Type of Class: Honors Geometry

Related VA SOL: G.1c

Time Frame: 90-minute block period

Objectives:

• Students will be able to create a Venn diagram from an argument or conditional statement.
• Students will be able to create an argument or conditional statement from a Venn diagram.
• Students will know how to reach a conclusion, valid or not, based on a given information.
• Student will be able to create a Venn diagram from sentences that start with “some” and “no”.

Materials:

• Colored chalk
• Paper and pencil
• Markers/colored pencils

Procedures:

1. Introduce an argument and draw a Venn diagram to represent it.
   1. For example:

If you like oranges, then you like apples.

2. Illustrate how if Betty likes oranges, then she is in the inner loop, which means that she is in the outer loop as well.

Therefore, we can write the following:

1. If you like oranges, then you like apples.            p ® q

Betty likes oranges.                                           p

\Betty likes apples.                                         q

3. Since this is equivalent to the original sentence, there is a valid conclusion.
4. Show how if Bobby likes apples, then you don’t know if he likes oranges.

Therefore, you can write the following:

a.       If you like oranges, then you like apples.    p ® q

Bobby likes apples.                                   q

\No valid conclusion                                p

5. Since this is equivalent to the converse of the original sentence, there is no valid conclusion.
6. Illustrate how if Sally does not like oranges, then she can be either inside of the outer loop or outside of the outer loop. (The shaded region demonstrates “not liking oranges”!)

Therefore, you can write the following:

a.       If you like oranges, then you like apples.     p ® q

Sally does not like oranges.                       ~p

\No valid conclusion                              ~q

7. Since this is equivalent to the inverse of the statement, there is no valid conclusion.
8. Show how if Peter does not like apples, then he does not like oranges. (The shaded region illustrates “not liking apples”!)

                   Therefore, you can write the following:

a.       If you like oranges, then you like apples.     p ® q

Peter does not like apples.                       ~q

\Peter does not like oranges.                  ~p

9. Since this is the contrapositive of the original statement, there is a valid conclusion.
10. Students can come up with their own Venn diagrams and have other students come up with the arguments or vice versa.
11. Give examples for more practice. For instance, create arguments and have students draw its Venn diagram and have them reach a conclusion, if any.
12. Other types of Venn diagrams:
    1. To illustrate “Some butterflies are yellow and red.

 

2. To illustrate “No cats are gray and orange.”

Assessment:

          Students should be tested on drawing Venn diagrams from the arguments and vice versa. Also, they should be assessed on what the conclusion is, if any, when given a statement.

Suggestions/Comments:

            This activity can be done in any level of Geometry. For more practice, students can go up on the board and draw Venn diagrams from an argument and see if there is a valid conclusion or not.