calculus vs. business calculus

There is a lot of common ground between business calculus, sometimes called “applied calculus” or “calculus for business and the social sciences”, and calculus or “calc 1”. However, the focus of the courses are different …


How many people do you need to get together before it will be reasonably likely that two share the same pin code? We know by the pigeonhole principle that given more than 10,000 we can be certain there is a repeat, but does the probability pass 50% much earlier?

It turns out that this is closely related to the famous “birthday problem” and that you actually need fewer than 150 people before it is reasonably likely two share the same pin. In fact, if you include just SOME of the data we have on actual pin codes people selected, the number is dramatically smaller! You can read the details over on my personal blog:

While many of you are students, I know that a few of you are math teachers. This update is just for you!

Over on my personal homepage (, I have started a series that will cover the tools and tricks I have learned about creating high quality math screencasts – the same screencasts I produce for this website and for my online courses. Part 1 of the series specifically talks about the software and hardware tools you would need if you were just getting started.

You can read that article here: Math Screencasting – Part 1: Tools

For better or worse, one of my main goals in college was to finish as quickly as possible. Yes I loved learning and yes I really loved taking every math course I could, but there were these other courses that… well… got in the way. Of course being a few years older now and having a bit more experience, I can see the value in every required course. But at the time? Let’s just say I was more focused on the time and money involved.

In reality, college is a bit of a game. There is the “love for learning” aspect where you develop as a person by progressing through these different courses and there is the “get the piece of paper and a job (or go to grad school, etc)” aspect that, while not ideal, is a big part of things. This second aspect is what initially drove my research into CLEP tests.

CLEP tests are given by the college board (the same group behind the SAT) and they have the test for a variety of subjects including college mathematics and college algebra for $77 each. Here is the biggie: Passing the test will result in credit for the corresponding course (depending on your college’s policy).

Think about that for a minute. A course is usually 3 credit hours. I don’t know about tuition near you, but near me the lowest I know of is about $150/credit hour and of course it only goes up from there. In other words, you pay $77 and get at LEAST 3 * $150 = $450 worth of credit. That doesn’t even account for the time saved! That is one less class to take and often it is a prerequisite to other courses. You can move on with your life!

I used CLEP tests to skip 6 credits of freshman composition courses and head straight to a really fun literature course (well, I didn’t think it would be fun but I KNEW I wasn’t going to pass a test to get out of it 🙂 ). At my college, this saved me over $1,500 and a full year of gen ed courses.

Anyone can study hard and pass a CLEP test. It’s worth researching the tests a bit to see if you can save some tuition here and there. Make sure you check with your prospective college as far as their rules for granting credit. Lucky for us, the college board made a sweet little tool you can use to find your college’s policy:

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One of the best things you can do as you are working through a course is to get a tutor. Tutors can guide your study towards problem areas and catch mistakes before they become habits. I think of a good tutor as acting like training wheels when you are learning to ride a bike: they keep you from falling over, but you are doing most of the real work!

There can be a problem though. If you ride your bike with training wheels all the time, you are going to get really good. You are going to get really good at riding your bike with training wheels.

“I worked with a tutor everyday and still bombed the test! I don’t know what else to do!”

It’s frustrating when you think you are doing all of the right things. When you are working side by side with a tutor for all of your studying hours you are missing out on a big part of learning: the fall (yup I’m back with the bikes thing). My area of expertise is in math, so I will use an example from there. In math, the fall is that painful part of working on a problem for an hour and ending up with the wrong answer. It continues as you go through all of your notes and your book to find a problem like it. Finally its over when you find SOMETHING that helps and you discover the issue was a mistake in line 3 of your 20 lines of work. Ouch. This may be incredibly frustrating at the time, but you are building a base of knowledge when you do this. Every mistake is part of the learning process.

It is true. If you had worked that same problem with a tutor, he or she probably would have caught the mistake before you even got to the next step. This feels great and you feel like you made tons of progress! But there is a skill to develop: The skill of catching your own mistakes. This really only comes from the process I described above. You don’t want your first time trying to find an error to be when you get a really bizarre answer on an exam! You want to be well practiced by then. In other words, you don’t want the exam to be the first time you have ridden your bike without training wheels.

Where does this leave tutoring? The tutor comes in when you make the same mistakes over and over again. The tutor comes in when you just finished lecture and have no idea where to start. The tutor comes in when you have put in the effort but really aren’t sure where you are going wrong. In other words, the tutor is a supplement to your own studying.

Make a habit of studying on your own before and after meeting with a tutor. This way, you can make the most of your tutoring time and you can practice on your own as well. Tutors are great and as I mentioned before, I would encourage anyone having trouble to hire one! However, they are a piece of the puzzle. This puzzle includes your own studying, working problems, reading the text, and getting the most out of lectures. All of it is needed to truly understand what you are doing.

PS – If you are a tutor, make sure you are letting your students fall here and there. Its hard, but don’t jump in as soon as you see a mistake. Let them see it or guide them to seeing it. Make sure you are asking just as many questions as they do (“why didn’t that work?” is a GREAT one).


Woohoo I LOVE email! My love of email may go so far as to be distracting, but that is a completely different topic. Just make a note: If you ever have any questions about doing well in math, send ’em my way! You can get a hold of me through (and NO I won’t do your math homework for you)

Last week, a reader asked if it was possible for him to do well in his calculus course even though he had never studied any trigonometry. According to him, his algebra skills are solid and so far he has been able to manage the course work.

There is a reason why a lot of high schools and colleges combine trig with algebra in a course like precalculus. If you are willing to put in a little bit of outside effort, many topics in trig are easy to pick up and there are really only a few key skills/ideas. Don’t get me wrong, there is a whole lot of memorization and things like solving a trig equation WILL come up in a calculus course. But, for someone who is able to learn math on their own, picking it up along the way is possible.

Will it be easy? No way. I warned our friend that instead of being able to focus on the new calculus topics by themselves like everyone else, his studying time will also be filled with learning the trig. Calculus is already a challenging course by itself!

For anyone else considering this, I recommend against it unless you are the type who generally can learn math on their own, and is willing to work through a book like schaum’s outlines at the same time as working through your calculus problems. Even then, it will be a tough road and you might not get the grade you would have if you had been able to focus on the calculus alone. It is all about how much time you are willing to put in and how well you use all the resources available to you.

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Everyone (yes, EVERYONE) who learns new math struggles at first and then slowly puts the pieces together. The only difference is that the people with more experience learning math know what to expect and how to push through it. This is the key to not only understanding math but really to doing anything tough in life.

Think about how it feels when you start a new topic in math. Can you identify these stages in your process? Do you let yourself get caught up in frustration before you push through?

  • Stage 1: Bewilderment – I’m sorry. The fact that no one uses this word anymore made it all the more important to me that I manage to use it in a sentence today. Anyway, back to our topic: This is the stage where you have literally no idea what is going on. All you know is that someone is writing symbols on the board and they might as well be making them up. You try to look in the book and it looks more made up than the stuff you saw on the board! Pushing through this stage means asking tons of questions and truly reading your math book. When I say reading, I mean sitting with a pen and paper and trying the examples right along with the book.
  • Stage 2: Stumbling – You know you have hit stage 2 when you can do problems in the book, but only if you have your notes. In other words, you get the first step or two and then get stuck. BUT — All it takes is a look at the notes or an example (and a few minutes of frustration) and you can figure out where to go. A lot of people get stuck here because they haven’t learned that the frustration they’re feeling is no different than “feeling the burn” when working out. Those little trip ups and times when you are trying to find you mistake? Those are exactly the moments your brain is building the connections it will need for the next problem.
  • Stage 3: Robotic Understanding – After a while of working problems and stumbling about you will find yourself at this stage. Things are starting to feel much different now. While you may not always know WHY you do certain steps, you can do problems on your own and are getting the right answers most of the time. You’re seeing connections between this and things you learned before even though some of the ideas are still fuzzy. This is a good place to be and a place to be careful! Plenty of people figure they got it at this stage and stop practicing/studying. Don’t fall for that! Keeping the work up now will pay off with…
  • Stage 4: True Understanding – Truly understanding a math concept or idea means that you can work problems that use it (even if you get stuff wrong here and there – that’s normal!), see how it is connected to other ideas, and even be creative with it (as in do a problem that is different from anything you have seen and apply this new idea to it). In my personal experience, there are probably 100 levels of “true understanding”. Even with all my experience, there have been times where I suddenly realize something about a topic I learned 10 years ago! If you are taking math class, you won’t necessarily reach this stage until you have had a few new topics that your mind can start to connect together. In my experience, the difference between accepting being in the “robotic understanding” stage and pushing to the “true understanding” stage is what separates the A/B’s from everyone else.
  • Your goal: Next time you are feeling overwhelmed and frustrated learning math, remember: It is part of the process. You will have to find your own way to push through it, but I am hoping just knowing this much will keep you motivated!

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Every semester I have a few students that have put themselves in a really tough situation. Dreading math and being convinced they’re no good at it, they put off taking any math requirement to their last couple of semesters. The moment I hear this, I imagine jumping in a time machine, finding them in their first semester – just as they are registering and saying:

Sign up for a math class now! Take your math courses in your first year! Please! Not only will you do better, you will be so happy you got it done!

a time machine

This is what a time machine looks like.

Math is a skill and like any other skill, it gets rusty over time. Even if your major only requires one or two math courses, you want to do well right? The idea is – take it while the math from high school is still fresh. Even if it has been a couple of years, that will be easier than when it has been four or even six years!

If you already have a math phobia, then making it the one thing that is keeping you from graduating in your last semester isn’t exactly helpful. If you are taking it in your first year, you can be more relaxed (knowing that you can drop if you must). Being more relaxed will make learning easier and probably help you do better than if you had all kinds of “my last semester” stress.

If you are a college or high school student, do me a favor. Right now, look at your schedule for next semester. Have you taken your math requirements yet? If not, make sure you are signed up for a math class next semester. Get it done.

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If you want to go to graduate school (not counting business or law school), chances are you will have to take the GRE or Graduate Records Exam. This test is designed to measure how much you have managed to absorb in four years of college but in reality, the math portion tests heavily on high school level math! The only difference between GRE math questions and the type you saw in high school is the level of trickiness or problem solving you are expected to do. From the GRE’s website:

The content in these areas includes high school mathematics and statistics at a level that is generally no higher than a second course in algebra; it does not include trigonometry, calculus or other higher-level mathematics

Alright, so no calculus or trig – then what can you expect?

  • Arithmetic – can you find percentages? add fractions? know how to work with absolute value and a number line? Can you simplify \sqrt[3]{16}?
    Specifically you should be familiar with:

    • the properties of integers: divisibility, prime numbers, prime factorizations, basic arithmetic, exponents, radicals, ratios and percents, using absolute value, the number line, and decimal representation.
  • Algebra – You find the usual stuff here such as simplifying expression and solving linear equations but also a couple of surprises as well.
    • rules of exponents, factoring and simplifying, relations and functions, solving both linear and quadratic equations, solving systems of equations, word problems (big!), dealing with the graphs of functions and inequalities.
  • Geometry – There are quite a few geometry questions on the GRE and since most people take this in high school, you should really take the time to brush up on this topic. However, you will NOT have to worry about constructing proofs – only problems such as:
    • congruence, similarity, special triangles, properties of parallel and perpendicular lines, polygons and quadrilaterals, area, perimeter, volume, the Pythagorean theorem, and angles.
  • Statistics and Data Analysis – If you recently took statistics, don’t worry. There is no hypothesis testing or linear regression on the GRE. Instead it focuses on the basics such as:
    • mean, median, mode, range, standard deviation (not calculating it!), quartiles, percentiles, reading from graphs and tables of all types, probabilities including compound events (like “or”, “and”), independent events, random variables, probability distributions, counting methods including combinations and permutations, and Venn Diagrams.

    A high score on the quantitative section of the GRE can mean the difference between getting into graduate school and making other plans – even if you are planning on majoring in a non-quantitative field. Don’t neglect this section, and start studying early. You can get a great score even if you aren’t a math whiz!


I realize that for a lot of people, taking math courses can be stressful especially if you are just used to “getting it” before and now having trouble. Personally, I think there is something we could do with the way we teach math that would help everyone relax and appreciate math more but that’s for another post on another day :).

To do my part to help, I’ve written about how you can do well in calculus, and even had a guest post filled with tips for NOT FREAKING OUT when you hit your first really tough math class.

Based on how much people are reading the calculus articles, I’m betting a lot of you guys are engineering or science majors. If so, you definitely should check out this article over on College Info Geek: Tips for Engineering Students. In fact, if you are taking any math class whatsoever, I would follow these tips – regardless of your major!

My favorite sentence in the whole article “Take an active interesting in learning.” ::nods::