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How To Master Quadratic Equations

Posted on by Denise Gaskins

G'Day Math logo

Feature photo above by Junya Ogura via Flickr (CC BY 2.0).

A couple of weeks ago, James Tanton launched a wonderful resource: a free online course devoted to quadratic equations. (And he promises more topics to come.)

Kitten and I have been working through the lessons, and she loves it!

We’re skimming through pre-algebra in our regular lessons, but she has enjoyed playing around with simple algebra since she was in kindergarten. She has a strong track record of thinking her way through math problems, and earlier this year she invented her own method for solving systems of equations with two unknowns.

I would guess her background is approximately equal to an above-average Algebra 1 student near the end of the first semester.

After few lessons of Tanton’s course, she proved — within the limits of experimental error — that a catenary (the curve formed by a hanging chain) cannot be described by a quadratic equation. Last Friday, she easily solved the following equations:

\left ( x+4 \right )^2 -1=80

and:

w^2 + 90 = 22 w - 31

and (though it took a bit more thought):

4x^2 + 4x + 4 = 172

We’ve spent less than half an hour a day on the course, as a supplement to our AoPS Pre-Algebra textbook. We watch each video together, pausing occasionally so she can try her hand at an equation before listening to Tanton’s explanation. Then (usually the next day) she reads the lesson and does the exercises on her own.

So far, she hasn’t needed the answers in the Companion Guide to Quadratics, but she did use the “Dots on a Circle” activity — and knowing that she has the answers available helps her feel more independent.

Continue reading How To Master Quadratic Equations

Math Teachers at Play #62: A Carnival with Books

Posted on by Denise Gaskins

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 full-size template.
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:

Continue reading Math Teachers at Play #62: A Carnival with Books

Quotable: The Adventure of Learning Math

Posted on by Denise Gaskins
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)

Homeschooling High School Math

Posted on by Denise Gaskins
photo by ddluong via flickr
photo by ddluong via flickr

Feature photo (above) by Sphinx The Geek 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:

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?

Continue reading Homeschooling High School Math

Moebius Noodles: New Must-Read Math Book

Posted on by Denise Gaskins

MoebiusNoodles2DCover

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 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.

Continue reading Moebius Noodles: New Must-Read Math Book

Math Teachers at Play #61 via Math Hombre

Posted on by Denise Gaskins
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:

Enjoy!

My Ebook: Kindle & Smashwords Updates Available

Posted on by Denise Gaskins

letsplaymathcover-mini

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:

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 your Content and Devices page. 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.