Monthly Archives: July 2016

Recap: Functions and Algebra, Day 5 (July 22, 2016)

The discussion on our reading in P to A on professionalism did not go far.  I am not sure everyone had a chance to read it – it has some very good information on how we can work toward collaboration, increasing our knowledge of mathematics and other things that were emphasized this week.  We tackled the last two standards (7 and 8) and watched videos of them being demonstrated in different classrooms.  Larry made a good point that the standard of structure could also relate to the structure of the classroom environment that he witnessed in the 5/6th grade room.

When the discussion was opened up as a wrap up there was talk of the benefits of hearing from teachers from different grade levels and being able to interact and discuss how math fits across the grades, it was mentioned that productive struggle came alive this week, we felt we could relate better to our students in terms of the rigors of the struggle and having long days of learning,   There was a recommendation that an opportunity for application would be valued in terms of time to spend creating a lesson using feedback from participants.  We are leaving with buckets full of new ideas, but it would be helpful to apply this to at least one lesson before leaving would cement things and allow for collaboration.

We took the post test and after lunch began our Bungee Barbie capstone activity.   We did our evaluations and then headed across campus for the Barbie drop.

Please make sure you use this space to communicate with us or each other as you process all you have learned these two weeks.  We are happy to be in community with you all.  Have a great rest of the summer.

Recap: Functions and Algebra, Day 4 (July 21, 2016)

We began talking about the vocabulary gallery walk and the benefits of using this type of activity both before and after covering a certain topic.  We then moved into talking about the concept of skills that translate with the ideas of Larry’s son at college and some of the number games in the power point that relate to input/output, functions (processes), and inverse operations.  We discussed the idea that these games would play out differently in the different grades and had two very different representations on the board for the idea of the inverse operations problem – “get me where I started”.

Charles then began with the pool membership problem and once we agreed on the additional information we needed, we worked on finding a “break even” point.  We had a great discussion about the 40% off versus the 10% then 30% and the 30% and the 10%.  There were several methods of approaching this idea.

Marina had us work in pairs on explaining the thinking behind the students on page FA5.6 and being precise in our explanations.  We saw benefits in all the types of thinking.

We ended sharing the website which has some great examples of mistakes students make but also affords the opportunity for collaboration about students’ mistakes.  We watched a video about a student’s misconception of the number 18 and place value behind it although she understood the idea in a different context.  We then discussed the benefit of thinking through our thinking and the thinking of others and watched some videos of what that can look like.

Math hw – do the three act problem Dandy’s Candies on FA pg 5.8

Pedagogy HW read P to A pages 99-108 on professionalism.

Recap: Functions and Algebra, Day 3 (July 20, 2016)

We began today with a funny video on how mot to use questioning in the classroom.  It elicited a lot of good discussion about funneling versus focusing and how we use that in our classroom.  We also talked about Standard 4 and looked at an example video.

Charles began with the material on functions.  We spent a lot of time understanding what makes something a function.  We learned the difference between range and co-domain. We connected the idea of function to the previous concept of process and then learned function notation.   There was a lot of good discussion about what makes a function one-to-one, onto and finally what one-to-one correspondence is.  Before lunch we began with the idea of functions relating to counting but got a bit hung up on the idea of what the domain and co-domain were and whether the domain changed when we skipped a stone when counting.

After lunch Marina began a discussion about measurement relating to functions but many found the idea a little too abstract for them to understand.    When we started the pattern problems many participants realized they related to last night’s homework.  The discussions were great and the presentations showed many ways to think about the pattern.  We revisited the idea of (n/2)(n+1) from the Days of Christmas problem and reinforced the reason that the formula makes sense.

The perimeter problem brought three very different and powerful approaches to y=4n and reinforced the idea that it could be repeated addition of n four times or repeated addition of 4 n times. There was also a very interesting question about why the candle burning problem was written as y = 12 -2x instead of in slope intercept-form of y = -2x + 12.  We ended with a good discussion of how to think about the f(x) notation.

We ended with a recap of our thoughts about questioning listed below:

  • A focusing question doesn’t lead to a one word answer.
  • A focusing question doesn’t just relate to this particular problem.
  • Ask a student to rephrase their thinking because that’s the basis for our questioning.
  • Let the students defend their positions (Socratic method)– issues and errors may just fall out.
  • Keep in mind the offense and defense model of adjusting on the fly.
  • It is important to know where the kids are going mathematically beyond what we are teaching them so we can question effectively.
  • It is also important to have thought about questions ahead of time so that we don’t have to come up with them on the spot. Keeping notes from year to year is helpful.

We also talked about the think-pair-share idea:  How do we deal with the fact that we can’t attend to every group at the same time, ways to ensure students remain on task and ways to reassemble them after.  We watched a video of a first grade class and Standard #5 – we saw that the paper plate activity modeled the idea of algebra but discussed other ways the teacher could have set it up and had the students more involved.  We ended with a gallery walk putting definitions, floating capacities, pictures, questions, etc. on the words – slope, y-intercept, x-intercept, proportion, and linear relationship.

By request I am listing some of the ideas here but please continue the gallery walk tomorrow morning in case things were added after you saw it.

slope:  y = kx,(underline k) y = mx+b (underline m), rise/run, change in y/change in x, + or -, horizontal line has zero slope, constant rate of change, think of a straight line, daredevil compare the steepness of a hill for sledding or comparing ski slopes with diamonds.

Linear relationship: predict where a point will be on a graph, constant rate, graphs like a straight line, one-to-one, not quadratic, + or – slope, comparing fees for phone companies, y = mx+b, each x has only one y, pairs and how they relate to form a line.

x-intercept: where the line crosses the x axis, picture drawn, phone bill minimum with no extra minutes, reserve a kayak(0 hours rented), ?? posed – what is its relation to domain??

y-intercept: Deposit on kayak, y = mx+b (circle the b), picture drawn, f(x)-range, point where x=0, monthly charges for phone bill,

Proportion: a:b, a/b, a to b, 2 equivalent ratios, line going through the origin, “parts of”, “k value”, one ration = to another, graphing lines

Math HW – finish page 4.18

Pedagogy – read P to A pgs 59-69 and in your journal use your textbook or curriculum guide to list the floating capacity for : doing and undoing process, rate, proportional reasoning, linear relationships.


Recap: Functions and Algebra, Day 2 (July 19, 2016)

We began today discussing our charts for the bus stop problem.  We talked about the benefit of titles, noting slope, noting “not drawn to scale”, pictures, color coding and whether or not scales on the x and y axes were helpful or not.  The graphs were amazing and I was so impressed with the process all the groups went through in drawing them.

We then discussed the idea of piece-wise functions and their use and moved into Standard 2 of CCSSM and watched a video of what this standard could look like in a 5th/6th classroom.

Charles picked up with Unit 3  material and reviewed the ideas of linear and proportional.  We had some good discussion about the exchange rate and how it works with rounding.  There as a question about when topics like this are handled in school.  Unit rates are in 6th grade and it builds to proportional statements in 7th grade.  One of the issues is using 10/9 or 1.11 and which is the numerator and which is the denominator.  There was also discussion about having students learn how to express ideas mathematically like “five plus seven” is 5+7.  Also we can talk about conversions younger grades discussed converting cups vs. ounces and understanding that both are valid representations of an amount and how we talk about how to decide on a scale when graphing. These are important building blocks to get to what they will be learning later on and we can use these ideas to connect to what they already know.   We also discussed the benefit of younger grade teachers seeing what types of skills seem to be stumbling blocks later on in order to inform their teaching of these concepts.  I was so excited by these questions and how participants were trying to think through problems like this that their age students could relate to.

Marina began talking about what makes a relationship linear and what makes it proportional.  We also talked about the equation of a line.

We ended the day by talking about Standard #3 and then discussed the concept of funneling and focusing questions.  We than moved into a questioning activity to practice the use of effective questions.  We will follow up with this in our homework tonight.


Math p. 3.16 #8, 10, 11, 12   Also determine why ax + by + c = 0 is more versatile than y = mx + b.

Pedagogy Read P to A 37, 39-40  Listen to your audio recording or think about the questions you asked in our activity.  Using pages 37, 39-40 to examine the questions you asked while playing teacher and respond to these prompts:

  1.  Were your questions funneling or focusing
  2. How might you change it now?
  3. What type of questions might be more productive for eliciting student thinking?  Write a few new questions – refer to pgs 36-37.

Percolating assignments:  When it is “ok” to use funneling questions.

Recap: Functions and Algebra, Day 1 (July 18, 2016)

We began today talking about norms for our time together.  We came us with:

  • Don’t be afraid to try a problem even if you think it is too hard
  • Be respectful of the speaker
  • Give space for trying problems on one’s own
  • Enjoy the productive struggle
  • Be patient with yourself

We then watched a video of a first grade class and watched the class explain the thinking of the presenter and talked about the importance of listening to other ways of solving problems and being able to understand them.  It is important that we pay attention to the speaker and be able to rephrase it.

Charles began the discussion of our materials with everyday examples of inverse processes.  There was a good discussion about why squaring is not an invertible process.  The idea of writing the composition processes was a new idea to some but we discussed using boxes for the operations and thinking of them as verbs and the input and outputs were nouns and not boxed.

After a break Marina talked about the idea of equivalent expressions.

After lunch we began the next unit and introduced graphing of ordered pairs.  We discussed the idea of inverse functions as well.  We liked the video representation of the chocolate milk with the extra scoop of chocolate and how can we fix it.  Charles wrapped up the graphical representation of the proportionality and approximation.

At the end of the session we discussed some of the standards from CCSSM and then worked on a problem of graphing distance versus time.  We will pick up with those graphs tomorrow.


Math:  Finish FA1.10 and challenge FA1.12 – pick one

Pedagogy:  Read P to A pages 35-41, 53-57  and the article Questioning our Patterns of Questioning.  In your journal respond to the following prompts:

In your notebook write a response to these 2 prompts:
1. in questioning small groups of students working on a problem, a teacher noticed that when she
asked “focusing” questions , the students continued to look at their own work and continued to
engage in gtheir own dialogue.  When she asked “funneling” questions, the students looked up at the
teacher.  Comment on these observations.
2.  Look at Maddie’s work and Gabe’s work shown in fig. 22 on pg. 55 of P to A.  How could Ms. Lewis
leverage the student’s representation to develop Maddie’s understanding of the problem?

Recap: Math as a Second Language, Day 5 (July 15, 2016)

We began today by discussion about what we want to praise in our classrooms and what our goals are for our students.  We watched a video about the power of praise when we are praising the effort and not the “intelligence”.  We brainstormed words we can use to do this in our classrooms.

We then continued with the study of fractions by talking about multiplying fractions in terms of the adjective noun theme (and related it to the idea of rates) and geometric representations for the multiplication.  This last topic yielded a lot of productive struggle when we jumped from proper fraction multiplication to improper fraction multiplication.  Several models were presented from the participants and we spent time at tables figuring out how they related to each other and why some did not seem to be valid.

After lunch we began the concept of division and we discussed the multiple ways of approaching it.  Before going on to the estimation topics we had each table become an expert at a particular problem from the problem set in the material.  We did a think, pair, share and then the groups created posters to represent their answers.  The presentations gave participants a chance to practice their vocabulary and mathematical reasoning and gave observers a chance to ask probing questions about the material.

The estimation page was a good way to look at ordering fractions and we worked through some issues with the list that was close to one.  The one fraction 26/51 made the pattern a challenge to find but we talked it through and came to a consensus.  Charles wrapped the section up with a discussion about inverse and the importance of not dividing by zero.

Have a great weekend.


Recap: Math as a Second Language, Day 4 (July 14, 2016)

We began today with a further discussion about the goal of this seminar in terms of deepening and broadening our mathematical understanding to be able to follow our students down their lines of thinking and determine if they are valid or explain why they are not.  We also watched the video about how one teacher does a formative check before moving to the next concept she is trying to introduce and related it to the iceberg model.

Charles then discussed an algebraic way to think about a negative times a negative and Jonathon added an idea of thinking of multiplication as repeated addition and how if we move from 8+7+(-3)+5 to 8+7+5 by “Taking away” the negative 3 our total ends up going up by 3.  And if you do that with 2 negative threes it goes up by 3 twice (or 6) .  He used this reasoning to explain why -1 times -3 is 3 and -2 times -3 is 6.  He also related it to a cable company mistakenly charging your bill for HBO $10 a month.  By the time 3 months have gone by they have mistakenly charged (double negative) you 30 dollars and will have to pay that back.

We then moved in to the discussion about division.  We talked about describing it as splitting up an amount to different individuals and even how to handle the “extras”!  We also discussed how it is the inverse of multiplication so the rules for signs are the same.

We spent a bit of time with the idea of taking a rectangle with length 8 and width 3 and transforming it into a rectangle with length 6 and width 4.  And talked about how the perimeter changes as the area stays the same.

Finally we discussed the ideas of the adjective and nouns in division and looked at the summary of these rules for addition, subtraction, multiplication and division.

Before lunch we began fractions.  We heard that “improper fractions” are also called “fractions greater than one” in some curricula.  We also discussed the adjective and noun theme with fractions.

We had multiple ways to view the fraction 2/3 by showing where it was in the interval 0 to 1 and then from the interval 0 to 2 and then moved to two dimensions by using area models and algebraic models and rates.

We talked a lot about simplifying fractions and why the word “cancel” and “reduce” can be unhelpful.

The pedagogy discussion involved talking about unhelpful words or phrases and then the Rules that Expire and why they expire.  Finally we ended with the difference between Ms. Flahive’s and Ms. Ramirez’s lessons and how they affect students’ learning.

Math hw p. 5.17 #9, pg 6.18 #3,6, p. 6.22 #6

Pedagogy hw : Read ‘Never Say Anything a Kid Can Say.” and respond in your notebook to :

As a community, we have focused on establishing a number of norms (ways of working) this
week.  We have also focused on a few messages about the learning and teaching of mathematics.
Please respond to the following prompts from youir perspective as a PMI workshop participant.
There are no right or wrong answers – we want you to respond from your perspecitve.
1.  Briefly describe 3 different norms that we have estabilshed about participating in the
workshop activitiies.
2. Briefly describe 2 different norms we have established about being a productive group member
3. Briefly describe 2 messages that you have taken from the workshop about what it means to
“know” and “do” math.
4. Briefly describe 2 messages that you have taken from the workshop about teaching math.

Recap: Math as a Second Language, Day 3 (July 13, 2016)

Today Andrea began with an informative talk about productive struggle and had us individually think about how we feel when we are productively struggling and when we are struggling unproductively.  We then went over the solution of the library problem and came up with different ways to approach it.

We did a gallery walk with the homework of coming up with a word problem involving subtraction.

Marina began our discussion of multiplication and we discussed how to view it as repeated addition and as an array.  We had a good discussion about the order of the numbers in the array and which means what.  It was determined that 2 x 3 means add 3 twice or 3 + 3 while      3 x 2 means add 2 three times or 2 + 2 + 2.  I realized that I was using the phrase “order does not matter” incorrectly – I should be saying that “the opposite order leads to the same answer”.

Before lunch we began finding the area of geometric shapes by only using the area formula for a rectangle.

After lunch we began discussing the distributive property and how it relates to factoring.  Many found that using boxes made the process of multiplication easier to see.

So 5(x + y) could be viewed as an area model and factoring could as well.

We also discussed the justifications for all the algorithms we use for multiplication.

We ended by using the Iceberg article to create icebergs for the procedures or skills that you determined in last night’s homework that you consider essential for your grade level to learn.  We did a gallery walk to make comments on the different icebergs and will pick up with that again tomorrow morning.

HW for tonight:

Math 4.9 and 4.12 – finish, pg 4.23 any piece of it.

Pedagogy – read P to A pages 48-52 and respond to the prompt: ”Review the ‘Beliefs about teaching and learning mathematics’  chart (p. 11, Obstacles).  What beliefs are evident in Ms. Flahive’s and Ms. Ramirez’s classrooms (see fig 21 on page 51)?  What impact do those beliefs have on students’ opportunities to grapple with the mathematical ideas and relationships in the problem?