Teaching Fractions to Students Who Have No Idea How to Do Them!

I wish I understood this!

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.

Procedure:

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 video lesson is included to help you.

Why Doesn't the U.S. Convert to the Metric System?

Did you know that there are only three nations which do not use the metric system: Myanmar, Liberia and the United States? The U.S. uses two systems of measurement, the customary and the metric. Yes, since our country does use the metric system, we have "given more than an inch, but we haven't gone the whole nine yards".

Today, when we shop for groceries, soda is sold in liters. Medicine is sold in milligrams, food nutrition labels are metric, and what about a 100-meter sprint or a 5K race? Still, we are the only industrialized nation in the world that does not conduct business in metric weights and measures. To be or not to be a metric nation has been a question of great consternation for our country for many years.

Here are some reasons why I think our nation should go to the metric system.

1. It's the measurement system 96% of the world uses. 
2. It is much easier to do conversions since it is based on units of ten. Water freezes at zero, not 32°, and it boils at 100, not 212°. 
3. Teaching two measurement systems to children is time consuming and confusing. 
4. It is the "official" language of science and medicine. 
5. Its use is necessary when you travel outside of the United States. 
6. Conversion from customary to metric is often fraught with errors. Because the metric system is a decimal system of weights and measures, it is easy to convert between units. 
7. There are fewer measures to learn. Once you learn the meaning of the prefixes, you can easily convert mass, volume and distance measurements. No further conversion factors need to be memorized except the specific power of 10. For the Customary System you have to remember 5280 feet = 1 mile, 4 quarts = 1 gallon, 3 feet = 1 yard, 16 oz. = 1 pound, etc. 
8. And just think, I would have less clutter in my kitchen since I wouldn’t need liquid and dry measuring cups or teaspoons and tablespoons! All I would need is a scale and liquid measuring cups!

So, while most nations use the metric system, the United States still clings to pounds, inches, and feet. Why do you think Americans refuse to convert? I’d be interested in your perspective and ideas.

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Get ready to gauge your students' proficiency and equip them for success in all things metric using this pre-assessment metric test. This math test is designed to assess your students' pre-existing knowledge of the metric system. Not only will your students gain a deeper understanding of the differences between metric and customary units of measurement, but with the help of visual examples, they will be able to remember those pesky measurements.

Using Mnemonic Techniques to Help Students Memorize

In my college class entitled Conquering College, we have been working on ways to remember for tests. Of course, mnemonic devices came up. Mnemonics connect new learning to prior knowledge through the use of visual and/or acoustic cues. Such strategies assist students in remembering and recalling larger pieces of information for tests. Included in mnemonics are acronyms, initialism, acrostics, rhyme, rhythm and song and association in addition to visualization using the loci and peg systems. Let's look at four of these categories.

1) Acronyms - A word formed from the first letters of each one of the words in a phrase.

• HOMES – The names of the 5 Great Lakes – Huron, Ontario, Michigan, Erie, Superior 

• ROY G. BIV – The colors in a rainbow – Red, Orange, Yellow, Green, Blue, Indigo, Violet 

• SCUBA - When you’re scuba diving, you’re using a “self-contained underwater breathing apparatus.” 

2) Acrostics – Sentences created from the first letters of key words.

• Please Excuse My Dear Aunt Sally – for the order of operations 

Parenthesis, Exponents, Multiplication, Division, Addition, Subtraction 

    **I personally prefer the phrase: Pale Elvis Meets Dracula After School. 

• My Very Earthly Mother Just Sliced Up Neptune.  – the planets in order from largest to smallest: 

Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune 

   **I particularly like this one since Earthly gives you a clue that the third planet is earth and Neptune is listed last. This means you only have to know 6.

3) Rhyme, Rhythm, Song – poems, limericks or silly songs – These work well for auditory learners.

• I before E, except after C and in sounding like A as in neighbor and weigh.
• In 1492, Columbus sailed the ocean blue.
• Twinkle, twinkle little star; circumference is 2 π r.    (I actually sing this for my students!)

4) Association – finding a common element. The association is usually coincidental.

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• Litmus Paper: Blue = Base – both begin with “B”. 
• Arteries: Artery = Away – both begin with “A”. 
• The principal is my PAL. Helps to distinguish from principle. 
• Affect = Action (a verb) Helps to separate it from effect which is a noun.

These ideas plus many more are in a resource called Mnemonic Techniques found on Teachers Pay Teachers. 

Games - An Important Part of Learning Math

I currently teach remedial math students on the college level. These are the students who fail to pass the math placement test to enroll in College Algebra - that dreaded class that everyone must pass to graduate. The math curriculum at our community college starts with Basic Math, moves to Fractions, Decimals and Percents, and then to Basic Algebra Concepts. Most of my students are smart and want to learn, but they are deeply afraid of math. I refer to them as mathphobics.

We all have this type of student in our classrooms, whether it is middle school, high school, or college. When working with this type of student, it is important to bear in mind how all students learn. I always refer back to the Conceptual Development Model which states that a student must first learn at the concrete stage (use manipulatives) prior to moving to the pictorial stage, and well in advance of the abstract level (the book). This means that lessons must include the use of different manipulatives.

I use games a great deal because it is an easy way to introduce and use manipulatives without making the students feel like “little kids.” I can also control the level of mathematical difficulty by varying the rules; thus, customizing the game to meet the instructional objectives my students are learning. However, as with any classroom activity, teachers should monitor and assess the effectiveness of the games. 

When using games, other issues to think about are:

1. Excessive competition. The game is to be enjoyable, not a “fight to the death”.
2. Mastery of the mathematical concepts necessary for successful play. Mastery should be at an above average level unless teacher assistance is readily available when needed. A game should not be played if a concept has just been introduced.
3. Difficulty of the rules. If necessary, the rules should be modified or altered in order that the students will do well.
4. Physical requirements (students with special needs). These should be taken into account so that every player has an opportunity to win.

In addition to strengthening content knowledge, math games encourage students to develop such skills as staying on task, cooperating with others, and organization. Games also allow students to review mathematical concepts without the risk of being called “stupid”. Furthermore, students benefit from observing others solve and explain math problems using different strategies.

Games can also….

1. Pique student interest and participation in math practice and review. 
2. Provide immediate feedback for the teacher. (i.e. Who is still having difficulty with a concept? Who needs verbal assurance? Why is a student continually getting the wrong answer?)
3. Encourage and engage even the most reluctant student.
4. Enhance opportunities to respond correctly.
5. Reinforce or support a positive attitude or viewpoint of mathematics.
6. Let students test new problem solving strategies without the fear of failing.
7. Stimulate logical reasoning.
8. Require critical thinking skills.
9. Allow the student to use trial and error strategies. 

Mathematical games give the learner numerous opportunities to reinforce current knowledge and to try out strategies or techniques without the worry of getting the “wrong” answer. Games provide students of any age with a non-threatening environment for seeing incorrect solutions, not as mistakes, but as steps towards finding the correct mathematical solution. 

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If you want a challenging but fun and engaging math game, try Contact. It is a fun and attention-grabbing way for students to review basic math facts and to use critical thinking without doing another “drill and kill” activity.