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## Math Goggles #11 – Just Listen

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All the Math Goggles challenges so far had to do with noticing math with your eyes. But for this week’s challenge, let’s try to just listen.

Let’s listen to the math in what our children talk about. I don’t mean like when we ask them what they did today in preschool, kindergarten or school. And I don’t mean like when we quiz them on how many teddy bears are in the room or what shape is the kitchen table.

Let’s listen to the math children bring up on their own. Our contributor, Malke Rosenfeld of Math in Your Feet, frequently describes such math chats on her blog. Here’s an example from her recent post:

Seven-year-old is pushing cart around the store, narrating as she goes: “Go forward, now one quarter turn to the right, now go forward, parallel park.  Okay, now turn half way around, go straight, one quarter turn…”

Here’s my six-year-old who is waiting impatiently for his first baby tooth to fall out, but it seems it won’t ever happen:

Mama, I have a tiny hope, and it’s quickly approaching zero, that this tooth will fall out soon.

Or David Wees’s “Decomposing Fractions” post, in which he retells a conversation with his son:

Daddy, I’m full. I had 1 and a half…no, one and a quarter slices of pizza which is the same as five quarters of pizza,” said my son at dinner tonight…

By the way, David’s whole project, Math Thinking, is about children sharing their mathematical thoughts.

So this week, let’s just listen. You might be surprised at how your child looks at things, at math ideas she explores on her own, and at mathematical reasoning behind what she says.

You may also share your observations here on the blog.

~

## What Would You Rather Have – Commutative Property Game

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Is 2*3 different from 3*2? My answer used to be “But of course! Don’t you know the commutative property?” Now, after following Malke Rosenfeld’s exploration of multiplication, I answer it with a lot more non-committal “it depends“. And I notice more and more examples when even though quantitatively, 3*2=2*3 and 8*1=1*8 and 3*5=5*3, qualitatively you sometimes get two distinctly different results.

Ironically, as I’m moving from quantity toward quality, my 6-year old is moving in the opposite direction. Consider these two examples:

Remember Maria’s review of Clap, Drum and Shake It by Marcia Daft? In particular, this part:

Do more multiplication. In particular, invite kids to multiply within pattern units. For example, how do you double the pattern unit “clap, clap, shake”? That is, how do you show 3×2 in the language of the book? “Clap, clap, shake; clap, clap, shake” is what the book does. You can also do “clap, clap, clap, clap, shake, shake”!

Two months ago I tried it with my son using 3 colors of PostIt notes instead of printed cards. He concentrated on the qualitative aspect (the patterns) rather than on the quantitative side (6 elements total) and insisted that the two strings had nothing in common.

Me: “But you said this one had 6 PostIts and the other one had 6 PostIts”.

Him: “But these are not the same 6s, Mom.”

Yesterday we finished listening to “Chitty Chitty Bang Bang” audio book. After giving the story some thought, my son asked me: “Do you know what it’d be if there were two Chitty Chittys?” He then explained that it’d be “Chitty Chitty Chitty Chitty Bang Bang Bang Bang”. He then went on to tell me what three Chitty Chittys would be like. You guessed it: “Chitty Chitty Chitty Chitty Chitty Chitty Bang Bang Bang Bang Bang Bang”.

Me: Can two also be “Chitty Chitty Bang Bang Chitty Chitty Bang Bang?”

Him:”Sure because you know, Mom, it’s the same thing”.

Which reminds me of a story we recently re-read. It was a chapter from the Karlson on the Roof by Astrid Lindgren. In it, a little boy says that he will have one birthday cake with eight candles on it (1*8); to which his friend adds that it would be a whole lot better to have eight cakes with one candle (8*1).

It also reminds me of a video in which Malke’s daughter is sharing her perspective on the quality vs. quantity issue.

What do you think? Share your examples of when you’d rather have a*b than b*a.

## Math Goggles #10 – Bagel Math

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This week’s Math Goggles challenge is a perfect excuse to go out and buy a dozen of freshly baked bagels. At least it was for me. (If you are not sure what is a Math Goggles challenge, read about it here). So the question is what do you get when you slice a bagel.

The answer depends on how you slice it. That’s one reason you need several bagels. Begin by cutting the first bagel horizontally. What will the cross section look like when you finish slicing?  Easy, right?

Next bagel! Instead of making a horizontal cut, slice this one in half vertically. Can you predict what the cross section will look like this time? Still easy-peasy! (And now you know the correct answer next time you see this little puzzle floating around on Facebook)

Time for bagel #3, inspired by James Tanton. Can you slice it in such a way (different from the first two) that the cross section will show two circles? Hint: choose your best-looking (most torus-like) bagel for this one.

The forth bagel can be cut into a trefoil knot which is super easy to do, but you sacrifice the ability to toast your bagel before putting cream cheese on, or rather in, it.

Four down, eight more to go. Time for the big one, George Hart’s interlocking bagel halves puzzle. I made a mistake of putting half my bagels into the freezer, so by the time I attempted this problem, I only had 2 bagels left. And I didn’t want to draw on any of them with a Sharpie. Let’s just say that I learned a few valuable lessons in the process and ended up with a failed, but perfectly edible experiment.

So, sharpen your knife, get out a tub of cream cheese, and keep your eyes open for math!

## Math Goggles #9 – Watch a Cartoon

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This week’s Math Goggles challenge is very simple – watch one of your child’s favorite cartoons and look for math in it. If it seems interesting, intriguing, strange, weird, and worth investigating, look further into it. Yep, that’s it! Here’s how it worked out for me.

My child’s most favorite cartoon of the moment is called Lego: The Adventures of Clutch Powers. I like it, but not enough to sit through all 88 minutes of it every other day. Still, I overhear snippets of dialogue as I go about doing my own things. One of the snippets that caught my attention was a trick question from the Troll Warrior to Clutch Powers:

Troll Warrior: Using nothing but standard eight-stud bricks, how many bricks would it take to build a spiral staircase as tall as three minifigs?

Clutch Powers: There are more than 915 million ways to combine standard LEGO bricks. But they will never connect on the diagonal.

Really? I mean, I understand the impossibility of connecting on the diagonal (had to explain it to my son a few times before), but 915 million combinations?! This sounds pretty amazing.

Turns out, the story is even more amazing than it sounds. The 915 million figure, or more precisely, 915 103 765, is the number of unique ways to combine just six two-by-four studded LEGO bricks of the same color. If this sounds unbelievable, check out the explanation of the LEGO Counting Problem. It is a great read and has just one formula that’s pretty easy to understand.

One of the best parts of the paper is in the “Why is this interesting?” section:

Such challenges are always important to drive mathematical research, and it often happens that methods developed to study a problem with no practical applications (like this one) are useful to study problems which do have an impact on everyday life.

Back to the original question of building a spiral staircase out of LEGOs… This is when the second part of my challenge started. And let me tell you, I totally cheated by looking at… my 6-year old’s recent builds.

He recently built a 3D treasure map using some unit blocks and LEGO bricks, complete with the “X marks the spot” signs which he made out of LEGOs, of course. That’s how I knew that, under some conditions, regular LEGO studs and plates can intersect at angles other than 90 degrees. So I used this idea and built a very basic spiral staircase in no time although it does not follow the  ”use nothing but two-by-four studs” rule.

Now it is your turn to find math in your child’s favorite cartoon!

## Math Goggles #8 – Make a Puzzle

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This week’s challenge is a bit late, but better late than never. And the reason it’s late is because I was too busy playing  exploring math with my son’s Legos. Anyway, are you ready for this week’s Math Goggles Challenge? If you’re not sure what I’m talking about, then here’s a helpful “what’s this all about” post.

This week, let’s work on some math and logic puzzles. If you do not like or avoid such puzzles because they make you feel anxious, nervous, stressed, harassed, tricked, lost, confused, insecure or otherwise remind you of a pop-quiz the day after you didn’t do your homework, relax. It’s not going to be like this. In fact, this challenge is not about solving puzzles (but if you do, that’s perfectly fine).

Here’s what to do this week. Find a math and logic puzzle that you’ve not seen or solved before. Now, build it with whatever it is you have handy – cardboard, wrapping paper and glue; modeling clay; marshmallows and toothpicks; building blocks. You might like the challenge of recreating a pen-and-paper puzzle with 3-dimensional objects. Or you might like the idea of taking a 3D puzzle and drawing it.

I got the idea for this week’s Challenge from MathFour’s 5-Room Puzzle post. I thought it’d be interesting to turn this puzzle into a little Lego adventure for my child. And so I sat down to build it. Admittedly, I didn’t do a very good job copying the puzzle exactly. But here’s what did happen. As I was building the puzzle, snapping Legos together, it occurred to me how I could check whether the puzzle I built had a solution. And that was a huge “AHA!” moment, I tell you and it felt great too!

So there you go. Find a puzzle that looks interesting, build it and concentrate on the process of building instead of on solving it. Enjoy!

## Math Goggles #7 – It is News to Me

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This week’s Math Goggles Challenge is inspired by Dr. Keith Devlin and his Introduction to Mathematical Thinking course on Coursera.org. By the way, if you are not sure what Math Goggles are and why wear them, here’s a good intro into the challenge.

Back to Dr. Devlin and this week’s challenge. In his very first lecture in the Intro to Math Thinking course, Dr. Devlin draws attention to the news headlines or, more specifically, to the rather common misuse of language in them. We, the readers, generally do not notice such misuse, unless it’s either a really glaring one or we are acting nitpicky. After all, we all know what the headline “Stolen Painting Found by Tree” means.

But in math precision is extremely important. Which explains, in large part, the time we took to craft definitions of the terms used in the Moebius Noodles book. Mathematicians have to be very literal. So this week, let’s look for ambiguous headlines. If you are pressed for time, you can always enter “ambiguous headlines” in the search bar of whatever Internet browser you use.

Otherwise, check out your daily paper or online news sites. Here are a few I spotted today:

Once you find the headlines, see if you can figure out a way to fix them to remove the unintended meaning. Don’t forget to share your finds with us.

## Math Goggles #6 – Let’s Read Math Story

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March is a great month for math holidays. First Wednesday of the month is the World Maths Day. Right after it comes the World Science Day. Then, of course, there’s the Pi Day. But first, at least this year, comes the Read Across America Day. So this week’s Math Goggles challenge is to pick up and read a fun math book. If you are not sure what Math Goggles Challenges are all about, check here first. If you are not sure how a math book can be fun, read on.

Math books can be boring, but so can non-math books. But just like non-math books can be fascinating, interesting, intriguing, beautiful, witty, uproariously funny, thoughtful, and in general unputdownable, so can many math books. The key is to find the book that speaks to you. Check out the LivingMath.net’s huge reading list, ask a librarian, do an online search or see what comes up on Amazon when you type in “math book” or “book about math” or search for a particular math term in their Books section.

Don’t limit yourself in your search. Don’t limit just to what you already know! In fact, there are some terrific books to be found if you search for “don’t know about math”. Most importantly, do not restrict your search to books for grown-ups. Why should you? Some of our most favorite and beloved books are the children’s books.

So what are you waiting for? This week, find your own great math book, enjoy it, and share with us! I’ll be reading Theoni Pappas “Fractals, Googols and Other Mathematical Tales“. And what about you?