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?

Battlefield Schools - A Summary

The article outlines two very different schools of thought in our education system, progressive and conservative. The conservative view on mathematics education focuses more on instructional and instrumental understanding of the material. This view sees that answers and facts are the way to teach math, treating it as a means to an end. In stark contrast, the progressive view sees that the journey to the answer is much more important and experimentation and understanding is the way that math should be taught. The rest of the article outlines how these two have “battled” for supremacy in how mathematics is taught in the K-12 level.

The progressive movement of the early 20th century saw a realization that experimentation and exploration would lead to more democratic citizens. Educators like Dewey found that, instead of dictating knowledge to the student, it was more effective to let students grapple with issues and problems and let them figure some things out for themselves.

In the 1960’s, a more abstract form of math, called the New Math, permeated the mathematics curriculum of the day. The launching of Sputnik into space meant that America was losing the space race and it was imperative that students in the secondary level ought to be trained to become future rocket scientists. This was seen as very radical and unhealthy for the state of mathematics education as it was conservative idea parading as a progressive one. It also did not account for the fact that not all students wanted to get into Mathematics and Science after finishing secondary school. Lastly, teachers could not teach the material at all because they were educated in a totally different than what the curriculum outlines.

In the latter part of the 20th century, the 80’s and 90’s, a third movement caused a great shift in mathematics education, the Math Wars. Conservative ideas of mathematics education were being branded as right-wing and radical while relational understanding and progressive ideas were seen as more desirable. These ideas are what we would see today happening in our current curriculum.