The key to understanding mathematics is to recognize that math is a logical system of patterns and relationships. Because of that, if a student knows what the pattern is (mathematical formulas are patterns) and knows the value of enough of the elements within the pattern, the remaining values can be calculated.

Mathematics is a Game of Logic

For instance, the area of a circle is equal to A=ℼr2. Letters are used in mathematics to identify specific elements. In this formula, A represents the area of the circle, r represents the radius of the circle (a radius is a line drawn from the center of a circle to one side of the circle). In this formula the value of r is squared, which means that r is multiplied by r. Some values in mathematics never change. These are called constants (because they never change). ℼ is the symbol for pi, a well known constant that is usually averaged to the value 3.14. Another well-known constant in mathematics (and science) c represents the value of the speed of light. These symbols are mathematical shorthand for their values.

Values in a formula which can change are called variables. Since circles can have different sized, the values of the area and the radius will change from circle to circle, but the relationship between the elements (A, ℼ, r) never changes, which means that, using this formula, anyone who knows the value of two of the elements can calculate the value of the third.

Formulas are the universal patterns of mathematics. Constants are the unchanging values of elements in mathematics. Variables are the changing elements whose values can be logically discovered by manipulation the values of varialbes using formulas.

Here is an example of how patterns and relationships can be used to calculate missing values. It also illustrates how some forumlas are discovered logically.

Parallel lines are lines where each point on the line is exactly the same distance from the same point on another line.

Pictured: Two vertical parallel lines approximately two and a half inches apart.

Now, imagine two parallel lines crossing these lines but running horizontally rather than vertically.

90°90°

90°90°

Pictured: Two vertical parallel lines approximate two and a half inches apart crisscrossed by two parallel horizontal lines approximately two inches apart to form a hash tag (#) with all right angles. The four right angles formed by the intersection of the top horizontal line with the leftmost vertical line are each marked as 90° (Right angles are 90° angles).

Each of the corners created would be 90°. The top left corners are marked, but all of the corners would be the same. A circle drawn around any of the four corners would have four 90° angles in it, which is why a circle is 360°. The sum of the two angles on either side of the vertical lines, then would equal 180° as would the sum of the two angles on either side of the horizontal lines.

The block formed inside all four lines is a rectangle. Since the area of a rectangle is equal to the length of the rectangle multiplied by its width, measuring a vertical side and multiplying that number by the horizontal side of the rectangle created by the intersecting lines would give the area of the rectangle (A=lw, where A represents the area, l represents the value of the length, and w represents the value of the width). If the length of the rectangle is 2.5 inches and the height of the rectangle is 2 inches, the area of the rectangle would be 2.5 inches x 2 inches, which is equal to 5 square inches (multiplying inches by inches is squaring the inches just like 3 squared (32) is 3 X 3).

Now imagine that each of the points where lines cross are pivot points. Tilting the vertical lines to the right would keep the lines parallel and would not change the area inside the block although it would now be a parallelogram. While the angles at the intersectons of the vertical and horizontal lines would no longer each be 90°, the sum of the four angles would still equal 360° where the sum of the angles on the horizontal line would equal 180° as would the sum of the angles on the the vertical lines.

To calculate the area of the parallelogram formed, a vertical line running at a right angle to the base line (the bottom horizontal line of the parallelogram) and intersecting the top left corner of the parallelogram would represent the original vertical line. This "height" of the parallelogram could then be multiplied by the length of the baseline (the original width of the rectangle). This would result in the same interior area as the original rectangle as expressed in the formula A=bh, where A represents the area, b represents length of the base, and h represents the length of the height.

As for the angles formed at the intersections of the horizontal and vertical lines of the parallelogram, knowing the value of any angle created by where lines cross would allow a person to calculate the value of every other angle (since all four angles would add up to 360°.). For instance, if one of the angles on a horizontal line is 45°, the angle opposite would equal 135° (180 - 45 = 135). The angle diagonal from the 45° angle would also be 45°, and the angle diagonal from the 135° angle would also be 135°. This would be true for all of the angles at every corner of the figure.

If a diagonal line is drawn from the lower left corner to the upper right corner of the parallogram, two identical triangles are created. In fact, any triangle can be mirror imaged to create a parallelogram. Since the area of any parallelogram is equal to the base times the height (A=bh), and any triangle is half of a parallelogram, then, logically, the area of any triangle is equal to ½ the base times the heigth (A=½bh).

Understanding the logic of mathematics makes math easier to understand. Developing math literacy means improving your ability to think and reason logically.

Math is Fun

I first realized that math was a game when I was studying fractions in fifth grade. When multiplying two numbers, it doesn't matter which number comes first and which comes second, so 2 X 3 is the same as 3 X 2. When multiplying fractions, the two numbers on top (the numerators) are multiplied together and the two numbers on bottom (the denominators) are multiplied together. Then the result of the multiplication of the two numerators is divided by the result of the multiplication of the two denominators to find the answer.

For example, 1/2 of 2/3 (one-half of two thirds) is equal to 1 X 2 (the numerators) divided by 2 X 3 (the denominators) which results in 2/6, and since 6 can be divided by 2, the result is 1/3 (changing 2/6 to 1/3 is called "reducing a fraction to its lowest terms").

I realized that I could create complicated looking fractions that would actually have simple answers. So, this problem 11/17 times 51/33, which looks very difficult, can be rewritten as 11 divided by 33 times 51 divided by 17 (11/33 X 51/17). Since 3 X11=33, 11/33 becomes 1/3. And since 51 is 3 X 17, 51/17 becomes 3/1. Multiplying 1/3 by 3/1 results in 3/3 which is equat to 1. Fun.

Solving math problems is like playing detective: examining the clues, exploring the evidence, and finding the criminal.

Recognizing that formulas are logical patterns means that a person can logically figure out formulas based on formulas already at hand. For instance, the formula for the area of a circle is A=ℼr2. Using some logic, it is easy to figure out what the formula is for the volume of a cylinder. A cylinder looks like a stack of circles. Since the base of the cylinder is a circle, the value of the base of the cylinder is A=ℼr2. Since the cylinder can be seen as a stack of circles, the volume would be the base times the height, giving the formula for the volume of a cylinder as V=ℼr2h, where h represents the height of the cylinder, all very logical.

Study Tips

Use Flash Cards to Memorize Key Formulas and Values

Despite the widespread use of calculators, it helps to memorize the values of common constants and formulas that might be used. Flash cards are the most effective way of doing that. This is also why grade school teachers use flash cards to help students remember the times tables. For more information on using flash cards, see Memorization Techniques.

Practice Looking for Patterns of Relationships

Knowing the formulas makes it easier to calculate missing values. Look for patterns of relationships within information that would represent a mathematical formula. I was on a committee once that was considering whether it would be practical to build slanted platforms on which to display large quilts rather than laying them on the floor. The quilts were 12 feet square. The higher the platform, the more likely the quilts would slide off, so the optimal height of the high end of the platform would probably be about 4 feet. I recognized this was a version of the Pythagorean theorem. I knew the width of the quilt. I knew how high off the floor the high end of the platform would be, and I needed to know how much floor spalce would be saved by building the platform rather than letting the quilt rest on the floor. The Pythagorean theorem is a formula for the relationships between the long end of a triangle (the hypotenuse) and the two sides. Our hypotenuse would be 12 feet. The vertical side would be 4 feet, so I needed to find how much of the floor would be covered. The theorem is a2 X b2 = c2. Using 4 feet for b and 12 feet for c and solving for a, I had 122 (144) minus 42 (16) equals b. Subtracting 16 from 144 leaves 128. I didn't need to do the square root of 128. 112 is 121, so our platform would still cover over 11 feet of the floor. It was not worth building these platforms to save a few inches. Knowing the formula and recognizing that the pattern fit the situation saved us a great deal of time and money.

Students often hate math word problems, but the purpose of these is to develop students math thinking skills so that they can recognize and apply math to real life situations.

More Tips

Translate Problems into English

Putting problems into words can help you understand what is being asked in the problem. When you study equations and formulas, you should put those into words, also.

Perform Opposite Operations

Use Time Drills

Practice working problems quickly, and time yourself. Then exchange problems with a friend.

Analyze Before You Compute

Make a Picture

Draw a picture or diagram if you are stuck or blocked.

Estimate First

An estimation is an excellent way to double check your work, and on multiple choice tests, may save you the effort of calculating the answer.

Check Your Work Systematically

When you check your work, ask yourself the following questions:

Resourcees

"16 Math Tricks for Quick Calculations." BYJU'S. 12 Nov. 2024. <https://byjus.com/maths/maths-tricks/>.

"Entire Maths Shortcuts and Tricks." OnlineCSK. YouTube. Videos. 12 Nov. 2024. <https://www.youtube.com/playlist?list=PLyvkZTryoM02YGS0Sg6sAgSHWp7XkBQ1C>.

Gibson, Brittany. "12 Easy Math Tricks You’ll Wish You’d Known This Whole Time." Reader's Digest. 27 June 2024. 12 Nov. 2024. <https://www.rd.com/list/easy-math-tricks-youll-wish-youd-known/>.

Helmenstine, Anne Marie. "10 Math Tricks That Will Blow Your Mind."Thought Co. 3 May 2024. 12 Nov. 2024. <https://www.thoughtco.com/math-tricks-that-will-blow-your-mind-4154742>.

"Learn Mental Math Techniques." Art of Memory. 12 Nov. 2024. <https://artofmemory.com/mental-math/>.

"Mental Arithmetic Shortcuts." Brainfit World. 12 Nov. 2024. <https://www.brainfit.world/mental-arithmetic-shortcuts/>.

Rockmaker, Gordon. 101 Short Cuts in Math Anyone Can Do. New York, Frederick Fell Publishers, Inc., 1965. <https://cymathkia.wordpress.com/wp-content/uploads/2012/06/101-shortcuts-in-math-anyone-can-do.pdf>.

"Studying for Math." Dartmouth Academic Skills Center. 12 Nov. 2024. <https://students.dartmouth.edu/academic-skills/learning-resources/studying-math>.

"Top 30 Math Tricks for Fast Calculations." GeeksforGeeks. 2 Jan. 2024. 12 Nov. 2024. <https://www.geeksforgeeks.org/math-tricks/>.