Category: Tales from the Classroom

Engaging students: Multiplying binomials

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Andy Nabors. His topic, from Algebra: multiplying binomials.

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A2. How could you as a teacher create an activity or project that involves your topic?

Multiplying binomials is an interesting concept because there are so many ways in which this can be done. I can think of five ways that binomials can be multiplied: FOIL, the box method, distribution, vertical multiplication, and with algebra tiles. I would incorporate these methods into one of two different ways. In either case, I would split the class into five groups.

  1. In the first way, I would assign each group a different method of multiplication. The groups would each be responsible for exploring their method, working together to master it. Then each group would be responsible for making a poster describing their method in detail. Then would then present their poster to the class, and the students not presenting would be taking notes. Already having one concept of binomial multiplication, the students would be seeing other methods and deciding which makes most sense to them.
  2. In my second idea, I would have five stations in the classroom each with their own method. The groups would rotate station to station figuring out the different methods collaboratively. The groups would rotate every 7-10 minutes until they had been to every station. Then the class would discuss the strengths/weaknesses of each method compared to the others in a class discussion moderated by the teacher.

These activities rely on the students being able to work and learn in groups effectively, which would present difficulty if the class was not used to group work.

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B1. How can this topic be used in your students’ future courses in mathematics or science?

I had the privilege of teaching a multiplying binomial lesson to a freshmen algebra one class in CI last spring. My partner and I focused on the box method first, and then used that to introduce FOIL. The box method was easier to grasp because of the visual nature of it. In fact, it looks a lot like something that the students will definitely see in their biology classes. The box method looks almost identical to gene Punnet Squares in biology. In fact, my partner and I used Punnet Squares in our Engage of that lesson. We reminded the students of what a Punnet Square was, and then showed them a filled out square. We went over how the boxes were filled: the letter on top of each column goes into the boxes below and the letters to the left of the box go in each box to the right. Then we showed them an empty Punnet Square with the same letters before. We inquired about what happens when two variables are multiplied together, then filled out the boxes with multiplication signs in between the letters. The students responded well and were able to grasp the concept fairly well from the onset.

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E1. How can technology be used to effectively engage students with this topic?

The internet is fast becoming the only place students will go for helpful solutions to school problems. This activity is designed to be a review of multiplying binomials that would allow students to use some internet resources, but make them report as to why the resource is helpful. The class will go to the computer lab or have laptops wheeled in and they will be given a list of sites that cover binomial multiplication. They will pick a site and write about the following qualities of their chosen site: what kind of site? (calculator, tutorial, manipulative, etc.), how is it presented? (organized/easy to use), was it helpful? (just give an answer opposed to listing the steps), did it describe the method it used?, can you use it to do classwork?, etc.

This is a sample list, I would want more sites, but it gives the general idea I’m going for. (general descriptions in parentheses for this project’s sake)

http://www.mathcelebrity.com/binomult.php (calculator, shows basic steps of FOIL of inputted problem)

http://www.webmath.com/polymult.html (calculator, shows very detailed and specific steps of FOIL of inputted problem)

http://calculator.tutorvista.com/foil-calculator.html (calculator, shows general steps of FOIL, not the inputted problem)

http://www.coolmath.com/crunchers/algebra-problems-multiplying-polynomials-FOIL-1.html (calculator but only problems it gives itself, more of a practice site)

http://www.mathwarehouse.com/algebra/polynomial/foil-method-binomials.php (FOIL tutorial site with practice problems with hidden steps)

http://www.themathpage.com/alg/quadratic-trinomial.htm (wordy explanation, lots of practice problems with hidden answers)

https://www.khanacademy.org/math/algebra/multiplying-factoring-expression/multiplying-binomials/v/multiplying-polynomials-2 (many tutoring videos, just the writing no person)

http://www.zooktutoring.com/now-available-my-very-first-instructional-math-video/ (many tutoring videos, tutor is seen with the work)

http://illuminations.nctm.org/Activity.aspx?id=3482 (algebra tile manipulator)

I will assume as a teacher that my students already look for easy solutions online, so I want to make sure they look in places that will help them gain understanding. I would stress that calculator sites are dangerous because if you just use them then you will not be able to perform on your own, but could be helpful to check your answer if you were worried. At the end of the lesson they would have a greater understanding of how to use internet sources effectively and have reviewed multiplying binomials.

 

Resources:

http://www.mathcelebrity.com/binomult.php

http://www.webmath.com/polymult.html

http://calculator.tutorvista.com/foil-calculator.html

http://www.coolmath.com/crunchers/algebra-problems-multiplying-polynomials-FOIL-1.html

http://www.mathwarehouse.com/algebra/polynomial/foil-method-binomials.php

http://www.themathpage.com/alg/quadratic-trinomial.htm

https://www.khanacademy.org/math/algebra/multiplying-factoring-expression/multiplying-binomials/v/multiplying-polynomials-2

http://www.zooktutoring.com/now-available-my-very-first-instructional-math-video/

http://illuminations.nctm.org/Activity.aspx?id=3482

How to Avoid Thinking in Math Class (Part 6)

Math With Bad Drawings had a terrific series on how students try to avoid thinking in math class. A quote:

I had a surreal moment this year. I’d almost finished a lesson when one boy, usually a hyperkinetic little bundle of enthusiasm, raised his hand.

“So, like, I don’t really understand anything you’re saying,” he informed me, “But I can still get the right answer.”

He smiled, waiting.

“Which part is giving you trouble?” I asked.

“Oh, you were talking about this extra stuff,” he said, “like the ideas behind it and everything. I don’t… you know… do that.”

I blinked. He blinked. We stood in silence.

“So is that okay?” he concluded. “I mean, as long as I can get the right answer?”

Here is Part 6: http://mathwithbaddrawings.com/2015/02/11/the-church-of-the-right-answer/

How to Avoid Thinking in Math Class (Part 5)

Math With Bad Drawings had a terrific series on how students try to avoid thinking in math class. A quote:

As for students, it can be frightening to start a math problem. You don’t know quite where it will lead. Will my approach be fruitful? Will it falter? Where do I even begin?

But unlike my desk-perching student, most kids don’t recognize that one rope holding them back is fear of the unknown. They just hesitate: too afraid to leap without a net, but never bothering to go in search of a net for themselves…

In all these cases, students are refusing to engage with their uncertainty. But if you’re uncomfortable with doubt, you’ll never break through to the other side. You’ll never have a “Eureka!” moment or an intellectual “Aha!” You’ll never… well… learn. After all, if you can’t bear to face the unknown, how will you ever come to know it?

I find that my desk-percher has it right. At times like these, the mere presence of an expert can supply the confidence you’re lacking.

Here is Part 5, introducing what happens when students get stuck getting started on a problem: http://mathwithbaddrawings.com/2015/02/04/fearing-the-unknown/

How to Avoid Thinking in Math Class (Part 4)

Math With Bad Drawings had a terrific series on how students try to avoid thinking in math class. A quote:

This speaks more to my naiveté as a first-year teacher than anything else, but I was shocked to find how fervently my students despised the things they called “word problems.”

“I hate these! What is this, an English lesson?”

“Can’t we do regular math?”

“Why are there words in math class?”

Their chorus: I’m okay with math, except word problems.

They treated “word problems” as some exotic and poisonous breed. These had nothing to do with the main thrust of mathematics, which was apparently to chug through computations and arrive at clean numerical solutions.

I was mystified—which is to say, clueless. Why all this word-problem hatred?

Here is Part 4, addressing students’ fears of word problems: http://mathwithbaddrawings.com/2015/01/28/the-word-problem-problem/

How to Avoid Thinking in Math Class (Part 3)

Math With Bad Drawings had a terrific series on how students try to avoid thinking in math class. A quote:

Every rule – even the craziest, most arbitrary mandate – has a reason rooted in this essential purpose. (Why leave the dishes with big particles? Because the person is still eating!) And so it is in math class. If you understand slope not as “that list of steps I’m supposed to follow” but as “a rate of change,” things start making more sense. (Why is it the ratio of the coefficients? Because, look what happens when x increases by 1!)

You get to work a lot less, and think a lot more.

Now, conceptual understanding alone isn’t enough, any more than procedures alone are enough. You must connect the two, tracing how the rules emerge from the concepts. Only then can you learn to apply procedures flexibly, and to anticipate exceptions. Only then will you get the pat on the back that every robot craves.

With my students on Friday, I garbled the whole analogy. I tend to do that.

But there’s a simple takeaway. Even if you don’t care about understanding for its own sake; even if you’re indifferent to the beauty and deeper logic of mathematics; even if you care only about test results and right answers; even then, you should remember that the “how” is rooted in the “why,” and you’re unlikely to master methods if you disregard their reasons.

Here is Part 3, addressing the importance of both computational proficiency and conceptual understanding: http://mathwithbaddrawings.com/2015/01/21/are-you-a-dish-washing-robot/

How to Avoid Thinking in Math Class (Part 2)

Math With Bad Drawings had a terrific series on how students try to avoid thinking in math class. A quote:

Occasionally, we teachers grow frustrated with our formula-thirsty students. (Okay, more like “often” or “weekly.”) Sometimes, we even denounce formulas altogether, deriding them as “brainless plug-and-chug” or “not real math.”

Of course, that’s going too far. The intelligent use of formulas is an important part of mathematics. But we’re right about one thing: there’s a lot more to formulas than just throwing numbers into a blender.

Here is Part 2, addressing students’ natural desire to mindlessly plug numbers into a formula without conceptual understanding: http://mathwithbaddrawings.com/2015/01/14/mmm-strawberry-rhuburb-root-2/

How to Avoid Thinking in Math Class (Part 1)

Math With Bad Drawings had a terrific series on how students try to avoid thinking in math class. A quote:

In teaching math, I’ve come across a whole taxonomy of insidious strategies for avoiding thinking. Albeit for understandable reasons, kids employ an arsenal of time-tested ways to short-circuit the learning process, to jump to right answers and good test scores without putting in the cognitive heavy lifting. I hope to classify and illustrate these academic maladies: their symptoms, their root causes, and (with any luck) their cures.

Here is Part 1, introducing the series: http://mathwithbaddrawings.com/2015/01/07/how-to-avoid-thinking-in-math-class/

Helping Mathematics Students Survive the Post-Calculus Transition

Every so often, I’ll publicize through this blog an interesting article that I’ve found in the mathematics or mathematics education literature that can be freely distributed to the general public. Today, I’d like to highlight Michael J. Cullinane (2011) Helping Mathematics Students Survive the Post-Calculus Transition, PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies, 21:8, 669-684, DOI:10.1080/10511971003692830

Here’s the abstract:

Many mathematics students have difficulty making the transition from procedurally oriented courses such as calculus to the more conceptually oriented courses in which they subsequently enroll. What are some of the key “stumbling blocks” for students as they attempt to make this transition? How do differences in faculty expectations for students and student expectations for themselves contribute to the “transition dilemma?” What might faculty incorporate into students’ learning experiences during the transition to help students better navigate the shift from procedural to conceptual, from concrete to abstract? This article offers some lessons learned in connection with these questions.

The full article can be found here: http://dx.doi.org/10.1080/10511971003692830

A veteran teacher turned coach shadows 2 students for 2 days – a sobering lesson learned

Last October, I read the following interesting blog post about a teacher who placed herself in the position of her students for a couple of days: http://grantwiggins.wordpress.com/2014/10/10/a-veteran-teacher-turned-coach-shadows-2-students-for-2-days-a-sobering-lesson-learned/

The lessons learned from this exercise partially explain why I’m an advocate for inquiry-based learning… under the firm presupposition that teaching with this method is an acquired skill, and that this technique can go south in a hurry if it’s not exercised properly.