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.

“They’ll Need It for High School” (Part 2)

So Part 2 was supposed to be about the big ideas in K-7 mathematics that students will need for high school. But that’ll have to wait for Part 3. Instead, more on times tables.

Three oft-used arguments for the importance of memorizing times tables:

  1. When learning higher levels of math, there just isn’t time to use calculators or strategies to determine basic facts.
  2. Besides, thinking taxes working memory which means by the time you’ve worked out the first part of the question, you will have forgotten the… Where am I?
  3. Because factoring.

1 & 2 are gospel. Well, so is 3; nevertheless, it’s the focus of this post. I have a couple of thoughts on times tables and factoring trinomials.

The reason some students struggle with factoring trinomials is not because they haven’t memorized products to 10 × 10. I can get away with this if we’re talkin’ Pythagoras. But factoring?! I mean, that’s all it is, right? To factor x² + 7x + 10, you just have to ask yourself, “What two numbers multiply to 10 and add to 7?”

HS math teachers, try this: give your students a quiz on factoring. Include both x² + 10x + 24 and x² + 25x + 24. Get back to me. For extra credit (yours, not theirs), throw x² + 6x + 5 in there. If your students are anything like mine, I bet x² + 25x + 24 gives them at least as much difficulty as x² + 10x + 24. What does this mean for these students? More practice multiplying by one?!

Of course, 1 × 24 falls outside most times tables. Recall of products to 10 × 10 gets us the factors of x² + bx + 60 – if b = 16. But x² + 17x + 60, x² + 19x + 60, x² + 23x + 60, and x² + 32x + 60 are fair game, right? Try c = 48. Or 72. Or 96. Or 100. What role does memorizing times tables play? What role does being flexible with numbers play?

My point, I think, is that these are different, albeit related, skills. In other words, the “it” they’ll need for factoring (trinomials) is factoring (numbers). And number sense. This has some implications for K-7: not necessarily more “What’s 4 × 6?” but more “A rectangle has an area of about 24 square units. What could its length and width be?” or even “The answer is 24. What’s the question?”; not thinking digits/standard algorithm but thinking – and talking! – factors/mental math strategies, e.g. 16 × 25 = (4 × 4) × 25 = 4 × (4 × 25) = 4 × 100 = 400 (via Sherry Parrish).

origami by @Mythagon nothing to do with post @k8nowak says put pictures in posts
origami by @Mythagon
nothing to do with post but @k8nowak says put pictures in posts

Say you’re still asking, “How am I supposed to teach them factoring when they don’t even know their multiplication facts?” When I introduced polynomial division in Math 10, some of my high school students didn’t even know long division. So I taught division of numbers and polynomials side-by-side, highlighting connections. Can the same miiindset (channeling my inner Leinwand) be applied to factoring trinomials and times tables?

And what about something like x² − 2x − 24? If that – asking yourself, “What two numbers multiply to -24 and add to -2?” – is all it is, why not factoring trinomials to teach multiplication (and addition) of integers?

Part One

“Under the M… the square root of 12”

On this blog, sometimes I share my thoughts about transforming math education. This is not one of those times.

Here, I’m using my blog as a digital filing cabinet.

One activity that my students enjoyed was MATHO (and its variations FACTO and TRIGO).

Have students select and place answers from the bottom of each column to fill up their MATHO cards. In some versions, I pulled prepared questions from a hat. In other versions, I translated answers to questions on the fly. For example, if I grabbed 2√3, I called out “Under the M… the square root of 12”. After a student shouts “MATHO!” ask potential winners to read aloud their numbers. (Remember to keep track of answers you have called.)

Nothing revolutionary here – just a fun way to review content.

Squares & Square Roots
Exponent Laws
Simplifying Radicals
Rational Exponents
Factoring Trinomials x^2+bx+c
Factoring Special Products
Trig Functions

By the way, if you are looking to read about changing things, please check out Sam Shah’s recent post, The Messiness of Trying Something New.