Saturday, June 4, 2016

Starting the summer with SMP 7

Back in December I attended Grace Kelemanik's closing session at CMC North in Asilomar California on unpacking the math practice standards. In her talk, she shared with us her framework on the standards for math practice.  She says that some of the standards take the lead in thinking and some do the supporting.  Standards 2(quantities and relationships), 7(structure), and 8(repetition) are the avenues of thinking.  Students with a strong foundation in these three standards will have a starting point with which to begin problem solving, and they can jump lanes if their starting strategy doesn't work.  This powerful talk has had me thinking about the SMPs ever since.

 SMP 7 is a particularly interesting standard to look at in the high school curriculum because we can find structure in expressions, diagrams, graphs and more. This year I worked closely with the Algebra teachers in my district while we implemented a new curriculum. One problem in particular from our new curriculum is below. ( From Engage NY Algebra Module 4 Lesson 8 (Student View) )

Students use the symmetry of a parabola to complete the graph (structure in the graph).  Later in the problem set students make note of the pattern in the table of values (structure in the table).

There were additional problems that made use of the structure of a parabola in order to solve.

  • Is f(4) greater than or less than -6? Explain
  • f(-4)= -13.  Predict the value for f(4) and explain your answer.

Towards the end of the lesson there was a section called Finding a Unique Quadratic Function.  Below are some of the prompts from that section:

  • Can you graph a quadratic function if you don’t know the vertex? Can you graph a quadratic function if you only know the 𝑥-intercepts?
  • Remember that we need to know at least two points to define a unique line. Can you identify a unique quadratic function with just two points? Explain.

At one of the schools that I work at we spent two days exploring the structure of parabolas and questions like the ones above using Desmos.

Day 1 Activity: Broken Parabolas

Day 2 Activity: 1-3-5-7 Parabola Challenges

In the 1-3-5-7 Parabola Challenges students explored the symmetry of parabolas by locating missing points given a set of points on a parabola. For this lesson students only looked at parabolas with an a value of 1. We didn't use that language since it was an introductory lesson, but it allowed students to use 1,3,5,7 (etc.) pattern when comparing the differences in y-values.The hope was that this informal exploration would set students up to look for and make use of structure when they graph parabolas by hand later in the unit. 

Having students reason through the problems using a pattern can also make for a great error analysis activity. Some of the student work from Activity 2 is below.  Each of these challenges started off with a set of blue points, and students were able to drag the red points to complete the parabola. They would then turn on the green function to check that it is going through all of the red and blue points. At the end there are also a couple of unique responses for challenges 1 and 2 from the 1-3-5-7 Parabola Challenges activity that would have been great to share with the class.

The original version of this activity had students entering coordinates into a table instead of dragging points. I thought this would help them focus on the patterns in a table of values as well, which could be an additional problem solving tool for them to utilize later in the chapter.  This turned out to be too challenging for the group of students that I initially worked with.  Adding points to a table of values using a pattern requires strong number sense, and is a different skill then having students use a pattern to add points to a coordinate grid.  Here is the original activity in case you'd like to take a look.

Focusing on SMP 7 in the planning of these activities shaped them in ways that led to increased reasoning and discourse.  Of course this is not the only way to increase mathematical reasoning, but definitely worthy of some focus this summer.

Footnote:  I also wanted to thank Silicon Valley Math Initiative for an inspiring end of the year session.  Their summer focus is also on SMP 7, and we launched this work in our last session by finding examples of SMP 7 in the K-12 curriculum.  I've been thinking about SMP 7 and seeing examples of it ever since.

Tuesday, March 29, 2016

Global Math Department Talk

Thanks to everyone who joined the webinar Desmos Activity Builder: Best Practices for Charging Up Lessons!  Below are the resources from the session.


Sunday, January 24, 2016

My Favorite Calculus: Crazy Integrals

While reading on my computer today I received a notification from Dropbox that a shared file called "Crazy Integrals" had been updated.  This was pretty exciting since I created this activity four years ago and even though I am no longer teaching Calculus AB, the teachers are still using this activity.  This also reminded me that I could share this activity for the Explore MTBOS Week 2 blogpost challenge.

Before we launched into our integration chapter I did some pre-work to help students gain a strong understanding of both the notation and the concept of integration.  For Crazy Integrals I started by directing their attention the the diagram below.

 We talked about how finding a definite integral is like finding the area between a curve and the x-axis.  There are some finer details that need to be covered, and we brought them in both in the course of the activity and throughout the chapter.  The first two problems that we talked through are below.  Students noted on their diagram that the top piecewise curve is f(x) and the bottom piece-wise curve is g(x).
Students then discussed the best way to break up each area using vertical lines.  This helped them to rewrite each definite integral as two separate definite integrals.  Then they found each area.  We talked about how in Calculus an area below the x-axis is counted as negative.  Students worked in groups of 3 or 4 to complete the rest of the problems.  

Below are the other diagrams used for the activity.

There were all sorts of great questions that students brought up throughout the activity, most of which called on them to use precise definitions and language to answer.  For example, students wanted to know what to do with the smile and eyes/nose from the clown picture.  We had to use our working definition of definite integral to determine that they were extraneous to the problem.  Students also wanted to know how to find the largest/smallest x-values for the clown's hair.  We had to interpret the associated integrals (below) to know that those parts of the hair were not to be included.  

For the fish diagram students wondered why I only asked them to find the definite integral of the top curve.  Great question, and one that they could probably answer in their own groups.  

Overall super engaging activity with lasting pay-offs in understanding for the remainder of Calculus.  Students didn't want to leave class without finishing this assignment.

Note:  After the first couple of problems students stopped rewriting the definite integral as a sum of definite integrals, and instead got really into showing their work for finding the separate areas.  I figured this was fine as long as they demonstrated the understanding of the skill in the first part of the activity.