Abstract

Students will learn how to use Venn diagrams to represent classifications of shapes and to use counting arguments to find the number of objects in a given set.

Standards

The activities and discussions in this lesson address the following NCTM standards:

Algebra
Use mathematical models to represent and understand quantitative relationships

• model problem situations with objects and use representations such as graphs, tables, and equations to draw conclusions

• classify two- and three-dimensional shapes according to their properties and develop definitions of classes of shapes such as triangles and pyramids

• represent data using tables and graphs such as line plots, bar graphs, and line graphs

Some of the optional exploratory questions are applications of the counting principle. Note that the NCTM standards don't include the counting principle, but one often needs it to determine the number of possible outcomes, which is needed to compute theoretical probabilities. Morevoer, some states, like North Carolina, do include the counting principle in their standards.

Student Prerequisites

• Technological: Students must be able to:
  • perform basic mouse manipulations such as point, click and drag
  • use a browser such as Netscape for experimenting with the activities

Teacher Preparation

Before class, draw on the board:

• A red triangle
• A red circle
• A red square
• A blue square
• A blue triangle

Students will need:

• Access to a browser
• Pencil and paper
• Copies of the supplemental materials for this activity
  • Shape Sort Game Scorecard
  • Shape Sort Work Sheet (if you choose to use the worksheet)

Lesson Outline

1. Focus and Review
   • Ask for several volunteers.
   • Have volunteers choose an object from the ones drawn on the board and describe it. What is its shape and color? How can you describe its size?
   • Record the description on the board under the object.
   • Have a second volunteer describe the characteristics of another object.
   • Record the descriptions on the board.
   • Continue the process until all the objects on the board have been described.
   • Circle some descriptions that apply to a lot of objects (red, blue, circle).
   • Ask the students which objects are elements of the set of RED objects. Ask which are elements of the set of BLUE objects.

2. Objectives

   Let the students know what they will be doing and learning today. Say something like, "Today we are going to learn about Venn diagrams, which are a way of showing how elements of different sets are related."

3. Teacher Input
   • Draw two large separate (disjoint) circles on the board. Label them BLUE and RED.
   • Have a couple students come to the board and draw the blue square, red circle, and other object you have been discussing in the appropriate circles.
   • After the students have finished placing all the items in the correct areas of the Venn diagram, draw 2 more separate circles and label them as RED and POLYGON
   • Ask the students where the RED SQUARE goes.
   • When the students say the RED SQUARE goes in both circles, ask how they can show that the RED and POLYGON circles intersect.
   • Erase one side of the one of the circles so you can re-draw it to intersect with the other, and put the RED SQUARE in the intersection.
   • Ask the student to help you put the rest of the objects in the right place in the diagram.
   • Explain that this method of showing sets and their intersections is called Venn diagrams (they were first used by John Venn, a 19th century British mathematician).
   • In some classes, you could ask where a BLUE CIRCLE would go -- this shows that outside the circles represents being in some universal set but not in the ones being represented on the Venn diagram just now.
   • Ask students for pairs of words that would go with intersecting circles (for example, PURPLE and FLOWERS) and others that would go with disjoint (separate) circles (RECTANGLES and TRIANGLES, for example)

4. Guided Practice
   • Say something like: "We are going to use the computers to practice classifying objects using Venn Diagrams, but please do not turn your computers on until I ask you to. I want to show you a little about this activity first."
   • Open your browser to Shape Sorter applet.
   • Explain how to use the applet. In particular, say they should set the buttons at the left on "MAKE THE RULE" and point out how to choose the version to use (one circle, interecting circles, disjoint circles).
   • Show how to choose a rule. Using the 1 circle version, set the rule to YELLOW. Show what happens when you try to move a SMALL RED SQUARE into circle in the Venn diagram. Show what happens when you try to move a LARGE YELLOW TRIANGLE into the circle of the Venn diagram.
   • In the game the students will be playing, the RED SQUARE is a wrong answer and the YELLOW TRIANGLE is a correct answer.
   • Divide the class into teams of 2 or 3 students, ideally with students of similar ability in each group.
   • Have the students open their browsers to the Shape Sorter applet.

5. Independent Practice
   • Have the students work with the Venn Diagram Shape Sorter applet and the worksheets (the Scorecard and the Exploration Questions).
   • Wander around the room to ensure that everyone is on task and understands how to use the applet.
   • Encourage students to find the number of RED objects by multiplying the number of different shapes by the number of different sizes (4x2) instead of counting.

6. Closure
   • Ask the students to close their computers.
   • Review the differences between intersecting and disjoint circles in Venn diagrams and the rules for putting objects in the right part of the diagram.
   • Review how to find the number of objects that meet one of the computer's rules by multiplying the numbers of various types of objects. For example, RED objects come in 4 shapes and 2 sizes, so there are 8 RED objects.
   • Review all pertinent terminology.

Alternate Outlines

This lesson can be rearranged in several ways.

• Instead of classifying figures drawn on the board, have students draw objects from a bag, as described in Venn Diagrams.
• Use the Shape Sort Race Track Sheet instead of the scorecard. The disadvantage of the racetrack is that the students do not practice adding. Students will need some kind of tokens to move around the track.
• Have students cooperate (try to get higher scores for one another) inestead of competing.
• If only one computer is available for the classroom:
  • The teacher may do ths activity as a demonstration. Choose the version and the rule(s). Allow the students to decide individually, or in groups, which object to move onto the diagram and where to move the object.
  • Groups may take turns practicing Venn diagrams with the applet for 10 minutes each.
  • In either case, students or groups can still complete the worksheet.

Suggested Follow-Up

• Recognizing Patterns explores the idea of patterns through the use of physical activities, class discussions, and computer applets.