Percents and Percent Change

Received via Twitter:
What % drop is 2,800,000 from 4,150,000?

THE SOLUTION:
To calculate percent change, we take this formula: and write the answer as a percentage (just move the decimal point two places to the right). If the resulting value is positive, we have a "percent increase." On the other hand, if it is negative, we call it a "percent decrease" (i.e. a "drop"). In this particular problem, the initial value is 4,150,000 and the final value is 2,800,000. Based on the way the problem is stated, this may not be obvious. The word to focus on is "from" because it tells you that you're starting at 4,150,000.

Now, if I were answering the original question, I would leave the negative value out. The problem asks for a percent drop so the negative sign is not necessary. We'd simply say, "The drop is about 32.5%."

A Note About Percentages:
"Percent" and "percent change" are two very different concepts, but people are often confused between the two. Let's say the original problem said, "What percent of 4,150,000 is 2,800,000?" You would use instead of the formula above. For this particular problem, that means .
Words like "drop", "decrease", "change" and "increase" are clues that you're dealing with percent change, whereas those involving a discount ("x percent off") or calculating one's grade are straight percentage problems.

Calculus rate problem

Received via Twitter:
A trough is 10 ft long and its ends have the shape of isosceles triangles that are 3 ft across at the top and have a height of 1 ft. If the trough is being filled with water at a rate of 12f^3/min, how fast is the water level rising when the water is 6 inches deep?

NOTE:
Before we get start, let me just say that this is outside the scope of what I envision for this blog. My primary intent is to help middle and high school students. Having said that, I haven't received any other questions yet, so I decided to work this one out and post it.

THE SOLUTION:
First of all, it's important to understand which values are changing over time and which are fixed. The volume (V) and the dimensions of the triangle (b and h) are changing as the trough fills with water. The length of the trough, however, remains constant, so .

I always find it helpful to draw a picture so I can see what information I have and what I'm trying to find:


The trough is a triangular prism, meaning the volume can be represented by the equation . Since we're interested in how fast the water is rising, we'll look for as your professor suggested. To do this, we start by rewriting the volume equation so h is on one side of the equation: .

Next, we differentiate the whole thing with respect to t:

From there, we can simplify a little since . We need to calculate the values for volume (V) and width (w) when h = 1.


Alternate Solution:
It was proposed to me that using the Quotient Rule was overly cumbersome (in fact, I myself forgot to apply the chain rule when I first posted the solution). Therefore, here's an alternate, slightly more elegant solution based on the product rule.

Equipping parents

Received via Email:
I did not do well in math in school and still have trouble with it as an adult. I would really like to enjoy it and to teach our son to as well. Can you please recommend a book or resource for me - kind of like a relearning of the basics, but the correct way? I think if I had a better foundation early on, I would have developed confidence for the subject as you wrote about in your blog. My whole life I have struggled with math, but I know that it can be fun; something I'd like to get a handle on.

My Response:
Unfortunately, I really don't know of any resources for people in your situation. I'm sure there are books out there because I know there are others struggling with the same thing. Here are some things I would suggest (in this order):
1) Take a deck of cards and remove the King, Queen, and Jack. Let the Ace represent 1. Shuffle the cards and have your son add the values as you flip them over, one-by-one. As he gains confidence, gradually increase the speed. This actually develops several critical skills. The first is pattern recognition. In my experience, most people undervalue its importance, but the ones who excel at higher level math are usually really good at recognizing patterns and arrays. Secondly, it teaches compliments to 10. For example, to add 6 + 8 = 14, you really want your son to do this in his head: 6 + 4 + 4 = 10 + 4 = 14. The ability to regroup numbers like that is vital and directly related to the pattern recognition skill. Lastly, as you are able to increase the speed at which you flip the cards, the quicker your son will be adding stuff up. Speed is an important confidence booster. As he gets older, you can vary it up by including subtraction (e.g. add the first two cards, but subtract the third) and then multiplication.
2) Grab a small handful of loose change and throw it on the table. Let you son figure out how much money there is (someone mentioned to me that they used to do this with their child and if the child got the answer right, he/she got to keep the money - a good extrinsic motivator). In addition to the obvious counting skills, the child will learn the denominations of the coins, which in turn, is a good precursor to understanding decimals.
3) An analog clock is a great tool for teaching time reading skills, fractions, and factors of 60. You could say something like 1:30 is half past 1:00. A half hour is 30 minutes because 30 is half of 60. If you actually have a clock, it's easy to see that the minute hand is halfway around the circle at 1:30.

I hope you find these tips helpful and understand that you don't really need a fancy, expensive curriculum to empower your child with the foundation they need.

A note to parents

In my experience as a teacher, I have noticed that parents of struggling students typically fall into one of two categories. They either grossly overestimate their child's abilities or they tragically underestimate them. Each of these has detrimental consequences for the student.

The Overestimating Parent
This is the parent who goes ballistic the first time their child brings home a C or D (and it's usually in algebra 1 or geometry). "What happened?!" they ask incredulously, first of the student, then of the teacher. Well, let's say this particular student averaged 85%, a solid B, all the way through pre-algebra. That's a grade that most parents would be satisfied with. Is it sufficient? As a teacher, I look at that as the student understanding 85% of the critical material covered in the class. Would you build your house on a foundation that was only 85% complete? Probably not, right? Then why are we expecting our kids to succeed when they only have 85% of the basic skills down?

The Underestimating Parent
As a teacher, I've heard parents tell me, "My child isn't going to major in math. I just want him/her to pass this class." It was so frustrating to hear that. It's like cutting off the child's legs before they even have a chance to learn to walk. How do you expect your child to succeed if you keep telling them they're a failure? I don't think many parents realize how critical self-confidence is in math. Students who excel, often do so because they are confident. Those that do poorly, are caught in a dangerous, downward spiral, circling between failure and self-defeat.

If you are either of these kinds of parent, take a step back and reassess your situation. If you truly understand your child's strengths and weaknesses, you'll be better equipped to anticipate and deal with potential roadblocks. Your child's success is heavily dependent on your perception.