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Properties that are damaged by fire, water, storms, smoke, or mold require the services of a professional.  This is a job for Nate Dawson, Restoration Hero and President of Sterling Restoration.  Read on to see how he uses math to restore damaged properties back to mint condition.

Can you explain what you do for a living?

Sterling Restoration specializes in emergency repair to real property whether damaged by fire, water, storm, smoke or mold. Sterling Restoration is trusted for high quality and comprehensive cleanup, mitigation, and restoration services for both residential and commercial projects. We are a locally owned company based in Springfield, Ohio serving the Miami Valley and Central Ohio areas. We take pride in knowing that our team of professionals and extensive network of resources have the expertise to return any property to its pre-loss condition as quickly as possible.

When do you use basic math in your job?

Basic math is used in all aspects of our business including our accounting, estimating and production departments. Our accounting department uses it to calculate payroll, receivables, and payables. Our estimators use math more than anyone in our business. During the estimating process for reconstruction, we use square footage formulas (L x W) for calculating materials used, for example:  subfloor framing, roof framing , insulation, drywall, painting, etc.. We use square yard formulas (L x W/9) for calculating vinyl floors and carpet. Basic algebra formulas are used for calculating rafter lengths based on the rise and run of roof slopes.

One of our most interesting uses of basic math, and one I will focus on going forward is with water mitigation (returning a structure to dry standard). Basically, drying a wet building! Once we determine the affected area we then use a cubic footage formula (L x W x H) along with the extent of saturation to know how much dehumidification is needed. Dehumidifiers are rated based upon how many pints of water they are capable of removing from the air within a specific amount of time (AHAM Rating). Therefore, depending on the type of dehumidification used and it’s rating, we are able to determine the number of dehumidifiers we need to dry a structure within the standards of our industry (S-500 ANSI approved standard). We also use the atmospheric readings to determine whether we are creating the desired conditions required to remove water from affected materials and to determine the effectiveness of our equipment. To do this we use the temperature and relative humidity to determine specific humidity (the weight of moisture p/lbs of air) and dew point (the temperature at which water vapor will begin to condense). The formula we use to determine the number of dehumidifiers needed is as follows:

Step 1 – Determine Cubic feet (CF).

Step 2 – CF/Class Factor(a low grain refrigerant dehu has a class factor of 40 in a class two loss) = # of AHAM pints needed.

Step 3  – AHAM points needed/Dehumidifier rating = number of dehumidifiers needed.

I know! It’s starting to sound a little complicated but it is all basic math.

Do you use any technology (like calculators or computers) to help with this math? Why or why not?

Absolutely! Even though we are in the building trade we are not in the dark ages. We use the most advanced estimating system designed specifically for the insurance restoration (property repair) business. After in-putting the dimensions into a sketch type format, this system automatically calculates all the square footages, cubic footages, and linear footages. The next step is to add a specific line item. For example, when you add drywall to your estimate  it uses a current square foot price to calculate how much to charge for hanging, taping and finishing the drywall in your project. It will also calculate how many sheets of drywall, how many fasteners are needed , how much drywall tape, and how much joint compound is needed. Finally, it will calculate the material sales tax and any state sales tax on the service.

How do you think math helps you do your job better?

I do not feel it’s a matter of doing my job better. I simply could not perform my job without math! As I stated earlier, we use math in every aspect of our business. I do not feel there are too many moments throughout the day that I am not using some form of math.

What kind of math did you take in high school?

During my high school years I completed algebra and some trigonometry. If I remember correctly, that was all that was offered (yes, I graduated high school 32 years ago). Once leaving high school I furthered my math education in mechanical engineering. In my opinion, the levels of math being taught in high school today are far superior to what was then taught.

Did you like it/feel like you were good at it?

I feel like there are individuals that have an aptitude for math and those who do not. Math will obviously come easy for those who have this aptitude. I would also say that if you are good at something, the chances of enjoying it are far greater than if you are not good. Having said that, I do not believe I had this aptitude. Therefore, I had to work a little harder than others, and, at best, I was average at math. Guess where I’m going with this…no I did not like it.

Did you have to learn new skills in order to do the math you use in your job? Or was it something that you could pick up using the skills you learned in school?

I had to learn how to use the math skills I had already acquired to accomplish the task at hand. For example, if you have the lengths of two sides and the angle of a triangle, you can calculate the length of the third side. It is crazy how much I use this algebraic formula; however, it took some time and experience to learn how many applications this formula has. Having said that, ninety percent of my daily tasks require math learned in high school.

Are you interested in learning more about restoration? Let me know and I will pass your information along to Nate.

Welcome to the first edition of Ask a Math Teacherwhich will feature real, live math questions from real, live people. How often will I do this? As often as I can. What kinds of questions can you expect? Whatever people ask. If you have a question, please post it to the Math for Grownups Facebookpage (after clicking “like” of course!) or email me at lelaing-at-gmail-com.

Today’s question comes from my friend and cookbook author, Debbie Koenig. You really should check out her blog and bookParents Need to Eat Too. Debbie posted this question on Facebook, which led to a two-day long post-a-thon. We finally got to the root of the question — and answer — and I thought you would like to hear about it.

My son’s math homework has me scratching my head–he’s supposed to “Draw a picture of 600 hundreds” in a space that’s maybe an inch and a half high. How does one draw 600 of anything in that small a space? And why is he drawing 600 of something? I have no idea how to help him. 

One thing that isn’t clear in this question is that her son is in second grade. This is a really important piece of information, because the answer is going to seem completely counter-intuitive to us grownups.

Some background: When children learn their numbers and then learn to count and then learn to write 3s, 7s and 4s (sometimes backwards), they are picking up teeny-tiny bits of number sense. When all of this information is put together, we call that numeracy. You can think of numeracy like literacy. It’s not just being able to count or add; it’s being able to understand how numbers work together in a much larger sense. As you can imagine, this is a big, hairy deal. It takes years and years to get to where we adults are. And most of us grownups take for granted the numeracy that we do have.

I say this because what this “hundreds” thing is getting at is place value, or the position of a digit in a number. Teachers can just tell students that the 4 in 9433 is in the tens place, or — and this is a muchbetter idea — students can learn a great deal more about numbers by really exploring this concept.

You see, place value is not some random construct. There are reasons that the first place to the right of a decimal is the ones place and the fourth place to the right of the decimal is the thousands place. Exploring this can help kids get better at multiplying or dividing and lay the foundation for decimals and even percentages.

So with that said, the first thing to do is ignore what you think hundreds means. Unless you’ve had some experience in math education, you’re probably not going to take the right guess. In second-grade math class, hundreds does not mean one hundred. It means the hundreds place.

The easiest way to get into this is by looking at a hundreds chart.

If you have one of these hundred charts, you have 100, right? How can you represent 600 then? I’ll give you a second to think about it…

Yep, with six of these buggers! Here’s a visual representation without the numbers:

So what Debbie’s son was being asked to do was draw something like the above. It’s important because it has to do with place value. Only most second graders don’t have a clue about that stuff yet. And what they have learned so far sounds like a big mistake to the rest of us — because the language being used is not what we expect. He’ll learn the word “place value” in due time and forget about these hundreds tiles and charts and suchlike.

Asking students to draw “600 hundreds” is helping students visualize place value and other important concepts. Teachers call these manipulatives, especially because they’re often real objects that students can pick up and move around. But on a homework worksheet, they’re a little harder to translate, especially for a parent who went to elementary school more than a few years ago.

So that’s the story of “hundreds,” at least as far as a second grader is concerned. I’d love to hear your thoughts! Do you know of other ways to get to the basics of place value? Do you, personally, think of place value differently? Share in the comments section.

P.S. I’m going to be speaking to parents of elementary-aged kids at my daughter’s school later this month. If you have questions that you think I should address, feel free to shoot me a quick note or post on the Math For Grownups Facebook Page. And if your school — in the D.C.-Baltimore area — would like to have me come down for a Math Chat, let me know. I’d love to meet you!

So yesterday, we reviewed some really basic stuff about percentages. Like: 10% is the same thing as 1/10 or 0.1. Easy peasy, right? Well, today it’s time to really put this stuff to work, finding percentages of numbers or the numbers, given the percentages. Oy. I can hear you groaning from here.

Most folks forget when to multiply and when to divide. So I’m going to show you a process that works no matter what kind of percentage problem you’re doing. For reals. It’s why it was important for you to know about turning percentages into fractions. Let’s start with an example.

You’ve had your eye on a gorgeous cashmere sweater for months and it’s finally on sale. But can you afford it? The original price is $125, but it’s now on sale for 30% off. Do you multiply or divide or what to find out what you’d be saving with this sale?

All you need for this problem — and pretty much all other percentage problems — is to set up a proportion. What is that, you ask? A proportion is made up of two equal ratios or fractions. The proportion you need for a percentage problem is this one:

If you can remember this proportion — and how to use it — you’re home free. So let’s dissect it a bit to help you remember. The fraction (or ratio) on the right of the proportion represents the percentage itself. You should recognize this from yesterday, when you learned to change a percent to a fraction, right? So in this problem, that ratio will be 30 over 100. That’s because the sweater is 30% off.

The ratio on the left is a little tricker, but not by much. It is the percent off of the sweater over the original price of the sweater: the part of the price over the whole price. Got it? The original price (or whole price) is $125. But we don’t know the discount (or part of the price). Let’s call that x.

DON’T PANIC! That little old x isn’t going to hurt you one bit. Promise. Just because you have an x in your math problem does not make it too challenging to solve.

But yes, you will need to solve for x. This involves two, very simple steps: Cross multiply and then get by itself. There are tons and tons of shortcuts for this kind of a problem, but for now, we’re going to stick with the more scenic route.

To cross multiply, just multiply the by 100 and then the 125 by the 30.

100x = 125 • 30Do you have to have the equation in that order? Nope. 125 • 30 = 100x works the same way. Heck you can even multiply in any order. Now, just start simplifying and getting x by itself:

Now, remind me, what is x? Is the price of the sweater? Nope. It’s what you would save if you bought the sweater at 30% off. The sale price of the sweater is $125 – $37.5 or $87.50.

That wasn’t so painful, was it?

But what if you needed to know what percent a number was of another number? Let’s say you just had lunch with your dad, who is known for being a bit stingy. He left a $7.50 tip on a $50 check. Was it enough? Well, set up that proportion, why don’t you?

What’s the whole? $50 or the total cost of lunch. And what’s the part? That would be the tip or $7.50. You are trying to find the percent, and 100 is always 100. Substitute, cross multiply, isolate x and voila!

Looky there, good old Dad did okay with the tip — 15%.

You can also use this proportion to find the whole, when you know the percentage and the part. Just substitute what you know, shove xin there for what you want to find and follow the same darned steps as the previous examples.

Seriously ya’ll, if you can remember this one proportion, percentages will no longer be a huge stumbling block. But I can hear a couple of you whining: “What about percent increase or percent decrease???” You’ll have to wait until Friday. (Promise. It’s not all that difficult either.)

This is a good thing to practice, so try out these problems. Remember: Identify the part, whole and percent before you use the proportion. (That’s not going to be as easy with these, because they’re not word problems.) Then cross multiply and get x by itself.

Questions about this process? Do you have any better ideas? (I’ll bet you do!) Share them in the comments section. Meanwhile, here are the answers to yesterday’s percent problems: 11/20, 41/50, 3/20, 0.04, 0.31, 1.4. How did you do?

Ever have one of those strings of bad ju-ju that just won’t quit? Welcome to the last two weeks of my life. From email woes to blog problems, it’s as if the electronic gods have cursed me. This is my way of explaining why there was no post yesterday. I’ll make it up to you today — as I wait on hold for the good folks at Comcast to answer my call about my email account. Wish me luck!

It’s the third week of our review of basic math. Time for percents. These little guys are everywhere — from the mall to your tax return to your kid’s grades to the nutritional label on your Cheerios. You simply cannot go a day without coming across a percent in one form or another.

(Try it. Just for today, notice the percents. If you’re so inclined, jot them down and post what you noticed in the comments section.)

So what’s the big deal? What are percents so darned ubiquitous?

Percents represent a part of the whole. We love to know what part of our extra-cheese, deep-dish pizza is fat or what part of the population is in favor of gun control. This information helps us make decisions and form opinions. And because of the way that percents are found, they’re not so challenging, actually.

First the basics: if you break down the word percent, you will immediately understand what it means. Per means every and (in the U.S.) a cent is 1/100 of a dollar. So percent literally means for every 1/100. Get it? (It should be noted that the notion of a percent came long before the U.S. penny, but the one-cent coin has its roots in Roman currency, which launched percents. Cool, huh?)

With this information, you can easily convert a percent to a fraction — which is a pretty darned useful thing to know. 10% is the same thing as 10 for every 1/100 or 10/100. The only thing left to do is simplify.

See what I did there? To turn a percent into a decimal, just put the percent over 100 and simplify. Works like a charm every single time.

But what about turning a percent into a decimal? That’s even easier. There are a couple of ways to look at this, but I chose 10% for a good reason. It’s the same thing as 1/10 or if you say it out-loud: “one-tenth.” And what’s another way of writing one-tenth? Put a decimal on it.

Think about what you learned in elementary school about decimals. One place to the left of the decimal point is the “tens” place. One to the right is the “tenths” place. Two places to the right is the “hundredths” place. And so on. If percents mean out of 100 or for every 1/100, really what you’re doing is thinking of place value.

10% = 0.10 = 0.1

All of this boils down to a really simple process. To change a percent to a decimal, move the decimal point two places to the right. Here are some examples:

Incredibly basic stuff, right? But it is important. We can use this information to help find the percent of a number or find the value of the whole, given the percent (which is a little bit harder). That’s up tomorrow and Friday.

Until then, how about giving these really simple problems a go?

Any of the above problems give you trouble? (Yep, I snuck in a few toughies, but I know you can do it. Just think it through.) Here are the answers to last Friday’s fraction problems: 2/3, 3/7, -3/14 (Yowza! That was a tricky one!), 5/9, 13/24.

We’re wrapping up a review of fractions today. If you missed Monday’s or Wednesday’s posts, be sure to look back to refresh your memory on multiplying and dividing fractions.

If you’re the product of a traditional elementary and middle school education, you likely spent many, many months (collectively) learning about adding and subtracting fractions. It is definitely one of the trickiest arithmetic skills to have, but it can also be quite useful. Now that you know how to multiply with fractions, you’re ready to unlock the secret of adding and subtracting them. And it all comes down to multiplying by the lowly, little 1.

This process is really easy — if the fractions in question have one important characteristic. Take a look:

Don’t solve the problem! Just look. What do the fractions have in common? You’re one smart cookie, so I’m sure you recognized that the denominators (the numbers on the bottom of the fractions) are the same — 5. And that’s the key in this process. Whenever you’re adding or subtracting fractions, you need to have common denominators. Then, all you need to do is add the numerators together and keep the same denominator.

If you took a few moments to run this through your brain, you probably wouldn’t have even needed to know this rule. And since we’re grownups, we can use this example: If you have 1 fifth of Jack Daniels and 2 fifths of Johnny Walker, how many fifths of alcohol do you actually have? Well, that would be 3 bottles or 3 fifths. (And believe me, while some of my high school students would have appreciated that example, I don’t think I could have gotten away with using it.)

Same thing is true for subtraction. Let’s say that the fraternity πππ (yeah, I made that up) is having a huge party. They’ve purchased 7 fifths of bourbon. But just before the gig gets started, one of the brothers knocks over the bar and breaks 3 of the fifths of bourbon bottles. How many are left? Well, that would be 4, right? Using this analogy, you can see that because the denominator was the same (5), all you needed to do was subtract the numerators (7 – 3) to get what was left (4).

And here’s where you can break even more rules. As a grownup, you can do these things in your head. If you need to add 1/8 yards of fabric to 1/8 yard of fabric, it’s pretty simple to see that you’re dealing with 2/8 yard (simplified, that’s 1/4 yard).

Yeah, things get a little trickier when you have different denominators. Let’s go back to that pizza example from Monday, shall we? Remember, we were figuring out how many pizzas to order, if we knew how much each person typically eats. Let’s say that you can eat 1/4 of a pizza, your sister can eat 1/3 and your brother can eat 1/2? In other words:

Notice something? Yep — no common denominator. So how do you get one? Well, there’s the short cut and then there’s the longer explanation. In case you’re curious, let’s talk explanation first.

You need a number that all three of these denominators will divide into evenly. That’s called a common multiple. In fact, it’s best if you have the least common multiple. (If you have a really good memory, you might remember that this is often referred to as an LCM.) So what’s the LCM of 4, 3 and 2? Turns out to be 12.

So the common denominator is 12, but do you just replace all of the denominators with a 12, adding 1/12, 1/12 and 1/12? No way, Jose. That won’t get you the right answer. What you need to do is change the numerator so that the denominator is 12. And to do that, you need to multiply by 1.

Remember 1 is the same as any fraction that has the same number in the numerator and denominator. So to change 1/4 to a fraction with 12 in the denominator, you’ll need to multiply by 3/3.

So, think ahead: what do you need to do to turn the other fractions into ones with 12 in the denominators? Multiply by 1. But which 1? You need to think about what number multiplied by the denominator will give you 12.

There’s another way to think about this, for sure. Think about the denominator you want: 12. What is one-fourth of 12? 3, right, so 1/4 is the same thing as 3/12. For some folks, that way of thinking is going to work much, much easier. But you can choose what works for you. Now we can solve the problem:

So in this case, you need a little more than one pizza. You can either ask your siblings to eat a little less (and get by on one pizza) or you can order two pizzas and put the rest in the freezer. (Personally, I’d choose the second option.)

Subtraction works the exact same way! Just find the common denominator and change the fractions. Then subtract, and finally, simplify your answer (if necessary).

Got it? If not, ask your questions in the comments section. And make sure you try out these practice problems to see how well you can really do! (Remember, no one’s grading anything, so what have you got to lose?)

If you have questions, don’t forget to ask them in the comments section. I also love to hear about different ways to approach these ideas. Don’t be afraid to tell us how you do things differently.

Here are the answers to Wednesday’s practice problems: 15/4, 7/16, 28/15, 30, 1/3.

New here? You’ve stumbled upon January’s Review of Math Basics here at Math for Grownups. This week, we’re doing a quick refresher of fractions.  Monday was multiplication, so if you missed that, you might want to take a quick look before reading further. Math concepts build, you know. 

Psst! Wanna know a secret? Sure you do. So here you go: There’s a debate among math educators about whether dividing with fractions is useful at all. There. I said it. But don’t tell your kids or they might rebel.

But yes, I’m being somewhat serious here. Among math teachers who really, really think about these things — perhaps too much and I’m often in that camp — dividing with fractions is pretty much unnecessary. Okay, so you might need to divide with fractions (like when you’re halving a recipe). But while the process is stupidly simple (trust me), there are other ways to think about it that may make more sense.

Let’s take a look at that rule:

Dividing by a fraction is the same thing as multiplying by its reciprocal.

If you know what all of those words mean, you can recognize that this is pretty darned easy. But if your days in elementary school are long past, you might have forgotten what the reciprocal is. Luckily, this is no big deal. The reciprocal of a fraction is formed when you switch the numerator and denominator. In layman’s terms, you turn the fraction upside down. Like this:

It couldn’t be easier, right? So let’s put it all in context with an example.

See what we did there? We turned the second fraction over and multiplied instead of divided. This is called the “invert and multiply” process. Now, all we need to do is simplify the answer.

Notice how the 4 and 6 are both divisible by 2? Well, that means the fraction can be simplified. On a 4th-grade math test, this means your teacher wants you to do more work. In the real world, it just means that the fraction will be easier to work with or even understand. (When you see the result, you’ll know what I mean.)

Doesn’t 2/3 seem a lot easier to understand than 4/6? Think of recipes. Do you have a 1/6-cup measure in your cabinet? (I don’t.)

So let’s consider how this works (or why, if you’d rather) by considering a really basic division problem: 1 ÷ 1/2.

How many ½s fit into 1? That’s the question that division asks, right? Think about those measuring cups. If you had two ½ cup measuring cups, you would have the equivalent of 1 cup. In other words:

Make sense? Now here’s another way to look at it:

Let me summarize: 2 ½s fits into 1. In other words, 1÷ ½ is 2. And that turns out to be the same thing as multiplying by the reciprocal of ½, which is 2.

That’s a lot to take in, and you don’t have to know it by heart – or even fully understand. It just explains why this crazy rule works. And here’s another secret – there are lots of other ways to divide fractions. You can do it in your head. (It’s pretty easy to solve this problem without any arithmetic: ½ ÷ ¼. Right?) Or you could even find a common denominator (more on that Friday) and then just divide the numerators. (I’ll leave that process for you to figure out if you’re so inclined.)

The thing is, there aren’t many times in the real world that dividing by fractions is really necessary. Here’s an example to explain what I mean. Let’s say I’m cutting a recipe in half. The recipe calls for ¾ cup of sugar. How much will I actually need? Well, I can look at the question in a couple of different ways. (See which one jumps out at you.)

I would bet – and I can’t prove it – that most of you thought about the second option. That’s because you’re cutting the recipe in half, not dividing the recipe by 2.

In short, dividing by fractions is pretty darned simple, compared to other things you have been required to do in math. Too bad it doesn’t show up much in the real world, right?

Just for fun, try these problems on for size – using whatever method works for you. (No need to show your work!) Bonus points if you can simplify your answer, when necessary. (And no, there are no bonus points, because there are no points.)

The answers to Monday’s problems: ⅓, 4/35, 15/8 or 1⅞, 5¼, 9⅔. How did you do? ETA: Me? Not so good. I made a careless error with the last problem. The correct answer is 3 ⅔, which is explained by the comments below. 

Continuing on in our review of basic math, I welcome you to Day 2. The answers to Day 1 questions are at the bottom of the post — along with new questions. But first, let’s learn how to multiply and divide integers.

Let’s say you have a bank account with a service fee of $15 per month. If that amount was deducted every single month, how can you represent the yearly amount for these fees? Well, you would multiply -$15 (the fee is negative because it’s taken out of the account) by 12 (the number of months in the year). But how the heck do you multiply negative and positive numbers? Let’s find out.

Remember integers — those negative and positive numbers that aren’t fractions, decimals, square roots, etc.? I like to think of them as positive and negative whole numbers (though most real mathematicians would argue against that classification). On Wednesday, you learned how to add and subtract these little buggers. (Check out the post here, if you missed it.)  Today, we multiply and divide.

Her’s the really good news: it is way, way easier to multiply and divide integers than to add and subtract them. First, though, it’s a good idea to understand how the rules work. When you first started multiplying numbers, you did things like this:

2 x 3 = 2 + 2 + 2 = 6

In other words “2 x 3” is the same thing as adding up three 2s. Get it? And because you started working with positive numbers when smacking a girl upside the head meant you “like-liked” her, you know without a shadow of a doubt that the answer is positive.

Let’s see what happens when you multiply a negative number by a positive number:

-2 x 3 = -2 + -2 + -2 = -6

Now to understand this, you need to either pull up your mental number line and count or remember the addition rules from Wednesday’s post. When you add two numbers with the same sign, add the numerals and then take the sign. So -3 + -3 is -6.

But what about multiplying two negative numbers? Admittedly, this is a little trickier to explain. It helps to look for a pattern using a number line. Let’s try it with -2 x -3.

-2 x 2 = -4
-2 x 1 = -2
-2 x 0 = 0
-2 x -1 = ?
-2 x -2 = ?

Based on the pattern shown on the number line, what is -2 x -1? What is -2 x -2? If you said 2 and 4, you are right on the money.

And now we can summarize the above with some rules. Believe me, this is one math concept that is much, much easier to remember with the rules. Still, if knowing why helps anyone get it, I’m all for pulling back the curtain.

When multiplying integers:
If the signs are the same, the answer is positive;
If the signs are different, the answer is negative.

Bonus: The same rules work for division. That’s because division is the inverse (or opposite) of multiplication.

When dividing integers:
If the signs are the same, the answer is positive;
If the signs are different, the answer is negative.

The only tricky part is this: Sometimes it seems that if you are multiplying or dividing two negative numbers, the answer should be negative. It’s a trap! (Not really, but you could think of it that way, if it helps.) The key in multiplying and dividing integers is noticing whether the signs are the same or different.

In fact, if you are doing a whole set of these kinds of problems, you can simply run through the problems and assign the signs to the answers — before even multiplying or dividing. (I tell students to do this all the time, because I think it helps them to remember the rules.)

4 x -3 → signs are different → answer is negative
-4 x -3 → signs are the same → answer is positive
-4 x 3 → signs are different → answer is negative
4 x 3 → signs are the same → answer is positive

Then all you’d need to do is the multiplication itself:

4 x -3 = -12
-4 x -3 = 12
-4 x 3 = -12
4 x 3 = 12

And like I said, division works the same way:

-24 ÷ -2 = +? = 12
24 ÷ -2 = -? = -12
24 ÷ 2 = +? = 12
-24 ÷ 2 = -? = -12

Got it? Try these examples on your own.

1. 5 x -6 = ?

2. -18 ÷ 9 = ?

3. -20 ÷ -4 = ?

4. 8 x 4 = ?

5. -2 x 7 = ?

Questions? Ask them in the comments section. Up Monday are fractions. If you can’t remember how to add, subtract, multiply or divide fractions or mixed numbers, tune in. 

Answers to Wednesday’s “homework.” (It’s not really homework, I promise.) -10, -4, 2, -15, -2. How did you do?

Welcome to Day 1 of our tour of basic math. If your New Year’s Resolution is to brush up on your math skills. You’re in the right place. 

Winter is really the perfect time to talk about integers.

But first, what are integers? It’s quite simple, really. They’re positive and negative whole numbers. These are integers: -547, 9, 783, and -1. These are not integers: 0.034, -0.034, √3, and -1/2.

You are very familiar with positive integers. For the first three years of your formal education, you probably worked exclusively with these little buggers — or as you called them, “numbers.” You learned to count them, tell time with them, add/subtract/multiply/divide them, and even write them out as words.

(Soon after, you learned about fractions and then decimals, which are not integers, but are still positive, so it was all good.)

If you’re like me, the part that completely blew your mind was when you first learned that numbers could be negative. Now that I think back, this was kind of a silly surprise in my world. I grew up in an area of the United States that gets pretty cold in the winter. This means two things: we measured temperatures with Fahrenheit and the temps got below zero. And those two things pointed to negative numbers. Duh.

Regardless, with a lot of work and determination, I finally understood integers, which included adding and subtracting negative and positive whole numbers. But before I show you how this is done, let’s take a look at the number line, which can help you visualize how this works.

The number line isn’t a real thing. It’s just a way to visualize how numbers work. And the key is the zero in the middle of the line. Notice what happens on the right — the numbers get larger, one by one, right? And what happens on the left? Yep, they get smaller.

Did you get that smaller part? If not, don’t worry. You’re just a little rusty. See, when two numbers are negative, the smaller one actually has the larger numeral. In other words -37 is smaller than -1, while 1 is smaller than 37.

(This is a good time to note something else that you may have forgotten. If a number has no sign, it is positive. The positive sign, +, is understood.)

If you can picture a number line, you can add and subtract integers, no problem. Here’s how:

-1 + 3 = ?

Start at -1 and count three places to the right. We’re counting to the right because we’re adding. What is the number on the number line? If you said 2, you’re right on target.

4 – 5 = ?

This time start at 4 and count five places to the left. That’s because we’re subtracting. What do you get? If you said -1, give yourself a gold star.

So this number line thingy is pretty cool, but it’s not all that useful if you need to find an answer pretty quickly. And what happens if the second number is negative? (Well, you change direction, actually, but that’s pretty clunky and somewhat confusing. So how about if we find another process?)

Once you understand the why of adding and subtracting integers, you can learn an algorithm that works every single time. It goes like this:

This is much easier to understand with an example:

-10 + 4 = ?

We’re adding two numbers with different signs. That means we need to ignore the signs, find the difference and take the sign of the larger numeral. But what does “find the difference” mean? It’s pretty simple, actually. Just subtract the smaller number (without the sign) from the larger number (without the sign). 10 – 4 is 6, and if we take the sign of the larger numeral, the answer is -6.

Another way to think of “difference” is the distance between the two numbers on the number line. So if you got back to the number line, it’s a matter of counting spaces between the two numbers. Then take the sign of the larger numeral. Make sense?

-10 + 4 = -6

Okay, let’s try a subtraction example.

-3 – 9 = ?

First step is to change the subtraction to addition and change the sign of the second number.

-3 + -9 = ?

Now all you need to do is follow the addition rule for numbers with the same signs. That means to ignore the signs, add, and keep the sign.

-3 + -9 = -12

So, no need to pull out a number line for these. Just practice with these rules, and you’ll have them down in no time at all. Here are a few additional examples to help you.

5 – 8 = 5 + -8 = -3

-7 – 4 = -7 + -4 = -11

3 + -3 = 0

-12 + 8 = -4

Now, try these out on your own. I’ll post the correct answers on Friday. And if you have questions, ask them in the comments section.

1. 15 – 25 = ?

2. -7 – -3 = ?

3. 10 + – 8 = ?

4. -3 – 12 = ?

5. -6 + 4 = ?

For nearly two years now, I’ve been harping on the same old message: Whether you like it or not, math is necessary. In other words, suck it up, cupcake. You’re going to have to do the math.

And yet, I still hear the same thing over and over again: I’m no good at math; I hate math; I just let my husband/wife/child/parent do math for me; I’ve never had to use the math I was forced to take in high school; etc., etc., etc.

That’s why January will be devoted to brushing up on your math skills. Week by week, we’ll look at a few areas of math that tend to trip people up.

Week One: Integers
How do you add, subtract, multiply and divide negative and positive numbers? And what’s the point of those stupid things, anyway?

Week Two: Fractions
I’ll remind you how to find a common denominator, perform various operations, and visualize fractions so that you can manipulate them easily.

Week Three: Percents
Do you remember how to convert fractions to percents or find a percent off or even the percent change? If not, relearn it.

Week Four: Statistics
If the recent election had you tied up in knots, perhaps a little statistics refresher is in order. Finally figure out the difference between a mean, median and mode. And learn what makes a good sample size or how to spot sample bias. It’ll be just enough info to help you analyze the news.

Week Five: Word Problems
Everyone’s favorite! Not. Here’s where I’ll show you how to dissect a problem in real life — so you can feel confident and eager to find a solution. We’ll also consider when estimation is a good idea (lots more than you might think) and when you should reach for a calculator or an online tool.

But wait! There’s more! I also have a few promises for you.

1. No grades. This is what we call low-stakes learning. As grownups, you aren’t required to study for tests or memorize formulas. At the end of each post, I’ll throw in some problems that you can choose to solve or not. But you won’t be graded or evaluated in any way. This is just for fun. Promise.

2. No trains or cantaloupes or pizzas. Stupid word problems make me crazy. Everything I offer will be grounded in reality, not constructed for a particular answer or to make a point.

3. No required algorithms. I’ll show you why these things work, so that you can find your own way to a solution. Don’t get me wrong — I’ll include a step-by-step process in most cases. But you might get alternative methods, too. And I’ll always encourage you to look for your own way.

4. No assumptions about what you do and don’t know. If we need a fancy vocabulary word, I’ll define it. If we need to follow a process, I’ll show it to you. I won’t talk down to you, but I won’t expect you to know everything.

So is it a deal? Will you join me in this New Year’s Resolution? If so, grab a No. 2 pencil (or crayon, fountain pen or stylus) and let’s get to work. Meet me back here on Wednesday, and we’ll talk integers.

What questions do you have about the topics I’m going to cover? Ask them in the comments section, and I’ll try to include them this month. If they don’t quite fit — or there’s not enough time — I’ll answer them here.