Hello again math folks! I've been in the middle of a major transition, moving between Asia and North America, so haven't really had time to post recently. Putting together this month's carnival was a nice opportunity to see what everyone else has been writing about and get some new ideas!

As you scan through the links I've highlighted, please don't get too fixated on the grade level splits. These are really approximate and I expect you will find worthwhile activities for all ages in every section.

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In Memoriam: Maryam Mirzakhani

On the 14th of July,

Maryam Mirzakhani passed away. She was the first woman to win the Fields Medal. It would be wonderful if you could do some exploration in her honor this month. One of her areas of research was on pool tables. Here are some places to get an idea of the way mathematicians have been inspired by this game:

If you find other kid-friendly projects related to Mirzakhani's work, please tell me in the comments!

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Some 110 facts

This was the best number carnival to be able to host, because:

- 110 = 10 * 11. That means it is
*pronic*, the product of two consecutive integers.
- 110 looks suspiciously like a binary number. Binary 110 = decimal 6. Decimal 110 = Binary 1101110, which I like to read as 110 1 110
- Because it has an odd number of 1s in its binary expansion, 110 is
*odious*
- 110 is a
*Harshad* number because it is divisible by the sum of its digits
- The element with atomic number 110 is Darmstadtium (Ds).
- 110 is the number of millions of dollars spent in March for a Basquiat painting, the highest amount paid at auction for a work by an American artist.

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A number talks picture that caught my eye

I'm not sure there is anything especially 110 about this picture, but there are a lot of mathematical questions to ask and things to observe here:

In a related vein, if you and your kids need some mesmerizing math gifs, take a look at

Symmetry.

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Elementary skills

Denise Gaskins has written a lot to help parents engage playfully and mathematically with their kids. In this blog post, she has collected highlights that are great with young students and worth remembering for older ones, too:

How to Talk Math with Your Kids.

I love board games and think there is still tremendous value in the physical games that electronic versions miss. Here's an example from Sasha Fradkin, where

*cleaning up after playing* gives us a chance to think about whether skip counting is just a chant or if the words mean something:

Skip counting or word skipping
While she's at it, Sasha Fradkin also has a

nice puzzle activity with Numicons. I would think of this as a progression step toward tangram and other dissection puzzles.

Which one doesn't belong is a math meme you should know already. If you don't, ask in the comments and I'll point you in the right direction. Christopher Danielson has recently introduced

Which Poster Doesn't Belong? While you are visiting his blog, enjoy his story about

The Three Year Old Who is Not a Monster.

Exploding Shapes is a catalyst for notice and wonder from The Math Forum. I really like this because here are many different directions to go and no single "right" answer. Also, let's give a cheer because it looks like this recent set of posts shows the math forum folks have returned to posting nice conversation starters.

Swine on a Line by Jim Propp is a nice game/puzzle that seems a great companion to James Tanton's

Exploding Dots. Hmm, maybe July 4th inspired me to look for lots of explosions...?

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Middle school(ish)

Rupesh Gesota starts with a nice puzzle and shows us how it was analyzed by several different students:

One Puzzle, Many Students, Many Approaches. I particularly like how the introduction to the puzzle encourages us to think of different methods.

Mike Lawler has done a huge number of really great explorations with his kids. Here are some recent projects with books from the Park City Mathematics Institute:

Playing Around. If you haven't been following Mike and his kids, I really encourage you to go through his past posts.This blog is fantastic for great projects and connections with other resources.

Curious Cheetah shows us several ways to

calculate square roots. I would say, like long division, the value isn't in memorizing the algorithms, but understanding how they work and using them to play with numbers.

Manan Shah has a couple of nice summer explorations. The first is an excursion into the digits of prime numbers:

Prime Numbers. The second is a coin flipping and gambling game to ponder during these warm vacation months:

Summer Excursion Coin Flipping.
There are other, problem-based, posts on Benjamin Leis's blog, but this one made me jealous of his recent purchase of the

A Decade of the Berkeley Math Circle.

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High school/more advanced

Thinking Inside the Box, Simon Gregg takes a new look at a familiar shape, the cube. His comment about the exploration really nicely captures something that is beautiful about mathematical exploration:

**"I came back to a familiar place from an unfamiliar starting place."**
A cute absolute value game now appears as a nicely animated game:

Absolute Value. I think this is a nice simplification and implementation of the

original game.

There's an improv game where the players have to switch between movie genres. Film noire or "hard boiled detective" comes up every time.

This TedEd video could introduce fractals and this film genre at the same time.

Michael Pershan puzzles over two measures of steepness in his trigonometry class:

When Measures of Steepness Disagree. I really like the questions he raises about how to use two different scales that measure the same concept, but are not linearly related.

Also, Michael links to the

New Zealand Avalanche Advisory, with a nice graphic showing a case where the greatest danger of avalanche is in the middle of a slope range: