Category: Math for Teachers

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Basic Math Review Math for Grownups Math for Teachers

Time on Your Hands: Translating base 60

A few weeks ago, a screen shot from Yahoo! Answers was floating around the interwebs, and a friend posted it on my facebook page asking if I would decipher it.

This is my last day of vacation, and because this question relates very well to bases (the topic of Monday’s post), I thought I’d take an easy route today and explain it. I’m guessing that many of you can see the fallacy right away, but the question speaks to how bases work (and don’t necessarily play well together).

Remember that our decimal system is in base 10. That means each place value depends on a multiple of 10: 10s, 100s, 1000s. This is also true for values smaller than 1: 10ths, 100ths, 1000ths. Got it?

Our system for measuring time is different. As the questioner correctly notes, there are 60 seconds in a minute. In fact, we measure time in base 60. Seems that this derived from the Babylonian’s astronomical calculations, a very elegant system. See, 60 is the smallest number that is divisible by the first six counting numbers: 1, 2, 3, 4, 5, 6. Neat, huh? It’s also divisible by 10, 12, 15, 20 and 30, making it an even more flexible number.

This in turn gives way to the analog clock, which is circular. Circles measure 360 degrees: base 60! In fact angles and circles are measured in base 60. (Check out this cool way to teach kids how to read an analog clock and understand circles.)

Okay, so time is measured in base 60. All that means is that 1 minute equals 60 seconds and 1 hour equals 60 minutes. (Forget the hours and days for now.) But remember, our decimal system is base 10.

And that’s where this questioner has gone wrong. You can measure time in base 10, but it won’t translate the same way as base 60. In other words, 120 seconds is not 1.2 minutes. Nope, it’s 2 minutes.

And this is exactly why it’s hard for kids to learn to read analog clocks. And why microwaves might burn brain cells along with the popcorn you were having for a snack. Even though we’ve spent our whole lives using base 60 to measure time and base 10 to measure practically everything else, sometimes it’s tough to switch back and forth. Darned Babylonians.

Have you ever gotten mixed up because time is measured in base 60? Share your story (especially if it’s travel related) in the comments section.

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Math for Grownups Math for Parents Math for Teachers Travel

How Far? Estimating metric distances

Earlier this month, I showed you how to convert currencies, when given the exchange rate. When you’re not using an online calculator, that process involves proportions, which are pretty simple to use, but do require a little figuring on paper. This same process works for any conversions, including miles to kilometers, liters to ounces, etc.

But while being exact with your money is pretty important, estimating how far you have to drive or walk is usually good enough. So instead of going into details about metric-to-traditional measurement conversions, let’s look at how you can find these distances with a little mental math. First, you’ll need to know a few facts:

1. A mile is longer than a kilometer. So, when you convert miles to kilometers, the answer  will be larger than the original amount. (mi –> km = larger answer)

2. A kilometer is shorter than a mile. So, when you convert kilometers to miles, the answer will be smaller than the original amount (km –> mi = smaller answer)

2. In fact, 1 mile equals 1.61 kilometers. And 1 kilometer equals 0.625 mile.

3. Those values are pretty darned close to 1.5 kilometers and 0.5 mile.

Remember, we’re estimating here, so you’re not looking for an exact answer. Forget what your middle school math teacher said about the precision of math. You don’t always need to getan exact answer. But there’s another fact you’ll need to consider:

4. The larger the value that you’re converting, the less precise your answer will be.

If you depend on the estimate 1 mi = 1.5 km and you’re converting 15 mi to km, your answer will be pretty close. BUT if you’re converting 1,468 mi to km, your estimate will be a lot lower than the actual answer.

Look, estimating is no big deal. In fact it’s a really, really powerful tool that can make your life much easier. You do need to know when estimation is in your best interests and when you should pull out the calculator. (See? Math really isn’t all that black and white!)

Let’s look at an example. Zoe has finally made it to London! She’s spending the summer studying Shakespeare and working part-time as a docent at the Tate Modern. And she’ll have some time to roam around Europe a bit. She’s rented a car so that she can chart her own path, and next Friday afternoon, she’s going to cross the channel to France, where she hopes to spend four days winding her way down to Paris and back.

But how long will it take her to get there? According to her map, the distance is 454 km. Since Zoe is used to miles, she’d like to convert the distance so that it makes more sense to her. She’s okay with a rough estimate, especially since she has no firm schedule. So she decides that knowing there are about 1.5 km in a mile is good enough.

To make the math even easier, she decides to round the distance as well: 450 is pretty close to 454. Now she can easily do the math in her head, but we’ll get to that in a minute. Let’s write it out first.

Because she’s converting kilometers (shorter) to miles (longer), her answer will be smaller than the original amount. That means she’ll need to divide.

450 km ÷ 1.5 = 300 mi

So she’ll travel about 300 miles to get from London to Paris — not a huge distance!

But how could she do this in her head? For that, she’ll need to remember a few things about fractions.

1.5 = 3/2

450 ÷ 1.5 = 450 ÷ 3/2

450 ÷ 3/2 = 450 • 2/3

(That’s because when you divide by a fraction, it’s the same thing as multiplying by its reciprocal — or the same fraction upside down.)

So in order to convert kilometers to miles in her head, she’ll need to multiply the value by 2 and then divide by 3 (which is the same as multiplying the value by 2/3. In other words:

450 • 2/3 = (450 • 2) ÷ 3 = 900 ÷ 3 = 300

Whew!

But once Zoe remembers this little trick, she can estimate these conversions quickly and easily.

30 km = ? mi

30 km • 2 = 60

60 ÷ 3 = 20

30 km = 20 mi (approximately)

Make sense? Try it for yourself: convert 75 km to mi and then use an online calculator to check your answer. Remember, if you’re using the process above, you’ll get an estimate, not an exact value!

So take a guess: If you’re converting mi to km, what process would you use? See if you can figure it out and then offer your explanation in the comments section. Feel free to choose a value to convert, if it’s easier to explain that way.

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Basic Math Review Math for Grownups Math for Teachers

The Metric System: What’s the big deal with bases?

I’m vacationing this week in sunny Radford, Virginia,and ike most parts of the United States, the metric system is not used here (to mark distances, anyway). But if you cross the border into another part of the world, there’s little doubt that you’ll be measuring kilometers rather than miles and grams instead of pounds. That’s because most of the world has embraced the metric system. (In fact, only two other countries — Burma and Liberia — have resisted the change along with the U.S.)

Ask any scientist or mathematician: the metric system is infinitely more intuitive and much, much easier to remember and understand. But why? The answer is simple: Base 10. What this means is even simpler: in base 10 the foundational number is 10. Take a look:

10 • 1 = 10

10 • 10 = 100

10 • 100 = 1,000

and so on…

Each time you add a digit in our number system, you are effectively multiplying by 10. That means that 99 is the last two-digit number in base ten, and 999 is the last three-digit number. In fact our entire decimal system is base ten. (But it wasn’t always like that.)

But here’s the thing — you don’t care (and you shouldn’t really care). We are so used to base 10 that we don’t even think about it any more. It’s like knowing how to ride a bicycle or drive a car; once you learn it, you don’t even give it a second thought, but if you’re asked about it, it’s hard (or impossible) to explain.

When you were in school, you probably were asked to convert numbers into different base systems — and this was probably pretty darned confusing. We’re not going to do that here for one simple reason: You don’t need to know how to do this. BUT it is important to know that different base systems are useful in a variety of situations and professions. For example, computers function in base 2 (or binary), which is simply a system of zeros and ones. Computer graphics depend on a hexadecimal system or base 16 (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F). Crazy, huh? Even less technical situations depend on a variety of bases — dozens and grosses are base 12 (one dozen is 1 • 12 and one gross is 12 • 12).

Compared to these other bases, base 10 is pretty darned easy, right? And that’s why so many mathy folks don’t understand why our country hasn’t embraced the metric system. Yep, unlike traditional measure systems, the metric system is base ten. Let’s compare:

Traditional system: 12 inches = 1 foot

Metric system: 100 centimeters = 1 meter

Traditional system: 5,280 feet = 1 mile

Metric system: 1,000 meters = 1 kilometers

Just a glance at these conversions and even the most math-phobic person would probably agree: the metric system is much easier to maneuver.

But agreeing that the metric system is easier doesn’t help you with conversions when you’re traveling, does it? On Wednesday, we’ll take a look at those conversions. I’ll show you some really easy ways to estimate the conversions. Because who wants to do math on vacation?

What other bases can you think of? How do you use them in your everyday life? Share your ideas in the comments section.

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Math for Parents Math for Teachers Travel

Kids in the Car: Keep ’em busy with math

Whether you’re flying across country or crammed in the mini-van for a trip to Grandma’s house, keeping a kid occupied on a long trip may mean you need a vacation at the end of it. And sure, we can plug them into movies or iPods or video games, but is that really what you want your children to remember about their trip to the Grand Canyon?

Being trapped in a car or plane or train for hours at a time will either kill you or make you stronger, and I’m rooting for stronger. You can look at this as an opportunity to hang out with your kids — and even sneak in a little math.

I know that sounds really, really geeky, but this was a real, live question that a parent asked me over at MSN.com’s Mom’s Homeroom where I’m the resident math expert. Since we’re talking travel this month, I thought I’d expand on the ideas here. The parent asked: “What are some fun math games that I can play with my 10 year old son and 7 year old daughter while on road trips?”

First and Last

This is a take on a game that I used to play with my daughter. She would say a letter, and I would say a word that began with that letter. Then she would identify the last letter of that word, and give me a word that began with that letter. For example: S prompted me to say spaghetti. She would say I and then igloo.

This can easily be adapted to math, which helps kids (and adults) practice their mental computation skills. For example:

First player: 16 + 3

Second player: 19

Second player: 19 – 10

First player: 9

First player: 9 • 3

and so on…

Set the rules of the game so that everyone can play. For example, no negative numbers, fractions or exponents, if your 13 year old is playing with his 8-year-old brother. Or tell them that they can only use even numbers or only addition and division. You might just find that your kids are getting really creative — and making some cool connections. (Did you know that when you add or subtract only even numbers, the answers will always be even?)

Road Sign Math

If you’re in the car, sometimes the only thing to read are road signs and license plates. But if you take a close look, you could find some math in there. In fact, someone has created a cool wiki devoted to this game. Take a look at the sign below.

Photo courtesy of Road Sign Math wiki

Do you see the math in there? It’s a very simple addition problem: 2 + 4 = 6.

These can get downright complex! But you can keep it easy for your younger kids. Look out for route numbers, license plates and billboards for more ideas. If you’re used to traveling the same road over and over, this is a particularly good way to pass the time. What’s old becomes new again!

I Spy

This perennial favorite can be adapted to all sorts of situations. For example:

“I spy with my little eye: a prime number!”

“I spy with my little eye: 17!”

I spy with my little eye: a fraction!”

Try this with a boring magazine on the plane. Keep the questions on grade level and offer encouragement for good — or close or creative — answers. Need to remember what a prime number is? If you’re not driving, do a quick search on your smart phone.

There are countless other ideas that can help you pass the time and inject a little math into the trip. Do you have suggestions? Offer them in the comments section!

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Math for Grownups Math for Teachers Travel

Keeping Current: Using proportions to convert currency

Last Friday, we looked at exchanging currency — how far will your money go in another country? In that post, I introduced you to online currency conversion calculators and helped you assess whether or not your answer made sense. Today, we’re going to look at doing these conversions by hand.

Out of every basic math skill I know and have taught, proportions are the most useful — and most often forgotten. You can use them to shrink photos proportionally (so that the Eiffel Tower doesn’t look squat and fat or that mime doesn’t resemble a human hericot vert), alter a recipe to feed an army or find unit price. With proportions, you don’t need to remember whether to multiply or divide. Get the numbers in the right place, cross multiply, solve for x, and you’re good to go.

But let’s back up for a second. What is a proportion? It’s simple, really. A proportion is merely two equivalent ratios. (Remember, a ratio is a way to compare two numbers, often written as a fraction.)

1/2= 2/4

The two fractions (ratios) in the above statement are equivalent: 1 out of 2 is the same thing as 2 out of 4. But that’s just an example. The key to setting up currency exchange proportions is knowing where each part goes.

There are four parts: the original currency ($1USD, for example), the currency exchange rate (the value of $1USD in the other currency), the value you are converting, and the value after the conversion (the answer or x). You want to be sure that all of your parts are in the right place.

But there is more than one right place! So, I suggest being consistent with these parts. That way, you can always, always use the same proportion for each conversion that you do.

($1USD)/(euro exchange rate) = (USD value)/(euro value)

That looks a little clunky, but it’s not really difficult to dissect. Look at it carefully, and you’ll notice a few things:

  1. The $USD amounts are in the numerators of the ratios.
  2. The € amounts are in the denominators of the ratios.
  3. The conversion exchange ($1USD to €) is in the first ratio, while the actual values are in the second ratio.

To use this proportion, you need three of the four values found in this proportion. What do you think they will be? One of them will always be 1, because it’s the base value of the currency exchange. If you’re converting $USD to €, you’ll use $1USD. If you’re converting € to $USD, you’ll use 1€. The second known value will be the currency rate. Last Friday, we used $1USD = 0.794921€, so let’s stick with that, making the second value 0.794921. The third value will always be the value you’re converting.

Let’s look at an example. You spy a gorgeous pair of boots in Paris for only 324€. You have $500USD budgeted for a special splurge. Are these special boots within your budget? Plug things into the proportion to see:

1/0.794921 = x/324

Before you let your nerves get the best of you, look at this proportion carefully. Which values have gone where? Now, do you think there is another way to set up this proportion? (Psst… the answer is yes.)

0.794921/1 = 324/x

Or even:

1/x = 0.794921/324

Notice that while the numbers themselves have changed places, their relative positions have not. The $USD values (1 and x) are still related (either in the same ratio or in the numerator or denominator), and the € values (0.794921 and 324) are still related (either in the same ratio of in the numerator or denominator).

But how do you solve this proportion? (In other words, “Holy crap! There’s an x in there, and it freaks me out!”) Take a deep breath and cross multiply. Choose one of the proportions above (I’m going with the first one), and picture a giant X on top of it. One segment of the X lies on top of the numerator of the first ratio and the denominator of the second ratio (the 1 and the 324). The other segment of the X lies on top of the denominator of the first ratio and the numerator of the second ratio (the 0.794921 and the x). Multiply the connected values, like this:

1 • 324 = x • 0.794921

Now you can simplify and solve for x.

324 = 0.794921x

Divide each side of the equation by 0.794921 (in order to get the x by itself).

324 ÷ 0.794921 = x

407.587672 = x

You’ve just discovered that 324€ is equal to $407.59USD. That’s within your budget, so you’re good to go!

Now, try the other conversions to show that they work, too. See? Flexibility in math! (Who knew?)

What did you think of this process? Scary? Easy? Too hard? Stupid, because you can always use a calculator? Do you have another way to convert currency (besides proportions and using a calculator)? Share your ideas in the comments section.

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Math for Grownups Math for Teachers Travel

Keeping Current: Converting currency right

You’ve booked that trip to Paris. Vive les vacances! Now that your credit card has borne the brunt of your plane tickets and hotel reservations, with just enough space for a couple of fantastic meals, it’s time to turn to the cash. How much should you bring — and even more importantly, how far will it go?

When traveling out of country, you need to consider the currency exchange rate. Only very rarely is this exchange equal. (In other words, one Euro almost never equals one U.S. dollar.) That means, you’ll need to use a conversion to find out how far your cash will actually go.

There are actually three things to talk about here: using an online conversion calculator, doing the conversions by hand and checking your answer to see if it’s reasonable. Remember, math is infinitely flexible, so there’s no reason you have to do this in one particular way. Next Wednesday, we’ll look at doing conversions with paper and pencil. Today, it’s all about online calculators and checking your answer.

First, the conversion calculators. Go ahead and use them! If nothing else, a reliable online calculator will give you the most up-to-date conversion rate with the click of a button. For example, using the XE currency conversion calculator, I found that $1USD is equal to 0.794921€ (as of Monday, July 2, 2:05 p.m.).  This means that one U.S. dollar is worth a little more than 75 percent of a Euro.

If you know the exchange rate, it’s really easy to exchange values of 10, 100 or 1000. In these cases, you can simply move the decimal point.

$10USD = 7.94921€

$100USD = 79.4921€

$1000USD = 794.921€

Notice that when there is one zero (as in 10), you move the decimal point one place to the right. When there are two zeros (as in 100), you move the decimal point two places to the right. And when there are three zeros (as in 1000), you move the decimal point three places to the right.

Of course, if you want to convert $237.50USD to Euros, that trick won’t work. In that case, you can plug $237.50 into the online calculator. If you have $237.50USD in your pocket, that’s 188.717€.

XE also has iPhone and Droid apps, so you can take the online calculator on the road with you. (Note: I don’t have any relationship with XE. It just looks like a good, reliable online currency calculator. Want to recommend something different? Feel free to respond in the comments section.)

The thing about online calculators is that they’re only as good as the information that you put in. If you think you’re converting $USD to €, but you’re actually doing it the other way around, well, your fancy pants calculator is not going to spit out the answer you were looking for. You have to know how to assess whether your answer is correct.

I’m the first to admit that I get this very confused. I have to stop and think really hard to be sure that I’ve done the conversions correctly. (And to be honest, this is one of the reasons I prefer to do it by hand.) But there are some simple rules you can consider that will help:

  • If the conversion rate is less than 1, the conversion will be less than the original amount.
  • If the conversion rate is greater than 1, the conversion will be greater than the original amount.

Let’s say that $1USD equals $1.26SGD (Singapore dollar). If you convert $USD to $SGD, will your answer be greater or less than the original amount? If you said greater — you’re right! But if you convert $SGD to $USD, the answer will be less than the original amount. Make sense?

The good news is that you can figure this out before you leave. Write it down or keep a note on your phone. Then you will always be able to check to see if your answer makes sense. Because the worst thing is to come home from a relaxing vacation to find that you’ve spent way too much.

Be sure to come back next Wednesday to get the deets on how to do these conversions by hand. It really isn’t that difficult — and the process is applicable in so many other situations, so it’s worth learning.

Where are you traveling this summer? Share your plans in the comments section below!

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Math for Grownups Math for Teachers

Where’s the Dollar? An answer to Monday’s riddle

On Monday, I posted the following travel- and math-related riddle. I’m guessing everyone was too scared to post their answers — or perhaps you’re all celebrating Independence Day a little early — because no one chimed in. But no worries, my feelings aren’t hurt in the least. Still, I promised the answer, so here it is.

First the riddle itself:

Three friends are traveling to their high school reunion together. They arrive at their hotel late at night, only to find that their reservations were lost.  There is only one room with three beds available. They have no choice but to share the room, which the hotel has discounted to $30. Each of them takes out a 10 dollar bill, which the clerk collects.

After the friends are settled into their room, the manager reconsiders the discount. (He feels terrible!) He decides to offer the room at only $25 and sends a porter upstairs with $5 for the three friends.

The porter starts thinking about how to divide the $5 into three equal parts. When he can’t figure it out, he decides to give $1 to each friend, and pocket the rest. The friends accept the $3 refund, and the porter heads back to his post, with the remaining $2.

Given their $3 refund, each of the three friends paid $9 for the room (3 • 9 = $27). The porter has $2 in his pocket, making the total $29 ($27 + $2 = $29). But the friends originally paid $30!

What happened to the $1?

If you’ve been around the block a few times, you’ve probably heard this riddle. And if you google “missing dollar riddle,” you’ll find thousands of results that outline where that dollar actually is. (Heck, there’s a Wikipedia entry about it!) Most of these talk about a logical fallacy, which is a perfectly reasonable way to describe things. In my mathy brain, there’s another way to explain it, using equations.

This is what we know:

In other words, the friends originally paid $30, but the manager decided to discount the room by $5. That meant that the clerk took $25 from the original $30 and the porter took $5 from the original $30.

$30 = $25 + $5

Then the $5 was split up — $3 for the friends and $2 that the porter pocketed.

$30 = $25 + $3 + $2

Clearly there is no missing $1. Here’s another equation to prove why. If you subtract $3 from each side of the equation, you get this:

$27 = $25 + $2

This works, because with their discount of $1, each friend paid $9 for the room, rather than the original $10. Another way to look at it is this:3 • $9 = $25 (the cost of the room) + $2 (the amount the porter pocketed)$27 = $25 + $2Get it? If not, take another look. It is confusing at first, but once you see it, it does make sense.Now if you subtract the $2 from both sides of the equation, you can see how the amount that the friends paid minus the amount that the porter pocketed equals the cost of the room itself.$27 – $2 = $25$25 = $25Make sense? Sometimes it does to me, and then my understanding floats away! But I do think it can be fun to look at these problems mathematically. I hope you did, too.Did you come up with the correct reasoning before reading this? Did you use math? If not, how would you explain that the dollar is not missing at all? Share your ideas in the comments section!

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Home Math for Grownups Math for Teachers

The Mighty Hexagon: Let bees help you garden

On Monday, Cristina Santiestevan of Outlaw Garden shared a post about the geometry of gardening, introducing us to the idea of “hexagonal spacing.” This was such a cool idea that I thought I’d explore it further. I wanted to know the math behind it. In other words, why are hexagons so darned special?

Let’s start by with the bees. In research for a magazine assignment, I’ve done some reading about bees lately, and once again, I’m in absolute awe. These little guys are the linchpins of our ecosystem in a lot of ways. Not only does their pollen-collecting insure the reproduction of a variety of plant species (and therefore the survival of critters that depend on these plants), but their colonies are efficient little factories that seem to mirror human manufacturing — from the dance the workers do to relay directions to the best pollen to the efficiency of their job descriptions.

And then there are the hives. If you think of the bees as efficient — and they are — you can deign why the hive is made up of tiny hexagons. (Remember, a hexagon is a six-sided figure.) Not wanting to waste any space whatsoever, the bees figured it out: instead of making circular cells, which leave gaps around the sides, they create a tessellation of hexagons, which leave no empty space at all.

Photo courtesy of wildxplorer

(A tessellation is the repetition of a geometric shape with no space between the figures. Think M.C. Escher or a tile floor.)

The same concept applies to gardening. Why waste space? As Cristina pointed out, choosing a hexagon-shaped planting scheme, you’ll get more plants in your beds.  And if you’ve got an outlaw garden, like Cristina, it’s best to make the most of your space! Here’s how:

In regular rows, you plant 6″ apart in only two directions, getting nice, even rows. But if you consider six directions, you’re replicating the hexagon, instead of a square — and as a result maximizing your space (just like the honey bees). Cristina describes it as planting on the diagonal. Or you can think of each plant at the center of the hexagon.  Then you can plant the others 6″ from the center in six directions — creating the vertices of the hexagon. (If you’ve ever looked carefully at a Grandmother’s Flower Garden quilt pattern, this idea might jump out at you. Not only is each plant the center of a hexagon, but it’s also the vertex of another hexagon.)

Drawing courtesy of Cristina Santiesteven

Did you see what I did there? Math can be described in a variety of ways! Look at the second diagram carefully, and see what jumps out at you — the hexagons or the diagonal rows?

So there you have it. We can learn a lot from a bee. And I can already think of times when this can be useful in other areas. How many more cookies can you fit on a cookie sheet, if you arrange them diagonally (or in a hexagon shape) rather than horizontal rows? What about kids desks in a classroom?

Where can you apply the hexagon to make your space more efficient? Share your ideas in the comments section!

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Current Events Math for Grownups Math for Parents Math for Teachers

How Hot Is It? Calculating the heat index

Lordy, it’s hot. And the heat makes me cranky. When I saw that the temps were creeping up to the 90s and beyond this week, I vowed to stay in the airconditioning. Trust me; it’s best for everyone involved.

So don’t even tell me what the heat index is. I really don’t want to know. But I have always been fascinated with how it is calculated. What are the variables that affect the heat index? Let’s take a look.

The heat index is how it really feels when the humidity is figured in. (Those of you who live in a climate with dry heat have no clue about this. Count yourselves lucky.) When the humidity is high, the heat index goes up, producing a hot, sticky mess that makes my hair frizzy and sours my otherwise lovely temperament.

The thermometer may say 95 degrees Fahrenheit, but if there’s significant humidity, it might feel like it’s 105. But of course meteorologists don’t guess at this number. There’s an actual formula that’s used to find the heat index.

Before we get to that, let’s consider the variables involved. According to the National Oceanic and Atmospheric Administration (NOAA), there are 20 (yes, twenty) variables that are used to calculate the heat index. These range from vapor pressure to the dimensions of a human to ventilation rate to sweating rate (ew). Because most of these are very specific to each person, a mathematical model was used to determine an appropriate range for each. This allows meteorologists to use a (relatively) simple formula for finding the heat index:

HI = -42.379 + 2.04901523T + 10.14333127R – 0.22475541TR – 6.83783(10-3T2) – 5.481717(10-2R2) + 1.22874(10-3T2R) + 8.5282(10-2TR2) – 1.99(10-6T2R2)

Pretty, right? It’s actually not that hard to understand, if you break down the pieces. First, let’s define the variables.

HI = heat index

T = ambient dry bulb temperature (in Fahrenheit)

R = relative humidity (integer percentage)

So there are basically three variables, one being what we are looking for — the heat index. If you were to use this formula, you would need to know two things: the ambient dry bulb temperature (which is merely the ambient temperature as measured by a thermometer) and the relative humidity.

If you put to work the logical part of your brain that notices connections and patterns (yes, you do have one), the math becomes clear. When the temperature and relative humidity go up, so does the heat index. How do you know that? Look at the equation. It’s full of addition and multiplication. In fact, aside from the negative exponents (which actually yield smaller numbers), the equation is based solely on increasing values.

(That is, unless you have negative values for T and R. But in that case, you wouldn’t be figuring the heat index, right? A negative T means a negative air temperature, which is really cold in Fahrenheit. And I’m not sure that relative humidity can be negative at all.)

Now, almost nothing is absolute in weather prediction and measurement, right? And this equation is no exception. As NOAA points out, this equation is created by multiple regression analysis, which means it is not exact. (Basically, in this process, the mathematicians are fitting points to the closest line. Think of a bunch of points on a graph and how you can draw a predictable line or curve that is closest to all of those points.) There is in fact an error of ±1.3 degrees Fahrenheit. But what’s 1.3 degrees when you’re looking at a heat index of 102? Either way, it’s still darned hot.

How do you manage the heat? Do you head inside or hide in a cool, dark place? Share your ideas in the comments section.

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Getting an Education in Student Loans

How about these scary statistics:

1. In the U.S. student loan debt is huge. Last year alone, students took out $117 billion in federal student loans. The Consumer Financial Protection Bureau estimates that the total U.S. debt has now exceeded $1 trillion. And this debit is not simply because new students are going to school. Nope, it’s also because folks with college degrees are behind in their loan payments, which increases the total interest costs. (The New York Federal Reserve estimates that 1 in 4 people with student loan debt is behind in their payments.)

2. The cost of a college education is rising fast. From the 1999 school year to the 2009 school year, tuition and room and board at public institutions rose 37% and at private insituations rose 25%(adjusting for inflation).

All of these statistics — and more — have some economists worrying that student loans are the new economic bubble. Like the tech and real estate bubbles, if this one bursts, the country could be in for another deep recession, this time with the federal government holding the bag.

So what the heck are colleges, parents and students doing to slow down this fast-moving train? Elgin Community College (ECC) in Elgin, IL is getting proactive, requiring financial aid counseling to students who are seeking federal student loans.

“The feedback has been positive,” says Amy Perrin, ECC’s director of financial aid and scholarships. “Students have expressed appreciation for educating them on the loan basics, budgeting, percentage interest rates and expected monthly payments.”

But student expectations are still a big issue. “We’ve had several students walk in with an inflated idea of what they ‘want’ to borrow — and walk out with a better understanding of what they ‘need’ to borrow,” Perrin says.

Student loans aren’t free money. And unlike other debts, these loans can follow a person forever, since they cannot be discharged in bankruptcy. It’s not just the math that trips students up.

“There seems to be a conflict between the Department of Education’s regulations and the student’s reality,” Perrin says. “The loan advising meeting covers many concepts, including creating a budget, interest rates, monthly payments, the student’s rights and responsibilities, and the consequences of default. After meeting with the staff, they should have a good understanding of the basic financial concepts of borrowing a student loan.”

So how can math help? A solid understanding of interest payments is critical here, and although there are online calculators that can help students estimate the total cost of these loans, students must have some basic math skills in order to use them. Perrin also suggests that parents and schools work harder at developing financial literacy skills.

“Parents can definitely play an important role in educating their children on basic financial concepts such as budgeting, how to open a checking account, why having a savings account is important and explaining ‘wants’ vs. ‘needs,’” she says. “Additionally, high schools should infuse financial literacy concepts into their classroom curriculum to further communicate the importance of wise financial decisions. High schools can partner with colleges to offer financial aid awareness events for parents and students.”

This student loan debt isn’t going anywhere any time soon. Unless we turn on our math brains and really deal with the numbers behind these scary statistics, our country could end up in another ugly economic place. Here’s hoping that other colleges require students to attend these programs–so that college degrees can actually mean something more than a monthly debt that must be paid off.

I’ll be the first to admit that my understanding of student loans is limited. So if you have questions, I completely understand! Post them here, and I’ll find the right expert to answer them. 

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Saving Lives with Math

Math Appreciation Month has finally come to a close. And I thought I would end with some math that could save your life. This is serious — and I think really interesting — stuff.

If you’re seen a recent “best college degrees” list, you probably wondered two things: Why the heck is Applied Mathematics on the list, and what is it? First off, applied mathematics is not about crunching numbers. Instead, these folks use higher level mathematics — from abstract algebra to differential equations to statistics — to solve a myriad of problems in a myriad of industries. And that, my friends, is why it’s on the list. In industries like energy, cell phone technology and medicine, math modeling and statistical analysis have been applied to solve really big problems.

Math modeling is one branch of this field that has become a very big deal. Let’s say a city planner wants to know how many snow plows to buy so that the city isn’t paralyzed by a winter storm. Modeling this problem using mathematics is one way to address this problem. The way I look at it, math modeling helps us understand things we can’t see — because they’re part of situations that haven’t occurred or are too far away or are too tiny and hidden.

That too tiny and hidden part that is what math modelers are honing in on with medicine. In this field — sometimes called bioinformatics or computational biology — mathematicians help medical professionals address problems that are under the skin. Here are two examples:

Fighting Cancer: Researchers at University of Miami (UM) and University of Heidelberg in Germany have created a math model that will help oncologists predict how a tumor will grow, and even if and how it will metastasize. There have been other math models that look at tumors, but this one is different. Instead of looking at each cell or all of the cells has a big group, this model creates a kind of patchwork quilt of areas of the tumor to examine. As a result, the doctor can create a tailored plan for treating the disease that is very specific for each patient. The promise is that with specialized (rather than generalized) treatment plans will offer patients a better chance at survival.

Treating Acetaminophen Overdoses: When a patient comes into the emergency room having overdosed on acetaminophen, the ER staff is faced with a really complex decision. Often these patients are hallucinating, unconscious or comatose. And since it’s relatively easy to overdose on the drug (it takes only five times the daily safe dosage, and acetaminophen is in many different over-the-counter and prescription medications), it’s sometimes impossible to determine when and how much of the drug was ingested. There is an antidote, but at a certain point, the doctor needs to skip that step and put the patient on the liver transplant list immediately. The trick is accurately identifying that point. University of Utah mathematician, Fred Adler, developed a set of differential equations that can better pinpoint the critical information needed to make these decisions.

In both of these cases, the math is pretty darned complicated, depending on a branch of calculus called differential equations. This approach is a step up from statistical analysis, which compares patient data to data collected from other patients. In other words, it assumes that tumors grow in the same way in all patients — which we know isn’t true. These dynamical math approaches allow doctors to offer treatments that are customized for each patient, based only on the information collected from the patient.

And the best part is that the doctors don’t have to know the math. If future studies bear out these new discoveries, a simple app can be designed for smart phones or tablets, allowing physicians to make diagnoses and treatment plans bedside.

I suspect these applications will continue to grow, as the medical community turns to mathematicians for insight into what we can’t see. That’s great news, because these advances can save lives.

I hope you’ve enjoyed what we’ve put together here for Math Appreciation Month. If you have questions, please ask them below. I’m always open to ides for future blog posts, so please share them!

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Ten Things Parents Wish Math Teachers Knew

We’ve gotten advice from math teachers to parents and from students to math teachers. But parents can also play a big role in how their kids learn math and succeed in school. So, I’ve decided to given them a chance to share their feedback with math teachers. (Besides, when I went looking for students to give me advice, parents just couldn’t help themselves!)

I’ve been on both sides of this equation, so I have lots of empathy for teachers and parents. Neither of you have easy jobs! In case it’s not clear, I wholeheartedly believe that most teachers are in the classroom because they love kids and want to make a positive difference in their lives. But we’re all human, and teachers can always strive to be better at their craft.

Here goes:

Help a parent out.

The language of math is different than it was when most of us learned it the first time. (For example, in subtraction many of us “borrowed.” Our kids “regroup.”) A cheat sheet or a website with information would go a long way in helping parents help their kids with understanding the concepts.

This goes double (or triple) for discovery-based math curriculum, like Investigations or Everyday Mathematics. These programs often don’t rely on the algorithms that many of us are used to using. To be fair, the curricula have parent components, but if the school or teacher doesn’t use them, parents are often left in the dark.

Know the kids.

Parents do understand that there are a lot of big stressors on teachers. Teachers are often told to do things that they wouldn’t choose to do (like teach to a test). They have large classes and short periods of time with the kids. But parents still expect teachers to know each child well. Teachers should know which kids have trouble with memorization and which ones struggle with understanding difficult concepts.

Give parents a homework estimate.

How long should students be working on an assignment? An hour? 15 minutes? Two hours? Kids work at different speeds, and parents need to know when we should be encourage our kids to pick up the pace or investigate whether our children are moving slowly because they don’t understand the concepts.  And while we’re on the topic of homework, parents told me that there was no point in sending home 50 of the exact same problems. One parent said: “Hours of pointless busywork make kids hate math.”

Mean what you say and say what you mean.

This doesn’t have anything to do with classroom management, though this is good advice here, too. Parents told me about very poorly worded questions that confused their kids. “My [child with Aspergers] is very literal,” said one mom. “This sometimes means he actually answers the question correctly but not the way the teacher intended. More than once I have had to ‘correct’ his homework and say, ‘Yeah, I know what you put is accurate, but that is not what the teacher meant by the question.’” One parent suggested having someone who is not an educator look at your materials to be sure that the questions are clear.

Update your materials.

Don’t pull old worksheets from old curricula that doesn’t apply to current pedagogy. And by all means, make sure that what you’re sending home with kids is what they’re learning about in class. It’s really frustrating for parents and kids to see homework that is not jibing with classwork.

Review tests and graded assignments.

Students need to understand where they made their mistakes and why. Parents need to know where students’ gaps in understanding are. Reviewing tests also reinforces the important idea that tests are a means for assessing understanding, not a big, red stop sign for learning. But don’t let students check each other’s work. “It’s demoralizing,” said one parent.

Don’t confuse computational errors with conceptual misunderstanding.

When a student makes a common addition error, that doesn’t mean she doesn’t understand the concepts behind the problems.

Introduce relevant and meaningful application (word) problems.

At the beginning of this school year, my sixth-grade daughter vented about a word problem she was given for homework: Carlos eats 25 carrots at dinner, and his brother eats 47 carrots. How many carrots did they eat in all? “Who eats 47 carrots?” she wanted to know!

If you don’t know what’s relevant to your kids, ask them. Or watch a television program they may like or talk to parents or search the internet. Along with word problems, parents want financial literacy introduced early and often. These problems can be included in a variety of places within traditional curricula.

When a child isn’t succeeding, ask why.

Sometimes this is because of misbehavior, but sometimes misbehavior occurs when a child is bored or confused or just feels unconnected to the class. Some kids give up easily. And others have undiagnosed–or unaddressed–learning disabilities. Get the parents involved as quickly (and often) as possible.

Don’t write our kids off.

Some kids struggle and some kids understand the concepts right away. Parents want teachers to stick with their kid, no matter what. Parents can tell when teachers have decided that a kid isn’t worth their effort. That’s heartbreaking to parents–and students.

Not all parents want or can be intimately involved in their kids’ math education, but I think it’s fair to give each parent a chance. Just as it’s fair for parents to give teachers the benefit of the doubt.

Parents, do you have any additional advice for teachers? Teachers, do you want to respond to any of these ideas? Let’s get a good conversation going!