Open Middle Math

In my previous post, I shared some of the principles that guided Marc and me when creating a series of math videos for parents (Mathematics 6 & 7; 8–10): make it visual, make it conceptual, and make it inviting. In this way, we also set out to make these videos representative of math class. It was our hope that they presented parents with a view into their child’s classroom (“window”). Further, we hoped that Surrey teachers saw their classrooms in what was reflected (“mirror”). In that spirit, several videos in this summer’s collection included an open-middle problem.

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
Visual depiction of problems with a closed beginning, open middle, and closed ending from Dan Meyer

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.

Polynomials

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 x2 + kx − 8 factorable? x2xk? See the answer animations below.

x2 + kx − 8 (solution)
x2xk (solutions)

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 x2 + 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.

set of equations

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 3x2 + 7x + 6. What could they be? Here’s a problem, adapted from Open Middle Math, that also tackles adding polynomials:

adding polynomials

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 (3x2) + (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., Two trinomials add to 3x2 + 7x + 6. What could they be?) bridges this gap.

Integers

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

adding and subtracting integers
order of operations with integers

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

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.

order of operations with integers (solution)

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.

Percents

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…

75% of 48

… and another that utilizes tenths.

80% of 50

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:

commutative percents

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”!

x% of y is equal to 2x% of y/2; x% of y is equal to kx% of y/k

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.)

percent discounts

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.

Get yourself 920 calories, playa

sakir-10

How many watermelons do you see?

You might be a math teacher if you answered five. Whereas a normal person sees produce, a math teacher sees fractions (4 × ½ + 4 × ¾) or perfect squares (3² − 2²).

onlyinmathproblems

As a student, I never asked Sally’s question. To me, math was like an action movie (minus the action). It required suspension of disbelief. I accepted that.

As a teacher, I want more for my students. I want my students to use mathematics to better understand the world around them (i.e., the real one).

So when world-class pool player and Korean War expert Dr. Tae craved a burger, I wanted to find a connection between mathematics and the real world.

The best that I could come up with:

Two burgers and one order of regular fries have 2020 calories. Four burgers and three orders of regular fries have 4660 calories. How many calories are there in each menu item?

There’s plenty not to like about this problem. As a real-world application of mathematics, like a Subway foot-long sandwich, it doesn’t measure up. For starters, it asks students to pretend that the total number of calories is known while the number of calories in each menu item remains a mystery. “We all use math everyday” meets “Yeah, right.” It would be easy to dismiss this problem. Pseudocontext. Full stop.

However, I also want to use the real world to have my students better understand mathematics. Whereas I’m critical of this problem presented as an application, I’m much more accepting of it as an investigation.

To introduce solving systems of linear equations, I have asked similar questions. My students would reason that if two burgers and one order of fries have 2020 calories, then four burgers and two orders of fries must have 2 times 2020, or 4040, calories. Comparing this with the number of calories in four burgers and three orders of fries means one extra order of fries adds 4660 – 4040, or 620, calories. Two burgers must then have 2020 – 620, or 1400, calories. Each burger has 1400 ÷ 2, or 700, calories. Students have solved a system of linear equations using the elimination method. All before having x‘s and y‘s thrown at them. My role was to help my students link their ideas within this context to this:

Solve using elimination.
2x + y = 2020
4x + 3y = 4660

Is that the real issue?

Hey, I just met you and I wanna rock your gypsy soul

Carly Rae Jepsen’s “Call Me Maybe” passed Van Morrison’s “Into the Mystic”.

I’m referring to my iTunes library, of course.

It wasn’t me. Meet the culprits:

First, “Van the Man”. On October 13, 2008, I added “Into the Mystic” to my library (‘Date Modified’ in iTunes). I’m calling this t = 0. I’ve played it 62 times. I last played this “song of such elemental beauty and grace” 1284 days later on April 19, 2012.

Jepsen’s up next. “Call Me Maybe” was added (not by me) on February 28, 2012. This is 1233 days after I added “Into the Mystic”. Seventy-five days later, on May 13, 2012, I listened to this sugary pop tune for the 63rd time. This is 1308 days after adding “Into the Mystic”.

NB: Screenshots of the iTunes Summaries for both songs would make a better first act. Here’s the summary for “Call Me Maybe”:

My initial questions were:

  • When did this happen?
  • Could I have predicted this?
  • How will the number of plays compare in the future?

I modelled this situation using a system of linear equations. For the Irish singer-songwriter, we get p = 0.05d, where p is the number of plays and d is the number of days. For the Canadian Idol, we get p = 0.84d − 1035.72.

Comparing slopes is an obvious discussion topic. The line for “Call Me Maybe” is much steeper than the line for “Into the Mystic”; the rate of change is 0.84 plays/day versus 0.05 plays/day.

This problem can also be used to explore unit rates. Unit rates can be expressed in more than one way. It’s about what one is one.

I wanted to express the equation p = 0.84d − 1035.72 in the form − 63 = 0.84(d − 1233). Slope-point form tells a better story than slope-intercept form in this situation but my GeoGebra skills are rusty.

Having students look at their own iTunes libraries might make a better investigation than practicing solving catch-up problems like this:

I assumed that this situation could be modelled using linear relations. For “Into the Mystic”, fair enough. I think this reasonably approximates the real data. Outside of perhaps when I was commenting on Michael Pershan’s blog, I didn’t go through a Van Morrison phase. Van Morrison is in my wheelhouse and “Into the Mystic” is just in the rotation. The number of plays per day is (almost) constant.

For “Call Me Maybe”, this assumption is likely incorrect. The song’s got legs but the instantaneous rate of change has to be decreasing, right? For my mental health, I hope it is. That many plays would surely take its toll.

And what if Carly has competition?

What if I modelled this using a logarithmic function? Check this out:

Note that ≈ 5½ years after first being added to my library, “Into the Mystic” can be expected to pass “Call Me Maybe”. The natural state of the universe is restored.

Update: I learned how to animate my GeoGebra construction. Also, I corrected a math mistake. (What was my misconception?)