That’s one reason your children are encouraged to memorize their multiplication tables. But over the years, educators have discovered that straight memorization is not always the best. In fact, when kids spend a great deal of time really unpacking what these math concepts mean, they’re far more likely to expand their understanding of many other concepts.

So are math “tricks” a good thing or a bad thing?

“Kids should have a way of figuring out the math fact that uses reasoning,” says Dr. Felice Shore, assistant professor and co-assistant chairperson of Towson University’s math department in Maryland. As an expert in mathematics education, Shore knows that when children’s natural curiosity is stimulated, they can make important mathematical connections that will deepen their understanding.

“But once kids can reason their way to the answer and understand various ways to do so, these ‘tricks’ can help them get answers quickly,” she continues.

The key is to introduce these tricks at the right age.

“I don’t think the third or even fourth-graders should learn tricks,” Shore says. “The important mathematics at those grades is still about building an understanding of relationships between numbers—the very reasons behind math facts. Once you show them the trick, it’ll most likely just shut down their thinking.”

But math tricks can be useful. If your fifth grader is still struggling with her multiplication tables, these can be a godsend. Even better is when they reveal something about the math that makes them work.

If you’re going to show your child a quick way to multiply, make sure that you help her understand why the trick works. Here are five cool examples—and the math behind them.

Multiplying by 4

This trick is so simple and logical, that it could hardly be called a trick. But it could come in handy for your budding Sir Isaac Newton. To multiply any number by 4, simply multiply it by 2 and then double the answer.

35 x 4
35 x 2 = 70
70 x 2 = 140
35 x 4 = 140

Why does it work?

This trick is based on a very simple fact:

2 x 2 = 4

That means that:

35 x 4 = 35 x (2 x 2)

And

35 x 2 x 2
70 x 2
140

The underlying lesson of this “trick” is that you can solve a multiplication problem by multiplying by its factors.

Multiplying by 9

Hold up both hands, with your fingers spread. To multiply 4 x 9, bend your fourth finger from the left. Count the number of fingers to the left of your bent finger—you should get 3. Then count the number of fingers (and thumbs) to the right of your bent finger—you should get 6. The answer is 36. This works when multiplying any number 1-10 by 9.

Why does it work?

Simple algebra can show that what you’re doing with your fingers boils down to this: When you multiply by 9, you’re really multiplying by 10 and then subtracting that number. But you don’t need to do the algebra. Some kids figure out that reasoning without the mysterious finger trick.

You can help your child extend her understanding of the number 9 by pointing out an important piece of this trick: in the 9s multiplication tables, the digits add up to 9!

4 x 9 = 36   —>   3 + 6 = 9

9 x 9 = 81   —>  8 + 1 = 9

Then you can prompt your child to notice other patterns. For example, 4 -1 = 3 and 3 + 6 = 9 and 4 x 9 = 36. The patterns in the 9s multiplication tables are endless and can lead to many other discoveries about numbers.

Multiplying by 11

Sure, multiplying a one-digit number by 11 is a cinch.

4 x 11 = 44
7 x 11 = 77

But did you know there’s a trick to multiplying any number by 11?  Here’s how using an example: 52 x 11.

The first digit of the answer will be 5 and the last digit of the answer will be 2. To get the digit between, just add 5 and 2.

5 (5+2) 2
572

You may have noticed that when you add the two digits together, you get a one-digit number. If you get a two-digit number, things are a little trickier.

87 x 11
8 (8+7) 7
8 (15) 7
(8+1) 57
957

Why does it work?

If you think of doing long-hand multiplication by stacking the two numbers, you’ll see right away:

But the more precise reasoning has to do with place value. What you’re really doing is multiplying 87 by 1, then multiply 87 by 10, and finally adding the two products together:

87 x 1 = 87
87 x 10 = 870
870 + 87 = 957

The trick itself is just a shortcut to the answer.

Multiplying by 12

Just like the previous track, you can multiply any number by 12 very quickly and easily. Let’s try it with 7 x 12.

First multiply 7 by 10. Then multiply 7 by 2. Finally, add them together.

7 x 12
7 x 10 = 70
7 x 2 = 14
70 + 14 = 84

Easy peasy. When this gets really impressive is with larger numbers.

25 x 12
25 x 10 = 250
25 x 2 = 50
250 x 50 = 300

Why does it work?

This trick works for the same reason that the 11s trick works. But there’s another way to describe it. Think of 12 as the sum of 10 and 2.

25 x 12
25 x (10 + 2)
(25 x 10) + (25 x 2)
250 + 50
300

Is a number divisible by 3? (Or in math terms: Is a number a multiple of 3?)

When a number is evenly divisible by another number it is said to be a multiple of that number. In other words: since 27 is evenly divisible by 3, 27 is a multiple of 3.

Turns out, there’s a nice little trick for this as well. Add up the values of the digits. Is that sum a multiple of 3? If so, the number itself is also evenly divisible by 3. Check it out:

Is 543 divisible by 3?
5 + 4 + 3 = 12
12 is divisible by 3
So 543 is divisible by 3

Why does this work?

Place value is key here, but there’s an easy way to show your child what’s happening before you even introduce the trick. Do this with something tangible, like M&Ms or pieces of cereal.

1. Start with 45 candies.
2. Have your child divide the candies into two piles based on the place value—one pile of 40 candies and one pile of 5 candies.
3. Now ask your child to divide the 40 candies into groups of 10 candies. (She should notice that there are four groups of 10 candies.)
4. Now ask her this question, “How can you change each of these groups often, so that the number is divisible by 3?” She should suggest that you take away one candy from each pile. (If not, coax her to that answer.)
5. Have her take one candy from each group of ten and move them into another group.
6. Point out that she has six piles of candies: four piles of 9 candies, one pile of 4 candies and one pile of 5 candies.
7. Ask her what happens if she combines the pile of 4 candies and the pile of 5 candies. She should notice that she’ll get 9, which is divisible by 3.
8. By now, she will probably notice that the 4 and 5 come from number 45. See if she can come up with the trick, after doing this with a few examples using the candies.

So what do you think? Are math tricks a good idea or not? Do you have any other tricks to share? And can you explain why they work? If you need help with your math, I have written these great books to help you learn the easy way.

The post Five Cool Math Tricks You Didn’t Know appeared first on Math for Grownups.

It was last Tuesday — a pretty regular day.

April 2, 2013

6:00 a.m.: Review to-do list, estimating the time that each item would take. Count up the number of hours estimated to be sure not to exceed eight hours, while leaving time for lunch and exercise.

7:00 a.m.: Track all Weight Watchers points that I expect to use for the day, by planning what I’ll have to eat for breakfast, lunch, dinner and snacks. Allow the online program to add everything up, but pay close attention that my breakfast and lunch are around 6 points each and that I’m using less than 8 points from my weekly extra points.

10:00 a.m.: Review invoicing for first quarter. Within bookkeeping program, look at the data in a variety of ways: bar graphs, showing income for each month, and tables showing the income for each client. Compare income to goals and adjust expectations where necessary.

11:00 a.m.: Set budget for new book postcard, using designer’s estimates. Compare costs of a fewer number of cards to the costs of a much larger run. Table the decision to think about things.

12:00 a.m.: Attend weekly Weight Watchers meeting, and learn that I lost 0.4 pounds last week. Spend meeting mentally calculating how that could have happened, given the fact that I didn’t stay within my allotted daily points for a few days. Remember that balancing the equation of caloric intake and output, with variables like water retention, is way too complex for mental math. Decide to just feel fortunate and proud.

1:00 – 3:30 p.m.: Outline online lesson about linear, quadratic and exponential functions. (Yes, this is where I and the rest of the world differs! But I wanted you to know that this curriculum doesn’t appear out of thin air.)

4:00 p.m.: Meet with potential photographer for our wedding. Count backwards from the start of the wedding to estimate the time necessary and the cost of a second photographer. Mentally calculate how much over our budget we’d go if we hired this photographer. (Everything goes over budget, I’ve found.)

6:30 p.m.: Meet a friend for drinks at a local restaurant. Scan menu for lowish-calorie drink, decide that since a cosmo is the same points as a glass of wine, why not have the pink drink in the fancy glass?

7:30 p.m.: Get the check. Find the tip by taking 10% of the bill and doubling it. Then split the check evenly since we got the same drink and shared an appetizer.

11:30 p.m.: Daughter can’t sleep. Mentally add up the number of hours of sleep we can each expect to get if she would just fall asleep right now. Finally she dozes off.

And there you have it — my math day. As you can see, the math was tucked into various nooks and crannies. If I hadn’t been paying attention, I wouldn’t have even noticed it. And most of it had nothing to do with the way I learned to do math at school.

So what about you? Here’s my challenge: Just for today, jot down when you’ve used math. Then share what you learned about yourself in the comments section. Did you find that you used math more than you thought? Did you discover that you’re using a kind of math that you never, ever expected? I want to know!

The post Daily Digits: My math day appeared first on Math for Grownups.

When last we met, percentages were the topic of discussion. I had promised to shed some light on the mysteries of percentage increase and percentage decrease. This is, by far, the most-often asked question from writers. From time to time, I’ll meet a freelancer who is trying to find the percentage decrease of a company’s profit over the previous year. Or a freelancer may want to know how to calculate the percentage increase of  her income over the previous quarter.

Trust me. This is not difficult. But it is confusing. So my challenge is to lay this out in a way that you can both understand and remember. Let’s go.

First a definition. Percentage change — which can be either an increase or a decrease — is simply a comparison of values. In this case, we’re comparing the new value to the old value and expressing that as a percent.  And here’s how you do that:

(new value – old value) ÷ old value

That’s it. But let’s break it down. The change is found by subtracting the new value from the old value. And the percentage is found by dividing that answer by the old value. In other words:

Change:new value – old value

Percentage:divide by old value

This should make sense, because change is often found by subtracting. If you pay for $15 worth of gas, using a $20 bill, your change is $5 — which is also $20 – $15. Likewise, percent means division. To find what percent 15 is of 20, you divide: 15 ÷ 20.

Let’s look at this with an example. The crowds at President Obama’s first inaugural were much, much larger than at his second. It is estimated that 1.8 million people were on the mall in 2009, while only 540,000 showed up two weeks ago. (It’s worth noting that no one can say for sure how many people attend any event on the Washington Mall. These are simply estimates, which can vary widely.) What is the percentage decrease of the crowds from 2009 to 2013?

Change: 540,000 – 1,800,000 = –1,260,000

Percentage: –1,260,000 ÷ 1,800,000 = –0.7 = –70%

So attendance at the second inaugural had decreased by 70%. (Notice that negative sign? Whenever the percentage change represents a decrease, the percentage will be negative.)

Follow the exact same process to find the percentage increase. Each year — no matter who is in office — the cost of inauguration events goes up. President Obama’s first inauguration had a price tag of $160 million. While we won’t know how much the 2013 inauguration cost for several months, we can compare 2009 to Bush’s second inauguration in 2005, which totaled $158 million. What is the percentage change from 2005 to 2009?

Change: 160,000,000 – 158,000,000 = 2,000,000

Percentage: 2,000,000 ÷ 158,000,000 = 0.01 = 1%

The cost of the inaugural increased by 1% from 2005 to 2009. (Because the answer is positive, we know the percentage change represents an increase.)

For percentage change problems, don’t worry about whether you’re finding the percentage increase or percentage decrease. The answer — negative or positive — will reveal that. Instead, focus on the two steps: 1) New number – old number; 2) Divide by old number; 3) Change the decimal to a percent.

Practice with these examples. I’ll post the answers on Friday.

Find the percentage change:

1) In 2011, a company posted profits of $305 million. In 2012, profits were $299 million.

2) When she was in the fifth grade, Sally was 54 inches tall. As a sixth grader, she’s 58 inches tall.

3) Springfield has begun a recycling program in an effort to reduce the trash collected in the city. The year before the recycling program was enacted, the city collected 160,000 tons of trash. The year after the program began, the city collected 151,000 tons of trash.

4) Since her son went off to college, Margo has noticed that her grocery bills have declined. In July, she spent $327 on groceries, while in September, she spent $213.

Questions about this process? Do you find percentage change differently? Share them in the comments section. Meanwhile, here are the answers to the last blog post’s percent problems: 27, 250, 20, 90, 140.

The post Smaller Crowds: Calculating Percentage Change appeared first on Math for Grownups.

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 x 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 x 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?

The post Finding Percentages and the Numbers That Go With Them appeared first on Math for Grownups.

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.

The post Parts Is Parts: Get a handle on percents appeared first on Math for Grownups.

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.

The post How Fractions Are Like Drinking: Adding and subtracting appeared first on Math for Grownups.

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. 

The post Halving a Recipe: Dividing with fractions appeared first on Math for Grownups.

When kids are first learning about fractions, teachers often turn to something that all but the lactose- or gluten-intolerant can appreciate — pizza! (And I can empathize with the allergy inclined. For you, imagine a dairy-free, vegetable pie with polenta crust — yum!)

This is for very good reason: Fractions are simply parts of the whole. When you cut a pizza into 12 equal parts you are creating twelfths. To count them, you’d start at one piece and count around the pizza (or in random order, makes no diff): one-twelfth (1/12), two-twelfths (2/12), three-twelfths (3/12)… all the way to 12-twelfths (12/12) or the whole pizza (1). Half of the pizza is six-twelfths (6/12) or one-half (1/2). A fourth of the pizza is three-twelfths (3/12) or one-fourth (1/4). Get it?

(Okay, so it’s really, really hard to write a blog post about fractions. In Word, I can depend on something call MathType to write fractions, which I’ll create for examples below. But in paragraphs, this doesn’t work so well.  So please bear with me!)

It might make sense to start with addition and subtraction, but in this case, multiplication and division is the better start. (Spoiler alert: You’ll use multiplication to add and subtract. Really.) But just like with integers, multiplying and dividing fractions are really, really easy.

So let’s go back to those pizzas. Let’s say your son is having a birthday party, and he wants to serve pizza. If each kid can eat 1/4 of a pizza and there are 12 kids at the party, how many pizzas do you need to buy? (Seriously, this is not as dorky a question as it might sound. I have had to figure this out IRL.)

Are you actually multiplying two fractions here? Why, yes. Yes you are! In fact, any whole number can be written as a fraction — just use the number itself as the numerator (the top number in a fraction) and 1 as the denominator (the bottom number in a fraction). So…

Now, here’s the multiplication rule. Just multiply the numerators together and then the denominators together.

How easy is that? But what does 15/4 really mean? This is called an improper fraction — which just means that it’s got a numerator that’s bigger than the denominator. But it has a much, much bigger meaning — improper fractions are bigger than one.

How many pizzas is 15/4? Well this is easy too.

Fractions mean division. So to turn an improper fraction into divide the denominator into the numerator. But 4 doesn’t divide evenly into 15. In fact, 4 goes into 15 three times, with 3 left over. (Or as your third-grade self said: 3, with a remainder of 3.)

The whole number is the number of times 4 divides into 15. The remainder becomes the numerator of a fraction, and 4 stays in the denominator. Like this:

Whew! What this is means is that you need 3 and 3/4 pizzas. I don’t know of any pizzeria that delivers in this way, so round up to 4 pizzas, and you should be good to go.

That’s a lot of information. So here’s a quick summary:

1. Any whole number can be written as a fraction. Just use the number as the numerator and put a 1 in the denominator.

2. To multiply fractions, multiply the numerators together and then multiply the denominators together.

3. To change an improper fraction to a mixed number, divide the denominator into the numerator. The whole number answer is the whole number in the mixed number. The remainder is the numerator, and the denominator stays the same.

Show me (or better yet, yourself) what you’ve got with these examples. I’ll have the answers in Wednesday’s post. Questions? Ask them in the comments section.

Answers to Friday’s challenge questions: -30, -2, 5, 32, -14. How did you do?

The post Pizza Anyone? Introducing fractions (+ multiplying) appeared first on Math for Grownups.

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?

The post Multiplying and Dividing — Integer Style appeared first on Math for Grownups.

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 = ?

The post Pluses and Minuses: Adding and subtracting integers appeared first on Math for Grownups.