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If your students don't understand FOIL, Try Multiplying Binomials 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!

A Go Figure Debut for a Special Ed Math Teacher Who's New!

Meet Brooks Jones. This is Brooks’ ninth year teaching; however, education is her second career. Currently she is a middle school special education teacher, co-teaching in math and English/Language Arts classrooms in grades 6, 7 and 8. She was drawn to the classroom after working in the corporate world, doing graphic design, marketing, public relations and writing. She wanted to do something more meaningful with her days and teaching has certainly provided purpose to her life.

She is licensed in elementary education, special education, reading specialist and high school math, but most of her years in the classroom have been spent as a special education math teacher, providing mostly inclusion services for students needing that extra boost to reach grade level standards. Many of the resources in her store are designed to help middle and high school students understand pre-algebra and algebra concepts. She uses a lot of visuals and color-coding, which helps them remember math basics. In addition, she tries to teach the underlying concepts, not just the algorithms, because she wants students to know why we rely on certain algorithms to solve math problems.

Brooks loves seeing students engaged and interested in math. She believes math can open doors for people, and so she tries to connect the content in the classroom with realities of daily life in order to make learning relevant for students. There is nothing more rewarding than seeing that glow of new understanding unfold on the face of a smiling student. Teaching is truly a miraculous career, and she wouldn't trade it for anything.

Last but not least, Brooks is married with three children: one is grown with two kids of his own; one just started college this fall, and one is still at home enjoying her senior year of high school. They have two cats and a dog. In addition, Brooks likes reading, knitting, playing the ukulele and singing when she isn’t working on new teaching resources!

Brooks store contains 55 products; two of them are free. Most of her resources are math related. Her free item is entitled “Laws and Exponents Poster/Anchor Chart.” Since working with exponents can be confusing, help your students master the properties and laws of exponents with this jam-packed one-page resource, which can be used as a cheat sheet, anchor chart, or classroom poster. Concepts included are the properties of multiplication, division, zero and negative exponents, as well as rational (fractional) exponents. Properties are color-coded, shown with variable symbols and explained in clear language.

Brooks featured paid resource is a math bundle of two resources about quadratic equations. It is an introduction to quadratic functions and the relationship between their equations and graphs using guided notes and practice activities. It is great for beginning a unit on quadratic functions as part of an Algebra I class, or for reviewing concepts before moving to more advanced content in higher-level classes (Algebra 2, Pre-Calculus). By purchasing the bundle, you save 15% vs. buying these resources individually!

Brooks also has a blog called The Educator's Lifeline. I highly recommend that you read the May 26, 2023 post on Math Education and AI.

As many of you know, I work with remedial math students on the college level. This year, we have more students than ever. Brooks’ resources are perfect for these struggling students in that she gives them more than just an abstract way of learning algebra. She uses visuals, color coding, etc. to help them learn and memorize. As a math teacher, I believe that is what we all should be doing; so, take time to check out Brooks’ store, and while you are there, download one of her two freebies.

Algebra - Using Two-Sided Colored Beans to Add and Subtract Positive and Negative Numbers

When it comes to adding and subtracting positive and negative numbers, many students have great difficulty. In reality, it is a very confusing and abstract idea; so, it is important to give the students a concrete visual to assist them in seeing the solution. This idea is based on the Conceptual Development Model which is important to use when introducing new math concepts. (See the July 26, 2023 for more details about this learning model.) As a result, when teaching the concept of adding and subtracting positive and negative numbers, what would fall into each category?

When using the two-sided colored beans, the concrete stage of the Model would be where two-sided colored beans are used as an actual manipulative that can be moved around or manipulated by the students. There are a few rules to remember when using the beans.
  1. The RED beans represent negative numbers.
  2. The WHITE beans represent positive numbers. 
  3. One RED bean can eliminate one WHITE bean, and one WHITE bean can cancel out one RED bean. 
  4. All problems must be rewritten so that there is only one sign (+ or -) in front of each number.
Sample Problem

1) The student is given the problem - 5 + 2.

2) Since -5 is negative, the student gets out five red beans, and then two white beans because the 2 is positive.

3) Since some of the beans are red and two are white, the student must match one red bean with one white bean. (I tell my students that this is barbaric because the red beans eat the white beans. They love it!)

4) Because three red beans have no partner (they're left over) the answer to – 5 + 2 = - 3. (See example above.)

After mastering the concrete stage of the Conceptual Development Model, the students would move on to the pictorial stage. Sketching a picture of the beans would be considered pictorial. Have students draw circles to represent the beans, leaving the circles that denote positive numbers white and coloring the circles that represent negative numbers red.

As an example, let’s do the problem 3 - +5. First, rewrite the problem as 3 - 5. Now draw three white beans. Draw five more beans and color them red to represent -5. Match one white bean to one red bean. Two red beans are left over; therefore, the answer to 3 - +5 is -2.

3 - +5 = 3 – 5 = -2 

When students understand the pictorial stage, then abstract problems such as the ones in textbooks can be presented. (Notice, the textbook is the last place we go for an introduction.) I have found that most of my remedial college students move straight from the concrete stage (beans) to the abstract stage without any problem. Many put away the beans after two or three lessons. What works best for your students as they master this algebraic concept is something you will have to determine.

If you would like a resource that gradually goes through these lessons, you can purchase it on Teachers Pay Teachers. It introduces the algebraic concept of adding and subtracting positive and negative numbers and contains several integrated hands-on activities. They include short math lessons with step-by-step instructions on how to use the beans, visual aids and illustrations, four separate and different practice student worksheets with complete answers in addition to detailed explanations for the instructor.

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Skip Counting and Learning How to Multiply Using Pattern Sticks

Most elementary teachers use a Hundreds Board in their classroom.  It can be used for introducing number patterns, sequencing, place value and more. Students can look for counting-by (multiplication) patterns. Colored disks, pinto beans or just coloring the squares with crayons or colored pencils will work for this. Mark the numbers you land on when you count by two. What pattern do they make? Mark the counting-by-3 pattern, or mark the 7's, etc. You may need to print several charts so your students can color in the patterns and compare them. I usually start with the 2's, 5's and 10's since most children have these memorized.

On the other hand, the Hundreds Board can also be confusing when skip counting because there are so many other numbers listed which easily create a distraction.  I have found that Pattern Sticks work much better because the number pattern the student is skip counting by can be isolated. Pattern Sticks are a visual way of showing students the many patterns that occur on a multiplication table.  Illustrated below is the pattern stick for three. As the student skip counts by three, s/he simply goes from one number to the next (left to right).

Martian Fingers
For fun, I purchase those scary, wearable fingers at Halloween time. (buy them in bulk from The Oriental Trading Company - click under the fingers for the link.) Each of my students wears one for skip counting activities. I call them the Awesome Fingers of Math! For some reason, when wearing the fingers, students tend to actually point and follow along when skip counting.

Most students enjoy skip counting when music is played. I have found several CD's on Amazon that lend themselves nicely to this activity.  I especially like Hap Palmer's Multiplication Mountain.  My grandchildren think his songs are catchy, maybe too catchy as sometimes I can't get the songs out of my mind!

Think about this.  As teachers, if we would take the time to skip count daily, our students would know more than just the 2's, 5's and 10's.  They would know all of their multiplication facts by the end of third grade. And wouldn't the fourth grade teacher love you?!?

IMPORTANT:  If you like this finger idea, be sure that each student uses the same finger every time to avoid the spreading of germs. Keeping it in a zip lock bag with the child’s name on the bag works best. (Believe it or not, when I taught fourth grade, the students would paint and
decorate the fingernails!)

To help your students learn their multiplication facts, you might like the resource entitled Pattern Sticks. It is a a visual way of showing students the many patterns on a multiplication table. It also teaches how you to use the pattern sticks to recognize equivalent fractions, reduce fractions, and to change improper fractions to mixed numbers.