How to understand math formulas
In a recent IntMath Poll, many readers reported that they find math difficult because they have trouble learning math formulas and an almost equal number have trouble understanding math formulas.
I wrote some tips on learning math formulas here: How to learn math formulas.
Now for some suggestions on how to understand math formulas. These should be read together with the “learning” tips because they are closely related.
a. Understanding math is like understanding a foreign language: Say you are a native English speaker and you come across a Japanese newspaper for the first time. All the squiggles look very strange and you find you don’t understand anything.
If you want to learn to read Japanese, you need to learn new symbols, new words and new grammar. You will only start to understand Japanese newspapers (or manga comics ^_^) once you have committed to memory a few hundred symbols & several hundred words, and you have a reasonable understanding of Japanese grammar.
When it comes to math, you also need to learn new symbols (like π, θ, Σ), new words (math formulas & math terms like “function” and “derivative”) and new grammar (writing equations in a logical and consistent manner).
So before you can understand math formulas you need to learn what each of the symbols are and what they mean (including the letters). You also need to concentrate on the new vocabulary (look it up in a math dictionary for a second opinion). Also take note of the “math grammar” — the way that it is written and how one step follows another.
A little bit of effort on learning the basics will produce huge benefits.
b. Learn the formulas you already understand: All math requires earlier math. That is, all the new things you are learning now depend on what you learned last week, last semester, last year and all the way back to the numbers you learned as a little kid.
If you learn formulas as you go, it will help you to understand what’s going on in the new stuff you are studying. You will better recognize formulas, especially when the letters or the notation are changed in small ways.
Don’t always rely on formula sheets. Commit as many formulas as you can to memory — you’ll be amazed how much more confident you become and how much better you’ll understand each new concept.
c. Always learn what the formula will give you and the conditions: I notice that a lot of students write the quadratic formula as
[−b ± √(b2 − 4ac)]/2a
But this is NOT the quadratic formula! Well, it’s not the whole story. A lot of important stuff is missing — the bits which help you to understand it and apply it. We need to have all of the following when writing the quadratic formula:
“The solution for the quadratic equation
ax2 + bx + c = 0
is given by
x = [−b ± √(b2 − 4ac)]/2a”
A lot of students miss out the “x =” and have no idea what the formula is doing for them. Also, if you miss out the following bit, you won’t know how and when to apply the formula:
ax2 + bx + c = 0
Learning the full situation (the complete formula and its conditions) is vital for understanding.
d. Keep a chart of the formulas you need to know: Repetition is key to learning. If the only time you see your math formulas is when you open your textbook, there is a good chance they will be unfamiliar and you will need to start from scratch each time.
Write the formulas down and write them often. Use Post-It notes or a big piece of paper and put the formulas around your bedroom, the kitchen and the bathroom. Include the conditions for each formula and a description (in words, or a graph, or a picture).
The more familiar they are, the more chance you will recognize them and the better you will understand them as you are using them.
e. Math is often written in different ways — but with the same meaning: A lot of confusion occurs in math because of the way it is written. It often happens that you think you know and understand a formula and then you’ll see it written in another way — and panic.
A simple example is the fraction “half”. It can be written as 1/2, and also diagonally, as ½ and in a vertical arrangement like a normal fraction. We can even have it as a ratio, where it would be written 1:1.
Another example where the same concept can be written in different ways is angles, which can be written as capital letters (A), or maybe in the form ∠BAC, as Greek letters (like θ) or as lower case letters (x). When you are familiar with all the different ways of writing formulas and concepts, you will be able to understand them better.
Every time your teacher starts a new topic, take particular note of the way the formula is presented and the alternatives that are possible.
Do you have any tips to add? How do you figure out your math formulas? Which formulas are hardest to understand?
Good luck with understanding math formulas!
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20 Feb 2009 at 1:42 am Link to this comment
Dear Sir
thank you for your nice and spiring articles. Thank you again
20 Feb 2009 at 8:22 am Link to this comment
Hello,
If you could help me review dosage calculation for nursing?
Any help would be appreciated.
Thanks,
20 Feb 2009 at 4:28 pm Link to this comment
Dear Sir,
Receiving News Letter concerning IntMath is wonderful. This will definitely enable me to provide valuable information to my students and help them to encourage in Maths.
Thanks once again.
20 Feb 2009 at 9:47 pm Link to this comment
Hello, Am very greatful , very impressed and more motivated by your article. I hope this will make me improve my interest 2wards mathematics. Thanks very much,
20 Feb 2009 at 10:48 pm Link to this comment
Thanks,,,,,because math formulas are easy to memorize but hard to use it….so its important to understand it….
its all hepfull…i’m so gratefull…..thnx*10
21 Feb 2009 at 2:53 pm Link to this comment
thanksssss…………
I am impresed to see the tricks of mathematics formula’s……….
maths is my faviourate sub. and 4 me obivously maths formulas are easy to understand………i am greatful to log on this wonderful site………
again thanks……..
21 Feb 2009 at 4:18 pm Link to this comment
hey thanx for the general info abt math formulae but can u just throw some more light on how to remember the formulae based on …….integration…….reduction formulae……..rememberin expansons like dat of ‘e’, ‘’sin’, log(1+x) etc….these get the hell outta me……..
pls
pls help me out wid these
24 Feb 2009 at 2:56 pm Link to this comment
I really thankful to IntMaths newsletter to advice me on how to understand maths formulae…..but i really very much oblidge to yours if you give us tips and points about short methods and tricks used in all type of exams…
Thanking You!
MMs
25 Feb 2009 at 2:15 am Link to this comment
I have a suggestion to add…. I found working through how to change the terms of the formula helped me to remember the general formula. A very simple way to explain is… speed = distance/ time and also distance = speed x time. but I use this for many of the rearranged formulae needed in say mechanics or with the identities in trig. They were hard at first but I make a game out of seeing how many different ways I can rearrange them
25 Feb 2009 at 8:53 am Link to this comment
Great suggestion, Carol.
Permit me to add to your suggestion. It’s important to understand the units for each of the permutations, since this helps us to understand what the formula gives us.
So speed = distance/time has units “m/s” and distance = speed x time has units “m”.
For complicated formulas, this is certainly worth doing.
26 Feb 2009 at 10:10 am Link to this comment
Great article thanks,
Understand how to derive and manipulate formulas inside out. Memorizing helps, but don’t bank on it, as there are simply too many. Of course, basic definitions and methods have to be known. And you would probably inadvertently remember a few others by using them repetitively.
Maths is always consistent, and there aren’t exceptions to the fundamental rules. A solid grounding in algebra and techniques is the key. When you do encounter a new technique or perspective, take note of it.
Then you can derive quite a lot in quadratics, trigonometry, series, etc. with relative ease, and you’ll start to gain new insight and confidence – Internalize your skills so it becomes secondary to problem solving.
26 Feb 2009 at 10:51 am Link to this comment
Hi Keerthi,
Perhaps the Taylor and MacClaurin Series would help you with those expansions? It looks more scary than it really is.
26 Feb 2009 at 4:44 pm Link to this comment
this site is the most understandable math equationi’ve ever seen..
thank you for being such blessed and in turn blessing others…
28 Feb 2009 at 6:28 am Link to this comment
Dear Sir,
I am more confuse at applying the fomular than remembering it especially in differentiation and integration, sometime you forget to add a one or divide by the power or using the wrong substitution. Thanks to IntMath picking it up slowly.
28 Feb 2009 at 9:16 am Link to this comment
Hi David
In a future “Tips” article I will talk about the issue of differentiation and integration formulas and processes. Yes, it’s not surprising that it becomes a gluggy soup after a while.
1 Apr 2009 at 3:20 am Link to this comment
I have to take a very important test tomorrow 4/1/ and it is on formulas. I am older and never had formulas and have no Idea what I am doing. Is there anybody out there who can shoot me a email on what the H—! a formula is and how to conquer it? or just a article that explains ………anything at this point.
Thank you so very much!!!!
1 Apr 2009 at 8:34 pm Link to this comment
Hi Jerelyn
There are thousands of formulas! Please give more information on the kind of thing that you are trying to understand and maybe I can help.
1 Apr 2009 at 10:02 pm Link to this comment
Zac
Thank you so much for the help. I have no idea what kind of math formulas I need. Just the basics on how to do one. My test is today so I do not have much time. With the information you sent I have a little more knowledge………………Or I just think I do.. Anyway thanks again. It is nice to know there are still people out there willing to help a stranger.
15 May 2009 at 6:02 pm Link to this comment
[...] give tips on How to Understand Math Formulas posted at [...]
31 May 2009 at 8:33 pm Link to this comment
This newslettter has been a Godsend to Me too. I wish some of the articles, like on how to study had come out a week or two before they did, but, when they did, they hit home. Thank You
for caring enough to try to help! God Bless You!
31 May 2009 at 8:38 pm Link to this comment
You’re welcome, Sharon. Glad it was useful to you!
29 Jun 2009 at 3:52 pm Link to this comment
Writing the formula again and again is the most effective way of familiarizing any formula. It works for me, it may work to others.
11 Aug 2009 at 1:18 pm Link to this comment
yes i do agree that in writing a formula one could remember the formula, furthermore if one knows and understand the principles behind such formula the better he/she has control of such a formula. Thats all thanks.
4 Sep 2009 at 6:33 pm Link to this comment
How can you understand this serious math?
13 Sep 2009 at 7:38 pm Link to this comment
how i solve (a+b)2 formula
13 Sep 2009 at 8:51 pm Link to this comment
Hi Fazalerabbi. Do you mean how to expand (a+b)2? See The Binomial Theorem, where it tells you all about it.
1 Jan 2010 at 3:04 pm Link to this comment
[...] of squareCircleZ offers some valuable advice for students in How to understand math formulas. This is a great follow-up to his article How to learn math [...]
25 Feb 2010 at 5:14 am Link to this comment
hey what is the 0 point on a kelvin scale?
25 Feb 2010 at 2:19 pm Link to this comment
0° Kelvin is absolute zero – the temperature where nothing vibrates, so there is “no heat”.
See Kelvin
28 Feb 2010 at 12:21 am Link to this comment
thanks for every tip you have given.