Dump and Divide - Converting Fractions to Decimals

When working with fractions, my remedial math college students are never quite sure which number to divide by. This same thing often occurred when I taught middle school and high school. So the question I had to answer was, "How can I help my students remember what number goes where?"

First, the student must understand and know the vocabulary for the three parts of a division problem. As seen in the problem above, each part is correctly named and identified.

Side Note: The symbol separating the dividend from the divisor in a long division problem is a straight vertical bar with an attached vinculum (you might have to look this word up) extending to the left, but it seems to have no established name of its own. Therefore, it can simply be called the "long division symbol" or the division bracket. I wish it were named something fancier, but sometimes plain and straight forward is the best!
Now let's look at a fraction that the student is asked to rewrite as a decimal. The fraction on your right is two-fifths and is read from top to bottom as two divided by five. That's easy enough, but when my students enter this into their calculators, many will put in the 5 first, and then press the

division sign, followed by the 2. Of course, they get the wrong answer. Now let's look at the dump and divide method.

First, dump the 2 into the calculator. Then press the division sign; then divide by 5. The answer is 0.4.

I am aware that many of students are not allowed to use calculators; so, let's look at how this method would work using the division bracket. We will use the same fraction of 2/5 and the same phrase, dump and divide.

First, take the numerator and dump it inside the division bracket. (Note: Use N side instead of inside so that numerator and N side both start with "N".) Now place the 5 outside of the long division bracket and divide. The answer is still .4.

Dump and Divide will also work when a division problem is written horizontally as a number sentence such as: 15 ÷ 3. First, reading left to right, dump 15 into the division bracket. Now place the 3 on the outside. Ask, "How many groups of three are in 15?" The answer is 5.

Try using Dump and Divide with your students, and then let me know how it works. You can e-mail by clicking on the page entitled Contact Me or just leave a comment.

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Something Else to Think About:  

Since many students do not know their multiplication tables, reducing fractions is almost an impossible task. The divisibility rules, if learned and understood, can be an excellent math tool. The resource, Using Digital Root to Reduce Fractions, contains four easy to understand divisibility rules as well as the digital root rules for 3, 6, and 9. A clarification of what digital root is and how to find it is explained. Also contained in the resource is a dividing check off list for the student. Download the preview to view the first divisibility rule plus three samples from the student check off list.

Yes or No? Stay or Go? Solving for the Unknown in a linear equation.


When my basic college algebra classes begin solving equations containing one unknown, I tell them, they are inquisitive detectives looking for the unknown. My students' greatest difficulty is deciding what stays and what goes in an equation. In other words, which term should be cleared by using the inverse operation and which term should stay where it is?

I  start by referring to the written equation as a teeter-totter or a see-saw which must always stay balanced. In other words, the equal sign is the pivotal point and both sides of that = sign must be the same.  We also discuss the importance of the"Whatsoever thou doest to one side of the equation, we must doest to the other". (Out of necessity, I admit that I was with Moses when he received the Ten Commandments, but it "fell upon me" to convey The First Commandment of Solving Equations to future mathematicians.)

One Unknown

We begin with very simple equations such as: x + 9 = 12. Here's the rub; a few of my students know the answer and do not want to show any of their work. Maybe some of you have this type of student as well. Since, after 40+ years, I am still unable to grade what is in their minds, I insist that all steps are written down. I explain that it's like riding a tricycle to ride a bicycle to ride a unicycle.

First, I instruct the students to look at the equation and determine which terms are out of place. (Side note: Because my students are easily confused, at the present, we keep all of the unknowns on the left side and all of the numbers on the right side of the equal sign.) Let's go back to our sample of x + 9 = 12. Because the x is already on the left side of the equation, the students write a "Y" over it for the word, "Yes". The 9 is on the wrong side of the equal sign, so the students write a "N" over it for "No".  Finally, they write a "Y" over the 12 since it is the correct place. They now have exactly what they want, a Y and N on the right side and a Y on the left side. They now must clear anything that has a "N" over it.  The students recognize they if they use the inverse operation of addition, they can clear the 9. They therefore subtract 9 from each side of the equation resulting in an answer of 3.

Many algebra teachers will have the students write the step x + 0 = 9.  You may wish to include this step in the process, but since my college students readily see that +9 and -9 make zero, they put an X over the two opposites to show that they cancel each other out or when added together, they equal zero.

What if the equation is: 3 = y - 4? This always freaks my students out; yet, if they do the yes/no process, they will discover that they have two "no's" and one "yes", not a yes, no = yes.  This means they can rewrite the equation as y - 4 = 3 to get a yes, no = yes. The problem can now easily be solved like the one above.



Unknown on both sides
of the equation

The next step is what to do when an unknown appears on both sides of the equal sign.   Usually, my students are sure they are incapable of solving such a difficult problem, but let's use the yes/no method and see what it looks like. 

Notice in the sample on the left that we have a yes, no = no, yes. We start by clearing the "N" on the left hand side of the equation by using the inverse of -9. We then go to the right side and clear the y by using the inverse operation of addition. (Yes, I am aware both can be cleared at the same time, but again simple and methodical is what is best for my mathphobics.) We then divide each side by 4 resulting in the answer of 3. When the problem is completed, my students are amazed and proud that they could solve such a long equation. (You might notice in the illustration, a dotted line is drawn vertically where the equal sign is. This helps my visual students to separate the two sides of the equation.)

If any of you try this approach with your students or have a different method, I would love to hear from you. Just leave a comment and a short statement of how this process worked for you or what process you use that is even better. That way, we can learn from each other.

I have made math tutorials for the college where I teach, and one of them goes through this process in detail. If you are interested and would like to hear me sing as well, go to:  Yes/No. 

Is zero an even number? Now, that's an "odd" question!

My daughter and her husband are heading to Las Vegas with his family to celebrate his parent's 50th wedding anniversary.  I guess when you are at the roulette table, (never been there or done that) and you bet "even" and the little ball lands on 0 or 00, you lose. Yep, it's true; zero is not considered an even number on the roulette wheel, something you better know before you bet.  This example is a non-mathematical, real-life situation where zero is neither odd or even.  But in mathematics, by definition, zero is an even number. (An even number is any number that can be exactly divided by 2 with no remainder.)  In other words, an odd number leaves a remainder of 1 when divided by 2 whereas an even number has nothing left over.  Under this definition, zero is definitely an even number since 0 ÷ 2 = 0 has no remainder.

Zero also fits the pattern when you count which is the same as alternating even (E) and odd (O) numbers.

Most math books include zero as an even number; however, under special circumstances zero may be excluded. (For example, when defining even numbers to mean even NATURAL numbers.) Natural numbers are the set of counting numbers beginning with 1 {1, 2, 3, 4, 5....}; so, zero is not included. Consider the following simple illustrations. Let's put some numbers in groups of two and see what happens.

As you can see, even numbers such as 4 have no "odd man out" whereas odd numbers such as 3 always have one left over. Similarly, when zero is split into two groups, there is not a single star that does not fit into one of the two groups. Each group contains no stars or exactly the same amount. Consequently, zero is even.

Algebraically, we can write even numbers as 2n where n is an integer while odd numbers are written as 2n + 1 where "n" is an integer. If n = 0, then 2n = 2 x 0 = 0 (even) and 2n + 1 = 2 x 0 + 1 = 1 (odd). All integers are either even or odd. (This is a theorem). Zero is not odd because it cannot fit the form 2n + 1 where "n" is an integer. Therefore, since it is not odd, it must be even.

I know this seems much ado about nothing, but a great deal of discussion has surrounded this very fact on the college level. Some instructors feel zero is neither odd or even. (Yes, we like to debate things that seem obvious to others.)

Consider this multiple choice question. (It might just appear on some important standardized test.) Which answer would you choose and why?

Zero is…
            a) even
            b) odd
            c) all of the above
            d) none of the above

Mathematically, I see zero as the count of no objects, or in more formal terms, it is the number of objects in the empty set. Also, since zero is defined as an even number in most math textbooks, and is divisible by 2 with no remainder, then "a" is my answer.

My Students Are Having Difficulty Memorizing Those Dreaded Math Facts!

Many of my college students come to me without knowing their math facts. Some do, but most do not. Since we use calculators in the class, it really isn't an issue.  It just takes those students longer to do a test or their homework. One day, the students in my Basic Algebra Concepts class (a remedial math class) were playing a math game to practice adding and subtracting positive and negative numbers. We were using double die (see picture) where a small dice is located inside a larger dice. (I have to keep an eye on these because they tend to "disappear." The students love them!)  I noticed one of my students continually counting the dots on the die. He was unable to see the group of dots and know how many were in the set.  It was then that I realized he could not subitize sets. (to perceive at a glance the number of items presented)

Subitizing sets means that a person can look at a grouping or a set and identify how many there are without individually counting them.  (i.e. three fingers that are held up)  When a child is unable to do this, they cannot memorize math facts since memorizing is associating an abstract number with a concrete set.  Many teachers as well as parents fail to recognize the root cause of this memorization problem.  AND no amount of practicing, bribing, yelling, or pulling out your hair will change the situation.  So what can you do?

First of all, the problem must be identified.  Use a dice and see if the child must count each dot on each face. Try holding up fingers or laying out sets of candy (M&M's - yummy!) or using dominoes. Put five beans in a container, and ask the child how many are in the box. (They may count them the first few times.)  Take them out, and put them back in.  Ask the child again how many there are.  If, after several times, s/he is unable to recognize the set as a whole, then s/he cannot subitize sets.

How do you help such a child?  If you have small children at home, begin subitizing sets by holding up various combinations of fingers.  My youngest grandson just turned four; so, we worked on holding up two fingers on one hand and two fingers on the other; then one and three fingers, and of course, four fingers. I also like to use dominoes. They already have set groupings which can be identified, added, subtracted, and even multiplied. A dice is great because the child thinks you are playing a game, not doing math.  Roll one dice, and ask the child to identify the set of dots. Try the bean idea, but continue to change the number of beans in the box.  My grandchildren love the candy idea because they are allowed to eat them when we are done.  (All children need a little sugar now and then even though their parents try to control the intake.  I love being a Grandma!)

Gregory Tang has written two wonderful books for older children, The Grapes of Math and Math for All Seasons, which emphasize subitizing sets. At times, I even use them in my college classes!  I was fortunate to attend two of his workshops presented by Creative Mathematics. He not only has a sense of humor, but his books can be read again and again without a child becoming bored. Check them out!