratic.

Mathematics and Civics

We need math. Math is all around us. We need math to function as citizens of our nation. Numeracy teaches us about logic, problem solving, and common sense. To be able to function as “good” citizens in our society, we need all these skills that math teaches us. We, as citizens, use the basic principles of what we learn in mathematics class regardless of whether we see it that way or not. I find it, as well, that people don’t see the math they do as math. If we can get people to understand that the math they use in counting money, figuring out travel time, and other life skills are, fundamentally, mathematical, then I think that people will want to study math more.

This article basically talks about what I’ve believed in for as long as I’ve been studying math. As we are constantly immersed in numbers, shapes, statistics, problems, and other mathematics related ideas, the skills taught to us in math class help us in our day to day lives. We *can* live as citizens of our society *because* we learned math.

In that note, I agree with the article provided for us. Mathematics is necessary in our tool kit of skills for life as citizens in the world. It's just that we need to help people realize this fact.

reflecting on WIN

The What-If-Not approach of problem posing interests me. It takes a commonly viewed notion and turns it upside down. It takes something true, and asks how we can change these attributes. This would lead to a great many questions that need to be asked, of varying significance. I think that this strength is also its weakness.

Asking what-if-not, as said earlier, leads to many new questions asked. In asking these questions, these lead us to new and profound ways of mathematical thinking. It sparks our curiosity because we ask questions that, in some cases, have nothing to do with the original idea. It is, however, bad as well in a sense. The depth of these questions relies significantly on how the problem poser thinks. Things might be left out. Even deep questions that let us form a new mathematical perspective may be missed because the problem poser didn’t see an attribute and change it accordingly. Also, changing an attribute a certain way could lead to a similar problem like this.

Given that, it would be very difficult to put this sort of thinking into factoring quadratics. But that wouldn’t stop me. I could ask questions like what if it wasn’t a quadratic? What if the exponent wasn’t an integer? What if it wasn’t just one variable?

These thoughts would all lead to very open ended discussions revolving the subject matter. I like the WIN approach. It challenges both the problem poser and the problem solver. These what if not questions can lead to deep and profound mathematical ideas that could very well reach outside of the mathematics curriculum and I think that would be totally awesome.

10 Questions on The Art of Problem Posing

1. I found the first section quite dry. But I do understand that there is an intended audience for this book.
2. How do you encourage students to pose their own problems?
3. In respect to the title of chapter 3, what does it mean to accept?
4. How appropriate is asking a broad question to students who have never really thought about math in terms of problem solving/posing?
5. How do I make the ideas outlined in these first 3 chapters of the book more relevant in my pedagogy?
6. With respect to internal and external exploration, which do you think is more effective?
7. When exactly is it appropriate to start a thought experiment such as problem posing?
8. Should I encourage students to pose problems for each other?
9. I love how they let us pose our own problems about a² + b² = c² and showed that there wasn't any problem, but we answered our own questions about the equation.
10. Overall, I think the rest of this book will be a very interesting read.

Think Ahead 10 Years. What Would Students Think of You?

Student who likes you:

Hey Mom

I had the most wonderful year learning Math 12 with the best teacher that I have ever had. He made sure that the mathematics that we were doing were relevant to our daily lives. He gave us clever stories about the mathematicians that 'roamed the earth'. And he made it fun. We didn't just do work individually in class; we did a lot of group work. We even got to make 3 dimensional models of things like cubes and 20-sided dice. I had loads of fun.

Also, he's so helpful when we're doing work because he walks around the classroom checking up on us every now and then. I find that it helps when he asks questions that lead to the answer and does not just tell me what the answer is.

Oh and his teaching style isn't always lecturing and having us do work. It's a combination of many things including video, interactive presentations, and all really cool stuff.

I really enjoyed my time in his classroom. Thanks for enrolling me into this school!

All the best

Student A

Unhappy student:

Hey Mr. Vicente

I figured that I would write you an e-mail discussing my opinion of your teaching style. In my opinion, I think that you speak too fast and cover the material inadequately. Sometimes you mumble and your jokes are never funny. You're almost never available after school because you're too busy with other school activities and I find it hard to ask for help in class. You always put me in a group with people who are smarter than me and I feel that I don't learn because they are always just doing all the work instead of me. I just wish that you would spend a little more time helping me with my difficulties.

On that note, I feel that when you ask a question, you always seem to pick on me when i look so confused. Is it that you want to humiliate me in front of my students? I don't appreciate that so much. Just give me a chance to understand whatever you say in the class before throwing me into the fire.

Sincerely

Student B

As a reply to Student A, I hope to be able to do whatever he or she mentions in the above. I think it'll be cool to have a dynamic and engaging classroom.

As a reply to Student B, I hope that whenever I single out a student, I hope that they understand that I do it so that they can accomplish something. When I put a weak student with good students together for group work, I hope that the good students teach the weak student, and hope that the good students don't do what they do to student b.

Teaching the Marked Case: In Reflection

The video we had watched in class spoke to me. I saw that simple repetition and building upon what was established in class worked effectively to teach the students the basic principle of the number line. Small steps helped ingrain the basic principles in the students and I think that the way the teacher taught was quite effective.

Using repetition, he was able to tell the students that steps generalized to more complex things. He was able to teach gr 9 students simple algebra by example and by asking the students what they would do. In effect, he was able to let the students flourish and figure things out by themselves by guiding them. The students, in effect, already knew what to do. He just let them know that they already knew. This, I think, would build confidence on the children.

As well, whole class participation helped the teacher see if there was confusion among the students. This also let the students reflect on what they had just said and allowed for them to correct their mistakes. I thought this was very effective rather than the teacher telling the students that they were wrong.

The teacher also used the classroom effectively to teach the students. He used the space outside of the blackboard to extend the 'number line' so that his students may understand that the number line infinitely grows.

Overall, I think that his teaching style works quite well to be able to teach students concepts and such. My question is, how would you scale this sort of teaching to higher levels like Math 11 or 12?