The Whole Equals the Sum of its Parts

Word problems can be tough even for the math-minded. The challenge lies in correctly converting words to the numbers and symbols of an equation. One method that helps is the concept that “the whole is equal to the sum of its parts.” Start with these three questions:

• What is the whole in this question? Is its value known or unknown?

• What are the parts in this question? Are their values known or unknown?

• What is the relationship between the whole and its parts? Which remains constant in the question? Which changes? How does it change?

By figuring out what are the parts and what is the whole, we can decide whether we need to perform a synthesis (“building up”) or an analysis (“breaking down”) to solve the problem at hand.

Synthesis:
If the whole is unknown, then the task is to build it up from its known parts:

• If the parts are equal, we multiply.

• If the parts are not equal, we add.

In other words, affirming that “the whole is equal to the sum (total) of its parts.”

Analysis:
If the whole and one or more of its parts are known, then the task is to find the remaining part(s) by breaking down the whole, using the known part(s):

• If the known parts are not equal, we subtract.

• If the known parts are equal, we divide.

Basically, “Each individual part is equal to the whole minus all of the other parts.”

By identifying which category the problem falls under, we can designate a relationship and determine the plan of attack; this is called the whole-part method. Below are a few examples of the method in action.

Ex. #1:

A box contains some marbles. 6 of the marbles are red, 5 are green, and 14 are orange. How many marbles are in the box?

In this question, the whole (the total number of marbles) is unknown. Since the parts (the number of red, green and orange marbles) are known, we can use the Key Synthesis Concept to find the whole:

total # of marbles = (# of red) + (# of green)+ (# of orange) = 5 + 6 + 14

= 25 marbles.

Ex. #2:

A train traveled 200 miles at an average speed of 50 miles per hour. How long did the trip take?

In this question, the distance traveled is the whole and is known. In each hour the train traveled an average of 50 miles, so:

time of trip = (distance traveled) ÷ (average rate of speed)

= 200 miles ÷ 50 miles per hour = 4 hours.

Ex. #3:

Another sandbox contains 100 pounds of mixed sand, 15% of which is brown sand. The rest is white. How much white sand must be added to make the mixture only 5% brown.

In this question, there are 15 pounds (15% of 100) of brown sand. As we add white sand, the whole (the total amount of sand) changes, but the amount of brown sand remains constant (at 15 pounds).

What we want is “15 out of the (new) total” to equal “5 out of 100” (5%, the desired outcome).

We can ask the following equivalent questions,

“15 is to what number as 5 is to 100?” or
“5 out of 100 = 15 out of what number?” or “5/100 = 15/what number.”

All of these methods yield the same answer: 300 pounds. Since the box already has 100 pounds of sand, 200 pounds (300 – 100) must be added.

The whole-part method allows us to identify the first step of a word problem bringing us one step closer to the solution and making math make sense.

- Larry Martinek 

18,000 Students looking forward to a Summer of Algebra, Angles, and Arithmetic

For most kids, summertime means swimming, sports and sun. But for 18,000 students around the country, the summer months will also be filled with algebra, angles, and arithmetic.

Yes, math.

Mathnasium is anticipating its largest summer enrollment ever this year, with 18,000 students ranging from elementary level to high school expected to take math classes at its franchise locations across the country. The reason? Many students – and their parents – are looking to prevent the notorious summer slide, during which kids lose math concepts and skills developed during the prior school year.

“Much like the muscles athletes use in competition, a student’s math muscles have to be exercised to remain in top form. Research has shown that during the summer months, students literally lose up to 2½ months of computational math skills developed during the year. However, 18,000 students across the nation have decided to fight the summer slide this year and work out their math muscles at Mathnasium. When school starts again in the fall, they’ll be well ahead of the game,” said Larry Martinek, Chief Instructional Officer at Mathnasium.

Summer math students typically spend two to three hours a week at their local Mathnasium franchise locations, working from a customized curriculum designed to mesh with their skill levels and needs. The sessions include specially developed math workouts and math games sessions designed to both motivate and educate.

The sizable enrollment in the Mathnasium summer classes demonstrates the importance of math skills to many areas of academic achievement at all educational levels, as standardized math testing is often used to determine advancement, class selection, and placement. Importantly, summer math at Mathnasium is equally applicable to students who need to address deficiencies in their math repertoire as well as those who wish to progress further than their normal classwork allows.

“I’m expecting about 70 students to sign up for summer math this year, which equals nearly half of our enrollment during the academic year. These students are committed to math for a variety of reasons, with some looking to fill gaps and others wanting to take advantage of the summertime to move ahead. One thing they all have in common is the desire to have a little fun but at the same time be challenged, and that’s something I hope all our students come to understand. Math can be both fun and exciting – and it’s something that everyone can learn,” said Alan Flyer, Owner of the Mathnasium franchise in Roslyn, N.Y.

Mathnasium’s summer math programs are being offered at Mathnasium’s more than 400 franchise locations across the U.S. and abroad.