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Reducing Fractions with Pattern Sticks!

When working with fractions, many of my students seem confident in performing the different operations, but a few are still unsure of how to reduce fractions. 

Although I have stressed learning the Divisibility Rules for 2, 5, 10, and the digital root for 3, 6, 9,  some still have difficulty since they do not know their multiplication tables. As a mathematics tool, I have the students make Pattern Sticks, a visual and kinesthetic aid, similar to a multiplication chart like the one on the left. Notice that an extra column (blue) has been added to the chart. (In this space, a hole is punched so that a 1" ring can be inserted to store all of the sticks in one place.)

On the right are the directions for making the Pattern Sticks using a multiplication chart. 

(Side note: My students cut out individual Pattern Sticks which I prefer over cutting a multiplication chart apart.)

I then give the students fractions such as 9/36 to reduce. Using the Pattern Sticks, they search for a column where a 9 and a 36 are lined up in the same column. They easily find it on the 1 strip and the 4 strip. They then take the two strips and line them up so that the 9 is over the 36. (see illustration above) By moving to the left, they discover that 9/36 is the same as 1/4. This is 9/36 in its lowest terms. Also notice that all the fractions in the illustration are equivalent fractions - fractions that have the same value. The Pattern Sticks can also be used to determine what number to divide by and to change improper fractions to mixed numbers.

If you are interested in learning more about Pattern Sticks and how to use them in your classroom, check out the resource entitled Pattern Sticks: A Math Tool for Skip Counting & Reducing Fractions at Teachers Pay Teachers.

FOIL - It Doesn't Always Work!

In more advanced math classes, many instructors happen to hate "FOIL" (including me) because it only provides confusion for the students. Unfortunately, FOIL (fist, 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 mehod.)

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 Buckeye Who's New!

The Caffeine Queen is my newest Go Figure Debut. We have a great deal in common, especially when it comes to THE Ohio State University…..Go Bucks! She has taught both regular education and special education. Like most effective teachers, she is always on the lookout for exciting new teaching strategies. She describes herself as a hands-on teacher who enjoys creating items that are kid friendly. 

She believes RESPECT should be a two way street in any classroom. She says her shining teacher moment occurs daily when she receives hugs from her students! She thinks a fun and welcoming classroom atmosphere, along with engaging and interesting lessons, is truly the recipe for success. The internet world has really brought her teaching to life, and she desires to share some of those ideas and insights with other teachers.

She currently has 54 products in her store, most priced under $4.00. Her store features many math resources for the elementary as well as for middle school. If you visit her blog, you can read interesting and motivating articles about how she teaches math. I particularly like her May 2nd article about multiplication and how she uses shapes to help those who are struggling with two digit problems that require regrouping. Even I can relate to her April 5th post about fractions because my college students still struggle with them!

Her featured free item is a one-page divisibility rules poster that can be used during math class when students are working on factoring, simplifying fractions, etc. A smaller version of the poster is included for students to use in their Interactive Student Notebooks (ISN) or binders so that it is always within reach. Finally, a short worksheet for individual students or partners is included so that they can work on the newly learned rules while committing them to memory.

Her highlighted paid resource is a graphic organizer designed to make teaching the standard multiplication algorithm of two digit multiplication a bit easier to understand. Several different versions of the organizer are included.

The first three ready-made worksheet pages are multiplication without regrouping, with answer keys included. (The standard algorithm is difficult enough for beginners to conquer without having to immediately worry about regrouping.) Once students are comfortable with multiplying without regrouping, they are ready to begin regrouping. Hopefully, regrouping will go more smoothly because of the time spent solving problems without regrouping, and since the students are now familiar with the process of two digit multiplication.

Right now take a few moments to investigate the Caffeine Queen’s store and blog. Once there, why not become a follower, or download something that is free, or better yet, pick up an educational resource for your classroom?

Skip Counting and Learning How to Multiply

Most elementary teachers use the 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 others 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).

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!)

Using Bloom's in Math Class

As one of their assignments, my college students are required to create a practice test using pre-selected math vocabulary. This activity prompts them to review, look up definitions and apply the information to create ten good multiple choice questions while at the same time studying and assessing the material. Since I want the questions to be more than Level 1 (Remembering) or Level II (Understanding) of Bloom's Taxonomy, I give them the following handout to help them visualize the different levels.  My students find it to be simple, self explanatory, easy to understand and to the point.

Level I - Remembering

 What is this shape called?

Level II - Understanding

Circle the shape that is a triangle.

Level III - Applying

       Enclose the circle in a square.

Level IV - Analyzing

     What shapes were used to draw this picture?

Level V - Evaluating

How is the picture above like a real truck?  How is it  different?

Level VI - Creating

Create a new picture using five different geometric shapes.
(You may use the same shape more than once, but you must use five different geometric shapes.)

As teachers, we are only limited by our imagination as to the activities we ask our students to complete to help them prepare for a test. However, we still need to teach and provide information so the students can complete these types of tasks successfully. With the aid of the above chart, my students create well written practice tests using a variety of levels of Bloom's. When the task is completed, my students have also reviewed and studied for their next math exam. I consider that as time well spent!
Using Bloom's in Math

If you would like a copy of the above chart in a similar but more detailed format, it is available on Teachers Pay Teachers as a FREE resource.