DRAFT DRAFT DRAFT DRAFT DRAFT DRAFT DRAFT DRAFT DRAFT

Note to the reader: as of December 2010, I have taught this unit and project.  I will keep my running feedback on ways to improve this unit at the top, knowing that this could use some MAJOR revisions.

Running list of things to change…

• De-emphasize or somehow change focus on the independence of motion.  Too many students got wrapped up in the two mini-experiments and did not spend enough time analyzing the motion-detector experiments.
• For horizontal motion experiment, need to find materials that are less susceptible to slow-down.  The motion detectors are quite accurate and getting kids to say that the motion was linear when the slope was obviously not constant was a bit of a stretch.
• For both horiz and vert data, get enough trials so that each student his her own data to analyze.  A summary of each of these analyses could make their way into the final paper, while each individual paper describes the data THEY analyzed.
• Further work needed to reach student understanding on linear and quadratic regression.
• If using the video, introduce video tools (like Tracker) right away.
• Have students make videos or choose videos from the internet to analyze.
• MOTION DETECTORS would be valuable from the start.
• Technology use in general should be better supported: putting graphs in Excel, performing formulas to analyze large pieces of data (ex: performing second differences on a large table), statistical measures to analyze physical variances and prove or disprove a rule.
• Use activities like Barbie Bungee to supplement ideas needed for the final project (error in measurement, regression and the like).
• Use toys, marbles, cars, and more ramps!
• Re-evaluate exactly what math was needed to solve these scenarios.  Radical equations?  NOT NEEDED!

http://www.physorg.com/news/2010-10-father-son-iphone-hd-camera.html

The unit developed below was greatly aided by my involvement in the Math for America PLT on Problem Solving/Project-based math.  Hip hip, hooray!

Timeline: This unit began on November 8th, 2010, and will conclude by December 23, 2010.  This includes time for initial activities, a smaller “model project,” and a longer final assessment project.

Lesson Planning Calendar: Always updating, always a “draft,” but you might find it helpful.  Includes a description of the lessons as they actually occurred.

Essential Questions

Sometimes I don’t know what to do with these — I’ve made ones where I say, “Wow, that’s a great EQ,” but then what does it do for me?  So, and this seems obvious now, why not just make the EQs the exact project questions I have in mind?  (Generally speaking, so as not to over-define the project, but still putting that goal in mind.)

EQ: How can you predict the movement of a flying object?

EQ: What methods do physicists (and mathematicians) use to solve such problems?

Related EQ1: What are the mathematical models for objects in free-fall?

Related EQ2: Why are there laws of motion?  How do we know they are true?

Specific Learning Objectives

(using semester-long numbering) [standards 12 and beyond have not been assessed in class yet]

• Standard 6.  Solve equations requiring fractions and decimals
• Standard 7.  Convert between decimals and fractions
• Standard 8.  Solve quadratic equations with no linear term
• Standard 9.  Simplify radical expressions [including fractions, radical in denominator, etc]
• Standard 10. Solve radical equations
• Standard 11. Perform operations on irrational numbers
• Standard 12.  Perform operations on imaginary and complex numbers [advanced topic]
• Standard 13.  Create linear and quadratic rules
• Standard 14.  Create linear and quadratic graphs
• Standard 15.  Solve quadratic functions

Additional learning objectives include physics methods, describing simultaneous horizontal and vertical motion, developing a physical model from motion detector data, using video analysis tools, and writing technical documents.

Assessments

Concept Quizzes: Each “Standard” is assessed via a cycle of quiz questions given on a weekly basis.  Students can take “reassessments” at any time to show that they have mastered a topic that they struggled on earlier.

Project: Students will produce a written document explaining their explorations and conclusions for their research question, and they will also be able to verbally defend this document.

Packet: Two “toolkit” packets will be used for classwork.

Summary of the unit

We begin with story graphs.  Then we are developing a “quadratics toolkit.”  This happens by Thanksgiving.  (9 class days so far)

Coming back from turkey day, we look at the Kobe video.  This sets up the need for a “physics toolkit.”  We do experiments with motion detectors to determine an appropriate models for horizontal and vertical motion.  Additionally, the Kobe video provides a pre-final project model of the kind of technical writing they will need to do for their final project.  They also get to develop their ability to work in collaborative groups.

In the final phase, they have their own exploration to pursue.  Each student will use their group as a resource, but every student is writing their own final product.  The basic assignment is to come up with an application of the quadratic models they developed in their groups (and for which the Kobe assessment serves as a check of the model).

Online Resources

Quadratics stuff (though sometimes it’s hard to draw the line, you know?)

1. Functions of time [language support activity]
2. Function notation [simple numerical activity]
3. Linear vs. quadratic [Section 5.2]
4. Characteristics of linear and quadratic functions [Section 5.3]
5. Graph transformations [Section 5.4] [edit: scope of this activity changed to reduce time needed; now it is a guide for graphing quadratics using technology]
7. Multiplying binomials / Solving quadratics with factoring [Section 5.8]
8. Supplement: Completing the Square [Section 5.10]
9. The Quadratic Formula [Section 5.12]
10. Quadratic Strategy Guide [Section 5.13]

The Physics Toolkit

1. What is Physics – The Kobe video / or the Rocket video
2. Horizontal and Vertical Motion [Pirate Treasure Hunt]
3. Dropping/shooting a bullet
4. Motion detector – models of horizontal motion
5. Motion detector – models of vertical motion
6. Parametric models of motion [Section 5.6 and 5.7]
7. Write Up Your Results [info on Technical writing]

notes: deleted sections for this time around are: #7: Video analysis [activity: path of a ball / Kobe], #8: Newton’s Laws, a specifics Physics glossary [just put this in the quadratic toolkit]

The Karplus teaching model

• Exploration
• Invention
• Application

Really, just read the post linked from above.  Can’t say it better myself.

Backwards Planning Template: How do we plan for problem solving success?

1.  What problem should we do?

Bob is flying through the air.  Where will he land?  (and variations of this scenario)

2.  What math content do students need to know before they can successfully complete this problem?

• Factoring and unfactoring quadratic expressions
• Use of the four representations: words, tables, graphs, and equations
• Writing and using equivalent quadratic models (polynomial/standard/vertex/factored)

3.  What thinking skills or habits of mind do students need to learn to successfully engage with the problem?

• Generate and pursue answers to mathematical questions
• Record data and interpret (paper and on calculator screen)

4.  What structure makes sense for my classroom?

Students will work in groups of about 4 students each.  These groups will be mixed with regards to skill level, language background, gender, etc. as much as possible.

5.  How will I assess student learning?  What is the final product?

Students will produce a written document explaining their explorations and conclusions, and they will also be able to verbally defend and explain this document.

Textbook Resources

COMAP: Modeling with Mathematics: A Bridge to Algebra II.  (Warning: I think this is from the Texas version)  Chapter 5 has all the resources!

Note: First I typed all of the section headers with summaries.  I then bolded the sections that are most necessary to the goals of this unit.

5.1. – using motion detectors (replace this with Dan Meyer model lesson on story graphs)

5.2 – Linear vs. non-linear functions

Assignment: Interpreting motion graphs (save this for later)

Assignment: finding quadratic functions in tables

5.4 Transformations on y=x^2

Assignment: Graphing calculator activity

5.5 Interpreting traffic graph

5.6 Graphing parametric equation on graphing calculator

Assignment: graphing parabolas, using vertex form

5.7 Connecting parametric equation to factored form of equation

5.8 Solving quadratic equations by factoring

Assignment: Using area model of quadratics

5.9 Finding vertex point, interpretation

5.10 Using inverse operations to solve quadratics

Assignment: Completing the square model

5.11 Strategy discussion

Assignment: practice

5.13 Three ways of representing a quadratic equation

Assignment: practice

5.14 Area problem

5.15 Motion with Pendulum

Assignment: Diagonals problem

5.16 Reflections using patty paper

Assignment: domain/range inverses

5.17 Inverses

Assignment: practice

5.18 Pendulum returns

Assignment

5.19 Rolling stone / using data

Assignment

MODELING PROJECT

5.9