Keira, Grade 4, asked me to show her “the nines trick” one morning last week before school.

If you don’t know it, watch Jaime Escalante/Edward James Olmos:

I did not show my daughter this trick. I am not the Finger Man. It’s like she doesn’t even know me!

Instead, we had a quick conversation. No time for manipulatives. Five minutes to brush her hair and pack her lunch before we had to hop in the car.

Me: You remember what a ten-frame looks like?

Keira: Yeah. Ten dots. Five and five. Array!

Me: Ok, what about nine? What does it look like?

Keira: One missing.

Me: What if there were two nines? How many?

Keira: Don’t ask me that one. I already know it’s eighteen.

Me: Ha! Ok, what about seven times nine?

Keira: I knew that you were going to ask me that one!

Me: What if you had seven ten-frames, each with nine dots? How many dots altogether?

Keira: Sixty… three?

Me: Why?

Keira: You start with seventy but you take seven away.

We did a few more together. Success!

Then she asked me to show her the nines trick.

For the purpose of this post, I quickly put together this slide (and video):

In the car, Keira asked me “Can you multiply decimals? Like seven times nine point five?” This reminded me of “I’m wondering if fractions only work with circles” from Annie Fetter’s (@MFAnnie) #NoticeWonderIgnite talk. (We showed it at a workshop the night before.) This also reminded me of what I take for granted. Her sister and I did some explaining, but I’m wondering about a better (?) approach:

How many do you see? How do you see them?

(Not my normal approach to multiplying decimals — the photo below probably had something to do with that.)

I’m not usually a fan of equations in math picture books. But I like 100 Snowmen by Jennifer Arena and Stephen Gilpin. On each page, students can use the mental math strategy of adding one to a double to determine basic addition facts to 19. Each number is represented as both a number to be doubled and one more than a number to be doubled. Take five. Here, students can double five and add one more to determine five plus six.

5 + 6 = (5 + 5) + 1 = 11

Here, five is not doubled, but one more than four, which is doubled.

5 + 4 = (4 + 4) + 1 = 9

Dot cards can be used to draw attention to the doubles plus one strategy. Ask “How many do you see? How do you see them?”

There are things my daughters say that make me feel proud to be their dad. From my 7-year-old:

“I have a lot of stuff. For my birthday party, can I ask each of my friends for a toonie instead of a present? I’m going to give the money to the SPCA.”

There are also things my daughters say that make me feel proud to be their mathteacherdad.

One day this week, we were talking math at the dinner table.

Being in Grade 2, Gwyneth is not yet learning about multiplication at school. However, her best friend knows about “timesing,” so she is curious and motivated. We’ve been discussing multiplication in terms of groups of. Don’t worry, we’ll have conversations about arrays later. Dropping in, mid-conversation:

Me: What do you notice? Gwyneth: Two groups of three is the same as three groups of two.

At this point, I could have said, “That’s right. Changing the order doesn’t change the answer.” I didn’t. Being a math teacher and her dad, I also could have said, “That’s because multiplication is commutative, Sweetie.” I didn’t.

Me: What about three times five and five times three?
Gwyneth: Three groups of five is … fifteen.
Me: How do you know?
Gwyneth: Well, two groups of five is ten and one more group makes fifteen.
Me: Okay, so what about five times three?

What she said next, after a brief pause, blew me away.

Gwyneth: Nine and six make fifteen.
Me: How did you get that?
Gwyneth: I took one away from six to make ten and …
Me: No, I get that. I mean where did the nine and the six come from?
Gwyneth: Well, three groups of three is nine and two groups of three is six.

I was asking my daughter questions to have her explore the commutative property and she drops the distributive property into our conversation! Any English teachers still reading this blog after my last post may question my use of an exclamation mark. Math teachers will not. Gwyneth understands, conceptually, that 5 × 3 = (3 × 3) + (2 × 3).

I asked her to draw this for me. She drew five groups of three dots.

Gwyneth: Three, six, nine, twelve, fifteen.
Me: Wait! What about the nine and the six?
Gwyneth: I said those. Three, SIX, NINE.
Me: Yeah, I heard you. But, before, you ADDED the six and the nine.
Gwyneth: Dad, I’ve got LOTS of strategies.

I was so proud to hear her say this that I didn’t even mind the eye-rolling.

In his book The Joy of x, Steven Strogatz writes about the counterintutiveness of the commutative law.

Whereas it is intuitive to Gwyneth that adding five to three should be the same as adding three to five, it is not intuitive to her that having three groups of five should be the same as having five groups of three.

Why is 5 + 5 + 5 …

… obviously the same as 3 + 3 + 3 + 3 + 3?

Strogatz makes the point that if we visualize 3 × 5 as a rectangular array with 3 rows and 5 columns …

and turn this picture on its side giving us 5 rows and 3 columns, or 5 × 3, …

then 3 × 5 must equal 5 × 3. The commutative law becomes more intuitive.

Strogatz, a frequent guest on Radiolab, goes on to give examples of real-world situations in which people forget, or refuse to accept, the commutative law.

My background is in secondary, but I have spent the majority of the past two years in elementary. This blog hasn’t always reflected that shift. This year, I plan to blog more about my experiences teaching math in K-7.

Often, I use picture books to launch math lessons. Picture books allow teachers to leverage literature-based methodologies. The plan is to make this a series of posts.

I classify math picture books into three categories:

mathematics is explained

mathematics is weaved into the storyline

mathematics is hidden

Books in the first category are, by and large, horrible. The reader is told that learning a particular mathematical concept is important and this concept is explained. Sometimes, art imitates life and a teacher-like character explains a topic to student-like characters. That’s just cheating.

There are some great picture books in the second category. In these books, math (not the characters’ learning about math) is central to the story. For example, in Bean Thirteen by Matthew McElligott, divisibility is introduced when the characters don’t want to get stuck with the unlucky thirteenth bean. In If a Chicken Stayed for Supper by Carrie Weston,part-part-whole relationships are explored when each fox counts the others and concludes someone is missing. Often, these books provide more questions than answers.

Books in the third category are the most difficult (and most rewarding– think #anyqs) to find. In these books, the author did not set out to write a math book. You won’t find these books in the math section of your local independent bookstore. But the math is there if the reader looks at the story through a mathematical lens. (More on this later.)

This week’s math picture book is Cats’ Night Out by Caroline Stutson. I’d place it in the second category. It’s a counting book and that might stretch your idea of ‘storyline’. (That’s fine.) Counting by twos from two to twenty, each page is illustrated with cats dancing in the city. Here are the pages for eighteen:

How did you see 18? I first saw 9 on each page (5 and 3 and 1). Students could draw their own pictures of doubles on folded paper. Also, on the two pages there are 9 white cats and 9 black cats. Kids will find two 9s in other places. There are 9 cats with bows and 9 cats without. Doubles can also be seen in rows across the pages. For example, double 5 can be seen across the bottom row. The use of doubles is a strategy for mastering addition (and multiplication) facts.

These 10 cats can be seen in another way. There are 6 white cats and 4 black cats across the bottom row. Students could be asked to find ways of making a different number of cats or different pages could be copied and students could look for different part-part-whole relationships. This, too, helps students master addition facts. For example, 9 + 3 can be thought of as 9 and 1 makes 10 and 2 more is 12; 6 + 7 can be thought of as double 6 makes 12 and 1 more is 13.

A recent “Shark Tank” episode featured two entrepeneurs pitching MiX Bikini, the world’s first interchangeable swimsuit. Here’s a sneak peek:

Two things piqued my interest.

Thing One: The Product

“It’s no secret women love to stand out, but there is nothing worse for [a] woman than being at the beach and seeing another girl in the same bikini,” one partner says.

Nothing? Really?

Here’s how it works:

First, assuming [a] woman is not offended by the claim above, she selects a style of bikini top (halter or triangle). Next, she chooses one of 40 colours/patterns for the bikini top. She does this twice (right and left). She then selects a style of bikini bottom (classic or ‘scrunchie’) and picks out one of 33 colours/patterns. (In the “Shark Tank” video, the second model switches out the back bottom. On the Mix Bikini website, the front & back of the bikini bottoms always match.) The bikini tops must be connected. Customers must choose between rings or strings. Rings are available in 10 colours, strings in 9. Of course, bikini tops also need neck strings (right and left). Double neck strings come in 9 colours, rings & strings in 10.

This begs the question… How many Frankenkinis (sp?) are possible?

The website advertises it is possible to create thousands of bikinis.

Thousands? Try millions.

What number do you get? What assumptions do you make? Is fuschia & leopard print different than leopard print & fuschia? I maintain it is. It is best that I not elaborate.

Thing Two: The Pitch

“We are seeking fifty thousand dollars in exchange for five percent of our business,” says the first partner.

“That means that you’re saying the company is valued at one million dollars,” says Daymond, one of the Sharks.

“It was ten percent we were asking,” interrupts the second partner.

“So half a million dollars,” Daymond clarifies.

Uh-oh. The budding businessmen are confused. Mathematically disoriented. The Sharks smell blood. SPOILER ALERT– all does not end well. How did this happen? What went wrong?

My guess? The Sharks have number sense. They have mental math strategies. Daymond understands 5% is equal to 1/20. Therefore, if 1/20th of the business is valued at $50 000, then the total value of the company can be calculated by multiplying by 20 (or, more likely, by doubling and multiplying by 10). If $50 000 is 10%, or 1/10th, of the company, then the Sharks can multiply $50 000 by 10 (or, more likely, halve $1 000 000, the original evaluation).

In the “Shark Tank”, the Sharks often counter with benchmark percentages– 5%, 10%, 25%, 50%, 75%. I suspect the Sharks have strategies for other popular percentages (eg, for 40% they may halve, halve, and multiply by 10).

Our pitchmen, on the other hand, do not have number sense. They do not have mental math strategies. The bikini guys have procedures. The bikini guys have this:

April 22, 2013: Uh, just to be clear… the first part (top half?) of this post is about two things: (1) large numbers can be counterintuitive, and (2) me starting to see that math is everywhere. It is not a lesson. Because images of half-clothed women, however engaging to students, do not belong in math classrooms. That should be obvious, right?

Uh-oh.

BTW, if you’re looking for a lesson on combinations, check out Pair-alysis from Mathalicious.