Is FOIL to difficult for your students? Try Using the Box Method.

I tutored a student this summer who was getting ready to take Algebra II. He is a very visual, concrete person that needs many visuals to help him to succeed in math. We worked quite a bit on multiplying two binomials.

There are three different techniques you can use for multiplying polynomials. You can use the FOIL method, Box Method and the distributive property. The best part about it is that they are all the same, and if done correctly, will render the same answer!

Because most math teachers start with FOIL, I started there. The letters FOIL stand for First, Outer, Inner, Last. First means multiply the terms which occur first in each binomial. Then Outer means multiply the outermost terms in the product. Inner is for "inside" so those two terms are multiplied—second term of the first binomial and first term of the second). Last is multiplying the last terms of each binomial. My student could keep FOIL in his head, but couldn't quite remember what the letters represented, let alone which numbers to multiply; so, that method was quickly laid aside. 

I next tried the Box Method. Immediately, it made sense to him, and we were off to the races, so to speak. He continually got the right answer, and his confidence level continued to increase. Here is how the Box Method works.

First, you draw a 2 x 2 box. Second, write the binomials, one along the top of the box, and one binomial down the left hand side of the box. Let's assume the binomials are 2x + 4 and x + 3.

          (2x + 4) (x + 3)

Now multiply the top row by x; that is x times 2x and x times +4., writing the answers in the top row of the box, each in its own square.  After that, multiply  everything in the top row by +3, and write those answers in the second row of the box, each in its own square.

Looking at the box, circle the coefficients that have an x. They are located on the diagonal of the box.

To find the answer, write the term in the first square on the top row, add the terms on the diagonal, and write the number in the last square on the bottom row. Voila! You have your answer!

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Terrible at factoring trinomials (polynomials) in algebra? Then try this method which never fails! It is the one most students understand and grasp. This step-by-step guide teaches how to factor quadratic equations in a straightforward and uncomplicated way. It includes polynomials with common monomial factors, and trinomials with and without 1 as the leading coefficient. Some answers are prime. This simple method does not treat trinomials when a =1 differently since those problems are incorporated with “when a is greater than 1” problems.

Math Patterns to Investigate!

Some people say mathematics is the science of patterns which I think is a pretty accurate description. Not only do patterns take on many forms, but they occur in every part of mathematics. But then again patterns occur in other disciplines as well. They can be sequential, spatial, temporal, and even linguistic.

Recognizing number patterns is an important problem-solving skill. If you recognize a pattern when looking systematically at specific examples, that pattern can then be used to make things easier when needing a solution to a problem.

Mathematics is especially useful when it helps you to predict or make educated guesses, thus we are able to make many common assumptions based on reoccurring patterns. Let’s look at our first pattern below to see what we can discover.

What can you say about the multiplicand? (the number that is or is to be multiplied by another. In the problem 8 × 32, the multiplicand is 32.) Did you notice it is multiples of 9? What number is missing in the multiplier?

 
Now look at the product or answer. That’s an easy pattern to see! Use a calculator to find out what would happen if you multiplied 12,345,679 by 90, by 99 or by 108? Does another pattern develop or does the pattern end?

 
Here is a similar pattern that uses the multiples of 9. How is the multiplier in this pattern different from the ones in the problems above? Look at the first digit of each answer (it is highlighted). Notice how it increases by 1 each time. Now, observe the last digit of each answer. What pattern do you see there? Using a calculator, determine if the pattern continues or ends.

Recognizing, deciphering and understanding patterns are essential for several reasons. First, it aids in the development of problem solving skills. Secondly, patterns provide a clear understanding of mathematical relationships. Next, the knowledge of patterns is very helpful when transferred into other fields of study such as science or predicting the weather. But more importantly, understanding patterns provides the basis for comprehending Algebra since a major component of solving algebraic problems
is data analysis which, in turn, is related to the understanding of patterns. Without being able to recognize the development of patterns, the ability to be proficient in Algebra will be limited.

So everywhere you go today, look for patterns. Then think about how that pattern is related to mathematics. Better yet, share the pattern you see by making a comment on this blog posting.

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Check out the resource Pattern Sticks. It might be something you will want to use in your classroom.

A Negative number times a Negative Number Equals a Positive Number? Are You Kidding?

Have you ever wondered why a negative number times a negative number equals a positive number? As my mathphobic daughter would say, "No, Mom. Math is something I never think about!" Well, for all of us who tend to be left brained people, the question can be answered by using a pattern. After all, all math is based on patterns!

Let's examine 4 x -2 which means four sets of -2. Using the number line above, start at zero and move left by twos, four times. Voila! The answer is -8. Locate -8 on the number line above.

Now try 3 x -2. Again, begin at zero on the number line, but this time move left by twos, three times. Ta-dah! We arrive at -6. Therefore, 3 x -2 = -6.

On the left is what the mathematical sequence looks like. Moving down the sequence, observe that the farthest left hand column decreases by one each time, while the -2 remains constant. Simultaneously, the right hand answer column increases by 2 each time. Therefore, based on this mathematical pattern, we can conclude that a negative number times a negative number equals a positive number!!!!

Isn't Mathematics Amazing?

Myths and Fun Facts about St. Patrick's Day

March 17th is St. Patrick’s Day; so, for fun, let’s explore some of the

myths surrounding this Irish holiday as well as a few fun facts.

Myths

1) St. Patrick was born in Ireland. Here is a surprise; St. Patrick isn’t Irish at all! He was really born in Britain, where as a teen, he was captured, sold into slavery, and shipped to Ireland.

2) St. Patrick drove all of the snakes out of Ireland. It’s

true there are none living in Ireland today, but according to scientists, none every did. You can’t chase something away that isn't there in the first place!

3) Since the leaves of a shamrock form a triad (a group of three), St. Patrick used it to describe the Trinity, the Father, the Son, and the Holy Spirit so that people could understand the Three in One. However, there is nothing in any literature or history to support this idea although it does make a great object lesson.

4) Legend says each of the four leaves of the clover means something. The first leaf is for hope; the second for faith; the third for love and the fourth leaf is for luck. Someone came up with this, but since a clover is just a plant, the leaves mean absolutely nothing.

5) Kissing the Blarney Stone will give you the eloquent power of winning or convincing talk. Once upon a time, visitors to this stone had to be held by the ankles and lowered head first over the wall surrounding the Blarney Stone to kiss it. Those attempting this were lucky not to receive the kiss of death.

Fun Facts

1) The tradition of wearing green originally was to promote Ireland otherwise known as "The Green Isle." After the British invasion of Ireland, few people wore green because it meant death. It would be like wearing red, white, and blue in the Middle East today. When the Irish immigrated to the U.S. because of the potato famine, few were accepted and most were scorned because of their Catholic beliefs. For fear of being ridiculed and mocked only a small number would wear green on St. Patrick’s Day. Those who didn't adorn green were pinched for their lack of Irish pride. This “pinching” tradition continues today.

2) Did you know that in 1962, Chicago, Illinois began dying the Chicago River green, using a vegetable dye? An environmentally safe dye is used in amounts that keep the river festively green for about four to five hours.

3) The Irish flag is green, white, and orange. The green represents the people of southern Ireland, and orange signifies the people of the north. White is the symbol of peace that brings the two groups together as a nation. 

4) A famous Irish dish is cabbage and corned beef which I love to eat!

It is estimated that there are about 10,000 regular three-leaf clovers for every one lucky four-leaf clover you might find. Those aren’t very good mathematical odds whether you are Irish or not!

Want some St. Patrick's Day activities for your classroom? 

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Check out these three resources.

• Pot of Gold Glyph for Grades 1-3
• March Fraction Word Puzzles for Grades 5-7
• St. Patrick's Day Crossword for Grades 6-8