U Teach Your Child Fractions 6: Addition

(In this note a / b represents the fraction whose numerator is “a” and whose denominator is “b” )

A Work Problem
Mary mows her rear lawn in three hours. With a comparable mower, her younger brother can complete the same job in four hours. Once the lawn is mowed, Mary and her brother plan to see a fireworks display in the park, a ten minutes walk from their house. Mother has given them permission to see the fireworks after the lawn is mowed. If they work together, will they be able to see the start of the fireworks show? The fireworks will start in two hours.

This discussion centers on the concepts to solve this problem and the presentation of its solution.

What is needed to solve this problem?
To solve this problem, one needs to know how to add fractions and how to find a particular equivalent fraction of the sum.

Counting and Addition
Counting is one of the first operations a child learns. When she, for the first time, recites one, two, three, etc…, our faces light with pride. Before long, she masters the counting procedure when she points to objects and sounding the number in the sequence, 1, 2, 3, … stopping with her assignment to the last object a number. For small counting chores, this method is effective. However, when totaling objects from two or more piles, this method is time-consuming and often not effective.

Addition is our method of accelerating this process. With it, one counts without pairing the numbers 1, 2, 3, … etc.  to individual objects, thereby making addition faster than counting.

How is it done?

Let’s take our lead from a child. A boy describes to his friends his exploits of a successful game of marbles. Proudly he says, ”I won 3 shooters and 12 marbles. Marbles are non-shooters. Shooters are special, so he counts them separately. What’s going on?

The boy is counting things that are alike in different piles. It is natural to group like things and count them among themselves.

Now consider this example.
The proprietor of a small pest control company owns 12 light trucks, each having pest fighting equipment, and four cars for his salesmen. When asked the size of his mobile inventory, he reclassifies the trucks and the cars in a common category with the answer, “I have 16 vehicles.”  The word, vehicle, strips the trucks and the car of their designated function and places them in a broader category that make them alike -- namely motor vehicles. Now that they are alike, they can be counted together. That is, they can be added.

These two examples highlight two principles in fraction arithmetic.

Principle 1: Only add things that are alike. And when things are alike, adding is simply counting.

Principle 2: When things are not alike, place them in a common category. Then add (count them) within the common category.

When Are Fractions Alike?
Fractions are alike when they have the same denominator.

Example: 5 / 6 and 3 / 6 are alike. But 1 / 12 and 4 / 5 are not alike.

NOTATION: We use the plus sign is used to represent addition.

Example: Explain 2 / 6  + 3 / 6.

2 / 6 is represented by a bar with two blue cells. 

3 /6  is represented by a bar with three blue cells.

In these two piles of blue cells, each cell represents one-sixth of the whole. To add the fractions, add the blue cells. Since the pilies contains things that are alike, addition is counting. So count the blue cells in the two piles and get  (2+3) blue cells. Since each cell is one-sixth, the sum is 5 sixes or 5 / 6.

So, 2 / 6 + 3 / 6 = 5 / 6.

Example: Explain 1 / 5 + 3 / 5

1 / 5

3 / 5  

The red cells are alike so we can count the red cells contained in the two piles. Npw each cell represents one-fifth of the whole, so the sum is 4 / 5.

So 1 / 5 + 3 / 5 = 4 / 5.

Problem: Explain 2 / 7 + 3 / 7.

Adding fractions with the same denominator

The sum of two fractions with the same denominator is the fraction whose numerator is the sum of the numerators of the two fractions and the denominator of the sum is the common denominator of the two fractions.

So how does one add fractions that have different denominators?

Principle 2 directs us.  To add fractions with different denominators, find a common category that contains the fractions, then add them within that common category.

In our study of fractions, a common catergory translates into, "get an equivalent fraction for each of the fractions to be added."

Example: Explain 2 / 3 + 1 /5.

2 / 3

1 / 5

To find a common category containing thirds and fifths, draw a rectangle with one side showing thirds and the other sides showing fifths.

Representation of 2 / 3 and 1 / 5 within a common category.

2 / 3

The  2 / 3  rectangle is divided into three horizontal bars. Two of them were shaded red.

1 / 5

The  1 / 5  rectangle is divided into five vertical bars. One of them were shaded yellow.

Note:

• The heights of the two rectangles are one unit long. Visually it does not appear that way-- so pretend.
• The red bar represents 2 / 3 of the the first rectangle.
• The yellow bar represents 1 / 5 of the second rectangle.
• 2 / 3 is equivalent to 10 / 15 in the common category, count the cells,
• 1 / 5 is equivalent to 3 / 15 in the common category.

Within this common category ( the rectangle with 15 cells), add the fractions. The sum is  (10+3) / 15 = 13 / 15.

Problem: Explain 1 / 3 + 1 / 2.

When learning or teaching new concepts it is an advantage to use illustrations. They are powerful aids to understanding. Make an attempt to solve problems using an illustration, it will help to build mental maps that will solidify the techniques for you. However, illustrations can be time consuming, so a summary of these procedures in a formula is provided.

Summary (unlike denominators)

The arrows signify a multiplication.

This summary is provided as a memory device for adding fractions that have unlike denominators.  “The numerator of the sum of two fractions is the sum of the multiplication of the down arrow (red arrow) and the multiplication of the up arrow( the blue arrow). The denominator of the sum of the two fractions is the multiplication of the horizontal across (fuscha arrow).
The product of the down arrow is in red. It is a times d.

The product of the up arrow is in blue. It is b times c.

The product of the horizontal arrow is in black. It is b times d.

Example:

Problem: Try 2 / 7 + 2 / 3.

Caveat

The addition formula will give an equivalent fraction of the sum. It will not necessary be in the simplest form. Getting the sum in its simplest form will appear in a future note. For instance, when adding 1 / 2 + 1 / 4 by the formula, the denominator of the sum is eight (8). A simpler denominator is 4.

Part 2: The Solution of the Work Problem.

Review: If you have a fraction and you wish to get the fraction that is equivalent to the given on whose numerator is one, divide the numerator and the denominator of the given fraction by the numerator.

To get the equivalent fraction to 6 / 17 whose numerator is 1, divide the numerator and denominator of 6 / 17 by 6,

get .

Mary can mow the rear lawn in three hours. So in one hour Mary mows 1 / 3 of the lawn.
Her brother mows 1 / 4 of the lawn in an hour. If Mary starts on the left side of the lawn and mows in an up and down serpentine fashion while her brother start on the right side and mow in an up and down serpentine fashion, at the end of one hour they will have mowed 1 / 3 + 1 / 4 = 7 / 12 of the lawn. A person who mows Mary’s rear lawn in 12 / 7 hours (1 hour 42 minutes and 51 seconds) will mow   (an equivalent fraction) of the lawn in one hour. But Mary and her brother working together mow 7/12 of the lawn in one hour. So they too can mow the lawn in 1 hours or 1 hour 42 minutes and 51 seconds also.

Something To Think About
Difficult1 1: Jack Nerva had a backyard pool installed for his kids. To fill the pool, Jack plans to use his backyard hose and run it continuously. It will fill the pool in 26 hours. Jack’s next-door neighbor has access to a water source that can fill the pool in 18 hours. Jack convinces his neighbor to combine their water sources to fill the pool. How much time is needed to fill the pool?

Difficult 2: Assuming they work together, how much time will Mary and her brother need in order to complete the mowing and arrive at the park in time to see the start of the fireworks show? Give your answer as a fraction.

Difficult 3: When the spillway north of Coryville, USA is full and the water is not released, its waters will drains into an underground aquifer in 84 days. The State’s Corp of Engineers began a project to accelerate the drainage into the aquifer to 53 days but the construction funds were slashed severely and the construction ceased with only half of the project completed. The spillway is full with winter rain. If it is not drained, how long will it take to empty the waters into the aquifer?

From these problems, do you see a pattern? Can you think of other situations where you might apply this analysis? Write your suggestion as a comment..