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Multiplying Negative Numbers with Rectangles
I love using rectangles as a model for multiplication. In this video, Mike & son offer a pithy demonstration of WHY a negative number times a negative number has to come out positive:
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Math Teachers at Play #63 via Math Jokes 4 Mathy Folks
Hooray for Friday! Let’s celebrate by visiting this month’s Math Teachers at Play blog carnival, featuring mathematical activities, lessons, and games for all ages:
Hmm… let’s see… now where did I put my notes? I know that this is supposed to be the Math Teachers at Play blog carnival… but which one?
Maybe the following puzzle will help. In the grid below, do the following:
- Circle any number, then cross out the other numbers in the same row and column.
- …
Click here to read the entire post at Math Jokes 4 Mathy Folks blog.
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What Do You Notice? What Do You Wonder?
If you want your children to understand and enjoy math, you need to let them play around with beautiful things and encourage them to ask questions.
Here is a simple yet beautiful thing I stumbled across online today, which your children may enjoy:
It reminds me of string art designs, but the app makes it easy to vary the pattern and see what happens.
- What do your students notice about the patterns?
- What questions can they ask?
I liked the way the app uses “minutes” as the unit that describes the star you want the program to draw. That makes it easier (for me, at least) to notice and understand the patterns, since minutes are a more familiar and intuitive unit than degrees, let alone radians.
For Teachers or Advanced Students
Noticing and wondering is not just for students:
- What connections can you find between these star patterns and other topics in mathematics?
- For instance, how do the patterns relate to Matt Henderson’s gears: When designing a machine…
- Or can you see what these stars have in common with Euclid’s Game?
There are probably other apps on the web that do similar things, and if anyone wants to recommend them in the comments below, I’ll enjoy playing with those, too.
Addendum
If you are not used to asking questions about mathematics, here is a short paper to give you a kick-start:
- Drawing Stars by Daniel Shapiro
As you spend time thinking about his questions, you will almost certainly wonder about something yourself. Be sure to capture the idea before you forget it!
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Summer School for Parents, Teachers: How to Learn Math
Here’s an interesting summer learning opportunity for homeschooling parents and classroom teachers alike. Stanford Online is offering a free summer course from math education professor and author Jo Boaler:
Boaler’s book is not required for the course, but it’s a good read and should be available through most library loan systems.
You can register now, and the lessons launch on July 15. Professor Boaler estimates that to fully engage with the course materials will take about 2-4 hours per week, but as with most online courses you can choose your own pace and level of interaction.
Quotable: Learning the Math Facts
photo by Mike Baird via flickr
During off-times, at a long stoplight or in grocery store line, when the kids are restless and ready to argue for the sake of argument, I invite them to play the numbers game.
“Can you tell me how to get to twelve?”
My five year old begins, “You could take two fives and add a two.”
“Take sixty and divide it into five parts,” my nearly-seven year old says.
“You could do two tens and then take away a five and a three,” my younger son adds.
Eventually we run out of options and they begin naming numbers. It’s a simple game that builds up computational fluency, flexible thinking and number sense. I never say, “Can you tell me the transitive properties of numbers?” However, they are understanding that they can play with numbers.
…
I didn’t learn the rules of baseball by filling out a packet on baseball facts. Nobody held out a flash card where, in isolation, I recited someone else’s definition of the Infield Fly Rule. I didn’t memorize the rules of balls, strikes, and how to get someone out through a catechism of recitation.
Instead, I played baseball.
Conversational Math
The best way for children to build mathematical fluency is through conversation. For more ideas on discussion-based math, check out these posts:
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Bloggers: We Want You!
[Photo by Olga Berrios via flickr.]
Do you have a favorite blog post about math activities, games, lessons, or hands-on fun? The Math Teachers at Play blog carnival would love to feature your article!
We welcome math topics from preschool through the first year of 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:
Don’t procrastinate: The deadline for entries is this Friday. The carnival will be posted next week at Math Jokes 4 Mathy Folks blog.
How To Master Quadratic Equations
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:
and:
and (though it took a bit more thought):
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.
Introduction to the Quadratics Course
Life Lessons from James Tanton
This is Kitten’s favorite James Tanton quote so far, from the video we watched on Friday:
… And I am going to panic, because we got 16, but the problem doesn’t want 16. It wants 15. So I have two choices right now: Give up, and cry, and just go home.
Or use my common sense. What would I like this to be?
A good piece of advice: If you want something in life to work out the way you want it to work, just make it happen.
I want that to be 16. How can I make that happen? Just add one. Bingo! It becomes 16. However, if you make changes in your life, you’ve got to deal with the consequences …
— James Tanton
Quadratics 2: The Algebra of Quadratics
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