by Robert Webb

Do you enjoy math? I hope so! If not, browsing this post just may change your mind. Welcome to the Math Teachers At Play blog carnival — a smorgasbord of ideas for learning, teaching, and playing around with math from preschool to pre-college.

Let the mathematical fun begin!

POLYHEDRON PUZZLE

By tradition, we start the carnival with a puzzle in honor of our 62nd edition:

An Archimedean solid is a polyhedron made of two or more types of regular polygons meeting in identical vertices. A rhombicosidodecahedron (see image above) has 62 sides: triangles, squares, and pentagons.

• How many of each shape does it take to make a rhombicosidodecahedron?

Click for template.

My math club students had fun with a Polyhedra Construction Kit. Here’s how to make your own:

1. Collect a bunch of empty cereal boxes. Cut the boxes open to make big sheets of cardboard.
2. Print out the template page (→) and laminate. Cut out each polygon shape, being sure to include the tabs on the sides.
3. Turn your cardboard brown-side-up and trace around the templates, making several copies of each polygon. I recommend 20 each of the pentagon and hexagon, 40 each of the triangle and square.
4. Draw the dark outline of each polygon with a ballpoint pen, pressing hard to score the cardboard so the tabs will bend easily.
5. Cut out the shapes, being careful around the tabs.
6. Use small rubber bands to connect the tabs. Each rubber band will hold two tabs together, forming one edge of a polyhedron.

So, for instance, it takes six squares and twelve rubber bands to make a cube. How many different polyhedra (plural of polyhedron) will you make?

• Can you build a rhombicosidodecahedron?

And now, on to the main attraction: the 62 blog posts. Many of the following articles were submitted by their authors; others were drawn from the immense backlog in my blog reader. If you’d like to skip directly to your area of interest, here’s a quick Table of Contents:

• Early Learning Activities
• Elementary Exploration and Middle School Mastery
• Adventures in Basic Algebra and Geometry
• Advanced Mathematical Endeavors
• Puzzling Recreations
• Teaching Tips

EARLY LEARNING ACTIVITIES

Mathematics depend upon the teacher rather than upon the textbook and few subjects are worse taught; chiefly because teachers have seldom time to give the inspiring ideas, what Coleridge calls the ‘Captain’ ideas, which should quicken imagination.

— Charlotte Mason
Toward A Philosophy of Education

• The Girl Who Painted Trees discovers that “5 minutes here and there” add up to a pretty full course of Kindergarten Math.

• Donna Boucher shares a simple Ten Plus Card Game for cooperative play.

• Alice challenges parents of young children to Explore Really Big Numbers with an Abacus.

• Margo finds a great use for dollar-store plates: Dish Up Some Multiplication on Partitioned Plates.

• Is different from ? Join Yelena McManaman and friends in playing What Would You Rather Have – Commutative Property Game.

• Christopher Danielson probes his children’s understanding of fractions in Zero=half and Not really ready for fractions. And I like Michael Paul Goldenberg’s comment: “The great thing is that your daughter has really good wrong ideas and is able to articulate them. If we could get all kids to articulate their wrong ideas and pursue them with other kids and knowledgable adults, 90% of our national difficulties in mathematics education would vanish.”

• Encourage your children to ask questions with Paul Salomon’s beautiful Stars of the Mind’s Sky. What do they notice? What do they wonder?

• Cindy updates her Math in Children’s Literature book list page. So many wonderful books!

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ELEMENTARY EXPLORATION AND MIDDLE SCHOOL MASTERY

The child may learn the multiplication-table and do a subtraction sum without any insight into the rationale of either. He may even become a good arithmetician, applying rules aptly, without seeing the reason of them; but arithmetic becomes an elementary mathematical training only in so far as the reason why of every process is clear to the child.

— Charlotte Mason
Home Education

• Malke Rosenfeld discovers once more that math is personal (and that sometimes Mom should keep her mouth shut) in Her Own Math, Not Mine.

• I love Lula B’s description of her family’s week in 5 Days of Maths Playtime. And the fun didn’t stop: her children continued to explore mathematical relationships in Pythagoras and the Knotted Rope.

• Measurement gives children an opportunity to use numbers in a practical task. Penny Ryder shares the downloadable activity guide, Comparing and Ordering Capacity, with the bonus puzzle of folding an origami cup with a specified volume.

• What are your goals in teaching math facts? Cindy rethinks Multiplication Fact Memorization.

• Sarah clearly demonstrates the process of Long Division with Manipulatives.

• Maria Miller shares a Prime Factorization forest game.

• William Emeny addresses a common student mistake in Decimal line zooming.

• John Golden tells how he developed a game for 5th graders just starting with fraction multiplication and adds several fun scenarios invented by the students: Find It!

• Ben Orlin answers the perennial question, Why can’t you divide by zero? (And don’t miss What It Feels Like to Be Bad at Math.)

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ADVENTURES IN BASIC ALGEBRA & GEOMETRY

We take strong ground when we appeal to the beauty and truth of Mathematics; that two and two make four and cannot conceivably make five, is an inevitable law.

It is a great thing to be brought into the presence of a law, of a whole system of laws, that exist without our concurrence — that two straight lines cannot enclose a space is a fact which we can perceive, state, and act upon but cannot in any wise alter.

— Charlotte Mason
Toward A Philosophy of Education

• Have your students experienced how surface area and volume change when a shape grows? Check out Tina C’s Volume Lesson, Crowd Sourced.

• Guillermo Bautista shows that The Product of Negative Number and a Positive Number is Negative and challenges the reader to prove a corollary.

• Jonathan Newman’s pre-calculus students are working so hard on new ideas that they forgot basic algebra: What Precal Students Can’t Do. Can your students explain how to figure it out?

• Many students have a hard time understanding exponents. I’m looking forward to trying Michael Pershan’s Visual Exponential Patterns with my daughter.

• Brian Miller offers advice for Teaching Mixture Problems With The Mixture Picture.

• Lois Lindemann comes up with a creative way to emphasize justification in beginning geometry proofs: Who knew that learning parrot fashion could be this much fun?

• Rodi Steinig introduces a series of explorations of proof to her Math Circle by not proving the Pythagorean Theorem in Aspect Ratios, the Golden Ratio, and Z’s TV. [Sessions #2 and #3 are now posted.]

• Jonathan Halabi is also teaching proof-based geometry, with an emphasis on logic and constructions: Shaking up traditional geometry.

• Josh ponders the relationship between Ferris Wheels and Euclid. For more on geometric constructions, did you notice Paul Salomon’s Stars of the Mind’s Sky, mentioned earlier? Your students may also enjoy exploring the Math Open Reference website.

• Textbook questions got you down? Fawn Nguyen shares what happened When I Let Them Own the Problem.

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ADVANCED MATHEMATICAL ENDEAVORS

In a word our point is that Mathematics are to be studied for their own sake and not as they make for general intelligence and grasp of mind. But then how profoundly worthy are these subjects of study for their own sake, to say nothing of other great branches of knowledge to which they are ancillary!

— Charlotte Mason
Toward A Philosophy of Education

• Studying Statistics? Check out Colleen Young’s long list of Statistics Resources.

• Alexander Bogomolny shows us some Jewels in the Bride’s Chair: for any triangle, there are three points where four straight lines meet forming successive angles of 45°.

• I love Patrick Honner’s Proof Without Words: Two Dimensional Geometric Series.

• Do your students have trouble with logarithms? Whit Ford hits all the key point of this important topic in Summary: Logarithms.

• Rebecka Peterson’s pre-calc class tries to create the best error in Mistakes mean we’re getting better, right?

• Calculus asks seemingly impossible questions, and limits give a strategy for answering “impossible” questions. See Kalid Azad’s Intuitive Introduction To Limits.

• Andrew Chambers considers the question, Does it Pay to be Nice? Game Theory and Evolution.

• Math Curmudgeon posts a series of problems to challenge your students (or yourself!): UVM 2013.

• And don’t miss the 98th Carnival of Mathematics!

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PUZZLING RECREATIONS

The Principality of Mathematics is a mountainous land, but the air is very fine and health-giving, though some people find it too rare for their breathing. It differs from most mountainous countries in this, that you cannot lose your way, and that every step taken is on firm ground. People who seek their work or play in this principality find themselves braced by effort and satisfied with truth.

— Charlotte Mason
Ourselves

• Gary Antonick reports on a coloring puzzle simple enough for children to understand, yet interesting enough to be featured in the current issue of The Mathematical Intelligencer: Triangle Mysteries.

• Not a puzzle, but a refreshing treat: Guillermo Bautista shares 10 Math Quotes That Will Make You Giggle.

• What can you see, and how many ways can you see it? Don Steward offers a fun set of angles in polygons by chopping.

• Looking for summer reading to inspire you mathematically? Sue VanHattum reviews Measurement, by Paul Lockhart and Math from Three to Seven, by Alexander Zvonkin.

• Mashka relates a variety of math circle puzzles: Tricky Pre-K Math Lesson 18 and Lesson 19—Everything Math.

• Glen Whitney transforms coffee stirrers and rubber bands into a 3-D puzzle in Math Monday: DIY Tensegrity.

• Justin Lanier shares a selection of interesting math explorations in Circling, Squaring, and Triangulating.

• Gene Chase explores the topology of a computer game screen in Flat Donuts.

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TEACHING TIPS

A child’s intercourse must always be with good books, the best that we can find… We must put into their hands the sources which we must needs use for ourselves, the best books of the best writers.
…
For the mind is capable of dealing with only one kind of food; it lives, grows and is nourished upon ideas only; mere information is to it as a meal of sawdust to the body.

— Charlotte Mason
Toward A Philosophy of Education

• David Wees lists eight Powerful ideas in math, based on Seymour Papert’s book Mindstorms. Then Shecky Riemann adds a short video about Ideas… The Stuff of Math.

• Katherine Cook ponders the intersection of Math education in two themes and warns parents and teachers about Processed Math: Don’t Eat This.

• Christopher Danielson contrasts two teaching styles in Skills practice [#NCTMDenver] and sparks a (mostly) thoughtful debate in the comments.

• Jason Dyer launches a wonderful discussion in Tiny Games, mathematics edition? Dan Myer brings the project to everyone’s attention with Tiny Math Games. And John Golden attempts to collect the tsunami of ideas in one place for easy reference.

• Nico Rowinsky explores one of math’s great mysteries in Some Equal Signs Are More Equal Than Others.

• Caroline Mukisa says, “Listening to podcasts is a great way of taking in new information and taking a fresh look at old information.” Listen Up! 8 Fascinating Podcasts to Spark a Love of Math in your Teen.

• Are you looking for a good homeschool math program? Check out Amy’s MEP Math Story.

• Most homeschoolers feel at least a small tinge of panic as their students approach high school. For my entry to this month’s carnival, I offer an assortment of links and tips on Homeschooling High School Math.

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FINAL CREDITS

The Robert Webb rhombicosidodecahedron graphic is from Wikimedia Commons, and the Charlotte Mason quotations are from Ambleside Online’s Annotated Charlotte Mason Series.

Most of the book covers link to Amazon.com, where you can read descriptions and reviews of these and many other living books for math (and where I receive a small commission through Skimlinks if you actually buy one of them). All the books included in this post — except for Moebius Noodles, which is too new — should be available through any well-stocked public library or library loan system.

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 sometime during the week of June 10-14 at Math Jokes 4 Mathy Folks. 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 information 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!

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

Would you like to create your own math holiday? Look here for tips and sign-maker links:

• Every Day Is Mathematics Day

Leave a link to your Happy Math Day post in the comments below, so we can all celebrate!

Get all our new math tips and games:  Subscribe in a reader, or get updates by Email.

Can you hear it? That’s the sound of the awesomeness approaching.

— Patrick Vennebush
Math Jokes 4 Mathy Folks

• Go to Patrick’s blog to read all the details!

Get all our new math tips and games:  Subscribe in a reader, or get updates by Email.

Math Teachers at Play blog carnival time is almost here. Are you ready?

If you’ve written a blog post about math, we’d love to have you join us! Each of us can help others learn, so in a sense we are all teachers.

Posts must be relevant to students or teachers of school-level mathematics (that is, anything from preschool up to first-year calculus). Old posts are welcome, as long as they haven’t been published in past editions of this carnival.

To submit an entry, fill out this form: MTaP Submissions.

Don’t procrastinate: The deadline for entries is this Friday. The carnival will be posted next week right here at Let’s Play Math.

• Math Teachers at Play Blog Carnival — Submission Form
• MTaP Blog Carnival Information Page

Explore the Other Math Carnivals

While you’re waiting for next week’s Math Teachers at Play carnival, you may enjoy:

• Last month’s MTaP 61 at Math Hombre
• Math Monday Blog Hop (preschool & up)
• Bagels & Blogs (elementary)
• Carnival of Mathematics (high school & up)
• Carnaval de Matemáticas

Would You Like to Host the Carnival?

Hosting the blog carnival can be a lot of work, but it’s fun to “meet” new bloggers through their submissions. And there’s a side-benefit: The carnival usually brings a nice little spike in traffic to your blog.

If you think you’d like to join in the fun, read the instructions on our Math Teachers at Play page. Then leave a comment or email me to let me know which month you’d like to take.

Get all our new math tips and games:  Subscribe in a reader, or get updates by Email.

Math mascot Moby Snoodles

As for mathematics itself, it’s one of the most adventurous endeavors a young child can experience. Mathematics is exotic, even bizarre. It is surprising and unpredictable. And it can be more exciting, scary and dangerous than sailing the high seas!

But most parents and educators don’t present math this way. They just want the children to develop their mathematical skills rather than going for something more nebulous, like the mathematical state of mind.

…

Children marvel as snowflakes magically become fractals, inviting explorations of infinity, symmetry and recursion. Cookies offer gameplay in combinatorics and calculus. Paint chips come in beautiful gradients, and floor tiles form tessellations. Bedtime routines turn into children’s first algorithms. Cooking, then mashing potatoes (and not the other way around!) humorously introduces commutative property. Noticing and exploring math becomes a lot more interesting, even addictive.

Unlike simplistic math that quickly becomes boring, these deep experiences remain fresh, because they grow together with children’s and parents’ understanding of mathematics.

— Maria Droujkova and Yelena McManaman
Adventurous Math For the Playground Set (Scientific American online)

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

photo by ddluong via flickr

Most homeschoolers feel at least a small tinge of panic as their students approach high school. “What have we gotten ourselves into?” we wonder. “Can we really do this?” Here are a few tips to make the transition easier.

Before you move forward, it may help to take a look back. How has homeschooling worked for you and your children so far?

If your students hate math, they probably never got a good taste of the “Aha!” factor, that Eureka! thrill of solving a challenging puzzle. The early teen years may be your last chance to convince them that math can be fun, so consider putting aside your textbooks for a few months to:

• Subscribe to Games magazine.
• Read Brian Bolt books, or work through Raymond Smullyan’s What is the Name of This Book?
• Design your own tessellation T-shirts for Christmas gifts.
• Remodel the house. From financing to floor coverings, that is real math in action.

On the other hand, if you have delayed formal arithmetic, using your children’s elementary years to explore a wide variety of mathematical adventures, now is a good time to take stock of what these experiences have taught your students.

• How much of what society considers “the basics” have your children picked up along the way?
• Are there any gaps in their understanding of arithmetic, any concepts you want to add to their mental tool box?

photo by Sphinx The Geek via flickr

Transition Math

Many homeschoolers take advantage of the free videos and computer-graded quizzes at Khan Academy to review and consolidate their students’ skills before tackling high school math. My daughter enjoyed Khan Academy for a couple of weeks, but then the mindlessly-repetitive nature of their randomly-generated problems began to irritate her.

I’d rather use a textbook such as Basic College Mathematics (choose from several versions by various authors) that has old editions available for pennies — there’s no need to pay new-book prices, since arithmetic hasn’t changed in centuries. A textbook offers better problems, carefully selected and designed to build on each other. You can use the chapter review problems as a pre-test and study only the topics your child needs. (Buddy Math is great for high school!)

But beware: many books and online resources present mathematics primarily as a follow-the-rules subject. That is fine for review, but if you find a concept that is new to your student, don’t be satisfied with just teaching the procedure. Memorization can give your students quick “success,” producing page after page of correct answers, but for most people, memorized rules will not stick in the mind well enough to support future learning.

If you don’t understand a math topic yourself, dig for more information about it. Ask why it is true and how it relates to other topics. Find out how it connects to the deep ideas of mathematics, such as symmetry, functions, change, etc.

One excellent way to explore the meaning of mathematics is through Herb Gross’s free online Math as a Second Language courses.

You and your student may need to search out several explanations and work your way through dozens of sample problems in order to wrap your minds around a new topic. Wrestling a math concept into submission can be hard! But the work is worth it, because when you are able to help your student truly understand what the math is doing, that concept becomes a rock-solid foundation on which to build.

graphic by Timboliu via wikimedia commons

The Meat of High School Math

If your children plan to go to college, they will benefit from a serious amount of mathematics in high school. Even those who don’t plan on college should learn as much as they can because we live in a technological society, and technology runs on math.

As mathematician and author Oleg Gleizer says, “Math is freedom. If we don’t know math, our choices are so limited.”

This does not mean that every student must follow the traditional algebra-to-calculus approach of school mathematics. Creative students who want to blaze their own trail of interest-led learning can master the basics of mathematics and then branch out to explore many non-traditional topics, such as logic, combinatorics, or game theory.

But first, what are the basics of high school mathematics? What math does every student need in order to be a math-literate member of our technology-based society?

Everyone Needs Algebra

Many people think that algebra is “the rules and procedures for doing math with letters,” and they can’t imagine ever using it in real life. But algebra is much deeper, richer, and more important than just rules and procedures.

Algebra is:

• Using the things you know to figure out something you don’t know.
• Thinking about the relationships between things, how changing one thing affects everything else.

Things we know, things we don’t know, and the relationships between them — such is the stuff of real life. Algebra gives us the tools to model, understand, and solve the problems we face at home and at work, from shipping to science, politics to planning retirement.

Everyone Needs Geometry

Geometry is not just formulas, two-column proofs, and memorizing stuff to pass a test. Like algebra, geometry is deeper, richer, and more important than most people realize.

Geometry is:

• Using the shapes and properties we know to figure out something we don’t know.
• Thinking about the relationships between things, how changing one thing affects everything else.

We live in a three-dimensional world, surrounded by length and height and breadth. From crafts to construction, marketing to medicine, property development to playing in virtual worlds — shapes and volumes and the relationships between them affect almost everything we do.

Everyone Needs Statistics

No one can be considered an educated adult in our society without a basic understanding of statistics, and especially of what the numbers do not mean. The modern world is an ocean of data. Waves of information buffet us from every direction. Our students need to learn how to obtain, analyze, synthesize, evaluate, and draw inferences from statistics.

photo by Karen via flickr

Online Resources for High School Math

The free resources in this section will appeal to unschoolers and independent learners who want to dabble a bit in the different areas of high school math before committing themselves to deeper study.

Other families, who prefer a traditional approach, may also find these links useful. There will be times when a textbook is not enough. Your student may read the text (or watch the video) several times and work through every sample problem, yet he or she may still have trouble understanding a concept. In cases like that, it helps to look up a different explanation that can give your student a new way of looking at the topic.

Whichever category describes your family, keep in mind what Albert Einstein said: “Learning is experience. Everything else is just information.”

In math, experience means wrestling with problems. Even the most useful website can only provide information. If we want our children to truly learn mathematics, they will have to build their own experience by working their way through lots and lots of math problems.

Algebra

Don’t let your students flounder in algebra: the web is overflowing with help. For instance, many students need hands-on exploration to help them wrap their brains around a new math concept. Henri Picciotto offers a selection of interesting activities at A New Algebra.

Stretch your mathematical modeling skills with the Visual Patterns blog. Pick any design you like and practice recognizing, describing, and predicting the pattern.

Do the explanations in your math book go right over your students’ heads? One of my favorite resources for quick explanations of pre-algebra and algebra topics is Purplemath.

Art of Problem Solving offers a terrific resource with their video lessons featuring Richard Rusczyk, who is always fun to watch. And be sure to take advantage of their online learning system, Alcumus, where students can practice their problem-solving skills.

Geometry

The best way to learn geometry is to play with geometric objects such as lines, curves, angles and shapes. The Maths Is Fun site offers a quick overview of geometry topics, including vocabulary and how to draw various shapes. Or check out the math software GeoGebra, with its wealth of user-created instructional materials for all ages.

For more than two millenia, people have struggled with and mastered Euclidean geometry as the foundation for higher math and logic. It’s still a great way for students to sharpen their thinking skills — as long as they approach each theorem as a puzzle to understand rather than as a rule to memorize. Your students can go straight to the source: Interactive Euclid’s Elements. Or they can work through the easier presentation at Math Open Reference.

The great advances of modern math and science began when geometry and algebra learned to dance together on a coordinate grid. Develop a beginner’s intuition about the meaning of graphs with Graphing Stories. Then explore the relationships between equations and shapes with the Desmos Graphing Calculator, and try your hands at some of the Daily Desmos blog challenges.

Statistics

Many homeschoolers have enjoyed the classic How to Lie With Statistics by Darrell Huff: “The crooks already know these tricks; honest men must learn them in self-defense.” The book is older than I am (in the first chapter, $25,000 is considered a salary that a Yale graduate might boast about), but it is still in print, and the information is as relevant as your daily newspaper. Maybe more so, since newspapers routinely commit many of the errors Huff warns against.

For a basic introduction online, check out Robert Niles’s Statistics Every Writer Should Know, a tutorial for math-phobic journalists. Your students may also enjoy browsing the Gallery of Data Visualization, which features several of the world’s best (and worst) statistical graphs. As the authors point out, seeing may or may not be believing, but “above all, data analysis involves visual, as well as statistical, understanding.”

And if your students want to dig deeper into the topic, Stat Trek offers a good Statistics and Probability Tutorial.

photo by CLF via flickr

Are You Ready for a Textbook?

A mathematics textbook can be a wonderful learning tool for students who know how to use it. A good textbook offers clear explanations of mathematical concepts and an assortment of worked-out sample problems to demonstrate procedures. It can help students become fluent in the language of math, how to represent their thoughts so others can understand them.

Most important, a textbook provides a rich source of practice problems, arranged in order of increasing difficulty, which give students plenty of opportunity to gain the kind of experience that builds true mathematical learning.

Don’t be fooled by your own experience of dry or tedious math classes: textbook mathematics is still math the mathematician’s way, as mental play. But it is no longer the play of a child dabbling in the shallows, nor the play of the couch potato who wants only to observe.

No, this is the play of the athlete, who works hard at training and enjoys seeing his muscles grow firm, who can’t wait to test himself against a new and challenging opponent. As the athlete does not simply memorize rules and procedures, but instead analyzes the game until it truly makes sense and practices every move, no matter how difficult, until it feels natural — so the student must work with each math concept until it becomes a part of himself.

Anything else, as Einstein said, is merely information.

Learning How To Learn Math

I won’t offer specific advice about choosing your high school math curriculum. Each family is different, and even within a family, what works for one child may fail miserably with another. After 25 years of homeschooling, I’m still reduced to guessing which program will interest my youngest daughter and then adjusting my expectations as we work our way through the book.

But whatever math program you end up using, here are a couple of valuable hints:

• Don’t read a lesson straight through, as if it were a novel.
• Do attempt each sample problem yourself, before reading the book’s solution.

For more tips on mastering high school and college mathematics, check out Stan Brown’s detailed instructions about How to Read a Math Book and How to Study Math at the Tompkins Cortland Community College website.

Working Yourself Out of a Job

If your students have a strong foundation of understanding elementary and middle-school math, and if they are mature enough to recognize when they need to ask for help, they might be able to do their high school math independently. You may find yourself serving more as occasional consultant than as daily instructor.

Even if you never studied trigonometry, for instance, a few Socratic questions will often lead your student toward the answer:

• Describe your problem to me.
• What do you know?
• What information do they give you?
• What do you want?
• What are you trying to find?
• What can you do?
• Tell me what you’ve already tried.
• Have you had a problem like this one before?
• Let’s look at the examples in the chapter …

When I first studied calculus, my dad helped me through at least one killer problem in every homework set this way, just by asking questions. He later admitted to me that he truly hadn’t known what I was supposed to be doing, but I never would have guessed that. His questions were enough to spark my memory of something we had studied in class or of an earlier problem that helped me work through the place I was stumped.

You will find more tips on solving high school math problems in my blog post The Case of the Mysterious Story Problem. For assistance with specific questions about high school math, be sure to check out the Forums Where You Can Ask for Help section on my Internet Math Resources page.

Best wishes to you and your children as you continue to pursue the lifelong adventure of learning!

This article is an excerpt from my book Let’s Play Math: How Homeschooling Families Can Learn Math Together, and Enjoy It!

Get all our new math tips and games:  Subscribe in a reader, or get updates by Email.

Related posts on Let’s Play Math! blog:

• How to Recognize a Successful Homeschool Math Program
• Thinking (and Teaching) like a Mathematician
• How to Solve Math Problems
• Who Killed Professor X?
• Why Study Mathematics?

Homeschoolers, after-schoolers, unschoolers, or anyone else: if you’re a parent with kids at home, you need this book. If you work with children in any way (grandparent, aunt/uncle, teacher, child care, baby sitter, etc.) you need this book. Or if you hated math in school and never understood how anyone could enjoy it, you need this book!

• Moebius Noodles: Adventurous Math for the Playground Crowd

Moebius Noodles is a travel guide to the Math Universe for adventurous families (and it has lots of beautiful pictures, too!) featuring games and activities that draw out the rich, mathematical properties of everyday objects in ways accessible to parents and children:

• A snowflake is an example of a fractal and an invitation to explore symmetry.
• Cookies offer combinatorics and calculus games.
• Paint chips come in beautiful gradients, and floor tiles form tessellations.

From the book’s online description:

How do you want your child to feel about math? Confident, curious and deeply connected? Then Moebius Noodles is for you. It offers advanced math activities to fit your child’s personality, interests, and needs.

Can you enjoy playful math with your child? Yes! The book shows you how to go beyond your own math limits and anxieties to do so. It opens the door to a supportive online community that will answer your questions and give you ideas along the way.

Learn how you can create an immersive rich math environment for your baby. Find out ways to help your toddler discover deep math in everyday experiences. Play games that will develop your child’s sense of happy familiarity with mathematics.

A five-year-old once asked us, “Who makes math?” and jumped for joy at the answer, “You!” Moebius Noodles helps you take small, immediate steps toward the sense of mathematical power.

You and your child can make math your own. Together, make your own math!

Read more and see sample pages at the Moebius Noodles page.

Other great resources from Moebius Noodles:

• Discuss early mathematics with people who value depth, connections, beauty and play at the Ask Moby Snoodles page.
• Explore the continuing adventure of mathematics with projects, activities, and inspiration from the Moebius Noodles blog.

Get all our new math tips and games:  Subscribe in a reader, or get updates by Email.

Come join the fun!

The Math Teachers at Play blog carnival is a monthly blogging round-up shared at a different blog each month, featuring posts from parents, teachers, homeschoolers, and students — anyone who is interested in playing around with school-level (preschool to pre-college) or recreational math.

This month’s edition is ready for your browsing pleasure:

• Math Teachers at Play 61

Enjoy!

Get all our new math tips and games:  Subscribe in a reader, or get updates by Email.

If you bought an early edition of my ebook Let’s Play Math, you can now update your copy to the latest version.

This update includes:

• typo whack-a-mole (fixing all I could find)
• toc.ncx navigation (the ebook magic that lets you skip ahead to the next chapter)
• additional living book and internet references in the appendix sections
• new quotations: insights from W. W. Sawyer, Malke Rosenfeld, and Maria Droujkova
• an expansion of the Homeschooling with Math Anxiety section
• How to Recognize a Successful Homeschool Math Program
• and several new sections in the high school math chapter, which I hope to publish on the blog as well (maybe next week?)

How To Update

If you bought at Smashwords, the latest update is always available for download at their site.

If you are an Amazon.com customer, you can get the updated version of this book by going to Manage Your Kindle. Find the book in your Kindle Library, click on the “Update Available” link next to the book’s title, and then follow the update prompts. After you do this, all of your Kindle devices that have the ebook currently downloaded will be updated automatically the next time they connect to wireless. If you tucked the book away in a folder, the update will replace it there, rather than cluttering up your home screen.

Also Available

Through the Smashwords distribution program, my ebook is finally spreading to other online booksellers:

• iTunes bookstore
• Barnes & Noble
• Kobo

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by Mike Licht, NotionsCapital.com

Activities, games, lessons, hands-on fun — if you’ve written a blog post about math, we’d love to have you join our math blog carnival!

Posts must be relevant to students or teachers of school-level mathematics (that is, anything from preschool up through first-year calculus). Old posts are welcome, as long as they haven’t been published in past editions of this carnival.

To submit an entry, fill out this form:

• Math Teachers at Play Blog Carnival – Submissions

Don’t procrastinate: The deadline for entries is this Friday night. The carnival will be posted next week at Math Hombre blog.

Explore the Other Math Carnivals

While you’re waiting for next week’s Math Teachers at Play carnival, you may enjoy:

• Last month’s MTaP 60 at White Group Mathematics
• Math Monday Blog Hop (preschool & up)
• Carnival of Mathematics (high school & up)
• Carnevale della Matematica
• Carnaval de Matemáticas

Would You Like to Host the Carnival?

Hosting the blog carnival can be a lot of work, but it’s fun to “meet” new bloggers through their submissions. And there’s a side-benefit: The carnival usually brings a nice little spike in traffic to your blog.

If you think you’d like to join in the fun, read the instructions on our Math Teachers at Play page. Then leave a comment or email me to let me know which month you’d like to take.

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Book Update

I’m still working on Let’s Play Algebra, the sequel to my Let’s Play Math book.

Here’s a quick taste of things to come…

Algebraic Manipulatives

In math education, a manipulative is a hands-on object that represents an abstract concept. Many homeschoolers are familiar with Cuisenaire rods, base 10 blocks, fraction circles, and other common elementary math toys. Now we need a hands-on object to represent a variable, an unknown number. Ideally, we want several small items about the same size, so they are easy to work with, in two different colors to represent positive and negative variables.

Teacher supply stores sell algebra manipulatives, and virtual manipulatives are available online, but I like the idea of making my own. In Vision in Elementary Mathematics, W. W. Sawyer uses a bag with stones inside, tied shut so we cannot see how many there are. I could cut out felt circles (in two colors), wrap them around a few pebbles, and tie them shut with rubber bands.

When Henry Borenson invented the Hands-On Equations system, he used board-game-style pawns. I have several old chess sets with missing pieces, and chess pawns come in two colors. If I look at a pawn from the side and squint my eyes a little, I can even imagine it looks something like a small bag. Maybe.

Or I could fold origami boxes for my variables.

But my favorite algebra manipulatives, which I will use for the Keep Your Balance game in this book, are acrylic octagon jewels like these. They come in a variety of colors, so I can choose my favorite color or pick up an assortment. I like the amber or topaz, which look like crystalline gold. For negative variables, I’ll use black gemstones as antimatter. My children love imaginative stories, so solving an equation will mean figuring out what our “pirate treasure” is worth.

Don’t Forget the Constant Numbers

We also need a symbol to represent known numbers: pebbles, coins, dry beans, Cuisenaire rods, or any other countable items will work. In an earlier edition of this book, I used Cheerios (with raisins for the negative numbers) because these are cheap and easy to find in large quantity, and because my children have always liked it when solving a math puzzle involved getting a snack.

But in keeping with the treasure theme, I think gold coins would make wonderful manipulatives, along with black coins for antimatter. I could buy plastic gold coins, but it’s even easier to spray-paint regular pennies. Don’t worry: it’s not illegal to deface pennies for educational use — just don’t try to pass them off as actual gold!

Treasure Map from Wikimedia Commons

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My all-time favorite Good Friday sermon. If the video doesn’t show, see it on YouTube.

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So I let her off early and pointed her toward these logical “algebra” puzzles instead:

Puzzle by Paul Salomon

Imbalanced Variables

The balance sets up inequalities, like a visual algebra. There are no numbers, so you cannot solve for the exact values of the unknown weights, but you should be able to figure out which shapes are heaviest and lightest.

See more puzzles here:

• Imbalance Problems
• More Imbalance Problems

After solving these, the natural next step is to make up puzzles of your own — but we got distracted because my husband called us outside to help push him out of the snow drift.

And it’s supposed to be spring!

The decimal exercises are still waiting for us, which means that Kitten will keep looking for an excuse to avoid them. Perhaps we’ll try making up imbalance algebra puzzles tomorrow…

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Related posts on Let’s Play Math! blog:

• Pre-Algebra Picture Puzzles
• Backwards Math
• Thinking (and Teaching) like a Mathematician
• The Game of Algebra
• Algebra: A Problem in Translation

Other Math Holidays Created by Bloggers

• Happy Mathy Day
• Happy today! It’s a special day!
• Happy 3/24 = 1/8 Day
• Happy 3/25 = 12/100 = 12% day

Make Your Own

Would you like to create a math holiday, too? Try one of these sign generators:

• Roosevelt Middle School Sign Generator
• Raceland Elementary School Sign Generator
• Custom Road Sign
• Wrigley Field Sign Generator
• University of Texas Stadium Scoreboard (Godzillatron) Sign Generator
• And plenty more to choose from

What kind of math will you celebrate?

Leave a link to your post in the comments!

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Mathematicians Ask Questions

If we want to teach our children to think mathematically, we need to model and encourage asking questions. For instance:

• What is the difference between the rectangular sounds and the round ones?
• At 5:20, the orange notes (violin) change to a different shape. Why? What change in the sound does this represent?

What questions does the video inspire for you? I’d love to hear your comments!

More Information

• Kids and the Music Animation Machine
• A Closer Look at Music Animation Machine Notation
• Music, Mind, and Meaning
• About the Piano Trio No. 2 (Schubert)

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