Gotta be 3 & 10! Or 4 &9. Balance and rotational symmetry.

Back in September, I shared Howie’s tweet with my daughters and am relieved to report that they, too, answered correctly. This is an ongoing thing with us. Whether eggs or cookies, what’s left should either (a) represent a pattern or (b) illustrate a mathematical concept. It’s these mathematical concepts that inform how I create or select a number talk image. There’s a purpose for each image.

Consider the following arrangement of macarons…

You might see six groups of six, each group its own flavour (left to right: crème brûlée, dulce de leche, pistachio, red velvet, chocolate & mandarin, chocolate). Or you might see six rows and six columns — an array. If I had 18 macarons left, I could place them in three rows of six or six rows of three, demonstrating the commutative property of multiplication…

You might see the remaining macarons not (only) as 18 but (also) as one-half. (See this tweet — similar to Howie’s — for one-half of a carton of eggs left.) Expanding from left to right or top to bottom introduces equivalent fractions: 3/6, 6/12, 9/18, … 18/36. Every second row or every second column also gets you one-half. So, too, does every second macaron, whether looking across rows or down columns…

This arrangement maintains the balance and rotational symmetry of my two-eggs-left choice above. (See Simon Gregg’s symmetrical eggs tweet.) There are many interesting ways of seeing eighteen here, including 2(1 +3 + 5) on the diagonals. If a particular strategy does not emerge from the class, I often “go backwards” (e.g., “I see 2(5 + 4). How do I see them?”).

If I had thirty macarons left, I’d remove one column or row, which introduces the distributive property of multiplication…

The number of macarons in these two photos can be expressed as 6(4 + 1) and (3 + 2)6, respectively.

Sticking with thirty, removing a diagonal can bring to mind part-whole relationships as well as the associative property of addition…

Here, five is composed of zero and five, one and four, two and three, and so on. Two plus three (pistachio) is equal to three plus two (red velvet).

If I had 20 macarons left, I could choose to emphasize multiplication as equal groups — a quincunx of squares (5 × 4) or a square of quincunxes (4 × 5) — and ask What is the same? What’s different?

Notice that if you still see rows and columns rather than groups, then you might count four rows/columns of four and two rows/columns of two. This can be expressed as 4 × 4 + 2 × 2, which calls on order of operations. So, too, does 2(4 + 4 + 2) if you take advantage of the line symmetry in each of these two arrangements.

Twenty-one macarons form a “staircase”…

Moving some of the macarons makes a “near array”: pairing chocolate with pistachio and mandarin-chocolate with red velvet produces 5 × 4 + 1 (or 6 + 5 × 3).

The Number talk images (aka “quick images” or “dot cards“) instructional routine continues to be one of my favourites. For teachers facing the challenge of facilitating this routine remotely, there are a few, albeit flawed, solutions within Microsoft Teams (SurreySchools’ supported platform).

I’m with Jonathan. In Desmos, it’s dead easy to create an activity in which students can mark up an image to show how they see a quantity, enter a number or expression to answer how many, and type within a text box to explain their mathematical reasoning. Also, the Teacher Dashboard allows teachers to take and present snapshots of students’ ideas to share and discuss with the whole class. See my sample Desmos activity. It’s intended to be a template, not a single never-ending number talk. Copy and paste screens as need be. The images above — and a few more food favourites — are also included in the slide deck below.

Monster Mash(-up)

The blog is going to be a 2013 version of itself for Halloween…

Act 1

Any questions?

  • How many different monsters can you make?

Here, a monster is made up of three cards: head, torso, and legs. In Bears vs Babies, a monster can be just a head or a head with one to four body parts. I’ve simplified the task to get at the fundamental counting principle.

Act 2

What information would be helpful to have here?

  • How many head, torso, and legs cards are there?

Students may want to act this out. Give them these cards. Encourage them to find a systematic way of counting the possibilities. How can the number of monsters be determined from the number of head, torso, and legs cards? Start with heads and torsos, if need be.

Act 3

The reveal…

Introduce tree diagrams. Connect this representation to students’ strategies. These might help:


  • You have about 50 monsters in your hand. How many head, torso, and legs cards might you have?

Krispy Kreme: Connecting Strategies and Models

Earlier this year, I wanted to share student work on Graham Fletcher’s Krispy Kreme three-act task with a group of intermediate teachers. When I last facilitated this task, many students thought of multiplication as repeated addition (only). Others used the standard algorithm — few successfully. At that time, analyzing student work revealed what students really understood (or didn’t). Further, the teacher and I discussed implications on practice going forward. (This prompted my last post.) But with my group of teachers I wanted to talk partial product strategies and models and these samples weren’t helpful. So Marc and I faked it and created some possible approaches:

What connections can you make between these students’ strategies?

I’m using approaches to include and differentiate strategies and models. Pam Harris defines strategies as “how you mess with the numbers” and models as how you represent your strategy. For example, I might use an open number line to model my adding up strategy for 2018 − 1984. The same adding up strategy can be represented with a different model (e.g., equation). The same open number line can represent a different strategy (e.g., keeping a constant difference).

We shared the approaches with the group and after some noticing and wondering invited them to find as many connections as they could. Some intended connections:

  • Students 1 & 5 thought of multiplication as repeated addition
  • Students 2 & 4 & 7 think place value to decompose 32 into two (or more) addends
  • Student 2 “splits” 32 symbolically; Student 7 partitions an open array
  • The partial products in Student 3’s algorithm can be seen in Student 4’s open array
  • Students 1 & 8 make use of the fact that four 25s make 100
  • Students 4 & 8 make use of halves and doubles

Teachers then discussed the placement of these approaches within a learning progression and how they might “nudge” each student.

Analyzing student work has become my favourite professional development activity. Here, what is lost in terms of authenticity is gained in terms of diversity of thinking. Still, I was excited to see this from @misskwiatkaski5‘s real students:

Krispy Kreme: Partial Products

How many doughnuts are in the box?

This Krispy Kreme three-act task above–from Graham Fletcher or YummyMath–cries out for partial products.

How would you partition the open array?

But more than once, the partial product strategies and models that I anticipated did not emerge. Not even close. 5 Practices-induced flop sweats. More on that in a future post. First, a progression of partial products across the grades, beginning with the basic multiplication facts:

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

Some students will see four rows of seven doughnuts and know that 4 ⨉ 7 = 28. Great. For students who haven’t yet mastered the basic multiplication facts, partial products are helpful. Have students use what they know. For example, they might break apart seven as five and two and then find the sum of two familiar products: 4 ⨉ 7 = 4 ⨉ (5 + 2) = (4 ⨉ 5) + (4 ⨉ 2) = 20 + 8 = 28. Or, they might double a double: 4 ⨉ 7 = (2 ⨉ 2) ⨉ 7 = 2 ⨉ (2 ⨉ 7) = 2 ⨉ 14 = 28. They might do both. They might even break a factor into more than two addends: 4 ⨉ 7 = 4 ⨉ (3 + 3 + 1) = (4 ⨉ 3) + (4 ⨉ 3) + (4 ⨉ 1) = 12 + 12 + 4 = 28. (Admittedly not the most useful relationship to help students derive this fact.) Mastery of the basic multiplication facts aside, playing with partial products–and open arrays–reinforces the big idea that numbers can be broken apart–or decomposed–in flexible ways to make calculations easier.

This idea extends to multiplying two-digit numbers by one-digit numbers:

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

Some students will understand that breaking apart by place value makes calculations easier: 5 ⨉ 12 = 5 ⨉ (10 + 2) = (5 ⨉ 10) + (5 ⨉ 2) = 50 + 10 = 60. Others might use doubles and double-doubles. Note that a factor can be broken into addends or smaller factors: 5 ⨉ 12 = 5(3 + 3 + 3 + 3) or 5 ⨉ 12 = 5(3 ⨉ 4). How students choose to express this will provide insight into their thinking.

Again, decomposing numbers in flexible ways extends to larger numbers:

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

Breaking apart both factors by place value is a common approach: 25 ⨉ 32 = (20 + 5) ⨉ (30 + 2) = (20 ⨉ 30) + (20 ⨉ 2) + (5 ⨉ 30) + (5 ⨉ 2) = 600 + 40 + 150 + 10 = 800. This approach might be too common if reduced to a procedure (i.e., the box method or FOIL). Again, it’s about flexible ways. Breaking apart just one factor by place value is an efficient mental math strategy: 25 ⨉ 32 = 25 ⨉ (30 + 2) = (25 ⨉ 30) + (25 ⨉ 2) = 750 + 50 = 800. A student who inefficiently decomposes 32 as 10 + 10 + 10 + 2 could be nudged towards 32 as 30 + 2. Or, a factor of 25 might spark thinking about 25 ⨉ 4 = 100, a familiar product.

The different varieties of doughnuts illustrate some helpful ways of partitioning the arrays. But each of these slides draws attention to a specific way of seeing the array. My preference would be to show the slides where all the doughnuts are the same. (Same goes for visual patterns.) Ask students how they see them. If students do not see a helpful way of partitioning the arrays, then corresponding slides with different varieties of doughnuts could be displayed. In a number string, 52 – 40 leads students to think about adjusting 39 in 52 – 39 to make the calculation easier. Similarly, a purposely crafted string of images could lead students to see fives, doubles, or place value–all useful relationships–in an original (glazed) array.

Related: The Math Learning Center’s Partial Product Finder

I’m Not The Finger Man

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.

james corden face palm GIF by The Late Late Show with James Corden

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 #NoticeWonder Ignite 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.)

Paint Splatter Arrays

This isn’t Splat!

Paint Splatter Arrays (.key)
Paint Splatter Arrays (.pdf)

In Steve Wyborney’s Splat!, the total number of dots is given and the number of dots under each splat is unknown. In my Paint Splatter Arrays, the total number of dots is unknown. My paint splatters do cover some dots but how many is beside the point. Also, Steve’s dots are scattered; mine are arranged in arrays. (More on that below.) Steve’s splats splat. My splatters are there from the get-go. See? Not the same.

h/t Andrew Stadel

Here’s why I created this activity…

Number Talks -- Dot Cards

T: “How many do you see?”

S: “Twenty-five.”

T: “How do you see them?”

S: “Two, four, six, …”

Every. Single. Time.

Not all students. Most students do see and use groups or arrays to figure out how many. Those strategies are described in this post. But some students don’t seem to make sense of others’ ideas. That’s a greater challenge than I’ll tackle here. (Recommended: Intentional Talk by Elham Kazemi and Allison Hintz.) Instead, I designed the activity above to (gently) shove students towards looking for and making use of arrays.

The first three are softballs. For example, the second:

Paint Splatter Arrays.004

Students can still see each dot and count all by ones or twos. But a more efficient strategy is to see 3 × 5 (3 rows, 5 columns).

The next several slides completely cover at least one dot, so students can’t count all by counting what they can see. In each, at least one complete row and one complete column is visible. For example:

Paint Splatter Arrays.010
4 × 6

Paint Splatter Arrays.028
6 × 6

I had some fun with the last two. In the next-to-last one, the middle column is completely concealed.

Paint Splatter Arrays.030
3 × 7

In the last one, most of the dots are hidden. A bit of estimation. How many?

Paint Splatter Arrays.032

How confident are you?

Paint Splatter Arrays.033
<shrinks> 6 × 8

What about now?

I test-drove these on my daughters. (Keira likes Booger Math! over Paint Splatter Arrays, by the way. It is catchier.) I’m looking forward to trying this out in Surrey classrooms. Feedback welcome!

And mine goes ding ding ding di di ding ding DING ding ding ding di di ding ding.

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…