A Key to Understanding Math

I’ve tutored in math, science, history and English; of these, I love math the most. One of my nephews recently needed help in working a “word problem” on his math homework. As we discussed the problem, it became apparent to me that he has been doing most of his math mechanically without truly understanding the core concepts underlying the problems that he is working, let alone correctly associating the meaning of words in the context of applying the principles of math. In my experience, this is far too common a problem, and one that is easily addressed with some remedial instruction.

One thing that I learned early on is that the primary problem in learning how to solve “word problems” isn’t the math—the calculations—itself; it’s understanding the language, the words, that’s used to express it. Word problems are just another way of describing facts and the relationships between them. With a solid primary school education, the arithmetic calculations are not the challenge; the challenge is knowing how to take ordinary, everyday situations and express them in the language necessary to describe them in a way that will allow the person to use the math that they know. The key to success in math is understating that math is a language that describes relationships, a language with it’s own unique terminology and contexts. It’s when we understand the concepts as applied to concrete examples that we can extend those concepts to abstract ideas.

For example, consider the following word problem:

Bob has four apples. Carol has six apples. Ted has twice as many apples as do Bob and Carol combined. How many apples does Ted have?

Most of us can easily perform the calculations necessary to answer this question. Let’s step through this.

In this problem, there are several key terms that indicate the relationships between the facts:

  • twice — two times
  • as many as — multiplication
  • combined — addition

If we were to exchange these words for their respective operations, we’d have:

Combined: Bob’s 4 apples plus Carol’s 6 apples, which equals 10 apples
Twice, as many as, combined: 2 times 10 apples, which equals 20 apples

Expressed in an sentence, the calculation is: “Twice as many apples as the sum of four apples and six apples,” or, “Two times the sum of four apples and six apples.” As an equation, which many of us can solve, this is 2 x (4 apples + 6 apples). Using the correct Order of Operations (PEMDAS or BEDMAS, whichever is easier for you to remember), we get:

\(2\times(4\text{ apples}+6\text{ apples})\\= 2\times10\text{ apples}\\= 20\text{ apples}\)

Ted has 20 apples, which is twice as many as apples as Bob and Carol have combined.

In explaining this problem to my nephew, I had to break it down into the smallest pieces, explaining the mathematical meaning of the key terms and the relationships described by them, translating them into operations and calculations. I had to use several different approaches with examples until I saw the light come on in his eyes; I was finally successful in conveying the knowledge through using wintergreen peppermints as counters as we talked through the problem, giving him a visual, concrete example of what the terms and amounts mean in the “real” world. His challenge wasn’t understanding how to perform the underlying mathematical calculations, it was his lack of understanding the relationships described in concrete terms and translating these relationships into mathematical calculations that confused him.

My nephew also had a word problem in which he had to calculate the area of two differently sized rectangles that shared a side. Helping him thought it, I thought that rather than duplicating it here, we could make this problem more real and applicable to our lives. Here’s an example of a situation that many of us have faced:

Alice wants to refresh the look of her dining room. With the help of a friend, she’s going to paint the old, faded 1960’s paneling antique white, visually opening up the space. She needs to know how much it will cost to do this, but she doesn’t know how much paint she needs. She’d like our help determining how much this remodeling will cost.

I’m glad that Alice came to us. There are several facts that she needs to gather and mathematical formulas—and remember, that means understanding the relationships between the facts—that she needs to know to determine the total cost. First, she needs to find out how much paint will be required to cover the room. In order to do so, she needs to measure the room to calculate the total surface area to be painted. She needs to know how much surface area that a given amount of paint will cover. With this and the cost of paint per gallon, she can calculate the cost of the paint. Finally, she needs to add the cost of the drop cloth, as well as the brushes and trays necessary for her and her friend.

To find the total surface area, Alice needs to calculate the surface area of each surface to be painted—four walls and the ceiling—knowing that the area of a flat surface is its length times its width. If there are any windows that she’s not intending on painting, she needs to find their surface area and subtract that from the surface area of the walls. If the windows are small and few in number and she doesn’t want to do the extra math, she probably doesn’t need to consider them: the effect on the required amount of paint will be negligible.

Since the room is a regular rectangle, she can either add the five surface areas:

\(\begin{align}\text{Ceiling area}\\+\:\text{Long wall area}\\+\:\text{Long wall area}\\+\:\text{Short wall area}\\+\:\text{Short wall area}\end{align}\)

or she can add the one surface area of the ceiling to twice the surface area of each of the long and short walls:

\(\begin{align}\text{Ceiling area}\\+\:2\times(\text{Long wall area})\\+\:2\times(\text{Short wall area})\end{align}\)

To ease the display of the calculations, we’ll use the slightly more complex later form.

Upon measuring, Alice finds that her dining room is 20 feet long, 16 feet deep and 12 feet high. The room has six windows but they are relatively small, so she isn’t including them in her calculations.

Let’s break this down. The ceiling, covering the length and width of the room, is 20′ long by 16′ wide. The long walls are 20′ across by 12′ high. The short walls are 16′ across by 12′ high. This is:

\(\begin{align}20\text{ ft}\times16\text{ ft}\\+\:2\times(20\text{ ft}\times12\text{ ft})\\+\:2\times(16\text{ ft}\times12\text{ ft})\end{align}\)

To find the necessary number of gallons of paint needed, Alice needs to know that the paint will cover so many square feet per gallon, dividing the surface area to be painted by the coverage rate of the paint.

Alice calls the local hardware store and is told that her desired brand of paint costs $12 per gallon with a coverage rate of 350 sq ft per gallon.

As previously noted, Alice will need to divide the surface area of the walls by the coverage rate of the paint to find how many gallons that she will need:

\(\begin{align}((20\text{ ft}\times16\text{ ft})\\+2\times(20\text{ ft}\times12\text{ ft})\\+2\times(16\text{ ft}\times12\text{ ft}))\\\div350\text{ ft}^2\text{/gal}\end{align}\)

We can also display this as a fraction:

\(\frac{20\text{ ft}\times16\text{ ft}+2\times(20\text{ ft}\times12\text{ ft})+2\times(16\text{ ft}\times12\text{ ft})}{350\text{ ft}^2\text{/gal}}\)

Now that Alice can determine exactly how much paint this will require, she has to multiply the number of gallons of paint required by the cost per gallon to find the cost of the paint.

Finally, Alice needs to add the cost of the required tools: a paint tray with a disposable liner; two roller brushes, one with an extension handle (only one because she wants to save as much money as possible); two hand brushes; a paint tray; and a drop cloth to protect the floor.

Alice prices the tools as follows:

  • Paint tray — $6
  • Disposable tray liner — $3
  • Roller brush — $8
  • Extension handle — $12
  • Hand brush — $7
  • Drop cloth — $4

To recap, Alice’s dining room is 20′ x 16′ x 12′. She has to paint every surface except the floor. The brand of paint that she wants costs $12 per gallon and covers about 350 sq ft per gallon. She needs the above listed tools.

The formula that represents this is: the sum of the surface areas of the roof and four walls, divided by the coverage rate of the paint, times the cost of the paint per gallon, plus the cost of the tools.

Being careful in the use of parentheses, the complete formula to determine the cost of painting Alice’s dining room is:

\text{Ceiling area})+2\times(\text{Long wall area})+2\times(\text{Short wall area)}}{(\text{Coverage rate})}\\\times(\text{Cost per gallon})\\+(\text{Cost of tools})\)

With this information, Alice can calculate the cost of painting painting her dining room. Plugging in the values, we get:

\(\frac{(20\text{ ft}\times16\text{ ft})+2(20\text{ ft}\times12\text{ ft})+2(16\text{ ft}\times12\text{ ft})}{350\text{ ft}^2\text{/gal}}\\\times($12\text{/gal})\\+($6+$3+2\times$8+$12+2\times$7+$4)\)

Remembering the Order of Operations, now it’s time to perform the calculations. (From this point on, I’ll only list the major steps, with the assumption that the basic arithmetic—the addition, subtraction, multiplication and division of whole numbers, fractions and decimals—is understood).

\(= \frac{320\text{ ft}^2+2\times240\text{ ft}^2+2\times192\text{ ft}^2}{350\text{ ft}^2\text{/gal}}\\\times($12\text{/gal})\\+($6+$3+$16+$12+$14+$4)\)

Recall that
\(1\text{ ft}\times1\text{ ft} = 1\text{ sq ft} = 1\text{ ft}^2\)
We don’t want to make any mistakes on the units!

\(= \frac{320\text{ ft}^2+480\text{ ft}^2+384\text{ ft}^2}{350\text{ ft}^2\text{/gal}}\times($12\text{/gal})+$55\\= \frac{1184\text{ ft}^2}{350\text{ ft}^2\text{/gal}}\times($12\text{/gal})+$55\)

\(\frac{1184\text{ ft}^2}{350\text{ ft}^2\text{/gal}}=1184\text{ ft}^2\times\frac{1\text{ gal}}{350\text{ ft}^2} \approx 3.38\text{ gal}\)
and the smallest quantity in which her chosen brand of paint is sold is a gallon, 4 gallons of paint are required. We input 4 into the equation in the place of 3.38.

\(= 4\text{ gal}\times($12\text{/gal})+$55\\= $48+$55\\= $103\)

It will cost Alice approximately $103 to paint her dining room, not including sales tax.

Alice is grateful to know how much painting her dining room will cost. After thinking about it, she wonders how much of a difference that not painting the windows will make in the amount of paint required. Since she really needs only 3.38 gallons, she wants to know how much paint that subtracting the surface area of the windows reduces from the required amount. She’d love to save the $12 on that fourth gallon of paint if at all possible. She measures the windows at 6 feet by 3 feet.

We already know how to calculate the area of the windows. Remembering from earlier that Alice’s dining room has six windows and plugging the values into the formula that we used earlier, we find that
\(6\times(6\text{ ft}\times3\text{ ft}) = 108\text{ ft}^2\)
which, when divided by the coverage rate of the paint:
\(108\text{ ft}^2\div(350\text{ ft}^2\text{/gal})\approx0.31\text{ gal}\)
reveals that she needs just over 3 gallons of paint (3.07 to be exact).

Given the fact that coverage rates are estimates that depend on how heavily the paint is applied in each coat, she might be able to get away with buying only 3 gallons, saving that $12; however, she just might run short. If this was a custom color of paint, she’d have a tougher decision to make because of the difficultly of matching the original mix, but she did choose an “off the shelf” color, so picking up another gallon if necessary is an option. While I’d recommend getting the fourth gallon up front and returning it if it’s not needed, the choice is hers.

If you’ve stayed with me through this entire post, thanks! I’ve demonstrated that a problem with Word Problems isn’t understanding the “math”; rather, it’s understanding the words—the language and concepts—as describing relationships between the relevant facts.

In summary, if we’re struggling with math, the first step is to know and understand how to apply the basic mathematical concepts to our situation, and then how to translate those relationships into calculations.

If you’re in need of a tutor in math or any subject, let me know. I can teach math from basic arithmetic through algebra, geometry and trigonometry. I’m available in the evenings and on the weekends. My price depends on the market and the distance driven, and I’ll work with you according to your situation.

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