Using technology not simply to do things better, but to do better things.

Professional Development Workshop
7 May 2003

Thinking About and Doing Problems

Thinking About Other Web Resources

Checking in at the Math Forum shows that in addition to a somewhat restructured Problem of the Week environment, there is a newly evolving feature - Math Tools - that invites visiters to find, explore, and give feedback on Web-based interactive sites and their associated resources. This will soon be expanded to other interactive technologies, including software and graphing calculators, and will provide an arena for people to share their experiences, opinions, lessons and materials, as well as a place for developers to get inportant feedback as they create tools to support teaching and learning mathematics.
For example, check out the tool and materials in
[Math 7 > Topic: Sequences & Series > Tool = Golden Rectangle], or
[PreCalc > Topic: Limits > Tool = Cantor's Comb].
Let's take about a half hour to explore the structure of the Math Tools site, as well as exploring some of the many resources available there. Work in pairs or threes and try to identify two or three discioveries worth sharing with the whole group.

Some of you may not know that the Lucent PoW is still active. Let's look briefly at the current problem, Hogania, since there is a terrific support page on conversions among units that I wouldn't want you to miss.

Thinking More About Work With Geometer's Sketchpad v4
  • Stimulated by my work with pattern blocks, I was led to investigate the properties of regular stars. Below is an image created with regular 5-pointed stars and rhombuses whose angles measure 72° and 108°.

    Let's define a regular star as a convcave polygon in which:
    1. all sides are congruent, and;
    2. vertices alternate between being a 'point' vertex (having an acute internal angle and creating a convexity) and a 'dent' vertex (having a reflex internal angle greater than 180°, and forming a concavity), and;
    3. all point angles are congruent, and all dent angles are congruent.

    GSP-4 can help us explore a curious relationship - the connection between the measures of the 'point' and 'dent' angles.
    1. Let's begin an exploration by creating a regular 3-pointed star whose angles are dynamically changable. You might start by constructing the radius in a circle; repeatedly rotating this radius and it's endpoint 60° will subdivide the circle into 6 sections.
    2. A smaller and concentric circle will intersect the radii. Use the segment tool to alternately connect points of intersection of the radii on the inner and outer circles to construct the star.
    3. Measure one of each type of angle, and calculate their sum. Watch the sum as you dynamically alter the shape of the star.
    4. Similarly, construct dynamic 4-pointed, 5-pointed and 6-pointed stars, and find the sum of a point and dent angle for each.

    In general, how could you describe the relationship of the sum of a point and dent angle for an N-pointed regular star?

  • We can explore algebraic relationships using sliders to manipulate parameters in equations. We will take advantage of tools that are already built-in to Sketchpad-4.
    1. Open the file: Sketchpad > Samples > Custom Tools > Sliders.
    2. Leave the sketch open, but minimize it. Open a new sketch. Note that holding down the Custom Tools button (at the bottom of the vertical toolbar) shows there are a variety of sliders available. Use the Basic Horizontal tool to create 2 horizontal sliders. Go back to get the select tool, then manuver the sliders so that their red points (origins) are lined up.
    3. Use the text tool to edit the label of each slider. You could label one as 'M' and the other as 'B'. You must also edit the label of the measurement for each slider so that they correspond.
    4. Use Graph > Plot New Function to get an equation editor. Note that you have choices for the type of equation. You could create the equation Y = MX + B; you click on the measurement to get the value in the equation editor.
    5. Use the select tool to click-&-drag the sliders to change the parameters of the equation, and the graph responds dynamically to these changes.
    6. Double-clicking on an equation allows you to edit it. Notice there are a set of built-in functions. Also, note that you can do complex functions like AX2 + BX +C, or AXsin(BX + C), etc.
    7. See Using GSP-4 as a Dynamic Function Plotter for more details and ideas.

  • The Folded Circle Construction Activity

Thinking About Mathematical Content

The Revised Core Curriculum Content Standards in Mathematics
( define the basic mathematical content that should be addressed:

(Numbered) Standards and Associated (Lettered) Strands
  • 4.1. Number and Numerical Operations
    • A. Number Sense
    • B. Numerical Operations
    • C. Estimation

  • 4.2. Geometry and Measurement
    • A. Geometric Properties
    • B. Transforming Shapes
    • C. Coordinate Geometry
    • D. Units of Measurement
    • E. Measuring Geometric Objects

  • 4.3. Patterns and Algebra
    • A. Patterns and Relationships
    • B. Functions
    • C. Modeling
    • D. Procedures

  • 4.4. Data Analysis, Probability, and Discrete Mathematics
    • A. Data Analysis (Statistics)
    • B. Probability
    • C. Discrete Mathematics--Systematic Listing and Counting
    • D. Discrete Mathematics--Vertex-Edge Graphs and Algorithms

  • 4.5. Mathematical Processes
    • A. Problem Solving
    • B. Communication
    • C. Connections
    • D. Reasoning
    • E. Representations
    • F. Technology

Evaluation Forms

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