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When students reach the point in their mathematical curriculum in which they learn to multiply binomial expressions, a typical way they are taught to remember how to multiply these binomials is through the FOIL method. FOIL, first, outside, inside, last, refers to multiplying the first term of the first binomial with the first term of the second binomial, then multiplying the first term of the first binomial with the "outside" term of the second binomial, and so on. While FOIL can be practical for students to first remember how to multiply binomials, it is not a great way to learn this concept because when the students then try to extend FOIL to trinomial expressions, they soon face confusion as it does not follow the same pattern that FOIL does. Hence, teaching students that multiplying binomial/trinomial expressions is just multiple uses of the distributive property is a much wiser way of teaching as they can extend their understanding of binomial multiplication to multiplying trinomial expressions and beyond (Karp et al., 2015).

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FACT #13: Example, Nonexample (Keeley & Tobey, 2011)

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Students would be instructed to write this table on a sheet of paper and take about 10 minutes to jot down examples and nonexamples of when they would use the FOIL method. During this assessment, I would be able to see how they determine if FOIL is a specific method for binomial multiplication and not a method for all multinomial multiplications. I like this FACT because it centralizes the students understanding of multinomial multiplication within the context of FOIL, which hopefully the student will realize is a very limited context at that. This would be turned into me after completion so that I can compare their responses at the end of the lesson in order to determine if conceptual change occurred.

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The idea of this problem is to get students to try apply FOIL to a problem for which it does not work. Students would be split up in groups and given about 5 minutes to compute this multiplication. After they have solved it or not, they will get one minute to explain how they solved it to the rest of the group. As the students are sharing, I would walk around and listen to the ideas that are floating around about how and when FOIL works; I would make sure to jot down the arguments I hear students make and make sure to use it for the next step.

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Following their group discussions, I will open the discussion to the entire class and ask the following question: Can you use FOIL to solve the problem I gave? I would expect that students raise their hands and share their reasonings. As the students share, I would make sure to question the mathematical validity of their statements in order to gauge their understanding of the concept of binomial/trinomial multiplication. I expect students to try to explain how they used FOIL to solve it when most likely they just used the distributive property multiple times. During these moments I would point out the use of the distributive property and how it is the agent at play when we are multiplying binomials and trinomials.

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After we have finished the class discussion, I will have the students return to their groups and give them a new problem of trinomial multiplication, most likely a small change to the previous problem as to not let students get hung up on the smaller details of the computation rather on the detail of using the distributive property. As the students solve this, they will write one to two sentences on how they used the distributive property to solve the problem. Once each student has solved the problem, they will once again share with one another how they solved it this time hopefully using the appropriate terminology i.e., distributive property.

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As the students are realizing their misconception about FOIL, I would circle back to the original FACT we used in the beginning. I would provide the students with a bank of varied binomial/trinomial multiplications so that they can determine whether FOIL is acceptable or not. I'd hope to see only binomial*binomial multiplications in the column of when to use FOIL. This would be individual and the students would turn it in so that I have evidence of conceptual change.

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FACT #24: I used to think... but now I know... (Keeley & Tobey, 2011)

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FACT #24: I used to think..., but now I know...

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FACT #24: I used to think..., but now I know...

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The purpose of this FACT is to get students to compare their previous misconception to their new understanding of multinomial multiplications. The goal of this activity is to help the students become aware of their misconception and verbalize how their new understanding was different from their misconception. At this point, students should realize that FOIL should be used sparingly and only when multiplying binomials. They should be using the concept of the distributive property to define binomial/trinomial multiplication as that is what is actually occurring when they perform these computations.

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Karp, K. S., Bush, S. B., & Dougherty, B. J. (2015). 12 Math Rules That Expire in the Middle Grades. Mathematics Teaching in the Middle School, 21(4), 208-215. doi:10.5951/mathteacmiddscho.21.4.0208