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:

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:

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

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:

Some students will see four rows of seven doughnuts and know that 4 â¨‰ 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 â¨‰ 7 = 4 â¨‰ (5 + 2) = (4 â¨‰ 5) + (4 â¨‰ 2) = 20 + 8 = 28. Or, they might double a double: 4 â¨‰ 7 = (2 â¨‰ 2) â¨‰ 7 = 2 â¨‰ (2 â¨‰ 7) = 2 â¨‰ 14 = 28. They might do both. They might even break a factor into more than two addends: 4 â¨‰ 7 = 4 â¨‰ (3 + 3 + 1) = (4 â¨‰ 3) + (4 â¨‰ 3) + (4 â¨‰ 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:

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

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

Breaking apart both factors by place value is a common approach: 25 â¨‰ 32 = (20 + 5) â¨‰ (30 + 2) = (20 â¨‰ 30) + (20 â¨‰ 2) + (5 â¨‰ 30) + (5 â¨‰ 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 â¨‰ 32 = 25 â¨‰ (30 + 2) = (25 â¨‰ 30) + (25 â¨‰ 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 â¨‰ 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 â€“ 40 leads students to think about adjusting 39 in 52 â€“ 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.

A few weeks ago, I took my daughter to the mall. Later, she complained that “Dad spent half the time taking math photos!” Five of one hundred twenty minutes is not half!Â¹

One of those photos:

I thought that this would make a great “Would You Ratherâ€¦?” math task. I considered a few approaches. My preference is probably to just display the offer and have students make up their own prices and riff on “What ifâ€¦?” That might be a tall order. I created a few combinations. (More on these in a sec.) But I wanted something more open.

The idea is that students would mix & match specific combinations of board games to justify their decisions.

For example, consider Carcassonne ($43) and Blokus ($40). With “buy one, get a second 25% off” the discount is $10 (25% of $40). Add Othello ($35) and with “buy two, get a third 50% off” the discount is $17.50 (50% of $35). It looks like the second option is the clear winner. But if we think about the (total) percent discounts, we get about 12% ($10/$83) and 15% ($17.50/$118), respectively. Proportionally, the gap shrinks.

What if we replace Othello above with Spot it! ($20)? Again, the discount is $10 (50% of $20). But it’s not a tie. Saving $10 on $83 is better than saving $10 on $103 (about 12% vs. 10%).

There are a couple of combinations where we can’t justify the second option. For example, consider Catan ($63) and Pandemic ($60). With “buy one, get a second 25% off” the discount is $15. Add Rock Paper Scissors ($6) and with “buy two, get a third 50% off” the discount sinks to $3.

Beyond making and justifying a decision using mathematics, I’d push students to generalize: When would you ratherâ€¦?

A couple more photos from the mall:

“Dad, stop taking photos of arrays! Are these like the paint splatter thing?” Yep. Partially covered arrays in the wild. Lack of fraction sense aside, it’s nice to know that she’s paying attention. And making connections.

Â¹BTW, I use Microsoft Office Lens to quickly crop, clean up, and colour these photos on the fly. An essential app for teachers using vertical non-permanent surfaces (#VNPS on twitter). Check it out.

March 2023

A few WYRâ€¦? prompts involving discounts (and more):

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.

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 TalkÂ by Elham Kazemi and Allison Hintz.) Instead, I designed the activity above toÂ (gently) shove students towards looking for and making use of arrays.

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

Students can still see each dot and count all by ones or twos. But a more efficient strategy is to see 3Â Ã— 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:

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

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

How confident are you?

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.

For fans of arrays (andÂ 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:

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