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Unlocking Fractions for the Confused and Bewildered - A New Approach to Teaching Fractions

I teach remedial math on the college level, and I find that numerous students are left behind in the mathematical dust if only one strategy is used or introduced when learning fractions. Finding the lowest common denominator, changing denominators, not changing denominators, finding a reciprocal, and reducing to lowest terms are complex issues and often very difficult for many of my students.

I classify my students as mathphobics whose mathematical anxiety is hard to hide. One of my classes entitled, Fractions, Decimals and Percents, is geared for these undergraduates who have never grasped fractions. This article encompasses how I use a different method to teach adding fractions so these students can be successful. Specifically, let's look at adding fractions using the Cross Over Method.

Below is a typical fraction addition problem.  After writing the problem on the board, rewrite it with the common denominator of 6.

1) Ask the students if they see any way to multiply and make a 3 using only the numbers in this problem.

2) Now ask if there is a way to multiply and make 2 using just the numbers in the problem.

3) Finally, ask them to find a way to multiply the numbers in the problem to make 6 the denominator.

4) Instruct the students to cross their arms. This is the cross of cross over and means we do this by cross multiplying in the problem.

5) Multiply the 3 and 1, then write the answer in the numerator.  *Note: Always start with the right denominator or subtraction will not work.

6) Next multiply the 2 and 1 and write the answer in the numerator. Don’t forget to write the + sign. *Note: One line is drawn under both numbers. This is to prevent the students from adding the denominators (a very common mistake).

7) Now have the students uncross their arms and point to the right using their right hand. This is the over part of cross over. It means to multiply the two denominators and write the product as the new denominator.

8) Add the numerators only to find the correct answer.

9) Reduce to lowest terms when necessary.

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It is important that students know the divisibility rules for 2, 3, 5, 6, 9 and 10. In this way, they can readily reduce any problem. In addition, it is extremely important that the students physically do the motions while they learn. This not only targets the kinesthetic learner but also gives the students something physical that makes the process easier to remember. The pictures or illustrations for each technique also benefit the visual/spatial learner. Of course, the auditory student listens and learns as you teach each method. 

I have found these unconventional techniques are very effective for most of my students.  If you find this strategy something you might want to use in your classroom, a resource on how to add, subtract, multiply, and divide fractions is available by clicking the link under the resource cover.

A Go Figure Debut for A Tutor Who Is New!

Caroline lives on the beautiful Wirral Peninsula, in between Liverpool and Chester, in the UK with her husband Ray. They have five grown up children and one grandchild. When they are not working. they love to travel and to spend time with their family and friends.

Caroline left the classroom around five years ago to start her own tutoring business; hence, the name of her Teachers Pay Teachers Store – The Booked Up Tutor. She worked for eight years as a teaching assistant at an inner-city primary school and while there gained a First-Class Honours Degree in Teaching. (An honours degree is a grading classification in the United Kingdom that distinguishes achievements of undergraduates. A degree with honours denotes a degree in a given subject that requires a higher level of academic learning and achievement.) She wanted the freedom to combine her years of experience with her interest in how we all learn differently while at the same time having the freedom to run her own business. She says she has never looked back and now help many students each week to improve their math and English. For her, to be fully booked with a permanent waiting list, is a wonderful feeling because she was just like the kids she now tutors. She could NOT do math and would sit and cry during lessons. Caroline never wants any child that she teaches to feel that way; she wants them to achieve what they can achieve and to be the best possible version of themselves.

Caroline of The Booked Up Tutor
By tutoring, she has become very adept at quickly identifying weak areas in math and subsequently helping students to improve greatly. She teaches her students quick mental math strategies that help them, and always starts by making sure that they know their number bonds (alternatively called an addition fact) to ten. Sadly, she has discovered that even some eight and nine-year olds don’t know them. (I teach college students who don’t know them!) Once they know these, it is a smooth transition to learn them to 100 and then 20 and then beyond that. She is also a huge multiplication fan. She believes times tables are the golden key to math in primary school and if a student doesn’t know them then obviously, they will struggle with division, fractions, prime numbers, area of shapes and many other things. (I see this all the time as I tutor college students in the Math Lab at the college where I teach.) 

Caroline has 385 diverse and assorted resources in her store. Her resources have all been developed from the ground up thanks to her students who range in age from 5 to 11 and sometimes older. Her products tend to share the theme of over-learning as that is what students often must do, but it works, and it helps her struggling students with reading and math.

One of her free items is called Number Bonds to 10. This FREE pack contains her number bonds rhyme that help students remember the number facts to ten plus color-coded support worksheets that teach the fact families. Help your students easily learn the addition and subtraction facts within ten by downloading this free item. 

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If you would like another resource for multiplication, check out her bundle called Multiplication Worksheets Bundle. Master multiplication worksheets, multiplication facts and division facts 1 to 12 with these 12 x 34-page booklets of multiplication worksheets. Self-paced times tables booklets (with answers) help students understand and secure basic math mastery of times tables and multiplication fluency through skip-counting, arrays, multiples, fact families and self-assessment. Use for extra practice, math homework, in remediation groups, for special education, math intervention or math centers.

As a late-comer to Math, Caroline really enjoys making sure no child is left behind with poor teaching, as she was. She never stops learning; her experience grows every single day and her students benefit from that. This is evident in the resources she creates; so, be sure to take some time to check them out.

Developing and Writing Effective Lesson Plans for Math

We often hear of research based strategies and how to use them in our classrooms. Having worked at two colleges in the past ten years, I have discovered that some who are doing this research have never been in a classroom or taught anyone under the age of 18!  (Sad but True)  Then there are others who truly understand teaching, have done it, and want to make it more effective for everyone. That's the kind of research I am anxious to use.  I came across the Conceptual Development Model while teaching a math methods class to future teachers. It was one of the first research models that I knew would work. 

The Conceptual Development Model involves three stages of learning: 1) concrete or manipulative, 2) pictorial, and 3) the abstract.  The concrete stage involves using hands-on teaching which might involve the use of math manipulatives or real items. Next, the pictorial stage utilizes pictures to represent the real objects or manipulatives. A visual such as a graphic organizer would also fit in this stage. Last, the abstract stage of development entails reading the textbook, using numbers to compute, solving formulas, etc. Let's look at two classroom examples.

Example #1:  You are a first grade teacher who is doing an apple unit.  You decide to have the children graph the apples, sorting them by color.

Concrete:  Using a floor graph, the children use real apples to make the graph.

Pictorial:  The children have pictures of apples that they color and then put on the floor graph.

Abstract:  The children have colored circles which represent the apples.

Example #2:  You are a fifth grade teacher who wants to teach how to find the volume of a cube or rectangular solid.

ConcreteBring a large box into the classroom, a box large enough for the children to climb inside, OR have the students build 3-D objects using multi-link cubes.

PictorialGive the students pictures of 3D objects which are drawn but shows the cubes used to make the solid. Have the students count the cubes to determine the volume.

AbstractHave students use the formula l x w x h to find volume.

Requiring my perspective teachers to think about this model and to use it when planning a math unit dramatically changed the quality of instruction which I observed in the classroom. 

Writing Math Lesson Plans
Now that I teach mathphobics on the college level, I am finding this model to be a crucial part of my planning.  Most of my students started math at the abstract level, "Open your books to page...." without any regard to the other two stages of development. Using manipulatives and graphic organizers have changed my students' ability to learn math, and some have even ended the semester by saying, "I like math". Maybe this is a model we should all consider implementing.

If you want more examples and suggestions about using this model to write math lesson plans, click on the link below the resource cover

Also look at the resource entitled Graphing without Paper or Pencil in which is appropriate for grades K-5 and is based on the Conceptual Model of Development: concrete to pictorial to abstract.

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We hope you and your students will enjoy our "Free Back to School Lessons" by The Best of Teacher Entrepreneurs Marketing Cooperative (TBOTEMC) for 2019. On each page, you will find links to a free resource as well as a priced resources created by a member of TBOTEMC.

Inside this 2019 edition of the "Free End of the Year Lessons" you will find resources like:
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