Construction Pod Game: Part A

Author:
Gerry Stahl
Topic:
Geometry

Welcome to the Construction Pod Game

The Construction Pod Game is a series of challenges for your pod to construct interesting and fun geometric figures. Many of the figures will have hidden features and your pod will learn how to design them. So put together your Construction Crew Pod with three, four, five or six people from anywhere in the world who want to play the game together online. The Construction Pod Game consists of several levels of play, each with a set of challenges to do together in your special online construction zone. The challenges in the beginning levels do not require any previous knowledge about geometry or skill in working together. Playing the challenges in the order they are given will prepare you with everything you need to know for the more advanced levels. Be creative and have fun. See if you can invent new ways to do the challenges. Each construction challenge has questions to think about and answer. These will help you to make sense of the challenges and your solutions. Your responses to the questions will help your team mates in your pod to understand what you discovered about the challenge and to know what you would like help understanding. Be sure to answer the questions and to read the answers from the rest of your pod. Try each challenge at your level until everyone in your pod understands how to meet the challenges. Then move on to the next level. Take your time until everyone has mastered the level. Then agree as a team to go to the next level. Most levels assume that everyone has mastered the previous level. The levels become harder and harder -- see how far your pod can go. Geometry has always been about constructing dependencies into geometric figures and discovering relationships that are therefore necessarily true and provable. Dynamic geometry (like GeoGebra) makes the construction of dependencies clear. The game challenges at each level will help you to think about geometry this way and to design constructions with the necessary dependencies. The sequence of levels is designed to give you the knowledge and skills you need to think about dynamic-geometric dependencies and to construct figures with them. Your construction pod can accomplish more than any one of you could on your own. You can chat about what you are doing, and why. You can discuss what you notice and wonder about the dynamic figures. Playing as part of a team will prevent you from becoming stuck. If you do not understand a geometry word or a challenge description, someone else in the pod may have a suggestion. If you cannot figure out the next step in a problem or a construction, discuss it with your teammates. Decide how to proceed together. Enjoy playing, exploring, discussing and constructing! ========================== List of Game Levels and Challenges PART A 1. Beginner Level Challenge 01: Play House Challenge 02: Play with Stick People Challenge 03: Play around with Points, Lines and Circles 2. Construction Level Challenge 04: Play by Dragging Connections Challenge 05: Play with Hidden Objects Challenge 06: Construct Polygons in Different Ways 3. Triangle Level Challenge 07: Construct an Equilateral Triangle Challenge 08: Find Dynamic Triangles 4. Circle Level Challenge 09: Construct the Midpoint Challenge 10: Construct a Perpendicular Line Challenge 11: Construct a Parallel Line PART B 5. Dependency Level Challenge 12: Triangles with Dependencies Challenge 13: An Isosceles Triangle Challenge 14: A Right Triangle Challenge 15: An Isosceles-Right Triangle 6. Compass Level Challenge 16: Copy a Length Challenge 17: Use the Compass Tool Challenge 18: Make Dependent Segments Challenge 19: Add Segment Lengths Challenge 20: Copy vs. Construct a Congruent Triangle Challenge 21: Construct a Congruent Angle PART C 7. Congruence Level Challenge 22: Combinations of Sides and Angles of Triangles Challenge 23: Side-Side-Side (SSS) Challenge 24: Side-Angle-Side (SAS) Challenge 25: Angle-Side-Angle (ASA) Challenge 26: Side-Side-Angle (SSA) 8. Inscribed Polygon Level Challenge 27: The Inscribed Triangles Challenge Problem Challenge 28: The Inscribed Quadrilaterals Problem Challenge 29: Prove Inscribed Triangles PART D 9. Transformation Level Challenge 30: Translate by a Vector Challenge 31: Reflect About a Line Challenge 32: Rotate Around a Point Challenge 33: Combine Transformations Challenge 34: Create Dynamic Patterns 10. Quadrilateral Level Challenge 35: Construct Quadrilaterals with Constraints Challenge 36: Construct a Rhombus Challenge 37: Quadrilateral Areas Challenge 38: Build a Hierarchy of Quadrilaterals PART E 11. Advanced Geometer Level Challenge 39: The Centroid of a Triangle Challenge 40: The Circumcenter of a Triangle Challenge 41: The Orthocenter of a Triangle Challenge 42: The Incenter of a Triangle Challenge 43: The Euler Segment of a Triangle Challenge 44: The Nine-Point Circle of a Triangle 12. Problem Solver Level Challenge 45: Treasure Hunt Challenge 46: Square and Circle Challenge 47: Cross an Angle 13. Expert Level Challenge 48: How Many Ways Can You Invent? Challenge 49: Dependencies in the World Challenge 50: Into the Future ====================================

Note to Teachers

The Construction Pod Game consists of 50 challenges that introduce the player to basic ideas of dynamic geometry as implemented in GeoGebra and teach the most important software functions. The challenges encourage thinking about geometric dependencies among points, lines, circles and polygons. The hope is that players will experience the excitement of mathematical discoveries and explore ways of deeply understanding and discussing geometry. The 50 challenges build step-by-step from doodling to major theorems of basic geometry. They provide hands-on involvement in problem solving and mathematical reflection. The sequence roughly follows Euclid and the US Common Core for geometry. The challenges were originally designed for use in the Virtual Math Teams project, in which small groups of middle-school students collaborated online, sharing a GeoGebra construction and a text-chat tab. The group of students worked together with no direct supervision, spending about an hour collaborating on each challenge. In the current Construction Pod Game, the challenges have been modified for use with the GeoGebra “Class” function, optionally within Zoom sessions. The new challenges can be worked on by individual students, with a teacher observing a dashboard of a Class of students progressing through the challenges. The Coronavirus has made it common for students to learn in online pods of about 5 students, rather than in traditional classrooms of about 30 students. This opens the opportunity for a more collaborative online learning experience. Although the GeoGebra Class mechanism does not allow multiple students to share a joint construction, they can work in parallel and discuss their work as they do it. The Class dashboard can be made available to all the students. If the work takes place in a face-to-face setting or in a Zoom session, the students can talk or chat with each other, as well as typing answers to the questions for each of the challenges and seeing what each other writes. The Construction Pod Game can also be used for an individual student in home schooling. Ideally, the student would find several other students (either acquaintances or online peers) to form a pod and collaborate. Although it is structured as a game, the goal should not be to compete, but to advance together as a united pod. Teachers who wants to use the Construction Pod Game should first make copies for themselves. They can modify their copies however they want, especially editing the text of the challenges or their questions to suit their teaching style, curricular goals or student characteristics. They can save their copy, publish it and press the “Create Class” button. Then they can invite a pod of students to the Class, both to work on the challenges and to view the dashboard. The Class can be embedded in a Zoom meeting and the meeting can be recorded by the teacher for review. Hopefully the students can collaborate among themselves with little or no teacher intervention during Game sessions. Students should be self-motivated to work through the levels of increasing challenges. The GeoGebra software provides extensive feedback about successful constructions, especially if students use the drag test. Pod mates can help each other in many ways. The teacher’s role can primarily be to integrate the sequence of challenges with complementary sessions of teacher-led classroom discussion(both introductory presentations before challenges and discussions of results afterward) and of individual student work (such as readings and homework). There can also be assignments such as reporting on Pythagoras, Thales, Euclid or Euler. The Construction Pod Game is divided into 5 Parts, each containing an average of 10 challenges. The GeoGebra resources for the 5 Parts are available at: Part A – https://www.geogebra.org/m/swj6vqbp Part B – https://www.geogebra.org/m/dnammypy Part C – https://www.geogebra.org/m/p7tx9vfp Part D – https://www.geogebra.org/m/vggypcdu Part E – https://www.geogebra.org/m/qhwajdzx Please let me know if you have any questions or to report on your experiences: Gerry Stahl – Gerry@GerryStahl.net -- August 2020 ==========================

LEVEL 1. BEGINNER LEVEL

Here is where you and your pod start to play with points, lines and circles.

Challenge 01: Play House

Questions. Please enter your answer to each question and read the answers of your pod mates.

How can you tell if a new point is placed on a line that is already there? Dragging a point with the arrow tool is called the DRAG TEST in GeoGebra. It is a very important way to make sure that you constructed what you thought you were constructing -- to be sure that things are connected properly. Always drag points you create to check them. If you want to construct a line segment, is it better to place the two end-points first and then make the segment go from one to the other, or should you just place the line and let it create its own end-points? If you want to create a circle, should you first create a point for its center and a point on its circumference, of should you just create the circle and let it create its own defining points?

Challenge 02: Play with Stick People

Questions.

Which points in the stick woman can move independently? Which points move the whole woman? Which points move parts of the woman? Why do some points move independently and others always move other points and lines? Can you tell what order the woman was created in? What was the first point, etc.? Can you create a stick woman that moves differently? Use the DRAG TEST to make sure your stick figure is working the way you want it to.

Challenge 03: Play around with Points, Lines and Circles

Questions.

How can you make a new point "stick" to an existing line segment? Can that point go off the ends of the line segment? How can you test to make sure that a point will always stay on a line segment? How can you test to make sure that one line segment always starts on another line segment? How can you test that a circle always has its center along a certain line segment? In the original construction, which points would you have to drag to test that end F of line segment CF always stays on the circumference of circle DE -- no matter how any other points in the construction are dynamically moved?

LEVEL 2: CONSTRUCTION LEVEL

At this level, you will play with geometric figures.

Challenge 04: Play by Dragging Connections

Questions.

What does each point in this construction control? Are there any points that cannot be dragged (except by dragging a different point)? Do they have different colors? What sequence of construction steps could have been used to build this?

Challenge 05: Play with Hidden Objects

Questions.

What is the difference between a Line and a Line Segment? What is the difference between a circle radius, a circle diameter and a circle circumference? Which steps did you have trouble doing? What is the difference between hiding an object and deleting that object? Which points are dependent on which other objects, even when those objects are hidden?

Challenge 6. Construct Polygons in Different Ways

Questions.

What are polygons with 3, 4, 5 and 6 sides called? What differences do you notice about the polygons constructed in these three different ways? Drag all the points around. What stays the same? What does this make you wonder?

LEVEL 3: TRIANGLE LEVEL

At this level you will explore dynamic triangles.

Challenge 07: Construct an Equilateral Triangle

Questions.

Did you construct your own equilateral triangle. Did you use the DRAG TEST to make sure it works properly? The equilateral construction opens up the world of geometry; if you understand how it works deeply, you will understand much about geometry. In geometry, a circle is defined as the set of points that are all the same distance from the center point. So every radius of a certain circle is the same length. Drag each point in your triangle and discuss how the position of the third point is dependent on the distance between the first two points. Is your triangle equilateral (all sides equal and all angles equal)? Why? How do you know? Does it have to be?

Challenge 08: Find Dynamic Triangles

Questions.

What kinds of triangles did you find in the figure? When you dragged the points, did any of the triangles change kind? For instance, can triangle ABF be a right triangle or equilateral? Discuss how this is possible. Are there some kinds of triangles you are not sure about? Why are you sure about some relationships? Does everyone in your pod agree?

LEVEL 4: CIRCLE LEVEL

At this level, you will start to explore circles.

Challenge 09: Construct the Midpoint

Questions.

Do you think that point E is in the middle of line segment AB? Do you think that point E is in the middle of line segment CD? Do you think your point J is in the middle of line segment FG? Can you prove that any of these are true (without measuring)?

Challenge 10: Construct a Perpendicular Line

Questions.

Compare this Challenge with Challenge 9. That construction of the midpoint also constructed a perpendicular. Challenge 10 extended the approach to construct a perpendicular through a point C that was not the midpoint of AB by making a segment DE that has midpoint C. Can you explain why this works? Can you extend the construction in this challenge to work through a point H that is not on line AB at all? Can you explain how your extension works? Does is still work when you drag point H all around?

Challenge 11: Construct a Parallel Line

Questions.

Do you see how to use the GeoGebra perpendicular line tool in the toolbar? It constructs something like you did in the last Challenge and hides all the construction lines and circles. Of course, you could also do the construction yourself. Most GeoGebra tools just automate constructions to save you steps. Do you prefer to do the construction yourself just using the elements of geometry: points, lines and circles? Did your new line (HI) stay parallel to your original line (EF) no matter what points you dragged? Explain why a perpendicular to a perpendicular is a parallel line. Imagine riding your bike in a city with a grid of streets. If you make two right turns, you will be riding a street parallel to your original street. Two more right turns (at right angles on the grid) might bring you back to your original street. If a right angle is 90 degrees, how many degrees is two right angles?

Continue to "Construction Pod Game: Part B"

Part B starts on Level 5: Dependency Level. Congratulations on mastering Part A! You now know how to construct basic geometric elements and relationships. In Part B you will learn how to make one element dependent upon another and how to copy lengths and angles that are interdependent.

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