Category Archives: Functions and Algebra

Day 5 of Functions & Algebra (7/29/16)

We started the morning with Andrea and Fran sharing resources. Andrea showed the Google Folder titled PMI 2016 Resources for Participants. Then she showed the NCTM website, and discussed benefits of membership (remember that K-8 institutions get a great deal!!). This segment ended with Andrea and Fran sharing some books (here are the covers):


Andrew and Matt spent a brief time discussing Unit FA-4 on Functions, which are really just a formalization of the Processes from FA-1 with a new notation.  We then dipped into the Fox’s Furniture Store sequence from FA-5, both from the perspective of solving problems and evaluating students work on its mathematical merits.

Next we formed a large circle where Fran led a discussion about the readings on Professionalism from Principles to Actions.

After lunch, the group took the post-test (a necessary evil).  After that wrapped up Andrew and Matt tied up some loose ends and lingering questions: the Chipmunk formula from Wednesday’s homework, a strategy for solving the Milk Problem, and a justification for the “add the digits” trick for recognizing multiples of 9

It has been a fantastic two weeks!  We will be writing a longer post later with a full summary of these workshops and the workshops at Greater Allegheny and Brandywine as well.


Keep up your enthusiasm.  Seek change, and be patient.

Keep the commitment you made to yourself as you progress through the school year.

Don’t shy away from digging for the why of the mathematics you teach.  It all hangs together to form a cohesive whole.  You are welcome to send burning mathematical questions to Andrew.

Look for good candidates to recommend to PMI next summer.

Keep us updated on any revelations or experiences in your classroom that you can attribute to PMI.

Day 4 of Functions & Algebra (7/28/16)

Today we began with a discussion about the symbolic rules that describe the tortoise and hare problem. Fran emphasized that writing symbolic rules can be supported by working through the reasoning in other representations. Andrea shared some data about young children’s misunderstandings of the meaning of the equal sign and practices to avoid. We also watched a video of a young child using relational thinking to solve an open number problem.

We worked on the candle burning problem, which illustrates how a negative rate of change effects a linear relationship. After debriefing, Andrew talked through consequences of linearity.  We ended the morning with analyzing 12 situations to determine whether they could be solved via a linear relationship, a proportion, or neither.

After lunch we wrapped up some of our big pedagogy ideas by reflecting on last night’s messages, generating a list of things that need to happen in order to implement the strategies. We discussed more about the Standards for Mathematical Practice , focusing especially on supporting students to persevere.

We constructed a big iceberg wall together, representing ideas at each grade level that 20160729_104509contribute floating capacity to the concept of linear relationships.


We closed with a debriefing of the 12 situations on “What isn’t a linear relationship?” and highlighted the important features that distinguish them from each other.

See you tomorrow for our last day together!


Pedagogy: Read the section on Professionalism in Principles to Actions (pp. 99- 108).

Math:  None.

Day 3 of Functions & Algebra (7/27/16)

We are hip-deep in linear relationships (and a few things that aren’t).

Gallery Walk

We started the morning with posting some of the extension problems that participants wrote last night and then did a gallery walk.  Andrew highlighted two strategies for getting students to move to systematic or efficient strategies: using uglier numbers that are less prone to “I just knew it,” and multi-step problems where the task requiring the strategy is repeated multiple times and is part of a larger whole.

The Milk Problem

Then Andrea, Matt, and Andrew led a discussion of the milk problem, trying to make 2% milk from 1% milk and 4% milk.  A mathematical take-away lesson is “proportions don’t add, but amounts do.”  A meta-cognition lesson is “sometimes you just need to try something to see why it doesn’t work.”

Currency Conversion

Then Andrew continued the Currency Conversion problem, where now there is a fee involved.  We drew a lot of nice lessons out of it already:  parallel lines correspond to lines with the  same slope, the conversion rate corresponds to the slope of the line, the benefits of writing both fee-schemes as “convert to euros then subtract fee.”

Upside down teaching, real-world problems

We then moved into a pedagogy session, where we began with a discussion of the rich problems teachers developed during HW. Then each table generated a list for tomorrow’s iceberg activity (also from HW prompt). We also had a discussion of upside down teaching (from last night’s reading). Finally, we watched Dan Meyer’s TED talk.

Andrew shared the site which has mathematically rich games and puzzles that students will happily engage in.

Terminology for Linear Relationships

Matt led a discussion putting names (slope, y-intercept, x-intercept) to concepts identified in the Currency Exchange with Fee problem.

The Tortoise and the Hare

We looked at the Tortoise and the Hare problem, wherein a scenario is presented and participants had to ask for certain information (e.g., the speeds) and not just handed it from the start.  Table-groups were then given different follow-up questions and asked to present their answers to the rest of the group.  Along the way we saw how many natural questions can be answered via a graph.

We finished the day be repeating the teaching episodes from Monday, but now practicing purposeful questioning and eliciting student thinking.

Homework for tonight:


Write formulas for the Tortoise and the Hare problem that can answer this question: After x minutes, how many feet has the tortoise/hare/chipmunk run?  The following information was provided during class:

  • In the first minute the tortoise ran 1200 feet
  • In the first minute of running the hare ran 3000 feet.
  • The tortoise and hare maintain a constant speed.
  • The hare starts with a 10 minute nap, then runs.  The tortoise starts right away.
  • (Follow-up #7): A chipmunk starts from 12,000 feet behind the tortoise and runs at 2000 feet/minute.


  1. Listen to the audio-recording of your “revised” teaching episode.
  2. In your notebook, write down the questions that you asked.
  3. Examine the questions you used while playing the role of teacher and write about what you learned about trying to use purposeful questions and eliciting student thinking.
  4. Read one of the messages from Smarter Than We Think. If your last name begins with A-M, read Message #14. If your last name begins with N-Z, read Message #16.
  5. Reflect back over all of the pedagogy discussions we have had in PMI and start a list in your notebook titled “Things I can do to help my students persevere when solving rich math problems.” Add at least three things to your list.

BTW: The “Messages” we’ve been reading come from the excellent book pictured below.




Day 2 of Functions & Algebra (7/26/16)

We began by reviewing last night’s math homework using the “Jigsaw” strategy.

Then we moved on in our Functions and Algebra notes, and we learned about Dan Meyer’s 3-act problems.  We looked at what makes proportional relationships special, and then pushed ourselves to solve problems based on proportions.

After lunch we discussed our responses to last night’s readings from Principles to Actions about questioning and eliciting student thinking. We used the reading to analyze the questioning practices of an 8th grade teacher leading students in the “Water Tank Task.” We both watched a video of her teaching and read a transcript episode.

We started our discussion of unit FA-3: Linear Relationships, but did not get past the first problem on currency conversion.  We will pick up there Wednesday morning.

Math Homework:

  1.  Create a “sequel” to one of the problems (1-5) on page FA2.12 that pushes the solver to use a systematic or efficient strategy to solve it.
  2. Revisit the milk problem on page FA2.12.

Pedagogy Homework:

  1. Look through your curriculum materials and identify a lesson that you might teach in the first few weeks of school. Refer to the picture below as a reference to find two “traditional problems” and rephrase them to make them “rich problems.” (Figure is from “Putting the Practices into Action” by O’Connell & SanGiovanni, published in 2013 by Heinemann).
  2. Also look through your curriculum materials and identify at least one lesson or topic that you teach that helps students build “floating capacity” for an idea we’ve discussed in the “Functions and Algebra” course.
  3. Read the “Upside-down teaching” article and think about how this connects to some of the time constraints we’ve discussed in class.



Day 1 of Functions & Algebra (7/25/16)

We are back for Week 2!

Walk the Graph

We began the morning with walking the graph.

FA-1: Processes

Andrew led discussion on the informal concept of “processes.”  This is like input-output machines that many teachers are already familiar with.  Our discussion took us many places, but some of the highlights were:

  1. Some processes are invertible, that is, they can be undone to return to the original input for any input.  Inverse operations generalize the idea of how addition and subtraction are inverses of each other, and how multiplication and division are inverses of each other.
  2. Some processes are not invertible, such as multiplication by 0.  Knowing that a number produced 0 as the output is not enough to know what the original number is.
  3. We can break more complicated procedures into smaller processes.
  4. When chaining together procedures, the order matters.  For example, “multiply by 2 then add 3” is different from “add 3 then multiply by 2.”
  5. We indicate the order of procedures by using parentheses in an algebraic expression.
  6. Two procedures are equivalent if they always produce the same output given the same input.  For example, “add 3 then multiply by 2” is a different process than “multiply by 2 then add six,” but they are equivalent in that they always produce the same output given the same input. In other words, they represent different journeys to the same destination.

Role-playing helping a student

We divided into pairs and took turns role-playing a teacher helping a student with a mathematical misunderstanding. We audio-recorded these so that we can use them for tonight’s homework.

FA-2: Proportions

Matt led discussion on Proportions.  We explored the relationship between the number of feet and the corresponding number of inches, and contrasted it with a length and the area of a square with that side length.  We will conclude the discussion and work on related problems Tuesday.



Complete problems 1, 2, 3 on page FA1.9–10.


Read Principles to Actions: “Pose Purposeful Questions” (p. 35-41) and “Elicit and Use Evidence of Student Thinking” (p. 53-57)

In your notebook, write responses to these three prompts:

  1. In questioning small groups of students working on a problem, a teacher noticed that when she asked a “focusing” question, the students continued to look at their work and continued to engage in their own dialogue. When she asked a “funneling” question, the students looked up at the teacher. Comment on these observations.
  2. Listen to your audiorecording from today. Use fig. 14 on p. 36 and fig. 16 on p. 39 to write a description of your question patterns.
  3. How might you change your questioning to elicit and then use evidence of your students’ thinking to move the student forward to the mathematical goal of the problem?