Geometry: Castles and ShadowsContent Guide - Andee Rubin
Supplies Needed for Workshop #3:
pencils, paper, scissors, rulers, calculators, tape, a variety of colored markers
About the WorkshopWhat is the theme of the workshop?
For some people, geometry was the most frightening part of mathematics; for others, it was the only part that made sense. We will try to shed some new light on the study of geometry by inviting you to use your hands and eyes to explore both familiar and unusual geometric objects.
Whom do we see? What happens in the videoclips?
We'll see students at a range of grade levels investigating relationships between two-and three-dimensional objects. In all of these classrooms, we'll see students working with their hands: building three-dimensional models, cutting and gluing paper shapes, making perspective drawings, and folding flat paper into solid objects.
What issues does this workshop address?
One issue we will explore is how teachers help students who are having trouble with a geometric task. What is the balance between helping students develop independence and guiding them - sometimes subtly, sometimes more explicitly - toward the mathematical goals of the lesson?
What teaching strategy does this workshop offer?
In two of the videos, the teachers have asked students to work in groups and then to report their work to the whole class. We will consider how teachers might comment on student work in these situations to make the experience meaningful both for the presenters and for the rest of the class.
To which NCTM Standards does this workshop relate?
In grades 1-4, this workshop is related to Standard 9: Geometry and Spatial Sense, and in grades 5-8 to Standard 12: Geometry. Both standards stress description, modeling, and comparison of geometric shapes; exploring the results of transformations on these shapes; and, in general, the development of spatial sense. You will also see Standard 1: Mathematics as Problem Solvingand Standard 3: Mathematics as Reasoning in Action.
Suggested Classroom ActivitiesDesign Your Own Instructions
Each student or group of students builds an object (e.g., a building) out of interlocking cubes. They then write and/or draw instructions for making the object and pass them on to another group. The second group follows the instructions and compares the results with the original object. This activity focuses on mathematics as communication and encourages students to think about the meaning of mathematical vocabulary.
More on Silhouettes
Each group of students builds a building with interlocking cubes, then draws the front, right, and top silhouettes. They trade their silhouettes with another group of students, who tries to reconstruct the original building. Interesting questions to discuss: Is there more than one building that fits a given set of silhouettes? What determines whether a set of silhouettes will fit only one building? How many buildings can you make that will have all three silhouettes look the same?
Do the pre-workshop activity with your class. Extend the activity by exploring all the nets for 2 x 1 boxes - there are a lot more. Can you (or your students) come up with a labeling system that makes it easy to determine whether any two nets are the same?
Drawing from Memory
Make transparencies of simple designs (some examples are given below). Use an overhead projector to show students one of the designs for a few seconds, then take it away and ask them to draw it. Put the design up again for a few seconds, then remove it and let students revise or complete their drawing. Finally, show the design once more - leaving it up this time - so that everyone can check their work. Ask students to describe how they remembered the picture: What shapes did they see? How did each drawing relate to the previous one? This activity can be done as early as kindergarten with very simple two-dimensional figures; it can be challenging for middle school students if the drawings are of three-dimensional objects.
Suggested StrategiesAll of the lessons we will see in the videos require additional materials other than the standard pencils, paper, rulers, etc. Consider how you use different kinds of materials in your classroom. How do they affect students' learning? What, if any, classroom management issues can arise when you use special materials, and how might you deal with them?
- Marco Ramirez spends a long time with one student clarifying the meaning of the word "side." How might you have handled the same situation?
- Language plays a major role in Marco Ramirez's bilingual classroom. In addition to the discussion referred to in Question 1, there are several situations in which Marco Ramirez encourages students to connect language and mathematics. For example, as students present shapes, he labels the shapes with their formal names. When students create an unusual shape, he allows them to name it (e.g., a Z with 2 heads). He encourages students to write the names of shapes in the best way they can, even if they don't know the exact spelling. How do you react to these techniques? Would you use them in your classroom? More generally, what do you think about the issue of mathematical vocabulary?
- There is great variety in the mathematical sophistication with which students in Nan Sepada's classroom classify the hexominoes. While we do not see her offer comments to any of the groups, how do you think she might have responded? What would you have done? In particular, how would you have responded to the group that used letters of the alphabet to classify the shapes?
Pre-Workshop Assignment for Workshop #4
Please conduct the following survey:
There are two groups of rectangles: Group 1 and Group 2. Show both groups to 20 people, and ask them to select one rectangle from each group which is the best looking or most pleasing. Record their responses on the tally chart (p. 52). You may also make a note of the age range of your respondents. Bring your results with you to Workshop.
There may be some surprises in your data. During the workshop, you'll can see how your results compare to results from teachers in other parts of the country.
Survey Directions Print out the Survey Grid and rectangles below. Show subjects the two groups of rectangles (Groups 1 and Groups 2). Ask them which rectangle in each group is the best looking or most pleasing. Mark one rectangle per group for each person you survey. Tally the results and enter them under Totals.
Survey Grid for recording results.
Mathematics: What's the Big Idea?