Welcome to the Carnival of Mathematics! We’ve got a full roster this time, including roller coasters, topological Turán theory, a mathematician arrested as a spy, a plane running out of fuel mid-flight, speed limits in Conway’s Game of Life, and much, much more…

The 59th Carnival of Mathematics features 59 blog posts (counting the multi-part posts and the carnival post itself) on a wide variety of interesting topics. Drop in and enjoy the browsing!

“Let’s give the governor a break,” says Williams College mathematician Edward Burger. “If nothing else, he’s encouraging math education.”

— Carl Bialik
Coincidental Obscenity Deemed Extremely Dubious

Check out the new math carnival to discover plenty of great fun for all ages:

• Math Teachers at Play #19

Are you wondering where MTAP #18 went? Here’s the story (contest-winning entry from Lisa Downing), and we’re sticking to it!

“The Odds were at odds with the Evens. It never seemed fair to them that two Odds made an Even but two Evens didn’t make an Odd…”

[Click over to MTaP #19 to read more.]

[Graphite drawing by Niner.]

Niner (pronounced Neener), who takes the photos for my blog header — which reminds me, we’re about due for a new one of those… — has started a new blog. She calls it 19 & Still Alive, because “the world doesn’t end when you’re 16 if you don’t go to Prom or don’t get your driver’s license. (I never went to Prom, and I didn’t go through Driver’s ED until 17, but I’m alive, amazingly.)”

There won’t be any math there, or at least I don’t expect to see any; the blog will be mostly her rambling thoughts about whatever catches her interest. But she does have a wonderful graphite drawing based on the photo Masarwa man. On her blog, you can click the image to see him up close and personal. Wow!

[Photo by *Irish.]

In my post Pre-Algebra Problem Solving: The Tools, I introduced word algebra as a way to help students think their way through a story problem. In the next two posts, I showed how the tool worked with simple word problems.

Now, before I move on to focus exclusively on bar diagrams, I would like to show how word algebra can help a student solve a typical first-year algebra puzzle.

A homeschooling friend who avoided algebra in high school, trying to help her son cope with a subject she never understood, posted: “Help! Our answer is different from the book’s.” Here is the homework problem:

Josh earned $72 less than his sister who earned $93 more than her mom. If they earned a total of $504, how much did Josh earn?

First Step: Translation

In the process of solving a word problem, the student must work his way through three steps:

1. Translate the words into a mathematical calculation or algebraic equation.
2. Do the calculation or solve the equation.
3. Interpret the resulting number in the context of the original problem.

When a student struggles with solving problems, most of the time it is step one that gives him trouble. He does not know how to translate the problem from English into “mathish.” If we want to help our students with their math problems, we need to teach them how to do this sort of translation.

In my friend’s problem, there are three facts that need to be translated. Let’s begin by separating these and isolating them from the rest of the words, so we know where to focus our attention.

Fact #1

“Josh earned $72 less than his sister…”

In word algebra, we use words from our problem to stand for whatever amounts we don’t know. In this case, we will use each person’s name to represent how much money that person earned. So this statement translates into math as follows:

Josh = Sister – $72
or
Sister = Josh + $72

If you are not sure whether to add or subtract the $72, stop to think: “Who had more money?” Then read the statement again, leaving out the number: “Josh earned less than his sister.” So we have to subtract from the sister’s amount to make it equal Josh’s total, or we need to add to Josh’s amount to find how much the sister had.

By the way, it is a good idea to practice putting the words into math both ways, with addition AND with subtraction. They both say the same thing, but sometimes one form will be more useful to you in solving your problem, and other times the other form will be more helpful.

Fact #2

“…his sister who earned $93 more than her mom.”

Again, we think, “Who had more money?” And again, it’s the sister who earned more. So this fact translates into word algebra as:

Sister = Mom + $93
or
Mom = Sister – $93

Fact #3

“If they earned a total of $504…”

You can ignore the “if” and just say they DID earn that much. All that the “if” in a math problem means is that, for this problem, we are going to assume the following statement is true. So…

“…they earned a total of $504…”

It translates into word algebra this way:

Josh + Sister + Mom = $504

What Are We Looking For?

“…how much did Josh earn?”

The final part of the word problem is the question. What did the problem ask us to find? It asked for Josh’s amount of money.

Now that we have translated the whole problem into word algebra, we are ready to start solving for the answer. In summary, let me list all the things we know, dropping the dollar signs so it looks more mathy:

Josh = Sister – 72
Sister = Josh + 72
Sister = Mom + 93
Mom = Sister – 93
Josh + Sister + Mom = 504

We have two options for our next step, and both involve a fundamental rule for solving algebra problems:

You cannot solve for two unknown numbers at once, so you have to say everything in terms of ONE unknown amount. That one unknown becomes the X that you will solve for.

Your unknown doesn’t have to be called X, but whatever you call it, it still marks the treasure, the solution to your problem.

First, Find Sister

Since Josh’s money is related to Sister’s amount, if we could find Sister’s amount, then we would know what Josh had. That is the easiest approach, because we have two equations like this:

Josh = (something about Sister)
Mom = (something about Sister)

So if we call the amount that Sister earned X:

Sister = X
Josh = X – 72
Mom = X – 93
(X – 72) + X + (X – 93) = 504
3X – 165 = 504
3X = 669
X = 223

If you do this, remember that you are not done with the problem when you find X. You still have to use X to find the amount of money Josh earned:

Josh = X – 72 = 223 – 72 = 151

Which means that Josh earned $151.

Or Go Directly to Josh

An alternate approach would be to solve directly for the amount we want to find, the money Josh earned. This method will take a little bit longer, but when we finally get to the “X=?” part, we will be done. We don’t have to remember to go back and do that extra step.

This time, let Josh be the X — except that we will use Y, because we just used X in the “Find Sister” section above, and we don’t want to get ourselves confused.

Remember that you can use ANY letter to name your unknown amount.

Josh = Y
Sister = Y + 72
Mom = Sister – 93
Mom = (Y + 72) – 93
Mom = Y – 21
Y + (Y + 72) + (Y – 21) = 504
3Y + 51 = 504
3Y = 453
Y = 151

So again, Josh earned $151.

In the end, it is “real” algebra that solves the problem, the same equation manipulations that the student has been using throughout his algebra 1 textbook. But algebra 1 word problems are primarily an exercise in translation, and the intermediate step of word algebra can help a confused student make the necessary translation from the words of his story problem to the equations he knows how to solve.

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Have more fun on Let’s Play Math! blog:

• The Game of Algebra
• Euclid’s Geometric Algebra
• Online Game: Math Caching
• A Mathematician for President
• How to Start a Homeschool Math Club

Math Mama is hosting the next Math Teachers at Play blog carnival (submit your post here) on Friday, and she writes:

The Math Teachers at Play blog carnival came out twice as #15. Since then we’ve had #16 and #17. We’d like to iron out the numbering, and so the upcoming issue will be #19. I am personally sponsoring a contest for the best little (ie, very short) story written about how the numbers got mixed up this way…

For more details, check out her blog post:

• Contest: Write a number story by Wednesday

Beginning next month, the MTaP carnival will change its schedule, coming out on the third Friday of every month.

If you are interested in hosting an upcoming carnival, please let me know! Carnival posts are a lot of fun to put together, you get to discover new (or new-to-you) bloggers, and you almost always get a bit of a traffic spike. And don’t worry if you’ve never hosted a blog carnival before — just ask for my very detailed “how to” email to help you get started.

[Bill Watterson identifies the trouble with math problems, through the eyes of Calvin and Hobbes.]

It’s time to revive and (hopefully!) finish my long-neglected series on solving word problems in elementary mathematics. I’ve been having fun making up the problems, so now I just have to write the posts. Coming up soon:

• 4th grade math in The Lion, the Witch and the Wardrobe.
• 5th grade math in The Hobbit.
• 6th grade math in… I haven’t decided yet. Any suggestions?

Since it has been more than two years since the last entry, however, I wanted to take a few minutes to recap our progress so far and to refer new readers back to the original posts:

• Pre-Algebra Problem Solving: The Tools
  An introduction to word algebra and bar diagrams.
• Pre-Algebra Problem Solving: 2nd Grade
  Math problems from Mr. Popper’s Penguins.
• Pre-Algebra Problem Solving: 3rd Grade
  Math Problems from Poor Richard.

Bar Diagrams are Visual Algebra

For the rest of the problem-solving series, I will focus on bar diagrams (also called rod models, or simply models) as taught in the Singapore Primary Math series. These diagrams are a pictorial form of algebra, with the length of each bar representing some quantity in the word problem, either known or unknown.

In creating a bar diagram of the word problem, the student must think through the same type of questions that are required in writing an algebra equation:

• What quantities are related to each other, and how are they related?
• How are the known quantities related to the unknowns?
• What is the basic unit (or variable) that I need to find?

The Whole is the Sum of its Parts

All bar diagrams descend from one very basic diagram showing the inverse relationship between addition and subtraction: The whole is the sum of its parts. If you know the value of both parts, you can add them up to get the whole. If you know the whole total and one of the parts, you subtract the part you know in order to find the other part.

When all the parts are the same size, students can use a (number of units) x (size of units) = total diagram to represent both multiplication and division problems, because multiplication and division are also inverse operations. Because the units are all the same, we only need to write a number in the first one.

Practice Makes Perfect

It is important for students to practice applying this method to simple problems, even though they may be able to solve the problems without a diagram. As with any tool, skill with bar diagrams must develop over time, through repeated use.

The 4th and 5th grade problems coming up will require relatively complicated diagrams with multi-step solutions. If you wish to teach this method to your students, be sure to step back to simpler problems (like those in the 2nd grade and 3rd grade posts) at first.

To get more practice creating bar diagrams, your students may enjoy these online tutorials:

• Thinking Blocks Addition and Subtraction Word Problems
• Thinking Blocks Multiplication and Division Word Problems

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Don’t miss any of “Let’s Play Math!”:  Subscribe in a reader, or get updates by Email.

Have more fun on Let’s Play Math! blog:

• Word Problems in Russia and America
• Reading to Learn Math
• Solving Complex Story Problems
• Solving Complex Story Problems II
• Story Problem Challenge Revisited

Beginning in January, I will teach a 4th-12th grade Blogging 2 Learn class through our local homeschool co-op. For now, here is my research blog, testing ideas and trying to imagine myself as a new blogger:

• Blogging 2 Learn

Have you used blogs with your students? If so, I would love to hear your suggestions and comments. And whether you are an experienced or a wanna-be blogger, please share: What do you think a “Blogging to Learn” class should cover?

Check out the action-packed blog carnival at mathrecreation:

• Math Teachers at Play 17

There is plenty of humor, a few movies, some great activities and explorations, and lots of math in Math Teachers at Play 17. Hope you enjoy reading these as much as I did….

I think this may be our longest edition yet, with plenty of mathematical fun for all ages. Enjoy!

Today’s Homeschool Freebie is an elementary math textbook from 1915:

• Everyday Number Stories

The book covers the basic arithmetic from addition and subtraction to simple multiplication and division, with quite a bit of measuring and fraction work, too. The concepts are taught through stories, and then the teacher is to give the student plenty of hands-on experience with measurements. Each lesson concludes with a page of practice problems.

If you miss the daily freebie, you may still be able to get the book through Google (but I’m not sure this works if you are outside the United States):

• Everyday Number Stories (Google)

All week long, CurrClick is offering a free “Make It Real Learning” math workbook:

• MIRL States by the Numbers – Wyoming

The States by the Numbers series features 4th-6th grade “math adventures” about your favorite states (the freebie is Wisconsin). The workbook covers place values, rounding, estimation, fractions and percentages using data from the Census Bureau’s 2008 Statistical Abstract of the United States. The workbook includes basic instruction and 80 practice problems, plus several “What’s the big idea?” journal pages which ask learners to reflect on the things they’ve learned. You may want to let your students use a calculator, since real data often means large, unwieldy numbers.