Centers of Rotation and Supports for ELL (?)
The following are a series of dynamic geometry environments possibly related to the Finding Centers of Rotation Task from the Mathematical Thinking curriculum.
The first is what seems to be intended by the image of the applet shown in the materials.
The other six were constructed simply based on the prompt given at this URL:
http://mathandlanguage.edc.org/mathematics-tasks/geometric-thinking/finding-centers-rotation-math-task
Relative to "Supports for English Language Learners" here are some questions to consider?
- for ELL?
- How so?
- What role for language?
- Software? (or pencil & paper; ruler & tape)
- Participation?
- Diversity of Responses?
- Discussion?
- Seeing Connections in the ‘real world’? Or to Beauty?
The related applet implemented in Geogebra: Click and drag B to rotate the segment.
Alternative introduction without Dynamic Geometry (or pencil and paper)
Task:
- Find TWO ways of rotating a segment of fixed length (e.g., a ruler) around a point (e.g., a piece of tape on the desk).
- Come up with a description of HOW TO DO each rotation.
- Share these descriptions with another person and see if he/she can use only these descriptions (without showing) to do what you did.
- Revise your descriptions of how to do each so that it is clearer what to do.
- Create a poster with the descriptions and drawings for each way.
- [Might introduce dynamic geometry here and have students implement their approaches and/or come up with new ones]
- Discuss possible applications or contexts for the rotations
- Discuss which might correspond to the definition of a center of rotation.
The following are SIX dynamic geometry environments constructed as possible interpretations of the original prompt (without first looking at the image related to the included applet).
So What?!?
THE POINT: As an alternative to simply illustrating a definition with a dynamic geometry environment (or with pencil and paper), the suggestion is we might allow for and even encourage a DIVERSITY of responses/interpretations so that we (students and the teacher alike) have something to discuss … using our languages … and as part of this discussion we might (also) make sense of what a “center of rotation” is (“supposed”) to mean. Language(s) would serve as a resource in doing mathematics instead of being something worked around (or fixed by using software to ‘give’ or to illustrate the definition).
[1]
[2]
[3] Move point C ... then turn on trace for D & E
[4] Move point D. Then turn on trace. Add midpoint.
[5] Move point C. Note that C moves in a circle. How is D moving?
[6] Move Blue point. As this point moves in a circle what is happening to the other points?
Do you see any of the rotations of a segment of fixed length around a point in the following animated GIF. Which ones?
OR
Can you come up with contexts or applications related to each of the examples above?
A Context for Rotating Segments around a Point: Which ones do you see/notice? Others?
PARALLELOGRAM PROBLEM :: Also from the Mathematical Thinking Currriculum
"A parallelogram has 3 of its vertices at the red, blue, and black points shown on handout A.
1. Draw a point on handout A that could be the fourth vertex of a parallelogram with the red, blue, and black points as the other three vertices.
2. How did you decide where to put the fourth point?
3. Why did putting the point where you put it form a parallelogram?
A parallelogram has three of its vertices at the red, blue, and black points shown on handout B. Work with your partner on the questions that follow.
4. Find and draw a point that could be the fourth vertex of a parallelogram with the red, blue, and black points on handout B.
5. Discuss why this point forms a parallelogram with the three points that are given."
(Note, the respective sets of three points can be turned on or off by clicking on the letters at the left).
>>Discussion of Parallelogram Problem(s)<<
The following environments were created to support discussions of the original task(s). Possible alterations to the task(s) are suggested in the second environment. The alternatives can be more readily explored using this first environment.
Points related to Handout A are shown. For the points from Handout B deselect A, B, C at the left and then select (click on) points C, D, & E
Possible Results and Follow-Up to Parallelogram Problem including Alternative Task(s)
TASK 6: COMPARING TRIANGLES :: Also from the Mathematical Thinking Currriculum
"Now it’s your turn to compare some shapes. Start with a piece of paper that is a non-square rectangle, and be sure it is a different size than your neighbor’s rectangle. Fold your paper so that point A is directly on top of point C as shown in the picture below."
Note, the environment below replaces the 'picture' in the original materials. Point C now corresponds to point D. The following also comes from the MT materials but The labels for the figure below are in parentheses.
"a) As in our picture above, you should see some triangles. Outline the triangles on your paper using a marker. How many did you find?
b) Compare triangles ∆EDC {∆G'BD} and ∆FGC {∆EFD} on your paper. How are they alike?
c) Your neighbor folded a different rectangle. Ask your neighbor what they noticed. What did your neighbor discover about how ∆EDC {∆G'BD} and ∆FGC {∆EFD}are alike?"
Text has been added for clarification and discussion purposes. The suggested alternative(s) might be seen to retain the focus -- suggested in the title above -- on comparing triangles that are formed (including, but not limited to, the ones referenced above).