[Photo by alicepopkorn.]

Therefore, as God’s chosen people, holy and dearly loved,
clothe yourselves with compassion, kindness,
humility, gentleness and patience.
Bear with each other and forgive
whatever grievances you may have against one another.
Forgive as the Lord forgave you.
And over all these virtues put on love,
which binds them all together in perfect unity.

Let the peace of Christ rule in your hearts,
since as members of one body you were called to peace.
And be thankful.
Let the word of Christ dwell in you richly
as you teach and admonish one another with all wisdom,
and as you sing psalms, hymns and spiritual songs
with gratitude in your hearts to God.

And whatever you do,
whether in word or deed,
do it all in the name of the Lord Jesus,
giving thanks to God the Father through him.

— Colossians 3:12-17

[Taken from the Holy Bible, New International Version, (c)1973, 1978, 1984 by the International Bible Society, used by permission of Zondervan Bible Publishers.]

[Photo by shonk.]

Welcome to the Math Teachers At Play blog carnival — which is not just for math teachers! If you like to learn new things and play around with ideas, you are sure to find something of interest.

Let’s start the mathematical fun with a couple of puzzles in honor of our 20th edition: First, the shape to our right is an icosahedron, one of the Platonic solids. Each face is an equilateral triangle — can you count them? For more fun, make your own model.

[Graphic from Wolfram MathWorld.]

A rooted tree consists of a fixed node (the circled dot) with branches growing from it. I think many of the “trees” look more like bushes, but mathematicians didn’t ask my opinion before assigning the name.

Anyway, if you have only one node, then all you have is a root, waiting to grow. Two nodes will give you the root with a single branch. With three nodes, you have two possible trees: one long branch, or two short ones. Etc.

Puzzle: How many nodes will you need in order to have 20 possible trees?

And now, on to the carnival itself. Below are a number of math-related posts, submitted by the bloggers or drawn from my overflowing blog reader, plus some suggestions for how you might contribute to our next edition, all tied together with some twists of visual humor from the FAIL Blog. Have fun browsing!

ELEMENTARY CONCEPTS

As Liping Ma showed, there is more to understanding and teaching elementary mathematics than we often realize. Do you have a game, activity, or anecdote about teaching math to young students? Please share!

• Quirky Momma points out some sneaky Ways Your Preschooler Can Learn Through Chores.

• Ruralmama shares some real life math in Grocery Math Count.

• Tom DeRosa gives instructions for the Ultimate Number Line Game: Number Sense on a Massive Scale.

• John Golden presents an upper-elementary game (adaptable to 1st or 2nd grade) that will give your students practice on comparison and decomposition, as well as strategy and adjustment for calculation: Pick On Someone Your Own Size.

• Have you looked at Mathwire recently? They offer a great assortment of elementary math activites and games, such as Pascal’s Pumpkins and other Seasonal Topics.

ARITHMETIC

This section is for arithmetic and number theory at the middle-school-and-beyond level. We would love to hear your favorite math club games, numerical investigations, or contest-preparation tips.

• Ashley Allain describes her family’s learning adventure in Just SCRATCHing the Surface of Fractions.

• Maria Miller shows us how to work Percentages with Mental Math.

• Also on the topic of percents, politicalmath explains Why Take Math? So Your Ignorance Isn’t Broadcast Nationwide on the AP Wire.

• Simon Job shares a fun video for Introducing Probability.

• I have been creating fantasy word problems lately, but my Narnia blog post isn’t quite ready to go. Instead, I’ll offer some insights on thinking through word problems. This is one of my favorite articles from the archive: Reading to Learn Math.

BASIC ALGEBRA & GEOMETRY

Can you explain why we never divide by zero, or what is wrong with distributing the square in the expression ? Struggling students need your help! Share your wisdom about basic algebra and geometry topics here.

• Rachel Lynette presents a geometric variation on the game Battleship in Symmetry Game: Guess My Grid.

• Maria Miller explains that organizing the information into a chart can be helpful when solving Mixture Problems — Algebra 1.

• Dan Greene shares a creative approach to coordinate graphing in XKCD Based Lesson: The Coordinate Plane.

• Dave Marain offers geometry and exponent puzzlers in The Return of the WarmUp Challenges!

• Zac links to a “real-world math tutorial” for algebra students: Calculating Takeoff Speeds.

ADVANCED MATH

Like most adults, I have forgotten enough math to fill several textbooks. I’m eager to learn again — but math books can be so-o-o tedious. Can you make upper-level math topics come alive, so they will stick in my (or a student’s) mind?

• Jonathan challenges his class with More Puzzle Extension: Expressing n as the Sum of Consecutive Integers.

• Mr. Sweeney revises a James Bond scenario in Sick of Security Camera Problems.

• Khalid Azad clears up students’ confusion with A Visual, Intuitive Guide to Imaginary Numbers.

• 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.

MATHEMATICAL PUZZLES

Recreational mathematics has a long and fascinating history. What kind of math do you do, just for the fun of it?

• MiaZagora plays around with logic in Word Puzzle Wednesday.

• John Cook wonders about Counterfeit Coins and Rare Diseases.

• Tanya Khovanova has some fun with assumptions in Hassan’s Horses.

• 10-Minute Math cooks up a puzzle: The Cone Problem.

• Exercise your mind daily with a problem from the American Mathematics Competitions (AMC-8, AMC-10, or AMC-12) at MAA Minute Math.

ABOUT TEACHING MATH

Other teachers’ blogs are an important factor in my continuing education. The more I read about the theory and practice of teaching math, the more I realize how much I have yet to learn. So please, fellow teachers, don’t be shy — share your insights!

• Brent Yorgey recommends Logicomix: An Epic Search for Truth.

• Kate Novak offers some guidance for Building a Better Worksheet.

• Jackie comes up with a creative way to check her students’ understanding of vocabulary in Formative Assessment?

• Dan Meyer passes on the cautionary tale of Clever Hans: “Take-home lesson: never underestimate your ability to fool yourself into believing your students understand something when really what they are doing is watching you.”

And that rounds up this edition of the Math Teachers at Play carnival. I hope you enjoyed the ride. The next installment of our carnival will open on December 18th. If you would like to contribute, please use this handy submission form. Posts must be relevant to students or teachers of preK-12 mathematics. Old posts are welcome, as long as they haven’t been published in past editions of this carnival.

Past posts and future hosts can be found on our blog carnival index page.

We need more volunteers. Classroom teachers, homeschoolers, unschoolers, or anyone who likes to play around with math (even if the only person you “teach” is yourself) — if you would like to take a turn hosting the Math Teachers at Play blog carnival, please speak up!

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

• Math Teachers at Play #1
• Math Teachers at Play #2
• Math Teachers at Play #5
• Math Teachers at Play #8
• How DO We Learn Math?

[Photo by nDevilTV.]

This month, our Math Teachers at Play blog carnival switched to a once-a-month schedule. We’ll publish the third Friday of each month — which is this week!

Teachers, homeschoolers, or anyone else who enjoys playing around with math: Wednesday night is the deadline to send in your thoughts, ideas, tips, or tricks about math using this handy submission form. Posts must be relevant to students or teachers of preK-12th grade mathematics. Old posts are welcome, as long as they haven’t been published in past editions of this carnival. Most bloggers have interesting gems buried in our archives. Why not dust one off and share it anew?

Hosts Needed

With the schedule change, I also need to rebuild the queue for hosting the Math Teachers at Play. If you blog about teaching or learning in grades K-12, and if you write about math at least occasionally, I would love to have you consider hosting the carnival sometime. Here’s how to sign up:

1. Check out the schedule on the “future hosts” tab at the MTaP home page,
2. pick a month that fits your schedule,
3. and then let me know.

And don’t worry if you’ve never hosted a blog carnival before — it’s easy, fun, and great publicity for your blog! I’ll send you a detailed email of instructions to get you started.

Hat tip to Dan at mathrecreation.

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.