StudentTeam

Math Teachers at Play #76

76[Feature photo (above) by U.S. Army RDECOM. Photo (right) by Stephan Mosel. (CC BY 2.0)]

On your mark… Get set… Go play some math!

Welcome to the 76th edition of the Math Teachers At Play math education blog carnival — a smorgasbord of links to bloggers all around the internet who have great ideas for learning, teaching, and playing around with math from preschool to pre-college.

By tradition, we start the carnival with a puzzle in honor of our 76th edition. But if you would like to jump straight to our featured blog posts, click here to see the Table of Contents.

PUZZLE: CRYSTAL BALL CONNECTION PATTERNS

K4 matchings

In the land of Fantasia, where people communicate by crystal ball, Wizard Mathys has been placed in charge of keeping the crystal connections clean and clear. He decides to figure out how many different ways people might talk to each other, assuming there’s no such thing as a crystal conference call.

Mathys sketches a diagram of four Fantasian friends and their crystal balls. At the top, you can see all the possible connections, but no one is talking to anyone else because it’s naptime. Fantasians take their siesta very seriously. That’s one possible state of the 4-crystal system.

On the second line of the diagram, Joe (in the middle) wakes up from siesta and calls each of his friends in turn. Then the friends take turns calling each other, bringing the total number of possible connection-states up to seven.

Finally, Wizard Mathys imagines what would happen if one friend calls Joe at the same time as the other two are talking to each other. That’s the last line of the diagram: three more possible states. Therefore, the total number of conceivable communication configurations for a 4-crystal system is 10.

For some reason Mathys can’t figure out, mathematicians call the numbers that describe the connection pattern states in his crystal ball communication system Telephone numbers.

TheWizardBySeanMcGrath-small

  • Can you help Wizard Mathys figure out the Telephone numbers for different numbers of people?
    T(0) = ?
    T(1) = ?
    T(2) = ?
    T(3) = ?
    T(4) = 10 connection patterns (as above)
    T(5) = ?
    T(6) = ?
    and so on.

Hint: Don’t forget to count the state of the system when no one is on the phone crystal ball.

[Wizard photo by Sean McGrath. (CC BY 2.0)]


TABLE OF CONTENTS

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

Tweet: Math Teachers at Play #76: a smorgasbord of great ideas for learning, teaching, and playing around with math. http://ctt.ec/5ei9W+

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EARLY LEARNING ACTIVITIES

  • Amy Tanner offers Four Super Simple Counting Games that help your child build number sense, learn to estimate, begin to think about addition and subtraction, and practice counting backward.
    [My favorite perk of hosting the MTaP carnival is discovering yummy new blogs. This one definitely belongs in my rss list.]
  • There may not be any numbers, but there’s a whole lot of math going on in Teacher Tom’s post, A Current Of Curiosity.
  • Joshua Greene tells how a simple pattern led to deep and interesting questions — and it only took “5 minutes in between other play”: Pattern Blocks (mini follow-up).
  • Sarah Dees adapted an activity from the Curious George PBS show in Composing Music with Math Activity for Kids. “Seriously, this was a lot of fun. The boys wrote many compositions, and couldn’t wait to perform them for Dad when he came home from work!”

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ELEMENTARY EXPLORATION AND MIDDLE SCHOOL MASTERY

  • Margo Gentile suggests practicing the math facts with Picnic Time Multiplication. If I were to modify it, I’d skip saying the equations and add the ABCs back in: “I’m going on a picnic, and I’m going to bring 3 apples, 6 buffalo, 9 candy canes, and…”
  • Stephen Cavadino’s class stumbles on what should have been an easy review problem, and he responds with “Aaargh Ruddy BIDMAS!
  • Bryan Anderson’s class creates a variety of graphs to compare different data sets in Human Histogram.

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ADVENTURES IN BASIC ALGEBRA & GEOMETRY

  • Fawn Nguyen’s students have fun investigating the relationship between a circle’s diameter and circumference in Friday Bubbles.
  • Sue VanHattum takes a break from book editing to explore Euclidean geometry in How I’m Playing With Math Today. “Geometry is my weakness in math, and I love trying to figure out how to do these constructions.”
  • Don Steward posts a grand collection of geometry puzzles in angle proofs. Each image can be printed landscape-orientation on a regular sheet of paper or added to PowerPoint for sharing with students.

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ADVANCED MATHEMATICAL ENDEAVORS

  • John Golden discusses how help students understand complex numbers in Complex Instruction, with a little help from GeoGebra. “One of the morals of the capstone class was that if mathematicians labeled a theorem as Fundamental, it’s worth your focus and understanding…”
  • Tina Cardone tweaks some Parametric Functions lessons to work on Desmos. “It was important to begin graphing by hand so students had an understanding of how parametrics work. Some students were concerned that the t value wasn’t showing up on the graph and tried to include it in some rather creative ways…”
  • Rebecka Peterson steals a favorite lesson and refuses to feel guilty because “this magic should be shared.” And so she does: Slope Field Activity.
  • As I’ve put my energy into working on my math books, my blogging has suffered. So I’ve started dipping into the past and bringing up oldy-but-goody articles to reblog. I especially enjoyed The Calculus Tidbits Collection.

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PUZZLING RECREATIONS

  • Fran Wisniewski shares one of my all-time favorite puzzle games: Tangrams. Print and cut out a set of pieces, or play online.
  • Shecky Riemann challenges us to try a Li’l Game From Martin Gardner. “Whoever does this gets all the money played, in cases of draws (no winner) you each take your money back. The question is, is there any strategy by which you could be assured a win?”
  • Julie’s family folds up some beautiful 3-dimensional math in Origami Icosahedron. “When the faces of solid figures protrude to form more complex solids, the shapes become star-like and are known as stellations. The icosahedron we created is the small triambic icosahedron…”
  • The Math Curmudgeon’s MathArguments180 is still going strong, bringing us some cool recreational puzzles to debate. What would your students do with 187: Spiral or 191: Walking the Labyrinth?
  • One of the great puzzles of mathematics is how to think about infinity. Along this line, Yelena McManaman and her son read the book Really Big Numbers in Infinity Is Farther Than You Think. And Vi Hart posts the latest in “a potential infinity of spinoff videos” in Transcendental Darts.

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TEACHING TIPS

  • Donna Boucher takes a look at one of my favorite elementary math curricula in What is Singapore Math? “Singapore Math is really a philosophy for mathematics instruction — it’s as much about how to teach as it is what to teach.”
  • Stephen Cavadino asks some important questions about assessment: “What is the big picture? What are we testing for? Should we be doing it?”
  • A friend asks, “I am doubtful that he will actually be able to solve this problem he’s puzzling through. What does a good teacher do in such a situation? You have a student who is really interested in this problem, but you know that it’s far more likely that he will hit a wall (or many walls) that he really doesn’t have the tools to work through.” Ben Blum-Smith offers wise advice in Hard Problems and Hints.

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GIVING CREDIT WHERE IT’S DUE

I found the pretty pictures at Flickr.com Creative Commons. John Riordan wrote about Telephone numbers in Introduction to Combinatorial Analysis.

And that rounds up this edition of the Math Teachers at Play math education blog carnival. I hope you enjoyed the ride.

The next installment of our carnival will open sometime during the week of August 25-29 at Math = Love. 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.

You can explore all our past MTaP carnival posts on our blog carnival Pinterest 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 want to take a turn hosting the Math Teachers at Play blog carnival, please speak up!


[Photo by Bob Jagendorf.]


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Ohio Jones 2

The Linear Inequality Adventures of Ohio Jones

Ohio Jones 1

Last week, Kitten and I reached her textbook’s chapter on graphing linear equations, and a minor mistake with negative numbers threw her into an “I can’t do it!” funk. It’s not easy teaching a perfectionist kid.

Usually her mood improves if we switch to a slightly more advanced topic, and luckily I had saved these worksheets on my desktop, waiting for just such an opportunity. Today’s lesson:

  • Some fun(ish) worksheets
    “For tomorrow, students will be graphing systems of inequalities, so I decided to create a little Ohio Jones adventure (Indiana’s lesser known brother)…”

I offered to give her a hint, but she wanted to try it totally on her own. It took her about 40 minutes to work through the first few rooms of the Lost Templo de los Dulces and explain her solutions to me. I’m sure she’ll speed up with experience.

So far, she’s enjoying it much more than the textbook lesson. It’s fascinating to me how the mere hint of fantasy adventure can change graphing equations from boring to cool. Thanks, Dan!


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Reimann-hexagon

Math Teachers at Play #70

800px-Brauchtum_gesteck_70_1[Feature photo above by David Reimann via Bridges 2013 Gallery. Number 70 (right) from Wikimedia Commons (CC-BY-SA-3.0-2.5-2.0-1.0).]

Do you enjoy math? I hope so! If not, browsing this post just may change your mind.

Welcome to the 70th edition of the Math Teachers At Play math education blog carnival — a smorgasbord of 42+ links to bloggers all around the internet who have great ideas for learning, teaching, and playing around with math from preschool to pre-college. Let the mathematical fun begin!

By tradition, we start the carnival with a puzzle in honor of our 70th edition. But if you would like to jump straight to our featured blog posts, click here to see the Table of Contents.

Click here to continue reading.

VisualPatterns-org

Algebra for (Almost) Any Age

VisualPatterns-org2

Fawn Nguyen’s Visual Patterns website just keeps getting better and better. Check it out:

In addition to the 115 puzzle patterns (as of this writing), the site features a Gallery page of patterns submitted by students. And under the “Teachers” tab, Fawn shares a form to guide students in thinking their way through to the algebraic formula for a pattern.

How can you use these patterns to develop algebraic thinking with younger students? Mike Lawler and sons demonstrate Pattern #1 in the YouTube video below.


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Carnival Parade in Aachen 2007

Math Teachers at Play #66

[Feature photo above by Franz & P via flickr. Route 66 sign by Sam Howzit via flickr. (CC BY 2.0)]
Route 66 Sign

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.

By tradition, we start the carnival with a couple of puzzles in honor of our 66th edition.

Let the mathematical fun begin!

Puzzle 1

how crazy 66

Our first puzzle is based on one of my favorite playsheets from the Miquon Math workbook series. Fill each shape with an expression that equals the target number. Can you make some cool, creative math?

Click the image to download the pdf playsheet set: one page has the target number 66, and a second page is blank so you can set your own target number.

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Rectangle algebra

Puzzle: Algebra on Rectangles

Gordon Hamilton of Math Pickle recently posted these videos on how to make algebra 1 puzzles on rectangles. As I was watching, Kitten came in and looked over my shoulder. She said, “Those look like fun!”

They look like fun to me, too, and I bet your beginning algebra students will enjoy them:

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RIMG0186 Satellite dish

How To Master Quadratic Equations

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.

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Math Teachers at Play #62

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:

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photo by Sphinx The Geek via flickr

Homeschooling High School Math

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?

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fryeburg-fair-by-alex-kehr

Math Teachers at Play #58

No 58 - gold on blue[Feature photo (above) by Alex Kehr. Photo (right) by kirstyhall via flickr.]

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. If you like to learn new things and play around with ideas, you are sure to find something of interest.

Let the mathematical fun begin…

PUZZLE 1

By tradition, we start the carnival with a pair of puzzles in honor of our 58th edition. Click to download the pdf:

How CRAZY Can You Make It

PUZZLE 2

A Smith number is an integer the sum of whose digits is equal to the sum of the digits in its prime factorization.

Got that? Well, 58 will help us to get a better grasp on that definition. Observe:

58 = 2 × 29

and

5 + 8 = 13
2 + 2 + 9 = 13

And that’s all there is to it! I suppose we might say that 58’s last name is Smith. [Nah! Better not.]

  • What is the only Smith number that’s less than 10?
  • There are four more two-digit Smith numbers. Can you find them?

And now, on to the main attraction: the blog posts. Many articles were submitted by their authors; others were drawn from the immense backlog in my Google Reader. Enjoy!

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Quotable: Why Study Algebra?

Blocks

[Photo by AlphaTangoBravo / Adam Baker via flickr.]

One reason to study algebra: because it’s a building block. And just as it was really hard at first to get those blocks to do what you wanted them to do, so also it can be really hard at first to get algebra to work. But if you persevere, who knows what you might build someday?

Algebra is the beginning of a journey that gives you the skills to solve more complex problems.

So, try not to think of Algebra as a boring list of rules and procedures to memorize. Consider algebra as a gateway to exploring the world around us all.

— Jason Gibson
Why Study Algebra?


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Abstraction in Language and in Math

photo by Robert Couse-Baker via flickr creative commons

Check out Dan’s interesting semi-philosophical discussion of the meaning and importance of abstraction:

The physical five oranges goes up the ladder to the picture of the five oranges which goes up to the representation of the five oranges as a numeral.

This points in the direction of a definition of abstraction: when we abstract we voluntarily ignore details of a context, so that we can accomplish a goal.

Dan Meyer

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Math Teachers at Play #52

[Photo by bumeister1 via flickr.]

Welcome to the Math Teachers At Play blog carnival — which is not just for math teachers! We have games, lessons, and learning activities from preschool math to calculus. If you like to learn new things and play around with mathematical ideas, you are sure to find something of interest.

Scattered between all the math blog links, I’ve included highlights from the Common Core Standards for Mathematical Practice, which describe the types of expertise that teachers at all levels — whether in traditional, experimental, or home schools — should seek to develop in their math students.

Let the mathematical fun begin…

TRY THESE PUZZLES

By tradition, we start the carnival with a couple of puzzles in honor of our 52nd edition. Since there are 52 playing cards in a standard deck, I chose two card puzzles from the Maths Is Fun Card Puzzles page:

  • A blind-folded man is handed a deck of 52 cards and told that exactly 10 of these cards are facing up. How can he divide the cards into two piles (which may be of different sizes) with each pile having the same number of cards facing up?
  • What is the smallest number of cards you must take from a 52-card deck to be guaranteed at least one four-of-a-kind?

The answers are at Maths Is Fun, but don’t look there. Having someone give you the answer is no fun at all!

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Math Teachers at Play #46

Welcome to the Math Teachers At Play blog carnival — which is not just for math teachers! Here is a smorgasbord of ideas for learning, teaching, and playing around with math from preschool to pre-college. Some articles were submitted by their authors, others were drawn from the immense backlog in my blog reader. If you like to learn new things, you are sure to find something of interest.

Continue reading

Understanding Algebra: How Many Roots?

In algebra 1, we spend a lot of time working with quadratic equations. Among other things, we want to know how many roots (solutions) an equation has and whether the roots are real or imaginary numbers.

One way to visualize this is by asking:

  • “Which values of x will make the equation equal to zero — that is, will make the graph cross the x-axis?”

I wish my algebra teacher had explained it like James Tanton does. It makes so much sense!


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Math Carnival Update, and an Algebra Puzzle

Oops! I misread my calendar last week. The Math Teachers at Play blog carnival will be this Friday at Maths Insider. That means you still have today and tomorrow to send in your blog post submissions using the handy submission form. See you at the carnival!

In the meantime, let me share with you this monster algebra puzzle from the Well-Trained Mind forum. Simplify:

[ \left ( {x}^{\frac {3}{2x}} \right )^{\frac{x}{9}} \times \left ( x^{\frac{9}{15}} \right )^{\frac{5}{18}}]^3

How would you explain this problem to a beginning algebra student who has just learned the exponent rules? Or to his non-mathy mom?

And Don’t Miss…

These other mathy blog carnivals:


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MAA Found Math for the week of June 21, 2010

Math Teachers at Play #39

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.

Several of these articles were submitted by the bloggers; others were drawn from my overflowing blog reader. Don’t try to skim everything all at once, but take the time to enjoy browsing. Savor a few posts today, and then come back for another helping tomorrow or next week.

Most of the photos below are from the 2010 MAA Found Math Gallery; click each image for more details. Quotations are from Mike Cook’s Canonical List of Math Jokes.

Let the mathematical fun begin…

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math notebooking clock, large

Math Teachers at Play #35

35 is a tetrahedral number

Welcome to the Math Teachers At Play blog carnival — which is not just for math teachers.

Do you enjoy math? I hope so! If not, browsing these links just may change your mind. Most of these posts were submitted by the bloggers themselves; others are drawn from my overflowing Google Reader. From preschool to high school, there are plenty of interesting things to learn.

Let the mathematical fun begin…

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Sept-Oct 2010 Math Calendars

As I was preparing for Wednesday’s Homeschool Math Club Games & Activities meeting, I remembered my old math calendars and thought, that would be a fun activity to offer. So I pulled up the files and discovered that the days of the week matched perfectly. What a cool coincidence!

So in case you missed the math calendars last year, or in case it’s been long enough that your children have forgotten, here are the “new” versions:

Addendum

Umm Ahmad created an easier version for young students:

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found math triads wall hanging

Math Teachers at Play #24

[Photo by internets_dairy.]

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 an arithmetic card game in honor of our 24th edition and a few number puzzles:

Continue reading

Math Teachers at Play #20

blue icosahedron, by shonk[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.

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Burnt Algebra

Algebra: A Problem in Translation

[Photo by *Irish.]

In my post Elementary 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?

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Math Teachers at Play #8

party-child-by-jaaron

[Photo by jaaron.]

Welcome to the Math Teachers At Play blog carnival — which is not just for math teachers! We accept entries from anyone who enjoys playing around with math, as long as the topic is relevant to students or teachers of preK-12th grade mathematics.

Some articles were submitted by their authors, other were drawn from the back-log in my blog reader, and I’ve spiced it all up with a few math jokes courtesy of the Mathematical humor collection of Andrej and Elena Cherkaev.

Let the mathematical fun begin…

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Kids’ Project: More Math Calendars?

tulips-by-kuzeytac[Photo by Kuzeytac.]

Several people enjoyed the April calendar and asked if there would be a May version. Unfortunately, my homeschool co-op classes are out until next fall, so I don’t have enough kids to make up problems for me. But if your children would like to send in some puzzles, I will be glad to put another calendar together. If we get enough participation, we could have calendars every month for the rest of the year!

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Factoring Puzzle3

Puzzle: Factoring Trinomials

Factoring Puzzle

My high school class ended the year with a review of multiplying and factoring simple polynomials. We played this matching game, and then I gave them a puzzle worksheet. I liked this idea, but I didn’t like the decoded answer. In my opinion, puzzles should give the student a “reward” for solving them — maybe a joke or riddle or something — but that answer seemed almost like nagging.

So I changed things around to make my own version:

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log-graph

Math Game: Logarithm War

[Graph created at Draw Function Graphs.]

Kate at f(t) took my popular Math War game to a new level by making a set of Logarithm War cards. Cool! Download a deck for yourself:

Homeschoolers

If your student has passed you up in math and has no classmates to play games with, he or she can still use these cards to practice logarithms. Just follow the instructions in this video.


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