On the PSAT, looking at Energy and Art in Math

The past week and a half have seen us focus as a class on three main threads.

The PSAT
Every one of our sophomores takes the PSAT as a means to practice and acclimate that kind of testing environment so that when they take it again as it Junior they are more accustomed to it. In our program, that means an additional layer of testing with multiple choices, since we spend more of our time immersed in deeper thinking about what we know of math and less on practice and rote problem-solving. As a result, the students and I agreed we should spend time just about every day just doing some practice problems which will serve a couple of purposes: give us a better review of the types of math that they are expected to know, cover distinct strategies about doing well on the test, fill in the gaps of areas that they think would help them do better on this first pass at the test. So that end, we’re using resources like the practice tests on majortests.com, the College Board website, and resources that we’ve accumulated to support the math side of the PSAT. We typically start by taking a small number of questions, and then debriefing them either as a class or in groups to better deconstruct the mathematical thinking that underpins those questions.

Energy Auditing
While we are waiting for our full set of equipment to arrive, we have been doing group work tied to understanding how to evaluate a space for electrical energy footprint. Perhaps a example will help in that regard:
Today, their challenge was to take the data from an air conditioner that had been running for 27 minutes, then look at its kilowatt hour usage, and estimate how much it costs to run that device for a month (current Hawaiian Electric cost: $0.34 per kWHr). Once they had that determined, they were then challenged with going to a building, counting how many air-conditioners are in that building, and estimating what the monthly electrical charge is to run the air conditioners in that building. I’ve attached Josh and Kris’s sheet that they gave me as an example, there is also a bonus questions that they were given to determine what the cost should be to run a fan instead – you’ll see that on the bottom:

The goal of all this initial work, is to create a more careful eye on their part, not just in being able to calculate correctly the energy usage of the device, but to also think about where these things are and how they might take a position with the client in helping them consider ways to save money and energy.

Math in Art

In my last blog post I talked a bit about the project we are undertaking to work with math and art. I just want to share a couple of artifacts that are still in their early stages, but they give a hint of the work that is to come. The pictures below is a shot of both a student hand-drawn art, and the mathematical models using Desmos that give an indication of how they can translate art into clear mathematical function and form.

***the skill and process of critiquing***

One of the things we believe fundamentally is the importance of critiquing and feedback in improving student work and developing a greater sense of autonomy/agency in student work. Once the students completed their first sketch, they used post it note pads to give each other feedback following our agreements development from Ron Berger:

Be Helpful
Be Kind
Be Specific

Examples of the process :

the students did blog posts last week in which they talked about their first round of drawings, the feedback that they received, and their directions. Here are two nice examples from? And? That show where they’re at and give some hint of where they’re going.

More updates to come in another week…

Mathematical Modeling, Authentic Math Activities and Standards

One of my goals for this year is to create (or borrow liberally or adapt) hands-on, minds-on engaging and authentic activities to teach and practice math concepts – both those in our Core Content (in my case Algebra II) and from the Common Core (which our Math Scope and Sequence aligns with).

For this past week I was designing around the following MPI core concepts:

Relation Properties,
Domain and Range,
Functions Properties,
Function Notation Direct Variation,
Slope,
Slope-Intercept form,
Graphing Lines

Which loosely corresponded to the following Common Core Standards:

Understand the concept of a function and use function notation (F-IF 1,2)
Create equations that describe numbers or relationships (A-CED 2)
Interpret functions that arise in applications in terms of context (F-IF 4, 5, 6)
Analyze functions using different representations. (F-IF 7, 8, 9)
Build a function that models a relationship between two quantities (F-BF 1)

My goal was to construct an experience that students would design and implement an experiment that would make them collect, analyze and model a behavior in the real world. This is what is looked like:

Mathematical Models and Modeling Functions
Goal: To design and practice the common core standards around linear model functions
Step 1. Define and design an experiment that you can conduct with existing classroom equipment that will generate a set of at at least 6 data pairs to examine the relationship between two variables of your choosing.
Your goal should be for this data to test a hypothesis that the relation is linear.
You must get permission for the teacher to move forward to step 2. You must define:
• your purpose
• your independent and dependent variables
• your anticipated domain and range
• your rationale for why you expect a linear relationship • your apparatus and procedure
Step 2: Conduct the experiment. Make sure to run AT LEAST 3 trials to minimize error.
Step 3: For your submitted report, you must complete the EVALUATION section of our standard lab report:
a table of calculated values
a graph that includes appropriate axes and labels as well as attempts to linearize your 
data
a statement of the relationship
a mathematical model of the data, including slope and intercept with correct units. 
(This should be in slope intercept form)
a brief discussion of the results and any divergent issues

The students proposed a variety of very interesting experiments (see below pics) – and although not all would actually lead to direct relationships, part of the ‘game’ was not pre-judging for them what might occur. That in itself led to some interesting conversations about best fit lines, the shape of data, and how we know what kinds of relationships are happening between variables in our experiments.

So how did it go? I think the notion of creating real opportunities to practice and implement our mathematical understanding are the places where real mathematicians and the habits of minds of mathematics at her. Certainly, it exposed areas where students were still unsure about something as simple as slope intercept form, or what kind of relationship a scatterplot of data that is not quite linear means. All in all, it lends itself to working and thinking as real mathematicians and scientists, which is the goal of a good STEM education. Here are two examples of submitted work from the activity (in pdf format):

slinky

Bounce Height of a Tennis Ball copy
So what’s next? The work that’s coming up next deals with inequalities, and so I’ve been inspired by some work that Alfred Solis did when he was at high Tech high on the connection between art and mathematical form. More to come on this in the next week… A little hint of where were going below: