If you are interested in the origins of geometry and algebra or if you simply like history, you should check out the BBC documentary series “The Story of Maths”. I stumbled across this series recently and was fascinated. The series shows how ancient people invented methods to solve advanced mathematical problems before they had today’s methods. It rekindled my interest in geometry by showing its historic origin. And it shed new light on some of the mathematical tools that we take for granted today. If this sounds interesting, you will not be disappointed!

Memorizing formulas is one part of math that even math nerds usually don’t appreciate. Certain formulas come in handy and make it easier and faster to solve problems. For example, the quadratic formula is complicated and not easy to memorize. However, knowing it replaces a multi-step process (completing the square) that is tedious and not always easy. So overall, using the quadratic formula makes a lot of sense.

There are other formulas taught at school, though, that are way less beneficial. Some of them are not necessary because they replace methods that are simple. Others can easily be developed each time and do not need to be memorized. Here are three examples.

The Point-Slope Formula

\(y-y_1 = m(x-x_1)\)

When students in 7th or 8th grade study linear equations, they learn three forms: the slope-intercept form, the standard form, and the point-slope form.

The slope-intercept form \(y=mx+b\) is probably the most important one, as it contains the information that we need to graph the corresponding line. m tells us how steep the line is, and b shows where the line crosses the y-axis. The standard form \(ax+by=c\) is the one used in higher math like linear algebra, so it makes a lot of sense to introduce it. But what about the point-slope form?

This form is introduced to set up the linear equation from a known point (x₁,y₁) and slope m. But that can also be done with the slope-intercept form without added difficulty: Substitute x and y by the coordinates of the point and solve for b. This process is conceptually much easier and a method that students will have to use in higher math classes anyway. On the other side, when using the point-slope form, students often do not understand the use of the additional constants x₁ and y₁ and how this process generates the constant b. So why do our students have to learn the point-slope form at all? My suggestion: get rid of the point-slope form.

The Midpoint Formula

\((x_m,y_m) = ({x_1+x_2 \over 2},{y_1+y_2 \over 2})\)

The midpoint formula states how to find the coordinates of the midpoint (x_m,y_m) between two known points (x₁,y₁) and (x₂,y₂). Even though the formula contains only two fractions, I have had students mix up addition and subtraction and confuse where to put x and y. If the origin of the formula is not understood, there are still many ways to make mistakes.

What does the midpoint formula do? It finds the average of the two points in x-direction and y-direction separately. For finding the average of two numbers, you add them and divide the sum by 2. Just do that for the x coordinates and the y coordinates, and you obtain the coordinates of the point in the middle. There is no need to memorize formulas.

The Distance Formula

\(d = \sqrt{(x_1-x_2)^2+(y_1-y_2)^2}\)

The distance formula finds the distance between two points (x₁,y₁) and (x₂,y₂). Every time my students see this formula for the first time, they get scared. Then I tell them: What you are seeing is just a different form of the Pythagorean Theorem.

How does the distance formula work? In order to find the distance between two points, you create a right triangle with horizontal and vertical legs and with the hypotenuse between the two points.

That means that in the distance formula you can replace the distance d by the length of the hypotenuse c, the length of the horizontal leg \((x1-x2)\) by a, and the length of the vertical leg \((y1-y2)\) by b.

\(c = \sqrt{a^2+b^2}\)

Then square both sides to obtain

\(c^2 = a^2+b^2\)

The distance formula is simply the Pythagorean theorem in disguise. So why not use that instead of adding another formula to memorize?

Formulas have their place: They can make it faster or easier or even possible to solve certain problems. But the risk in teaching formulas is that students do not understand the underlying math and make mistakes using them. Limiting the formulas to the essential ones and teaching the underlying concepts of others instead might not only reduce the number of mistakes, but also make math more fun.

Every morning at 8 am Mr. Pi walks to his office that is 6 miles away. His dog Willy accompanies him. Willy knows the way and runs ahead with twice of Mr. Pi’s speed. When he reaches the office, he turns around and runs back to Mr. Pi. Once Willy reaches Mr. Pi, he turns around and runs ahead to the office again, where he turns around and runs back to Mr. Pi again, and so on. They both reach the office together at 10 am. How far did Willy run?

(modified from Physik, Gerthsen 7th Edition)