Foil the Fool: The Vertical on Polynomial Multiplication

What we are teaching

Last night I had a great “nightowl #mathchat” — to quote @mathheadinc — with @davidwees, @malynmawby, @mathheadinc and @reflectivemaths about how to teach math. From the tweets I have since received, others have continued this conversation. Its unresolved gist is that we teach a curriculum full of insular skills (little–m math) at the expense of teaching, and even exploring, overarching problem solving schema and Math processes and patterns (big–M Math) often presented in cover stories authentic to students and their cultures. (I talk about little–m and big–M in the Math Labs section of my Teaching Resources website and in another post.)



Though until recently I never owned the terminology to describe the difference, since I first entered junior high I realized the distinction between solving and solution, and proving and proof. It seems obvious that the solution and the proof are the cleaned-up, teacher-readied products of solving and proving processes.

The solution and the proof are what we teach to precision and grade our students on. They are obviously what we value. Just ask your students. And why not? After all, if you can not communicate mathematically, what is the point?

Yet the solving and the proving are where the Math lives! It is through these that problems are solved. You can follow formulae and instructions to produce solutions and proofs, but someone has to create those formulae and instructions, and program those calculators and computers. Someone has to learn and work with the Math and realize its underlying ideas. The whole point of the changing curriculum is to produce leaders and creators, thinkers, not followers.

It is in the rough work where the interesting Math lurks — the messy scribbles, the barely geometric doodles, the scratches and cross-outs and circles and AHAs! It is in the inked or graphite sweat scrawled in lines and blocks and columns on toss-away paper. It is in the work, in what the teacher and the consumer do not see, where Math guides and learning grows. It is (in) this that we should assess. For it is in this that problem solving and true demonstration of problem solving reside.



The solution and the proof summarize the Math. They are not the Math, just our way of communicating its product. If we want to teach Math, we need to change how we teach, value and grade math.

When accountability trumps Math

Which brings us to interesting mathematical (little-m) facts such as 1+2+3+…+(n-2)+(n-1)+n equals (n2/2)+(n/2). Do you know how (big-M)?



Did you know that if the longer series has an odd number of terms it simplifies to n(n+1)/2 for reasons different than why the series does so if it has an even number of terms? What insights and information, patterns and processes, lie within the differences between the oddly and evenly termed series?

Just going with my statement that the two expressions are equivalent, and without considering the Math that equates the two, you lose the following information.


  1. How each series is structured.
  2. How the first series when odd-termed compares and contrasts to itself when it is even-termed.
  3. How the two series are related, or if they even are.
  4. How the two series differ and congrue in their structure and relationship when the first series is odd, then even.


  1. How to approach the first series to simplify it into the second one, in both the even and odd cases.
  2. How to use and apply this Math to solve problems that are different from the original.


  1. How the series and their relationship can be (usability) used and are (applicability) used.
  2. How the series and their relationship can be recognized as relevant to different problems.
  3. How the series and their relationship can be tailored to suit different problems.
  4. How the series and their relationship were first discovered, by whom, and with what related, back and implication stories.

With increased need to account for skill and knowledge in a packed curriculum, math, the minimalist calculation, takes front stage over Math, the engine of engagement and problem solving. Almost all Math is lost in received wisdom, to coin Hewitt (1999).

New wisdom can not form when the Math is forgotten and ignored.

And that is my problem with FOIL.

Through FOIL illiteracy

FOIL is an acrostic mnemonic (standing for First Outside Inside Last) designed to help students apply the distributive law during the specific case of double binomial multiplication. Only it is often taught as a general method of polynomial multiplication, rather than as a learning aid specific to multiplication of a pair of binomials. This is because most curricula only require students be able to multiply binomials. In fact, foiling is used as a verb erroneously synonymous with polynomial multiplication ( 2011).

Problems involving polynomials with more than two terms in one of the factors are not often encountered in schools, so the practice of distributive multiplication beyond FOIL is uncommon. Students I have subbed rarely encounter the limitations of FOIL and the underlying Math and methods of distributive multiplication. In fact FOIL impedes the learning of this Math ensuring continued illiteracy and closing problem solving opportunities for students.

Let’s foil 12 groups of 36 and 23 groups of 456, then move on to algebraic polynomials. Let’s also multiply polynomials vertically, a method of distributive multiplication alternative to foiling that most students I teach understand and favor.



Many students have trouble understanding, and retaining learning of, algebraic polynomials. A lot of this has to do with uncertainty with algebra itself. But additionally many students who learn how to multiply polynomials using FOIL have a problem recognizing and understanding FOIL as a method of distributive multiplication. This clouds student comprehension, leading to Mathematical illiteracy.

The restrictions of FOIL, tabulated in the Prezi, further confuse students when they try to use this case-specific mnemonic to multiply general, non-binomial polynomials together, as they must in the exploratory Math Lab, Strange Dice.

Strange Dice

Strange Dice is a challenging recreational inquiry designed to engage students in the Mathematics of several aspects and levels of polynomial multiplication. The students use technology, imagination, math and Math to:

  1. find how traditional Western and Chinese dice mathematically function, and
  2. create new dice with the same probability distribution as these traditional dice, within given constraints.

Ultimately, the students explore the patterns and processes of the Mathematics underlying the math of probability and polynomial multiplication — how for instance multiplying and factoring polynomials can construct and analyze different and congruent probability distributions and what each polynomial factor contributes to dice probability distribution and die face value.

Additional polynomial multiplication exercises can be found in Polynomial Multiplication of Urban Teaching Resources.

References (2011). Foil.

Hewitt, Dave. (1999). Arbitrary and necessary part 1: A way of viewing the mathematics curriculum. For the Learning of Mathematics 19(3): 2-9.

Urban, Shawn. (2005). Math lab 5: Multiplying polynomials with strange dice. Urban Teaching Resources.

Prezi images (in order of appearance)

Dawson, Fred. (2007). Ball and Chain.

beast love. (2007). Fox Hounds.

Dunn, Natasha C.. (2009). laundry {post}.

Hanchanahal, Nagaraju. (2009). Morning Fog, Nandi Hills, Karnataka.

Wallis, Caro. (2010). Garlic Bread.

Tanaka, Hisako. (2007). Maple.

Arutemu. (2008). Rapier Guard.

Gimpert, Adam. (2006). Genius.

Frangipani Photograph. (2008). Matryoshka Doll.

Orlando, Giovanni. (2008). NEW “Hay Bale” Version.

S., Rishi. (2008). Friends.

Ballez, Romain. (2010). Stairway to Heaven.

Post inspired by

Urban, Shawn. (2005). Math lab 5: Multiplying polynomials with strange dice. Urban Teaching Resources.

Wees, David. (2011). Flipping fractions. Reflections of a Math Teacher Candidate.

6 thoughts on “Foil the Fool: The Vertical on Polynomial Multiplication

  1. Pingback: Wiki-style maths | cartesian product

  2. Interesting post, Shawn.
    I’m surprised you didn’t take it the next step and combine the horizontal and vertical into a grid. In the UK, gridding is usually taught rather than long multiplication (the ‘vertical’ form, as you call it). Rob Eastaway (@robeastaway on Twitter) wrote an article about this last year (2010):

    I also used it as the principle behind: Algebra – (a + b)2 – Why is ab green? ( This principle can then be extended to any pair of polynomials you need to multiply together, and has the added advantage that you are unlikely to forget to multiply one pair of elements.

    • Hi Colin,

      I am aware of the grid method of multiplication and was indeed debating whether to discuss it as well in this post. But I decided the post was long enough as is.

      I had nurtured a couple of points I wanted to make already and felt that adding a third would dilute the post too much.

      On the other hand, I did include (links to) lessons on gridding, or boxing, in my polynomial multiplication lesson on my Teaching Resources site.

      Perhaps I will include a discussion on gridding in a future math (I believe you say maths) post. I rather enjoyed making this Prezi and can see myself making another. (It feels great to be creative!)

      I will take a look at the article you suggested. It sounds interesting. Your question about why ab is green in your lego tiles is intriguing. Students who grasp that topic are sure to have a deeper understanding of the implications of polynomial multiplication.

      It is interesting to see how different North American (Canadian and US) and European (UK) treatments of polynomial multiplication (PM) are. We are getting out of algebra tiles and gridding while you are using at least gridding as the main method of teaching PM. Meanwhile, we are stuck in FOIL, which from what I gather from your comment you have defeated (thankfully). I think the answer is sound exploratory explanation, regardless of method or methods taught. One learns through concepts not procedures.

      Thank you for your interest in this post and your welcome comment.


  3. Pingback: Using Prezi: A review « Digital Substitute

    • Hi Linear Fix,

      Thanks again for commenting and your encouragement. I appreciate both.

      Did you watch the new version or old version of the prezi?

      I made some changes to address the problem of swinging and to clarify some presentation. I was wondering how you regard each version and the changes I made.

      Ha, ha, ha. So I take it you were not too impressed with the introductory etymology of foil in my presentation. I actually added that to emphasize the restrictiveness of FOIL. I like how the unflattering definitions and origins of foil play off the use of the FOIL mnemonic as a “method” of teaching generalized polynomial multiplication. (This is also why I ended the prezi with the same etymology.)

      At least you liked the main presentation and the point that made. I am glad you enjoyed them.


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