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Conditional Drawing with Properties

Lesson Overview

This course builds on the concept of the condition "if ... then ...", prompting students to explore how the relationship between moving point and fixed point(s) can result in different geometry shapes. Various properties of conic section will be involved. Students are also encouraged to free exploration of more creative relationships.

Learning Objectives

  • Build up “scan the whole stage”, and “draw when meeting certain conditions” as a drawing paradigm.

  • Explore the condition of distance to [fixed point] = (<, >) constant.

  • Explore the condition of distance to [two fixed points].

  • Free exploration of more relationships between two distances, i.e., multiple fixed points or not constant

Task Description and Resources

Task 1: Scan and Condition

Objective: Sprite move all over the stage and “pin” on certain points.

Key Concepts: conditions; repeat in repeat; coordinate system.

Suggested Steps:​ Program a sprite to move all over the stage systematically (and once). First pin on every movement (result in traces covering the entire stage), then add the “conditional pinning”, only pin on certain position.

Reference Code:

20.png
Painting .gif

Link to Scratch: https://scratch.mit.edu/projects/912815256/

Task 2: Circle: Track of Points Whose Distance to Fixed Points Equal to a Constant

Purpose: Explore the track of points whose distance to fixed points equal to a same constant and explain why.

Key Concepts: Circle and circular area.

Suggested Step: Try out the condition [distance to] fixed points = constant.

Reference Code:

21.png
Painting 1.gif

Link to Scratch: https://scratch.mit.edu/projects/1000401268/

Task 3: Distance to Two Fixed Points

Purpose: Use Scratch to graph all the points whose distances to two fixed points are equal. Then graph all the points with other arithmetic relations for the distances, including sums and differences. Drag the fixed points to different positions and observe how the graphs changed.

Objective: Explore various relationship between the distances of a same point to two fixed points and the tracks of points under different relationships.

Key Concepts: Bisector of a segment, Ellipse.

Suggested Steps:

  • Program a sprite to move all over the stage systematically (and once).

  • Pin at the points required using the “if … then …” condition and the block “Condition1 and Condition2”.

Reference Code:

22.png
Painting 3.gif

Link to Scratch: https://scratch.mit.edu/projects/1000401280/

24.png
Painting 4.gif

Link to Scratch: https://scratch.mit.edu/projects/1000401292/

Task 4: Distance to Two Fixed Points

Objective: Free exploration of relationships between distances from one point to two or more fixed points (e.g., distances from constant to variables; sum of distances to two fixed points è three fixed points)

Suggested Steps:

  • Program a sprite to move all over the stage systematically (and once).

  • Pin at the points required using the “if … then …” condition.

  • Try out condition y = (??)*x; y = (??) * x + (bb); x+y+z<??.

Reference Code:

25.png
Painting 5.gif

Link to Scratch: https://scratch.mit.edu/projects/1000401310/

Summary

This course guides students through conditional drawing with the relationship of a moving point (the drawing sprite) and various fixed points. Through the exploration, students will experience: (1) such relationships represent certain geometric shapes, and (2) the properties of circle, ellipse, and hyperbola. In addition, students can use some mathematical paradigm such as change constant to variable to do creative mathematical exploration.

Acknowledgement 

The author would like to thank Zhi Hao CUI for designing this lesson and appreciate all the anonymous teachers and students who participated in this research.

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