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 Emma White. Her topic, from Algebra: multiplying binomials.

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

When it comes to multiplying binomials, there are various activities that can make this topic interesting and fun. Furthermore, I believe this activity will make the topic of multiplying binomials stick in the students’ heads. For those reading, note that when I refer to FOIL, this is a method that lets you multiply two binomials in a particular order. It stands for: First, Outer, Inner, and Last (for more information on this concept, “Multiplying Binomials by the FOIL Method” by Professor Dave Explains on YouTube does a wonderful job of explaining the concept. The link is down below and skip to time stamp 1:00 for binomials). One resource that makes multiplying binomials more tangible is “FOIL Bingo”. In the resource provided below, a teacher took the time to create various bingo cards with two binomials in each square. The students would have to solve the binomials and when the teacher calls out the product of the binomials, the students would cover that spot and so forth. It is like regular bingo where you want to get a certain amount in a row, blackout the card, get a certain design, etc. The choice is up to the teacher. Another way to do this game (if you’re wanting to conserve time), if give the students a bingo sheet for homework the day before or even as an entry ticket the day of. Then the student could solve the binomials prior to playing the game and will have the answers in front of them instead of having to wait for each student to solve the problem during the game. This could eliminate the risk of going too slow and having students get bored or going too fast and having those who need more time to solve left behind. Lastly, the layout provided in Excel can be altered. Therefore, you could change the values if you wanted to do this activity with your class more than once.

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What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

Around 600-700 AD, the Hindu mathematicians had taken the Babylonia methods of approaching equations a step further when it came to introducing unknowns, sometimes more than one unknown in a single problem. It wasn’t until the Medieval times did the Islamic mathematicians discuss the variable x and how important it was. This is when the binomials theorems evolved. Furthermore, the Islamic mathematicians were able to use many operations on polynomials and soon binomials, such as multiplication, division, finding roots, and more! One thing I find highly fascinating is the Islamic mathematicians advanced the study of algebra, which “flourished during the golden age”. Evermore so, private collections were found in a lost Islamic library, which was destroyed in the 13th Century. These private collections “altered the course of mathematics.” An example of a concept that was furthered studied was the Fibonacci sequence (which is, in my opinion, one of the most fascinating things in math history and how it relates to the world and finding mathematics around us, but that is for another time…). All I can say is the Babylonians, the Hindu and Islamic mathematicians were a driven and mathematically inclined people and it blows my mind how far these people brought the world of mathematics.

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How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

When it comes to finding ways to use technology for multiplying binomials, I truly believe visuals are essential. I’m a little biased since I was introduced to a way of multiply binomials just last semester in one of my teaching classes and it BLEW MY MIND. I wish I knew how to do this earlier in high school!  Essentially, this online source allows the student to use algebra tiles without having them physically in front of them. Therefore, they can use this source if they have technology capable of doing so (such as a phone, computer, tablet, etc.). This source is visual and easy for students to understand and manipulate. The student starts by placing the corresponding tiles for one binomial across the top like a table (would be 4 x-tiles and 2 1-tiles). Along the left side, the other binomial is represented (long ways/up-and-down). You then multiply corresponding values and where they meet in the open area (example: where an x-tile and another x-tile meet, it would become since x times x is ). Algebra tiles can also be used for upcoming topics the students would learn, such as completing the square. For a student who may have trouble grasping the idea of multiplying binomials and struggling to understand the concept of abstracts, using algebra tiles will hopefully help with the misunderstandings and confusion. All I’m saying is if this concept of online algebra tiles assisted a college student and made the topic MUCH easier to visualize and explain, I’m sure most high school students will find the use of technology in their math class interesting. Who knows, some students may come to love math more because of it!

Reference(s):

“Multiplying Binomials by the FOIL Method” by Professor Dave Explains:

https://www.youtube.com/watch?v=RTC7RIwdZcE

“FOIL Bingo”: http://www.teachforever.com/2009/03/two-review-games-multiplying.html

“FOIL Bingo” Direct Link to Download: http://teachforever.googlepages.com/bingoalg1foil.xls

“History of Polynomials”: https://polynomialshistory.weebly.com/history.html

“How modern mathematics emerged from a lost Islamic libray”: https://www.bbc.com/future/article/20201204-lost-islamic-library-maths

Algebra Tiles: https://technology.cpm.org/general/tiles/

Algebra Tiles Example for Multiplying Binomials: https://technology.cpm.org/general/tiles/?tiledata=b5____gWaRx__boy__aabuEq3abvMq3abwUq3abx_q3aby5q3auAbq2auAHq2auBFq2aatdswaatdtEautcveautcvKautcwcautcuMautcxaautcwIauA9q2aeuEssaevMssaewUssaeuEtAaevMtAaewUtAaex_tAaex_ssaey6ssaey6tAaaAessaaAetAaaAKtAaaAKssaaBctAaaBcssaaBItAaaBIssabuEuIabwUuIabx_uIaby5uIabvMuIabuEvaabwUvaabx_vaaby5vaabvMvaaby4v8abx-v8abwTv8abuDv8abuDvGabwTvGabx-vGaby4vGabvLvGabvLv8aby3w6abx+w6abwSw6abuCw6abuCwEabwSwEabx+wEaby3wEabvKwEabvKw6auAdvaauAcvGauAcv8auAbwEauAbw6auAduIauAHw6auAHwEauAIv8auAIvGauAJvaauAJuIauA9w6auA9wEauBav8auBavGauBbvaauBbuIauBGw6auBGwEauBHv8auBHvGauBIvaauBIuIhFCounting%20up%20the%20%22like%20terms%22%2C%20we%20get%3A%0A%0A10%20x%5E2%27s%0A38%20x%27s%0A24%201%27s%0A%0AFor%20an%20answer%20of%3A%0A%0A10x%5E2%20%2B%2038x%20%2B%2024__CwtChF2x%0A%0A%0A%0A%0A%0A%0A%0A%0A%2B%0A%0A%0A%0A%0A%0A%0A%0A%0A6__tXsohF%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%205x%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%2B%20%20%20%20%20%20%20%20%20%20%20%20%204%20%20%20%20%20%20%20%20%20%20%20%20%20__uzrQ

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