Two-Legged, Four-Legged, Winged, Finned: Patterns from Indigenous Art

Back when we were all together, I’d often stop on my way in or out of DEC to play with the 3-D printed First Nation shapes on display. These manipulatives were a collaboration between Nadine McSpadden (Aboriginal Helping Teacher), Eric Bankes (ADST Helping Teacher), and the Bothwell Elementary community (Bea Sayson, Principal). Like others who passed by, I just had to rearrange them to create repeating patterns or symmetric designs.

Photo: Nadine McSpadden

Before having students explore mathematics using these materials, it’s important to first teach the cultural significance of Indigenous works of art. In Surrey, we work and learn on the unceded shared territories of the Coast Salish. We acknowledge the Katzie, Semiahmoo, and Kwantlen First Nations who have been stewards of this land since time immemorial. 

Students should understand that, although there are similarities, not all First Nation art is the same. Both Coast Salish and Northwest Coast art reflect a worldview of connection to the land and environment. There are differences in design: Coast Salish artists use three geometric elements — the circle (or oval), crescent, and trigon — whereas Northwest Coast artists use formline — the ovoid and U-shape. The use of circles, crescents, and trigons is unique to the Coast Salish! These elements suggest movement and make use of positive and negative space. In his video covering Coast Salish design, Shaun Peterson invites viewers to “imagine a calm body of water enclosed by two borders and dropping a pebble in to create ripples that carry the elements away from the centre.” Just as there is diversity within both Coast Salish and Northwest Coast peoples, there is diversity within both Coast Salish and Northwest Coast art (e.g., compare the Northwest Coast styles of the Haida and the Tsimshian).

Patterns play an important role in aboriginal art and technology. Coast Salish art could provide opportunities for students’ across the grades (and into Pre-calculus 12!) to expand their ideas about “what repeats.” Dylan Thomas is a Coast Salish artist from the Lyackson First Nation whose work in silkscreen prints, paintings, and gold and silver jewelry is influenced by Buddhist imagery and M.C. Escher’s tessellations (see Mandala or The Union of Night and Day or Salmon Spirits or Ripples or Swans or…). Share this video in which Dylan Thomas talks about connections between geometry, nature, and art as well as the importance of noticing and wondering (4:00-4:40) with your students. In Mandala, Pythagoras — or a ruler — tells us that the ratios of successive diameters of circles or side lengths of squares is √2:1. Have your students investigate this relationship. This illustrates that sometimes it’s the repetition of a rule that makes a pattern a pattern. To learn more about the artist’s interest in mathematics, I recommend reading his essay on the topic. Now is a perfect time to remind students of protocols: students should not replicate a specific piece but can instead create their own piece that is “inspired by…” or “in the style of…”; if displayed, an information card acknowledging the artist, their Nation, and their story should be included.

I’m really interested in geometry and the reason I think I am is geometry is nature’s way of producing really intricate and beautiful things. I hope that when someone sees one of my pieces they see the correlation between what I designed and what you see in nature, these sacred geometries that have shown up in nature since life evolved. And I’m hoping that when they can look at my piece they can take that wonder into their everyday life and start noticing the things that I notice and the things that inspire me.

Dylan Thomas

My numeracy colleague, Jess Kyle, recently created a lesson around the 3-D shapes above to teach students about Coast Salish culture and repeating patterns with multiple attributes (shape, colour, orientation). I wanted to expand on this lesson and zoom out from these shapes to the animal forms seen in Coast Salish art. These animals — two-legged, four-legged, winged, and finned — are connected to the land. I’m imagining these math investigations within a classroom where learners understand that animals were and continue to be an important part of the lives (and art) of First Peoples. For example, see Maynard Johnny Jr., Coast Salish, Kwakwaka’wakw, talk about his work Ate Salmon, its past-tense play-on-words title displaying humour while addressing the effects of overfishing and stock depletion on Indigenous communities (3:00-4:00). In many First Nations, certain animals are significant or sacred to the teachings, histories, and beliefs of that Nation. Each will have their own protocols around ways in which these animals are portrayed. In some parts of British Columbia animals appear on crests and regalia while in other parts of Canada animals are sacred gifts from the ancestors.

The City of Surrey has commissioned several public Indigenous works of art. Four Seasons, by Brandon Gabriel and Melinda Bige, Kwantlen First Nation, is located in the Chuck Bailey Recreation Centre. 

Photo: City of Surrey

I have some mathematical noticings and wonderings but, again, it’s important to first teach the cultural context and meaning.

Throughout this cancellation of in-class learning due to COVID-19, Surrey’s cultural facilitators have been creating and sharing videos to show and discuss with your students. Chandra Antone, Squamish First Nation, shares her teachings about drumming with us in the videos “Honour Song” and “Animal Hides.” As well, Surrey’s Aboriginal Learning Helping Teachers have generated sets of questions to ask your students about each of these videos.

Display images (below) of the four drums and ask “What do you notice? What do you wonder?”

Students might notice the blues, greens, yellows/whites, and reds/oranges; they might wonder if these colours represent winter, spring, summer, and fall. They might notice the moons (“Why just two?”), two wolves, four salmon, and trees/leaves and wonder how they tell the story of the four seasons. They might also wonder “How big are they?” (30”), “What are the drums made of?” (buffalo hide) or “Who is the artist?” Introduce your students to Mr. Gabriel through this video:

We wanted to make sure that we captured the essence of the space that we were in, that Surrey didn’t begin as Surrey, that its beginnings are much more ancient and go back many more years than the current incarnation of it. This place is very special for Indigenous people — it was also home to multiple Indigenous communities that were established here for thousands of years — so we wanted to make sure that we were honouring those people in a way that was respectful and dignifying to them. We thought, what can we use as part of the narrative that we’re going to tell with these drums that not only speaks to the Indigenous community that’s always been here but to the people who now call this place home?

Brandon Gabriel

Students may also make many mathematical observations. For example:

  • in the winter drum, there is line symmetry
  • in the summer and fall drums, there is rotational symmetry
  • in the spring drum, there is line symmetry in (just) the moon and rotational symmetry in (just) the surrounding running water design
  • in the summer drum, there are two repeating yellow-white patterns (salmon and border)
Line Symmetry
Rotational Symmetry

Again, students should not replicate Four Seasons but can instead draw their own symmetric piece that is “inspired by/in the style of Brandon Gabriel and Melinda Bige, Kwantlen First Nation.” Challenge students to use pattern blocks to build designs that satisfy mathematical constraints such as:

  • has more than three lines of symmetry
  • has rotational but not line symmetry
  • has oblique — not horizontal or vertical — lines of symmetry
  • order of rotation is three/angle of rotation is 120°
inspired by Four Seasons, Brandon Gabriel and Melinda Bige, Kwantlen First Nation

For more symmetry in Surrey Public Indigenous Art, seek out and visit:

Like night following day (or moon following sun), the cyclical changing of the seasons is something that young children can connect to when introduced to the concept of patterns. With changes in the seasons comes changes in their own lives. This is an opportunity for students to learn how seasonal and environmental changes impacted the village of qəyqə́yt (now known as Bridgeview) and continue to impact the lives of First Nations peoples today.

We Are All Connected to This Land by Phyllis Atkins, Kwantlen First Nation, is installed on a small bridge on King George Highway spanning Bear Creek. The design features three salmon (one male, one female, one two-spirited), a sun, an eagle, a moon, and a wolf, cut from powder-coated red aluminum and mirrored on both sides of the bridge.

Phyllis Atkins at blessing ceremony for We Are All Connected to This Land
Photo: Surrey Now-Leader
Photo: City of Surrey

The animals are described on the artwork’s page on the City of Surrey website:

Salmon are resilient creatures that make an arduous journey to return to their freshwater spawning grounds, such as Bear Creek, to give new life and sustain eagles, bears, wolves, and people. The wolf represents the teacher and guide of the Kwantlen People while the eagle flying closest to the sun is carrying prayers to the Creator. The inclusion of Grandfather Sun and Grandmother Moon contrast day and night and indicate the passage of time.

Teachers should avoid giving “meaning” to each animal as it often leads to appropriating spirit animals. Instead, ask “Can you think of characteristics of each animal that might be important?”

What if these figures were the core of a pattern? What if, like Nadine’s 3-D shapes at the top of this post, we could pick up and play with these figures? We could create repeating patterns like salmon-eagle-wolf or finned-winged-4legged. We’re not limited to left-to-right patterns arranged in a line. Different displays of patterns will bring to light different patterns. For example:

AB

Maybe this example better illustrates this idea:

ABBC three ways

In the second and third arrangements I interrupted the black-red-red-white pattern core in the first row to offset the pattern in subsequent rows. What (new) patterns can you find? What would the fourth arrangement look like? What’s the pattern in the patterns? Like the idea of patterns as “ripples that carry the elements away from the centre” above this structure provides us with new ways of thinking about the core of a pattern: we can think in terms of repeating vertical columns just as we would if we were bead looming. (To learn more about bead looming, please register for Nelson’s Culturally Responsive Math webinar series. It’s free!)

Teachers can use First Nation rubber stamps — available from Strong Nations — to explore repeating patterns of animal images. While we strive to embed local content, this is not always possible so we may blend Coast Salish and Northwest Coast art.

AAB

A playful approach is to begin a pattern — say wolf, raven, … — and ask “What comes next?” Some students will suspect an AB pattern and predict wolf. Others will suspect that you’re trying to trick them by not revealing the entire pattern core; they might predict raven (ABB) or orca (ABC). Ask “How confident are you?” Repeat this a few times. Suppose that you’ve revealed wolf, raven, wolf, raven, wolf, raven. By now, students will be very confident that wolf will come next. Mess with them: add bear instead. Ask students “What’s my pattern rule? Would you like to revise your thinking?” and have them share their conjectures. Next, add eagle. Can students identify the pattern as 4legged-winged? And what if we throw colour or orientation into the mix? Multiple attributes can add ambiguity to pattern tasks. Invite students to use these stamps to create their own repeating patterns.

Beginning in Grade 2 (and continuing into Grade 10), students learn about increasing patterns. In Grade 2, it is expected that students describe the salmon pattern below as “start at 3 and add 1 each time”; in the upper intermediate grades, students describe the pattern as n + 2; and in Foundations of Math and Pre-Calculus 10, this is formalized as slope (or rate of change) and y-intercept (or constant).

3, 4, 5, …

Presenting only the first and second terms of a pattern is another way to add ambiguity. (For example, “Extend the pattern 5, 10, … (or▲◾ …) in as many ways as you can.️”) I’ve been playing with this approach to visual patterns. Take a moment to consider the pattern below. What comes next? What else might come next?

You might have noticed that three tiles were added and imagined a linear pattern — 3n as either n groups of three or three groups of n:

You might have saw this as doubling and visualized an exponential pattern — 3(2)ⁿ ⁻ ¹:

Or you might have spotted squares and pictured a quadratic pattern — n² + 2:

In later grades, these more complex patterns (quadratic, exponential, triangular numbers, Fibonacci) can be introduced. Again, there’s a chance to spotlight First Nations art. Here’s a different arrangement of 3, 6, …

What comes next? What else might come next?

(If there’s a way to see a quadratic pattern in this arrangement, I can’t make it out.)

I’m more than a bit apprehensive about sharing these last two examples. They feel inauthentic: swap in dots for the images of animals above and the task remains the same. However, in using these images and first teaching their cultural significance, I’m hopeful that this communicates my respect for First Nations culture, especially to Indigenous learners (and outweighs my concerns about curriculum design).

Huy ch q’u Nadine McSpadden and Heidi Wood for continuing to help me make connections between the cultural practices and perspectives of First Peoples and the teaching and learning of mathematics.

Egg, Head, …

Take a moment to think about the following image:

#whatrepeats? #patternchat

What comes next? What comes before? How do you know?

You might have sensed (the start of) a repeating pattern. Whether you considered the materials that make up the egg cups (glass, porcelain, …) or the position of the eggs (down, up, …), it’s a simple AB pattern. Or rather, like 🍀💎🍀💎🍀💎…, two synchronous AB patterns. If you were to extend the pattern, you’d get this:

AB; AB

Not so fast. Check out the video in the following tweet:

An AB pattern is maintained in the materials: still glass-porcelain. But the video hints at a new possible pattern–an ABAA pattern–with respect to the elliptical “dome”: egg-head-egg-egg.

AB; ABAA

Again, not so fast. These first four elements may not be what repeats; they may not be the pattern core. What if the pattern core were instead egg-head-egg-egg-egg-head (all the while still maintaining glass-porcelain)?

AB; ABAAAB

Patterns repeat. Repetition is what makes a pattern a pattern. Sometimes items repeat, sometimes a rule (e.g., add 3 each time) repeats. How would you describe what repeats in the following pattern?

All of these possibilities illustrate that without knowing what repeats, you can’t know for certain what comes next. For example, consider the following open question: Extend the pattern 5, 10, … in as many ways as you can. Common classroom responses include: 5, 10, 5, 10, 5, 10, …; 5, 10, 25, 5, 10, 25, …; 5, 10, 15, 20, 25, …; 5, 10, 15, 25, 40, …; 5, 10, 20, 40, 80, …; etc. (Variation: Extend the pattern ▲◾️… in as many ways as you can.)

The two attributes in the egghead examples–container and “contents”–made the task more interesting. In the classroom, this plays out by looking at repeating patterns with multiple attributes (i.e., colour, shape, size, orientation). Consider the pattern below:

What’s missing?

What’s missing? If you focus on colour, it’s an ABC pattern; it must be teal. If you focus on shape, it’s an AABB pattern; it must be a triangle. If you focus on orientation, it’s an ABBA pattern; it must “sit” on a vertex. If you hold all three asynchronous patterns in your mind, it must be a teal square resting on a vertex (a/k/a diamond). But I’m not looking for one right answer. In the classroom, I’d happily accept a teal triangle (or circle) from a student who sees a teal-orange-green pattern; an orange (or purple) square from a student who spots a triangle-triangle-square-square pattern; etc. If the claim is true, the answer is correct.

Pattern Fix-Its present another opportunity for students to examine patterns involving multiple attributes. Here, a pattern is messed with by adding or removing an element, changing one or more attributes of an element, or swapping the order of two adjacent elements. The math picture book Beep Beep, Vroom Vroom by Stuart J. Murphy provides a context: Molly plays with her big brother’s toy cars and must put them back in the right order before he returns. Using this context, I swapped the last two cars in a big-small and yellow-blue-green pattern:

Can you fix it?

Press Here by Hervé Tullet also includes some mixed-up pattern pages. That probably inspired my shaking effect here:

Like Which One Doesn’t Belong?, these questions allow all students to confidently contribute to and benefit from the discussion, whether they notice one or many patterns, whether they attend to simple (colour and shape) or more challenging (orientation) attributes, or whether they examine single or multiple attributes at a time.

* * ** *** ***** ********

I’d be remiss not to include Marc’s tweet somewhere in this post:

A Star Wars Tomatometer Story

I wrote this post two years ago but decided against hitting publish. With the final film in the “Skywalker saga” opening next week, now is as good a time as any. And yes, we have tickets!

Like The Force Awakens and Rogue One, my daughters and I saw The Last Jedi on opening night. It’s become a bit of a Hunter holiday tradition. Gwyneth loves the stories; Keira loves the Porgs. As much as the movies themselves, Gwyneth loves watching and discussing YouTubers’ takes on them — reactions, explanations, theories. She shared this one from New Rockstars with me, which begins with this:

“… Star Wars: The Last Jedi is the most polarizing film of the year, with one of the biggest gaps between critics ratings and audience scores for a major film ever. What the hell is going on here? Why are some people so annoyed with it, saying it ruined what made the original trilogy and The Force Awakens so good? Why are others fanboy crushing so hard over it, calling it the best Star Wars film ever made?”

New Rockstars

This reminded me of another passion of mine: the fundamental meanings of the operations. More specifically, subtraction as difference/comparison rather than take away/removal.

Here are the Rotten Tomatoes scores for The Force Awakens:

Tomatometer -- Episode VIII

Episode I | II | III | IV | V | VI | VII | Rogue One | What is the Tomatometer?

What’s the meaning, in context, of 50 – 90?

We’re measuring the gap between the percentage of professional critics (“Tomatometer rating”) and Rotten Tomatoes users (“Audience Score”) who rate the movie positively. We’re talking comparison, not removal. There’s a difference of 40%. Moreover, the difference here is negative (albeit my minuend/subtrahend decision is kinda arbitrary). This means that The Last Jedi is far less favourable among moviegoers as a group than among professional movie critics. We can compare this gap with that of others in the Star Wars franchise:

Episode IV: A New Hope (1977) → 96 – 93 = +3
Episode V: The Empire Strikes Back (1980) → 97 – 94 = +3
Episode VI: Return of the Jedi (1983) → 94 – 80 = +14
Episode I: The Phantom Menace (1999) → 59 – 55 = +4
Episode II: Attack of the Clones (2002) → 57 – 65 = -8
Episode III: Revenge of the Sith (2005) → 65 – 79 = -14
Episode VII: The Force Awakens (2015) → 88 – 93 = -5
Rogue One: A Star Wars Story (2016) → 85 – 87 = -2
Episode VIII: The Last Jedi (2017) → 50 – 90 = -40

Some patterns emerge. For example, all three films in the original trilogy received positive reviews from critics and audiences alike; all three are Certified Fresh. A greater percentage of Rotten Tomato users than critics liked A New Hope, The Empire Strikes Back, and Return of the Jedi: Audience Score – Tomatometer rating > 0. The Force Awakens and Rogue One received similar positive reviews, again from critics and audiences alike. However, these recent movies rated a little lower among audiences than among critics: Audience Score – Tomatometer rating < 0.

We can use absolute value to measure agreement between the two groups. For A New Hope, The Empire Strikes Back, The Phantom Menace, The Force Awakens, and Rogue One, |Audience Score − Tomatometer rating| ≤ 5. Rotten or fresh, there’s consensus. For The Return of the Jedi and Revenge of the Sith, |Audience Score − Tomatometer rating| = 14. Still, a relatively small difference of opinions.

The Last Jedi breaks this trend. Professional critics place it alongside fellow Disney films The Force Awakens and Rogue One. RT users score it lower than the prequels. Below Binks!

Movies may be more engaging than the usual contexts for integers — a diversion from temperatures and bank balances. Thinking about this data graphically may have more potential.

https://www.desmos.com/calculator/zzplnokx7f
https://www.desmos.com/calculator/zzplnokx7f

It’s very similar to my take on the food graph, with movie critics in place of nutritionists in the role of expert. Gwyneth played along as I asked “What’s going on in this graph?”. We predicted where some of our favourite movies would land. We explained our reasoning. We compared our predictions with Rotten Tomato data. And then we shut down the laptop and rewatched The Empire Strikes Back.

The data for eight of these nine movies hasn’t changed much in two years. The outlier? Yep, The Last Jedi. The difference is now up to — or down to? — negative forty-eight (Audience Score: 43; Tomatometer rating: 91).

Look-Alike Photos

This summer, Marc and I created a series of videos designed to help parents support their children in Math 8 and 9. As best we could, we tried to have parents actively “do the math” rather than passively consume content. The explorations were meant to simulate the classroom experiences of their children. Here’s one of my favourites…

Display the original photo and five enlargements.

Ask “Which of these photos look the same as the original?” This phrasing is intentionally vague. Have students talk about what it means to “look the same.” Introduce labels — it’ll make conversations easier.

At this stage, no numbers are given. I want learners to use their intuition and get a “feel” for the problem. Tell them not to worry about making an incorrect choice — they’ll get a chance to revise their thinking later on. Likely, they’ll rule out photos B and D. Photo B looks like a square; it looks like photo D has been stretched more horizontally than vertically. Photos A, C, and E are contenders. For example, students might suspect that the dimensions of E are double those of the original. Ask “How confident are you?”

Now is the time for numbers.

Ask “Would you like to revise your thinking? How confident are you now?” The numbers confirm this hunch about photo E (and C). They can also determine close calls, like photo A. Here, scale factors of 0.75 (height_original : width_original) versus 0.8 (height_A : width_A) or 1.25 (width_A : width_original) versus 1.33 (height_A : height_original) prove that photo A is not a true enlargement of the original. (Note that this might surface if students are making absolute rather than relative comparisons: after all, adding 1″ to both the width and height of the original gets us photo A.)

This context can also be used to explore strategies for determining a missing value in a proportion. What if the photo were “posterized”?

Although these videos were designed for parents, we’re hopeful that teachers find them helpful.

Recommended reading: Tracy Zager’s Becoming the Math Teacher You Wish You’d Had (Chapter 9: Mathematicians Use Intuition)

Recommended activity: Desmos’ Marcellus the Giant

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:

Sequels

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

A&W Math

IMG_1951

“That’s a lot of smiles,” Keira (10) said as we waited for our Teen Burgers.

“Yeah. How many?” I asked. “A lot” wasn’t going to fly with a “real-world” number talk in front of us.

“Sixty-three and nineteen is… hold on,” Keira said. She wanted to add tens and ones: three twenties is sixty and one and two make three. She knew that the nine in nineteen would make this strategy more challenging. So she took advantage of the associative property and (wisely) punted.

After a few moments Keira offered eighty-two. She explained that sixty-three and twenty make eighty-three so sixty-three and nineteen make eighty-two.

Her sister Gwyneth (13) used a different strategy. “I took one from the twenty-one and gave it to the nineteen,” she said. “That’s four twenties–ha!–and two more.”

At Graham Fletcher’s session at the Northwest Mathematics Conference in Whistler, he shared a story of one student using this strategy after engaging in his Bright Idea task: “Numbers are just Skittles now,” she said. Similarly, Gwyneth decomposed twenty-one, taking and giving one to create two landmark or friendly numbers. To Gwyneth, numbers are just smiles.

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

Wanted Parabola

As much as I love mathematical modelling, so much of Math 10 to 12 is contextless stuff like this:

Determine an equation of a quadratic function with vertex at (-5, 3), passing through the point (-7, 15).

Lately I’ve been looking for activities that address this sort of naked math yet engage learners in processes similar to those in a mathematical modelling cycle.

Consider the exercise above. What questions could you ask? If I were to ask a student about their equation, I’m likely to hear play-by-play, not colour commentary: “… and then I plugged -7 and 15 in y = a(x + 5)² + 3. Negative seven plus five is two…”

Instead, I could have students try to figure out a quadratic function that satisfies a set of criteria, gradually revealed to them as “clues.” Throughout, students would check their quadratic functions and make changes when necessary. This is the gist of Wanted Parabola, my adaptation of Cathy Marks Krpan’s Wanted Number:

I started with a very general clue: the direction of opening. I anticipated a variety of parabolas, which I got when I tried this activity out with math teachers in my district. When I tried this activity out in the MathTwitterBlogoSphere (#MTBoS), I got a bunch of y = x²s. The biggest difference was that my colleagues were invited to draw a parabola (on whiteboards) whereas my tweeps were asked to write an equation (in a Desmos activity). It’s interesting to think about this activity in terms of freedom and constraints. When I revealed the next clue, it pushed my colleagues’ thinking together. However, from my tweeps, it triggered new and diverse ideas, simulated here:

I like this as a blank-page (or whiteboard) activity but a Desmos activity (1, 2, 3) does provide the opportunity to talk about some interesting overlays. If using vertical non-permanent surfaces (#VNPS), I’d stop partway through to hold a “board meeting” where students would share possible parabolas.

In general, I progressed from providing more general to more specific clues. For example, “vertex in QII” divulges p < 0 before “axis of symmetry x = -5″ gives away p = -5. Most clues add new information and move students closer to the Wanted Parabola. Some confirm earlier decisions. For example, “vertex (-5, 3)” before “axis of symmetry x = -5″ and “minimum value of 3.” This last clue is anticlimactic. An earlier clue, “never enters QIII,” is much more interesting. It might feel like new information. But it must be true given preceding clues; a parabola that opens up and has no x-intercepts cannot contain points in QIII (or QIV).

You can play with the order of the clues. A second Wanted Parabola:

Here, the direction of opening clue is revealed midway through the set. It doesn’t add new information but is reasoned to through “two x-intercepts” and “vertex in QI.” I meant to delay students determining the direction of opening a bit, hoping to surprise them after a few clues. In a third Wanted Parabola, “passes through” is the first clue; I anticipate that some students will place the vertex at this point.

Instead of “How did you find a?” you could ask “Which clues were helpful? Which clues were necessary?” In my mind, helpful ≠ necessary. A clue might be helpful if it pushes students in the direction of the Wanted Parabola despite not providing the values of a, p, or q. Or a clue might be helpful if it tells students that they’re on the right track. In the way that the first Wanted Parabola plays out, three pieces of information are necessary (to determine three unknowns): “minimum value of 3,” “axis of symmetry x = -5,” and “passes through (-7, 15).” If some students don’t argue that only two clues are necessary — “vertex (-5, 3)” and “passes through (-7, 15)” — you could ask “What is the fewest number of clues you need?”

This activity helps students develop an understanding of the different attributes parabolas can have. It provides an opportunity for students to solve problems, reason, explain, justify, and connect mathematical ideas in ways that “Determine an equation…” does not.

Wanted Parabola (.pptx) (.pdf)

References

Marks Krpan, Cathy (2013). Math expressions: developing student thinking and problem solving through communication. Toronto, ON: Pearson Canada.

Update

A Function of Freedom and Constraints

In June, a colleague invited me into his classroom to teach a Desmos modelling task — Predicting Movie Ticket Prices — in his Math 12 class. Students experienced exponential functions earlier in the course. We were curious about whether his students would apply what they knew about exponential functions to a task situated outside of an exponential functions unit — a task not having to do with textbook contexts of half-life, bacteria, or compound interest. They did. And they deepened their understanding of how change by a common ratio appears in exponential equations (vs. change by a common difference in linear equations). They did this within 45 minutes of a 75-minute class. So my colleague let me try out another, less sexy, task — one adapted from MARS. This task, like much of Math 12, is about naked functions; no real-world context here. Nat Banting’s closing keynote at #NWmath reminded me of it. Watch Nat’s talk; view his slides.

functions

The original MARS task above is closed: two functions, one linear and one quadratic, each passing through four points. I wanted to open it up so I changed the prompt: “A set of functions pass through the points shown. What could the equations for the functions be?” Also, I removed one of the points — (5, 3) — to allow for different solutions of two functions. The thinking is that open questions encourage a variety of approaches. And then, from fifteen pairs of students:

fA

I anticipated this. The points scream linear and quadratic. They are sources of coherence. I had lowered the floor but no Rileys entered y = 5, y = 7, y = 8, y = 9. The problem wasn’t problematic. I had raised the ceiling but no one wrestled with equations for sinusoidal or polynomial or radical or rational functions. The freedom within my open question didn’t bring about new and diverse ideas. To support creativity — mathematical creativity! — I had to introduce a source of disruption, a constraint“A set of nonlinear functions pass through the points shown. What could the equations for the functions be?”

A student could have used the linear nature of absolute value functions to get around my nonlinear constraint — a bit of a Riley move? — but no one did.

fC

Instead, some students picked up on the symmetry of two new possible parabolas:

fB

Writing the equation of the second parabola — finding the parameters a and q — presented more of a problem.

Others bent the line; they saw the middle of its three points as the vertex of a cubic function that had been vertically stretched and reflected:

fE

Some saw four compass points and wrote an equation of a circle. This led to a function vs. not a function conversation: “Does that count?” Others saw a sine function that passed through three of these four points. There were “close enough” solutions — great for Coin Capture but not quite passing through the given points:

I didn’t anticipate this. Students weren’t as constrained by “pass through” as I was. Also, they were motivated to capture the points using only two functions, as before.

With more time, I could have shifted constraints again: “A set of functions pass through the points shown. What could the equations for the functions be? (P.S. The graph of at least one of them has an asymptote.)” This would have triggered exponential and logarithmic or rational functions. (Even without introducing this constraint, we noticed at least one student playing with rational functions at the end of class.)

Above, there’s evidence to support Nat’s #NWmath conjecture: “Shifting constraints triggered new mathematical possibilities.” My (more) open question didn’t cut it. The student thinking — and conversations — that I had hoped for only emerged when freedom “sloshed against” constraints.

Desmos activity

Update

Edited & published in BCAMT’s Vector 62(2).