Marveling At The Historical

Math Oldies But Goodies

  • About This Blog

    This blog is mostly about math procedures in textbooks dated from about 1825-1900. I’m writing about them because some of the procedures are exquisite and much more powerful, and simpler, than some of the procedures in current text books. Really!

    I update this blog as frequently as possible ... every 2-3 days. And, if you are a lover of old texts and unique procedures, you might want to talk to me about them, at markdotmath@gmail.com. I’m not an antiquarian; the books I have are dusty, musty, brown-paged scribbled-in texts written by authors with insights into how math works. Unfortunately, most of their procedures have vanished. They’ve been overcome by more traditional perspectives, but you have to realize that at that time, they were teaching the traditional methods.

Visualizing Fraction Operations with a Rectangle

Posted by mark schwartz on April 6, 2016

Introduction

The four basic operations with fractions can be demonstrated with a rectangle; one rectangle. It provides a visual representation of the operations, in essence demonstrating how the rules are generated. Visualizing these operations also leads to demonstrating a method for adding and subtracting fraction without needing to find a lowest common denominator.

The Story

Let’s use 3/4 and 1/2 to start the discussion. Using the denominators of both fractions, make a 4 by 2 rectangle. You can use nice colored chips; you can also use any objects that approximate a rectangle, or just draw the figure, as in Figure 1.

   
   
   
   

Figure 1

This rectangle is divided into 8 equal parts – 4 rows of two parts each and 2 columns of four parts each. Visualizing the rectangle this way is helpful for seeing how to demonstrate the 4 basic operations with fractions.

Let’s address addition first.

This rectangle idea is reinforced by talking about fractions in terms what a fraction means. The fraction 3/4 means that we are to take some thing (the above rectangle), divide it into 4 equal parts (which has been done horizontally) and do something with 3 of those 4 equal parts. The fraction 1/2 means that we are to take some thing (the same rectangle), divide it into 2 equal parts (which has been done vertically) and do something with 1 of those 2 equal parts. In my view, there are several keys regarding understanding of operations with fractions.

One is the above concept of dividing into equal parts and the second is realizing that when an operation is done, for example 3/4 + 1/2, it is with reference to the same object being divided into x equal parts and y equal parts at the same time. This view can be expanded to mean the same for the other basic operations as well. For example, if the expression is (3/4)(1/2) the question is “3/4 of what times 1/2 of what?” There may be a host of ways to interpret what this means but one very useful way is that the 3/4 and 1/2 are length and width of a rectangle. This will be addressed later.

… and now back to the rectangle … continuing with 3/4 + 1/2, some cells in the rectangle have been labeled to help with the discussion and it looks like this:

 a  
 b  
 c  
 d     

Figure 2

The fraction 3/4 says to divide something into 4 equal parts and do something with 3 of those 4 equal parts. The rectangle has 4 ‘equal’ rows, so taking 3 of these 4 equal rows gives 3 groups of 2 cells. This may seem a strange way to describe this, but it’s central to the idea of how to add fractions. Reading this as 3 groups of 2 cells is consistent with writing it in factored form as 3•2. Give students time to absorb this idea; most of them are familiar with the factored form but less familiar with the connection between what the rectangle shows and the factored form.

The problem then states that we are to add this to 1 of the 2 equal parts. Using the same approach, 1 of the 2 equal columns (the column with a, b, c, d) gives 1 column of 4 units each, or 1•4. .And to complete this, the third expression is that of the ‘area’ of the entire rectangle, which can be written in factored form as 4•2.

These 3 expressions form the ‘rule’ for adding (and also for subtracting two fractions) and in this case the whole problem can be seen as:

3/4 + 1/2 = (3⦁2 + 1⦁4)/(4⦁2)

The pattern that emerges is:

The numerator is: the sum of the numerator of the first fraction times the denominator of the second fraction and the numerator of the second fraction times the denominator of the first fraction.

The denominator is: the product of the two denominators.

Algebraically, given the fractions a/b + c/d,   the solution is   (ad + bc)/bd

This metthod for adding and subtracting two fractions is not a revelation. It is in the Algebra text currently used at our community college. I also found this method in 4 texts in my collection of old texts. These 4 texts are dated from 1858 to 1881. In a lot of cases, using this approach won’t give the answer in lowest terms but that won’t be an issue because reducing fractions is typically presented before operations with fractions.

It may seem that presenting this to a college numerical math class overrides a lot of other essential information that college numerical math students are expected to know, like prime factoring, ‘building’ a lowest common denominator, generating equivalent fractions, etc. but the point of this is that those related operations can be presented in other contexts.

After working with this idea, students have asked ‘How do we add 3 fractions?’ and this leads to exploring and then realizing that the formulation above can be done with 2 fractions, then taking that answer and use the formulation again. However, allowing discussion of this point has, on occasion, resulted in students collectively and correctly determining the formula for the addition of 3 or more fractions! It was a real “aha” moment.

One interesting outcome of using this concept is what can happen when subtracting two fractions. In most college numerical math texts, operations with signed numbers is typically the last chapter, given there as a transition from college numerical math to pre-Algebra. But, if the students learn how to handle signed numbers before working with fractions, then they can manage what happens when subtracting fractions.

For example, if the problem is 1/2 – 3/4, using the generic formula above, would give …

(1⦁4 ̶ 2⦁3)/(2⦁4) = ( 4 ̶ 6)/8 =   ̶ 2/8 = ̶ 1/4

Circumstances like this do appear in some Algebraic problems, but typically not in arithmetic math, but I do believe students would benefit by seeing such problems.

One more thing before moving on to multiplication and division. In the example above, the answer 2/8 is reduced to 1/4. But it could be reduced in the factored form before doing the indicated multiplications. When reducing fractions to lowest terms, the procedure is to cancel any common factor in the numerator and the denominator. This rule is still true even if there is more than one term in the numerator (or denominator). To demonstrate, look at this:

10/15 = (5 + 5)/15 = (5⦁1 + 5⦁1)/(5⦁3) = (1 + 1)/3 = 2/3

Notice what happens between the 3rd and 4th term and the denominator – the common factor of 5, which is in both terms in the numerator and is also a factor in the denominator, is cancelled out of all the terms. This is a demonstration in what I call “slow motion” math – showing every step, but when students do the work they need not do it in slow motion.

And now to multiplication. Using the problem 3/4 x 1/2 and using the rectangle again, it is divided into 2 equal columns and 4 equal rows. The notation for multiplication used here, “x”, is very intentional and aids students to see how multiplication works, and here’s how. Mark three of the four rows with “ “ and 1 of the two columns with “\”. The rectangle now looks like:

 x   
 x   
 x   
 \  

Figure 3

Now just count: how many of the squares contain an “x”? That’s the numerator. How many cells are there in the rectangle? That’s the denominator. You’re done. The rule: multiply the numerators together and multiply the denominators together, and reduce the answer to lowest terms. You may reduce either or both fractions before multiplying if there are any common factors. In fact, I tend to demonstrate the importance of this with by noting that, for example, 3/7 x 2/3   is the same as (3 x 2)/(7 x 3) and that the common factor of 3 can be cancelled, leaving 2/7. I then present a more vivid example for practicing this idea:

25/36 x 18/35 x  7/15   and ask the class to do the indicated operations.

and then there’s division …

The traditional rule is to state that in order to divide two fractions, multiply the first fraction by the reciprocal of the second (or other wording). This is stating the rule and some students dutifully apply the rule. Most don’t. So, the rectangle can again demonstrate the relationship. This isn’t the only way to do it, but it seems to make sense to students.

Using the problem 1/2 ÷ 3/4, we again have the rectangle divided into 2 equally parts vertically and 4 equal parts horizontally. Note that if the problem were written as a complex fraction, 1/2 is the numerator and 3\4 is the denominator. Taking one of the two columns (1/2) as the numerator – which is 4 − and 3 of the 4 rows (3/4) as the denominator – which is 6 −, the answer is 4/6 (“n” for column, “d” for row). Several examples are demonstrated, and then they are examined for the pattern between the problem and the answer and this demonstrates how the rule works.

 n  d   d 
 n  d  d 
 n  d  d 
 n

Figure 4

Summary:

The key is to demonstrate in slow-motion math ways of visualizing the operations and the origin of the rules. Repetition of the demonstration and having students use the rectangle to actually do the operation gives a visual reference point. The rules are no longer simply presented as something to memorize. At first it may seem confusing to have all 4 operations based on the same rectangle, but the hidden agenda of the rectangle is that it is the lowest common denominator for addition, subtraction, and division. But, this isn’t voiced until later or unless students ask about it, in which case it is definitely discussed … and it was a fun discussion.

 

 

 

 

 

 

 

 

 

 

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