In *Open Middle Math*, Robert Kaplinsky describes what makes a math problem an *open-middle* problem:

[M]ost math problems begin with everyone having the same problem and working toward the same answer. As a result, the beginning and ending are closed. What varies is the middle. Sometimes a problem’s instructions tell students to complete a problem using a specific method (a closed middle). Other times, there are possibly many ways to solve the problem (an open middle). Problems with open middles tend to be much more interesting and lead to richer conversations.

Robert Kaplinsky

This use of open-middle to describe *problems* has always irked me. There, I said it. To me, open- vs. closed-middle is not a characteristic of a problem itself. Robert argues that a problem’s *instructions* can close a problem’s middle. Agreed! But I go a step further. There are other ways through which students are told to use a specific method. It’s *us*. For example, consider a boilerplate best-buy problem. The middle is wide open! Doubling, scaling up, common multiples, unit rates — dollars per can or cans per dollar — and marginal rates are all viable strategies. However, *we* close the middle when we give this task after demonstrating how to use unit prices to determine best deals (i.e., “now-you-try-one” pedagogy). If students — and teachers! — believe that mathematics is about plucking numbers to place into accepted procedures then they are unlikely to experience the rich “open-middleness” of this task, regardless of its instructions. It’s no accident that the book’s introduction is titled “What Does an Open Middle *Classroom* Look Like?”

Most of the problems posted on the open middle site involve number — or digit — tiles. But I get why Robert didn’t go with “Number Tile Math.” The boxes in 25 × 32 = ⬚⬚⬚ and 63 − 28 = ⬚⬚ give each a fill-in-the-blanks answer-getting feel. The routine nature misses the problem-solving mark (despite their open middles). So, “open-middle” as an adjective for problems it is. Besides, math class could use more openness, which needn’t come at the end for problems to be interesting and conversations to be rich.

When I look at an Open Middle problem from the site, the mathematical goal of the teacher who created the problem isn’t always clear to me. (The same is true, by the way, of wodb.ca.) What is the deep conceptual understanding that they anticipate their students will develop by working on the problem? What ideas will emerge? What misconceptions might be addressed? Throughout *Open Middle Math*, Robert describes how Open Middle problems can give us X-ray vision into our students’ mathematical understanding. Similarly, he provides readers with X-ray vision into *his* thinking during the process of creating these problems. Below, I’ll share a few of the open-middle problems from our video series (plus some that ended up on the cutting room floor) as well as a peek behind the curtain into my thinking.

Towards the end of the Math 10 Factoring Polynomials video, I present two open-middle/number-tile problems. Teachers will recognize these as familiar “find *k*” problems: For what value(s) of *k* is *x*^{2} + *kx* − 8 factorable? *x*^{2} − *x* − *k*? See the answer animations below.

I think that the number tiles add an element of *play* to these problems. The tiles are forgiving. Make a mistake? No biggie, just move ’em around. (The decision to show an initial misstep in the first animation above was deliberate.) This upholds our third principle: make it inviting.

These two sample tasks above highlight the role of students’ prior knowledge in solving open-middle problems. My assumption here is that teachers have not “proceduralized” these problems — that students have not been provided with predetermined solution pathways (e.g., “First, list all the factors of the constant term *c*. Then, …”). Note the open *end* of the second problem. The intent of my animation is to convey that there are infinitely many solutions. The problem presents students with an opportunity to generalize.

Each of these problems can be classified as Depth of Knowledge Level 2 (Skill/Concept). In both, students need a conceptual understanding of factoring *x*^{2} + *bx* + *c* where *b* and *c* are given. The second requires pattern-sniffing (or logic). I created a third problem that asks students to think about these two equations — and a third — *simultaneously*.

Note that *x* − 4 *could* be a factor of each trinomial. However, students need to determine where to put 4 so that the other digits can be placed in the remaining boxes. This twist might not be enough to raise it to DOK 3 (Strategic Thinking). Roughly speaking, Robert’s DOK 2 problems involve making statements true. Sometimes it’s satisfying an equation, sometimes it’s satisfying a condition (e.g., a system of equations having no solution). Robert’s DOK 3 problems call for optimizing a result — least, greatest, closest to.

In my Math 9 Polynomials video, I pose the following open question in the style of Marian Small: *Two trinomials add to 3x ^{2} + 7x + 6. What could they be?* Here’s a problem, adapted from

Both tasks can help reveal students’ understanding of combining like terms and manipulating coefficients and exponents. (In Task 1, I specify that the two polynomials be trinomials. This rules out responses such as *(3x ^{2}) + (7x + 6)* that sidestep like terms.) Task 2 is much more likely to show what students know about additive inverses, although a small tweak to Task 1 (e.g.,

I include one open-middle problem in the application section of each of my Math 7 integer videos:

Note that the first is DOK 2 whereas the second is DOK 3. Maybe. I don’t want to quibble. What matters more than the differences between DOK 2 and 3 is that these questions require a deeper understanding than DOK 1 problems such as *Evaluate: (−9) + (+3)* or *Evaluate: (+3) − [(−5) + (−4)] × (+5)*.

In the first video, I ask “How might finding one solution help you find more? How are some of the solutions related to one another?” These questions aren’t answered in the video — an exercise left to the viewer. Here are just some of the ideas that I would anticipate to emerge in an Open Middle classroom discussion:

- addition and subtraction facts are related
- e.g., (+6) + (+3) = +9 and (+9) − (+3) = +6 belong to the same “fact family”

- this relationship extends from whole numbers to integers
- e.g., (−6) + (−3) = −9 and (−9) − (−3) = −6 also form a fact family

- subtracting a number can be thought of as adding it’s opposite
- e.g., if (+6) + (+3) = +9 makes the equation true then so, too, does (+6) − (−3) = +9

- swapping the number being subtracted (subtrahend) and the number it is subtracted from (minuend) changes the sign of the result (difference)
- e.g., (+9) − (+3) is equal to
*positive*6 whereas (+3) − (+9) is equal to*negative*6

- e.g., (+9) − (+3) is equal to

Order of operations is a natural fit for optimization problems. In the second video, the intent of my answer animation is to communicate my mathematical reasoning. Once more, note that I show a couple of slight missteps and revisions to my thinking.

In addition to performing the operations in the correct order, students must think about how to maximize sums and minimize products. They must consider how subtracting a number *increases* the result. See one of Marc’s Math 7 decimal videos for another open-middle order of operations example.

I did not include an Open Middle problem in my Math 7 Percents video. Rather, I chose to present a percents number talk: *Estimate 78% of 49*. Note that I show two strategies: one that makes use of quarters…

… and another that utilizes tenths.

Because there are many ways to solve this problem, it can be thought of as a having an open middle despite it not having number tiles. A third, interesting, solution pathway can be taken. Instead of 78% of 49, we can estimate 49% of 78: 50% — or one-half — of 78 is 39. The idea that *x*% of *y* is equal to *y*% of *x* should emerge from the following:

It’s for this reason that I did not add the constraint *Each number can be used only once*. You can always add it later, which should bring about doubling and halving — and tripling and “thirding”!

I like that the double number line problem below incorporates estimation; both 20 and 25 per cent are perfectly reasonable estimates. Also, it embraces our first principle — make it visual — which is largely missing from my other open-middle examples.

I wrestle a bit with whether or not to include the “only once” constraint. Does it enhance the problem above? I guess that it necessitates more *practice*; disqualifying *25% of 64 is 16* does compel students to seek out *25% of 68 is 17* or *25% of 76 is 19*. But concentrating on unique tens and units digits of parts and wholes is irrelevant to percents. Again, you can always add this constraint later. *(Update: Check out this slick Desmos activity from @TimBrzezinski!)*

What might a DOK 3 Open Middle percent problem look like? Below is a possibility — or three! — that uses Robert’s optimization approach. (I haven’t played with the dollar value amounts so treat them as placeholders.)

*Open Middle Math* is a must read that will help you implement these engaging tasks. Whether you’re new to Open Middle problems or think you know all about ’em, you’ll love the glimpse into how Robert *designs* opportunities for students to persevere in problem solving and for teachers to gain insights into what students *really* understand.

In this post, I’ll share some of the principles that guided us when creating the videos. This’ll be a peek behind the curtain of interest more to educators than to parents.

*Math* is visual. *Videos* are visual. So *math videos* should be visual. It is disappointing how often makers of digital content fail to take full advantage of visual aspects available to them. The animation of symbolic representations–line-by-line equation solving or drawing little arrows to show the distributive property–should not be the extent to which content is presented visually. It’s ballsy to sell this to educators as visual if that’s all you’ve got. By “make it visual,” I mean include images and animations that help viewers make sense of the mathematics at hand or the context in which the mathematics is situated. For example, we *show* that 2:3 is equivalent to 8:12 by repeatedly extending a black-red-red-red-black pattern of beads; we don’t describe two candles with different heights and different rates at which they burn, we *show* it–so long as we can figure out how to do it in Keynote.

In our videos, we make use of virtual manipulatives–or virtual virtual manipulatives?–like pattern blocks, colour tiles, counters, multi-link cubes, base ten blocks, algebra tiles, tangram-like puzzles, Solo cups and paper clips, etc. We employ other pictorial representations such as hundred charts, decimal/percent grids, number lines, *double* number lines, factor trees, factor rainbows, tables of equivalent ratios, graphs, etc.

I once watched a short video to fix an issue with my dishwasher. I carefully followed the directions, pausing the video at each step along the way. But no one is ever going to mistake me for a handyman! I don’t really *understand* how dishwashers work. I couldn’t connect the problem to any knowledge of the machine’s mechanical or electrical systems. If the solution shared online didn’t work, I was hooped. And even though I was successful, my procedure for fixing my dishwasher was useless for fixing my washing machine, let alone a different make and model of dishwasher. This skill–long-forgotten, by the way–didn’t transfer from one household appliance to another.

But it didn’t matter. I set out that morning to make one small repair, not become an appliance repair technician. Mathematics is different. The emphasis in math class must be on *sense-making*, not *answer-getting*. The same should be true of math videos. In our videos, we attempt to always address the why.

For example, we answer “*Why* is a negative divided by a negative a positive?” by revisiting what it means to divide whole numbers and then applying these two fundamental meanings to dividing integers. Later, a thermometer example reinforces dividing as measuring.

Conceptual understanding means seeing mathematics as a coherent whole rather than isolated procedures. Digital content can support students in developing conceptual understanding by creating opportunities for them to connect models and representations. For example, we ask learners to connect multiplying binomials to what they already know about multiplying two-digit numbers (i.e., an area model, partial products, the distributive property).

Of course, procedural fluency is important. Effective mathematics teaching focuses on the development of *both* conceptual understanding and procedural fluency. However, procedural fluency depends and builds on a foundation of conceptual understanding.

For example, percents are presented as fanatical comparisons to 100. No part-whole-percent triangles or is-over-of-equals-percent-over-100 cross-products here. Relating percents to decimals does not appear until grade 7. So, in our Mathematics 6 video, the emphasis is on equivalency and benchmark percents (i.e., 50%, 25%, 75%; 10%, 20%, … , 90%).

Procedural fluency includes the ability to apply procedures *flexibly*. Throughout each video, multiple strategies are discussed. For example, see the strategies–and representations–used in this proportional pizza problem.

To bridge conceptual understanding and procedural fluency, we try to build on learners’ own mathematical ideas. For example, before the elimination method for solving systems of linear equations is introduced, viewers are first encouraged to solve a puzzle using their intuition. Then, they are presented with a pictorial representation of a solution to a problem. It’s at *this* time that algebraic symbols and notation appear:

This process plays out when solving equations in one variable (boxes of doughnuts; algebra tiles) in Math 7 and when solving systems using substitution (scale puzzle; two types of tickets problem) in Math 10.

Of course, we can build on viewers’ own ideas only to the extent to which they engage with and participate in the learning experiences that we design. This segues into our last principle…

I think that this is the right adjective. To me, *inviting* goes beyond *accessible*.

One way in which we make an effort to invite parents to “do the math” is to use open questions. Sometimes, this means open-*ended*. For example, *Show me one-quarter in as many ways as you can*, *What could the numbers be?* and *What comparisons can you make?* (and later *What is being compared in the ratio 1:2?*) are open-ended; they allow for many correct answers and signal that a range of responses are valued.

Other times, we used open middle problems. An open middle problem may have one correct answer but multiple ways of getting it. For example, there is an arrangement–or two–of number tiles that maximizes the expression below. The animated placement of the number tiles is meant to model *one* strategy and includes me making missteps and backtracking as I went along.

(Here’s a number tile factoring task from Math 10 that has both an open end and an open middle.)

In addition to openness, we try to hold off on introducing formal symbols and notation early. For example, * Two numbers add to 12. What could they be?* comes before

Sometimes, when a task is not a soft place to start, we may still present it up front and then return to it later, after we’ve built up some knowledge. In this way, we hope to “make it inviting” by piquing the curiosity of viewers. For example, asking parents to pick two numbers that differ by two and multiply them is accessible whereas asking them to explain why this product is one less than the square of the number between them is not. A similar approach was taken with a gas vs. electric vehicle application of linear systems; it serves as a *hook* in the video’s introduction.

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

We believe that learners of mathematics should be active participants rather than passive spectators. There’s a tension between this belief and video. In our videos, we put a lot of trust in parents pressing pause when prompted. It’s in these moments that they “do” math, that they play, notice and wonder, solve problems, visualize, look for patterns, make conjectures, generalize, reason, explain, connect ideas, take risks, etc. We were limited by the medium–or our tech skills within this medium.

In a mathematics classroom–be it face-to-face or remote–this tension can be resolved. And this is one reason why we’re just as, if not more, excited about teachers using these videos. At the moments when we ask viewers to pause, students could be placed in visibly random groups or breakout rooms. Teachers are not limited by our prompts–or these moments. They can observe and adapt to what’s happening with their learners in the moment and ask *How else might you have solved the problem/represented your thinking? What does this remind you of? How are these the same? How are they different? What would happen if… ?* etc. There’s no need to “fake it” coming out of a pause as we had to do (e.g., “You might have noticed that…”).

These videos were intended to capture the big ideas or enduring understandings or key concepts of a topic–a whole chapter or unit. Although each clocks in at about twenty minutes, it would be inappropriate to have students experience an entire video in one sitting. Instead, a task or two clipped from a downloaded video could make up one day’s learning experience.

If you find this video series helpful, we’d love to hear from you. Drop a comment, question, or complaint in the comments.

]]>… Jazz is based on the emulation of human dialogue. When you’re trading fours, you’re having a conversation. And improvisation that it comes from, it’s really the American aesthetic on display. Freedom within form. You have to honour the form of the music but, as a soloist, you have the right as an individual to go as far as you want to go. We’re a nation of laws but as an American we celebrate individuality. Honour the form, honour the laws but be yourself, be free. It’s a finite amount of notes with an infinite amount of combinations. Improvisation. And then that taught me how to act. Because it’s a finite amount of words but with an infinite amount of ways of saying them and an infinite amount of ways of having those words affect you. And that’s jazz and that’s what the American aesthetic is, unique to our experience…

–Wendell Pierce

I love this! It reminds me of the relationship between freedom and constraints in mathematics. In fact, Pierce goes on to connect freedom within form to two different proofs that land on the same truth in Trigonometry class. He argues “the two can coexist” when talking about Shakespeare or stand-up. Or, more importantly, when talking about the Constitution and John Lewis or police reform. It’s quite the “coffee run” and starts at about 1:04:20 below:

But a less significant section of his riff grabbed my attention. I learned why Led Zeppelin’s “Kashmir” has always sounded “off” to me (1:06:30). John Bonham’s drums are in 4/4 time. Jimmy Page’s guitar is in 3/4 time. The effect is that the guitar (and strings) seem to be ahead and then behind the beat. Listen for yourself below:

They meet up on twelve, the lowest common multiples (LCM) of three and four! Play with the slider below:

It’s like a variation of FizzBuzz. FizzBuzz is a child’s game — or drinking game — in which players count around a circle, replacing any number divisible by three with “Fizz” and any number divisible by five with “Buzz.” Numbers divisible by both (i.e., numbers divisible by 15) are replaced with “FizzBuzz.” If a player hesitates for too long or messes up, they’re out.

This podcast came along at a time when Marc and I had wrapped up a series of Math 6/7 videos for parents. Factors & Multiples was one of mine. A few activities, like the following, ended up on the cutting room floor:

The gist of this activity is that students are challenged with determining two mystery numbers as their factors (or non-factors) are gradually revealed. Like Wanted Parabola, with each new “clue,” students must assess their thinking. For example, suppose that after 9, 4, 3, and 7 are placed, a student believes that *a* and *b* are 18 and 12, respectively. The placement of 1 isn’t helpful; it’s a common multiple of *a* and *b*, no matter their values. But the placement of 2 means that *a* cannot be an even number like 18; this student must revise their thinking. (Their choice of *b* can remain the same since any multiple of four is also a multiple of two.) Throughout, students can be asked “How confident are you?” Notice that with the placement of 5, *b* must be 60 (or some multiple of 60); the subsequent placement of numbers in this circle adds nothing new. (Maybe it’s worth tinkering with the order in which the clues are revealed?) After, students can be asked to reflect on their mathematical reasoning: “Which clues were most helpful? Which were unnecessary?”

I haven’t tried out this exact task with real students… yet. I’d welcome any feedback from anyone who’s able to test-drive the task for me during this time.

This activity was cut from the video for a couple of reasons. I questioned the task’s accessibility. And slow and patient disclosure of information just plays out better as a classroom activity. More on the design principles behind these videos in an upcoming post…

Woah! Joe Schwartz shared this video about Bonzo from Polyphonic. Check out “Kashmir” at 5:00!

*This is a great video explaining how Bonham’s drumming genius integrated into the band, including what happens in Kashmir. #2 for me behind When the Levee Breaks.*

*Originally tweeted by Joe Schwartz (@JSchwartz10a) on November 18, 2020.*

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.

]]>Baby back ribs are on sale for $11.00/kg but it’s the price per *pound* that’s front and centre. Same for strawberries. So you might think that imperial units are the norm at the grocery store. But sockeye salmon is on sale for $3.29/100 g. Note that the unit in this unit price is not 1 kilogram but 100 grams. Okay, so meat and produce are (prominently) imperial and seafood is metric. Why this difference? Maybe, to the Canadian consumer, $4.99 per pound *feels* less than $11.00 per kilogram. Then why not go all the way and advertise $1.10/100 g? The next stop is the deli where corned beef goes for $2.99/100 g. This time, meat is metric. Potato salad, too. To summarize, (fresh) meat and produce are imperial whereas seafood and (deli) meat and salads are metric. Potatoes are priced per pound — unless in a picnic salad. And this is just at the grocery store.

The Fresh St. flyer is typical of everyday life in Canada. Today, social distancing signs remind citizens to stay 2 metres, or 6 feet, apart. Canadians are fluent with both. This is not true of linear measures across all contexts. When driving or walking, I think kilometres or metres. After all, all roads signs were metric a good ten years before I began to drive. One lap around my school’s track was–still is–400 m, my “How far of a walk is it?” referent. Two metres and six feet are interchangeable when it comes to social distancing but a person’s height requires more precision and, thus, smaller units. So centimetres and inches it is, right? Nope. Feet and inches. Metres and centimetres are reserved for driver’s licenses and other official documents. Similar for a person’s weight, by the way.

Volume? *Litres* of gas but *gallons* of paint. In a restaurant or pub, a *pint* (20 imperial fluid ounces) or a bottle (341 millilitres) of beer, a five-*ounce* glass or a half-*litre* of wine. Temperature? *Celsius* for weather, *Fahrenheit* when cooking.

There are generational and provincial differences but I’m in line with the majority of Canadians.

This system of two systems of measurement can partly be explained by our proximity to the US. But there’s more to it. Conversion means new benchmarks and referents. And the old ones are sticky…

When thinking about a person’s height, six feet means something to me whereas 183 cm does not. Right away, I connect 6′ with the short side of tall or the tall side of average. Not so for 1.83 m. One hundred eighty-three centimetres could go about becoming a new benchmark in one of two ways: like milk or like butter. Milk is sold in 4 L jugs. Milk didn’t just slap a 3.89 L label on a gallon jug and call it a day. A new number, a new referent. (Remember milk bags? Blame metrification.) Butter, on the other hand, is sold in 454 g (i.e., still 1 lb) blocks. I can’t think 1 lb without also thinking 0.454 kg. I don’t foresee that happening to 6′ and 1.83 m. Two–in place of 1.83–metres isn’t helpful outside of the NBA. One hundred eighty centimetres has a nice ring to it and, at what we think of as 5’11”, would probably a better barometer of short or tall.

I will never not know that a National Hockey League (NHL) rink is 200 feet long, a measure set in cement by the phrase “a bad penalty 200 feet from his own net.” No Canadian hockey fan would ever use metres when talking about The Game. The length of a Canadian Football League (CFL) field is 110 *yards*, a noticeably non-metric number. This places midfield at the fifty-FIVE-yard-line, which is just awkward. Here’s the thing: 110 yards is 100.58 metres. That’s *so close* to one-hundred. Mathematically, I’d like to see a metric field. Even in a game of inches, a change from 100.58 to 100 metres would be negligible. Differences in downs would be more significant: 1st and 10 (metres) would be 1st and 10.94 (yards). (Still *three* downs to gain them.)

I can move fluently between outside temperatures of 20 ℃/68 ℉ or above; below I’m subtracting thirty and dividing by two. Zero–not 32–is my freezing temperature. But those baby back ribs above? 275 degrees *Fahrenheit*, low and slow. Pizza? 450 ℉. These numbers resist being replaced — they are “baked” in.

For a long time, Canadian curriculum was mandated to be metric. **Imperial units were part of everyday life but not school life, real-world math but not school math. **Fortunately, the imperial system of measurement has found a place in BC’s mathematics curriculum. Unfortunately, that place is Workplace Mathematics 10. This means that not all students in BC are taught imperial alongside metric. Learning imperial serves a purpose greater than “talking with Americans”; it’s part of *Canadian* culture.

Many teachers, however, still believe that all curricular materials must be metric. If, in a workshop, I pose a problem with imperial units, I promise that at least one colleague will point it out. But there are much more important questions to ask when evaluating tasks than “Are the units metric?”:

- Is the task engaging?
- Is it aligned with grade-level content?
- Does the problem require problem solving?
- Does it help students develop conceptual understanding or procedural fluency?
- Does it have multiple solutions or allow multiple approaches?
- Does it encourage discourse?

Consider Andrew Stadel’s File Cabinet three-act math task (surface area; Grade 8). Six enthusiastic yeses to the questions above. Live with the inches in Act 2. Or take the classic maximize area given a fixed perimeter/minimize perimeter given a fixed area tasks (Grade 5). Why give your students 1-inch square colour tiles to model these problems but not talk about the dimensions of the manipulatives? Be flexible in earlier grades. Say “they’ll need it for high school” if you get any pushback. (You won’t.)

Tasks that ask students to make comparisons or justify decisions lend themselves to using mixtures of units. For example, take a moment to think about the heights of the following action film actors.

Like your students, you probably placed the Toms–Cruise and Holland–on the left, Dwayne Johnson and Jason Momoa on the right. I can provide you with measurements and ask “How confident are you *now*?”

Note that it’s easy to make two comparisons: Dwayne Johnson is taller than Tom Cruise; Jason Momoa is taller than Tom Holland. (If you–or one of your students–lacked this bit of pop culture knowledge, you’re now up to speed.) These comparisons involve converting *within* systems of measurement: inches to feet and centimetres to metres (or vice versa). Comparing Johnson with Momoa and Cruise with Holland is more challenging; it demands converting *between* systems: metres or centimetres to feet or inches (or vice versa). **This elicits an essential mathematical understanding: using the same unit to measure different objects makes it easier to compare these objects.**

Would you rather…? math tasks also provide opportunities for students to learn imperial alongside metric. For example,

Would you rather…

(A) run a half-marathon or

(B) take part in a 20K run?

Would you rather…

(A) drive from Vancouver to Montreal (4554 km) or

(B) drive from San Francisco to New York (2903 mi)?

Would you rather…

(A) buy a vehicle with a fuel economy rating of 7.7 L/100 km or

(B) buy a vehicle that gets 33 mpg?

Would you rather…

(A) get gas at home or

(B) go to the US for gas?

Happy Canada Day!

]]>Next to your comrades in the national fitness program

Caught in some eternal flexed-arm hang

Droppin’ to the mat in a fit of laughter

Showed no patience, tolerance or restraint

If, like me, you were a kid in Canada in the ’70s or ’80s, you remember the “national fitness program.” To The Hip’s fanbase, this is not some obscure cultural reference. Millions participated. Operated by the Government of Canada, the Canada Fitness Award Program consisted of six events: flexed-arm hang, shuttle run, partial curl-ups, standing long jump, 50 m run, and endurance run. These events were proxies for strength, power, speed, agility, and cardiovascular/aerobic endurance — not to mention “patience, tolerance, or restraint.” The flexed-arm hang test timed how long you could hang still from a pull-up bar with your chin at or above the bar. Some lasted seconds, some over a minute. I excelled in this event despite never being mistaken for having “arm and shoulder girdle strength.” Did this test favour the scrawny?

I remember these signs. Even today, I could draw you a map of my elementary school gym, placing each event in the right spot.

This was a national fitness *test*. Results were mailed to the government who, in turn, sent awards to schools. Badges — bronze, silver, gold, and excellence — or <shudders> plastic participation pins were distributed to my “comrades” and me.

I remember getting silver. No muddiness in that memory. I remember being disappointed with bronze. I remember gold, too. But that might have been my brother.

I always suspected that this program was a response to the Summit Series — a close-call that threatened our national identity. It is, after all, the backdrop to “Fireworks”: “If there’s a goal that everyone remembers, it was back in ol’ 72.”

I completely forgot about Canada’s Sputnik: The 60-year-old Swede.

But this is a math ed blog, not a personal journal. What does the Canada Fitness Award Program have to do with teaching and learning? There are some takeaways with respect to my district’s “priority practices”: curriculum design, quality assessment, instructional strategies, and social and emotional learning.

Today I learned the flexed-arm hang was discontinued and replaced by push-ups. “The revised program was distributed to all schools in March 1980.” Not all, not all, not all! Not *my* school. When I do the math, the flexed-arm hang should never have been part of my fitness evaluation. Seven-year-old me should have just missed it. But I didn’t. Year after year, I took this test. So why did it persist? The easy answer is that my teacher missed the memo.

It’s not that easy.

*Curriculum* is weird in that it can mean standards, resources, or “the lived daily experience of young people in classrooms.” Here, I’m going to interpret curriculum as learning resources (aligned with learning standards and impacting students’ learning experiences). Take a moment to look at the textbook below. In what year was it published?

1987.

The photo on the cover threw me. I estimated earlier — “You could say I became chronologically ‘Fucked-Up’.” I’m confident that someone somewhere in BC is using *Journeys* today. Right now. Not in a you-can-find-good-problems-in-every-textbook kind of way but as the *core* math resource in their classroom. Ten years ago, stories of teachers hiding them over the summer break so that their administrators wouldn’t take and replace them with *Math Makes Sense* or *Math Focus* were common. Some student, somewhere in BC, has been assigned a (long out of print) textbook that could have been assigned to their parents.

This speaks to the difficultly in implementing change in education (and is not intended to pile on an individual teacher’s choice of instructional materials). It’s never *just* about new learning standards or resources. Swapping out a laminated flexed-arm hang poster for a push-up one is one thing. Changing PE practices is quite another. (Knowing that adding and subtracting fractions has moved from Math 7 to Math 8 while multiplying and dividing integers has moved from Math 8 to Math 7 is today’s equivalent to swapping station signs. This knowledge is one click away.)

With new curriculum comes new instructional strategies for teachers to learn. These new strategies may butt up against entrenched models of teaching (e.g., flipping the “I Do, We Do, You Do” script). Teaching is a *cultural* activity. New curriculum may usher in new values and beliefs. Pushback is predictable. A teacher who see mathematics as answer-getting will resist resources aimed at mathematics as sense-making. Lasting change often takes a teacher reframing their own relationship with and understanding of mathematics. Teachers need patience, time, and support to think about new ideas and put them into practice.

By 1992, the Canada Fitness Award Program was discontinued in part because its focus on performance and awards discouraged those it most intended to motivate (i.e., those lacking physical fitness or those deemed to be overweight). Such a program wouldn’t fly today. Now that we know better, we do better, right?

You can probably guess where I’m going with this…

BC’s Ministry of Education lays out four levels of proficiency in terms of evaluating students’ learning: *Emerging*, *Developing*, *Proficient*, and *Extending*. *Initial*, *partial*, *complete*, and *sophisticated* are supposed to be helpful descriptors.

Most teachers comment that this is far too vague. Many of those teachers are creating descriptions of these proficiency levels for both curricular competencies (or their headings) and content. Consider the following example from my colleague, Marc Garneau. For “use strategies to solve problems,” he takes into account differences in the two parts of this standard: use of strategies and types of problems.

These efforts are designed to focus on *learning*. These teachers want to communicate learning to students and their families in meaningful ways. But they are also tasked with reporting letter grades (or percentages). So they hold their noses and find imperfect work-arounds. Still, it can be a real challenge to convince students that *Emerging*, *Developing*, *Proficient*, and *Extending* are anything but badges.

*Note: This topic deserves a deeper dive. See our Numeracy Support During COVID-19 site for a start.*

Another (smaller) assessment takeaway…

I remember the Canada Fitness Award Program norms being almost arbitrary. I couldn’t find any flexed-arm hang details but the Royal Canadian Air Cadets adopted the (revised) program. Check out the graphs below for partial curl-ups. What do you notice? What do you wonder?

I notice a lot of ups and downs. At 13, the total number partial curl-ups required for boys to achieve Gold or Excellence is at a minimum; at 15, a maximum. The opposite is true for girls. Probably puberty except these differences don’t always appear at the same ages — or at all — when looking across events. Thirty-seven partial curl-ups would have placed a 12- and 15-year-old girl one shy of the 38 cut-off for silver and excellence, respectively. This difference feels as arbitrary as the difference between 86 (A) and 85 (B).

Play with the graph here. I’m curious what students might notice and wonder. The FitnessGram PACER Test (or “Beep Test”) will be more relevant in their real worlds. How does it compare? (Confession: all this time I thought it was the “Fitness Graham Pacer Test.”)

My sense is that PE classes have changed a lot since the days of flexed-arm hangs. There seems to be a greater emphasis on personal health and fitness — noble goals of the Canada Fitness Award Program — and less of the sportsball rotation of volleyball, basketball, badminton, rugby, repeat. No more pedagogy of winning-team-stays-on (and losing-team-takes-a-seat). There seems to be more personalized ways to achieve success.

Again, what does this mean for the teaching and learning of *mathematics*? In terms of a long-held practice with a lasting legacy, the “Mad Minute” is the closest match from math class. Its focus on speed discourages those it most intends to motivate (i.e., those without quick recall of the basic facts). We know that timed tests cause math anxiety. We know that there are more effective — and less destructive! — ways to develop fact fluency. See here, here, here, here, or here.

Each year, I’m invited to present a workshop to a teacher candidate cohort at a local university. And each year, they share their figurative vaccination scars left by math class. For every PE teacher who was awarded a red and black badge, there’s a math teacher who was an “Around the World” champ. “But *some* kids like it” misses the point and one more “*I* turned out okay” isn’t going to change my mind. I’ve been down this road before; it’s my eternal flexed-arm hang.

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.

Dylan Thomas

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 thatwonderinto their everyday life and start noticing the things that Inoticeand the things that inspire me.

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.

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:

Brandon Gabriel

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?

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)

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°

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

*Guardian Spirits*by Trenton Pierre, Katzie First Nation*snəw̓eyəɬ: Nature’s Gods (Nature’s Teachers)***by Wes Mackie, Katzie First Nation***Eight Salmon*by Leslie Wells, Semiahmoo First Nation*Retro-Perspective*by Drew Atkins, Kwantlen First Nation*Under the Double Eagle*and*Elder Moon*by Leonard Wells and Leslie Wells, Semiahmoo First Nation

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.

Photo: Surrey Now-Leader

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:

Maybe this example better illustrates this idea:

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.

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).

Presenting only the first and second terms of a pattern is another way to add ambiguity. (For example, “Extend the pattern 5, 10, … 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 — 3*n* 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 — *n*² + 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, …

(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.**

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:

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.

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)?

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? 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:

*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:

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:

“…

New RockstarsStar Wars: The Last Jediis 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 bestStar Warsfilm ever made?”

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*:

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.

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).*

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

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