Math Project Saturday Seminar - September 16, 2017

  • Posted on: 28 September 2017
  • By: Debbie


Have you seen “Open-Middle Tasks” from Robert Kaplinsky at At the last Saturday Seminar, we explored some, and then created others that can be used to review or practice skills.  I’ll describe how we launched the tasks, and a couple ways to prompt argumentation.

We didn’t use the images below to pose the problems, showing all info at once. Instead, we first drew the boxes on the board, then put in the fraction bar, and then said each box must use one of the digits 1 through 9. We let folks ask a clarifying question, “Can a numeral be used in more than one place?” To which we responded, “Not for this challenge, but maybe that could be an interesting place to go some other time.”  We then asked folks, “Raise a hand if you have an idea about what we could set as a goal for this challenge.”  We got various answers, including the goal of making the fraction as close to 1 as possible.  We also asked folks to come up with a way to compare the fractions they were testing without a calculator—to reason it out.

We pitched the other 2 tasks below in a similar way, giving everyone a chance to think about possible goals for the challenge. This didn’t take long—so don't think slow motion! It’s a sneaky way to get people to consider what is mathematically interesting—and then the hook is set, and they can’t help but start testing numbers and thinking about strategies!

The task above is also great for getting people (students are people too!) to ask spin-off questions as they go. For example, on the quest to determine which is greater, people were asking their group-mates, “Which is closer to 1, one version of a fraction, or its reciprocal? How can we tell?”

You can use a deliberate mistake to motivate them to make an argument. For example: “Since they are just reciprocals, it must be that neither is closer to one—they are both the same distance away from one.”

Or for the exercise shown below: “Raise a hand if you think I should be using the large numbers, 6, 7, 8, 9?  [All raise hands to agree.] Okay, then this must be it: 67 x 89.  Raise a hand if you think that will be the greatest.  Talk at your tables about how you know that won’t produce the largest answer when we multiply.”

 Another question was asked to instead prompt the construction of an argument:  “How can you tell which will be greater without multiplying it all the way out, 97 x 86 or 96 x 87?”  Some were thinking visually (using an area model) to consider that, while others were thinking about using a related problem (eg: 96 x 86) to reason about each of those.

The 3rd task is a tweak of Kaplinsky’s about finding 3 decimal numbers with a sum as close to 1 as possible (same constraints about digits as above).  Kaplinsky challenged folks to get it in less than 3 tries!

We did these 3 tasks in sequence, so that after folks had time to do the math themselves, we could have a discussion about the teaching.  But for students, one possible way to use such tasks would be to drop them in at just the right time in order to set up connections to the new material you need to teach.

CREATING MISSING NUMBER PROBLEMS…Deb’s and Scott’s and Kim’s Collaboration
You can create similar tasks for things you need to teach.  For example, suppose you are a high school teacher and you need your students to review what they learned about slope. How could you turn the review of slope into a puzzle? Rather than asking students to calculate the slope between two given points, you could give students some information, leaving other information out, and ask them to find a slope closest to 1, or closest to 0, or the steepest slope, or….

Deb’s First Thought…

Then Scott thought of some adjustments to the question.  He thought that some people could think of “flat” as meaning vertical, like a flat wall.  Which is true.  So someone could pursue this knowing that the word flat is ambiguous, expecting a clarifying discussion to ensue, and then get more specific about the prompt with the students.

Or he suggested perhaps alternative wording below:

Then Scott thought perhaps the word “level” would convey it better to students than “horizontal”. 

He also added, “There's a possible follow-up question about how many possible best slopes there are, and about how many possible optimal pairs of points there are.”

If you use the slope one, let us know how it goes!   I am curious if some students would graph, and reason about which are actually flatter lines, and if others would aim to get a numeric value as close to 0 as possible.  

Good luck stealing numbers out of problems to create your own—please share!