Math Picture Book Post #6: Fika

For fans of arrays (and those with OCD), there’s much to like about Fika, the Ikea cookbook. Each recipe spans two pages: the ingredients on the first, the finished product on the second.

A sample:

Fika 1

Fika 2

My daughters and I have been talking skip counting, equal grouping, repeated addition, arrays, multiplication, etc. “How many? How do you know?”

We got in on the act:

Cookies 1

Cookies 2

Our “family recipe”

Math Picture Book Post #3: Miss Lina’s Ballerinas

Miss Lina’s Ballerinas by Grace Maccarone is about “teamwork, making new friends, and the pleasures of ballet.”

It’s also about math.

In my previous post, I wrote about multiplication in terms of groups of and arrays. Both models can be explored in Miss Lina’s Ballerinas. Eight ballerinas–Christina, Edwina, Sabrina, Justina, Katrina, Bettina, Marina, and Nina–dance in four groups of two

Miss Lina's Ballerinas Groups

and four lines of two¹.

Miss Lina's Ballerinas Array

What happens when a new girl, Regina, arrives? Spoiler alert: three rows of three. What if there were ten dancers? Eleven? Twelve?

If you are playing alongMiss Lina’s Ballerinas falls into my third category; the math concept is between the pages but the author did not intend to write a math concept book.

¹ This bugs me. Should it?

Kitchen Table Konversations

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.”

“Dad, that new song by The Sheepdogs sounds a lot like The Black Keys, don’t you think?”¹

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.

Once again, I have taken a page out of Christopher Danielson’s playbook with this post.

¹ I just learned that The Sheepdogs’ album was produced by The Black Keys’ Patrick Carney. Impressive kid, eh?

Multiple Multiples

“Can you show me another way?”

Multiple representations show students there is more than one correct way to do the math. This is an important message in itself.

Multiple representations also allow students to learn new mathematical concepts and procedures.

For example, division can be thought of as sharing or grouping.

8 ÷ 2 = 4 can be thought of as:

  • I have 8 items. I share them equally between 2 people. Each person gets 4 items.
  • I have 8 items. I put them in groups of 2. I can make 4 groups.

I prefer the adjective flexible over multiple. Adaptability of, not number of, is what is important.

To learn how to divide integers and fractions, students must be able to visualize both representations.

For example, -8 ÷ 2 can be thought of as sharing equally between 2 groups. Each group contains four negative counters. However, -8 cannot be put in groups of +2.

Alternatively, having a negative number of groups does not make sense. However, -8 can be put into groups of -2.

Think about why 6 ÷ ½ is 12 (without simply applying the invert and multiply rule). Having a fraction for the number of groups doesn’t make sense. However, students can explore how many halves there are in 6 using pattern blocks or number lines.

3 × 2 is more than simply 3 groups of 2.  An understanding of an area model of multiplication helps students to learn two-digit multiplication.

An understanding of this model will help students make connections between multiplying binomials and multiplying two-digit numbers.

As a secondary department head pointed out a meeting last year, teaching how to multiply binomials may be easier than teaching how to multiply two-digit numbers – in algebra, there isn’t the added complication of place value.

Now if only we would stop using the term FOIL…