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When Multiplying Polynomials, FOIL Doesn't Always Work!

Using FOIL
In more advanced math classes, many instructors happen to hate "FOIL" (including me) because it only provides confusion for the students. Unfortunately, FOIL (an acronym for first, outer, inner and last) tends to be taught as THE way to multiply all polynomials, which is certainly not true. As soon as either one of the polynomials has more than a "first" and "last" term in its parentheses, the students are puzzled as well as off course if they attempt to use FOIL. If students want to use FOIL, they need to be forewarned: You can ONLY use it for the specific case of multiplying two binomials. You can NOT use it at ANY other time!

When multiplying larger polynomials, most students switch
to vertical multiplication, because it is much easier to use, but there is another way. It is called the clam method. (An instructor at the college where I teach says that each set of arcs reminds her of a clam. She’s even named the clam Clarence; so, at our college, this is the Clarence the Clam method.)

Let’s say we have the following problem:

(x + 2) (2x + 3x – 4)

Simply multiply each term in the second parenthesis by the first term in the first parenthesis. Then multiply each term in the second parenthesis by the second term in the first parenthesis.

I have my students draw arcs as they multiply. Notice below that the arcs are drawn so they connect to one another to designate that this is a continuous process. Begin with the first term and times each term in the second parenthesis by that first term until each term has been multiplied.


When they are ready to work with the second term, I have the students use a different color.  This time they multiply each term in the second parenthesis by the second term in the first while drawing an arc below each term just as they did before.  The different colors help to distinguish which terms have been multiplied, and they serve as a check point to make sure no term has been missed in the process.

As they multiply, I have my students write the answers horizontally, lining up the like terms and placing them one under the other as seen below. This makes it so much easier for them to add the like terms:


This "clam" method works every time a student multiplies polynomials, no matter how many terms are involved.

Let me restate what I said at the start of this post: "FOIL" only works for the special case of a two-term polynomial multiplied by another two-term polynomial. It does NOT apply to in ANY other case; therefore, students should not depend on FOIL for general multiplication. In addition, they should never assume it will "work" for every multiplication of polynomials or even for most multiplications. If math students only know FOIL, they have not learned all they need to know, and this will cause them great difficulties and heartaches as they move up in math.

Personally, I have observed too many students who are greatly hindered in mathematics by an over reliance on the FOIL method. Often their instructors have been guilty of never teaching or introducing any other method other than FOIL for multiplying polynomials. Take the time to show your students how to multiply polynomials properly, avoid FOIL, if possible, and consider Clarence the Clam as one of the methods to teach. 


A Go Figure Debut for a Science Teacher Who is New


Meet Katherine Thomas from Michigan. Currently she is an 8th grade science teacher and the coordinator for the International Baccalaureate program at her school. She just completed her 12th year of teaching and her 7th year as the coordinator. She has taught middle school science for all 12 years, ranging from grades 6-8. 

As a science teacher, it is her goal to make her class as hands-on as possible. She believes that learning science is best done through experience; so, activities, labs, experiments, and demonstrations are the focal point of her classes. Her class periods are 80 minutes long, so she tries to make sure every class has some form of a hands-on learning experience. It can be challenging to come up with this many activities; so, Katherine makes sure to share her ideas and lessons with others through her TPT store while, at the same time, trying to encourage others to find simple yet impactful ways to enhance the learning in their classrooms.

Katherine’s favorite part of teaching is the relationships she builds with her students. She feels that when she takes the time to learn about her students, who they are, what they like, and where they come from, they are more receptive and responsive to learning. It is her goal to know something about each of her students in order to engage in personal conversations whenever possible.

Outside of school, Katherine spends time with her husband and three sons, ages 17, 13, and 10, and her Bernedoodle, Dolly (a cross between a Bernese Mountain Dog and a Poodle). Her family enjoys spending time on the lake in their boat and at their family lake house, which she finds to be the perfect place to catch up on reading throughout the summer.

Her Teachers Pay Teachers store, Maude in the Middle, currently contains 178 products, 12 of them being free and 166 being paid resources. Katherine focuses on IB support materials, science labs and activities, and general middle school activities. She also has classroom d├ęcor available in her store that is perfect for the middle school level.

Katherine’s featured free item is called Candy Bar Density Activity.  This quick, easy to set up activity is a way to review or teach students how to find the density of an object using delicious candy bars! Students use a scale or balance, calculate volume using the water displacement method, and use the density equation to find the candy with the greatest density. It is appropriate for grades 6-8 or homeschool students.

$4.25
Katherine’s highlighted paid resource, Heat Transfer Popcorn Lab, teaches students about the three types of heat transfer (convection, conduction, radiation) with a fun, edible activity. Using popcorn, students will learn first-hand how to differentiate the different forms of heat transfer by using a microwave (radiation), a hot plate (conduction), and an air popper (convection). This allows students to see real life examples of these concepts, and the students are more engaged if they get to eat the activity! It is designed for 6th and 7th grade students; however, it could be adjusted to work for 5th or 8th.

Most everyone knows that students learn by doing. Hands-on science encourages the children to DO something – observe, ask questions, touch, smell, experiment. Allowing children to really get into science gives them the chance to make discoveries on their own. That is the goal of Katherine's science resources; so, head on over to her store to determine which ones you can use in your classroom.

The Golden Ratio - Finding Patterns in Nature

 As stated in a previous blog post, we come across Fibonacci numbers almost every day in real life. For instance, my husband and I were at the Wonders of Wildlife Aquarium in Springfield, Missouri. (If you haven't been, you should go because it is spectacular.) He was noticing how the herrings were swimming counter clockwise and discussing the Coriolis effect with the guide. When we got to the lower levels, where the sharks were, they were all swimming in a counterclockwise direction as well. I asked my rocket scientist husband why this was and again he said, with a straight face, "The Coriolis Effect."

Inside of a Nautilus Shell
I then spied seashells and started talking about Fibonacci numbers and the Golden Ratio. (I know the visitors around us were wondering just who we were!) On the right, you will see a picture of the inside of a Nautilus Shell taken by me! It clearly shows the Golden Ratio. (The Golden Ratio is a special number equal to about 1.6180339887498948482. The Greek letter Phi is used to refer to this ratio. Like Pi, the digits of the Golden Ratio go on forever without repeating.) Many shells, including snail shells and nautilus shells, are perfect examples of the Golden spiral.

Are you still not sure what I am talking about? Have you ever watched the Disney movie entitled Donald in Mathmagic Land? (It's an old one that
The Golden Ratio
you can find on You Tube.) Well, in the movie they talk about the Golden ratio. This is a proportion that is found in nature and in architecture. The proportion creates beauty. And that proportion is the Fibonacci sequence! If you divide consecutive Fibonacci numbers you will always get the Golden ratio. Try it! Start with the big numbers. If you divide 89 by 55, you get 1.61. If you divide 55 by 34, you get 1.61. If you divide 34 by 21, you get 1.61, and so on. You can look up the Golden Ratio and explore it more. It’s fun!

As I close, think about these two questions and try to answer them.
  1. Where is the Golden Ratio found in the human body?
  2. Why is the golden rectangle important in architecture and art? 

Examining Fibonacci Numbers

Fibonacci
Even if you were taught about the Fibonacci number sequence in school, you probably don’t remember much about it. As with other higher levels of math, many aren’t sure how Fibonacci could possibly be relevant to their real lives; so, why should they even attempt to remember him or his sequence? In reality, Fibonacci numbers are something you come across practically every day. Even so, let’s go back and start at the beginning.

The Fibonacci number sequence is named after Leonardo of Pisa (1175-1240), who was known as Fibonacci. (I love to say that name because it sounds like I know a foreign language.) In mathematics, Fibonacci numbers are this sequence of numbers:
As you can see, it is a pattern, (all math is based on patterns). Can you figure out the number that follows 89? Okay, let's pretend I waited for at least 60 seconds before giving you the answer….144. By definition, the first two numbers in the Fibonacci sequence are 0 and 1, and each subsequent number is the sum of the previous two. For those who are still having difficulty (like my daughter who is sitting here), it is like this.
  

The next number is found by adding up the two numbers that precede it.
  • The 8 is found by adding the two numbers before it (3 + 5)
  • Similarly, 13 is found by adding the two numbers before it (5 + 8),
  • And the 21 is (8 + 13), and so on!
It is that simple! For those who just love patterns, here is a longer list:

 
Can you figure out the next few numbers?


The Fibonacci sequence can be written as a "Rule “which is:   xn = xn-1 + xn-2   The terms are numbered from 0 forwards as seen in the chart below.   xn is the term number n.   xn-1 is the previous term (n-1) and xn-2 is the term before that (n-2)

Sometimes scientists and mathematicians enjoy studying patterns and relationships because they are interesting, but frequently it's because they help to solve practical problems. Number patterns are regularly studied in connection to the world we live in so we can better understand it. As mathematical connections are uncovered, math ideas are developed to help us be aware of the relationship between math and the natural world. 

As I close, here are two questions to think about:
  1. How might knowing this number pattern be useful? 
  2. What kinds of things can the numbers in the Fibonacci sequence represent?