In the Loop
By Michael Naylor

In the mid-1800s, Englishman John Venn invented a type of diagram to help visualize logical relationships. A Venn diagram is simply a rectangular box with circular loops in it that overlap to show how objects are related. These activities with Venn diagrams are a fun way to sharpen your students' logic skills and develop number and geometric relationships.

Button sort (Grades K-2)
This activity works with any small manipulative materials that have different ways of being grouped. I've found that buttons work well. Divide your students into small groups and give each a set of two different-colored buttons (say red and black) and some of each with two holes and four holes.

Give each group of students two loops of string and then write the words "Red" and "Black" on two cards. Ask the kids to lay the loops on the floor and place the cards inside. They should then sort the buttons by color into the two loops.

Now, trade the "Red" card for a card that says "Four Holes" and have the kids sort the buttons again. The question will arise: What do we do with the buttons that are red and have two holes? Discuss and then allow your students to overlap their loops so that some buttons can be in both groups.

Ask them to come up with their own way to sort the buttons into groups.

Class Venn (Grades K-5)
On a big piece of paper, draw two or three loop Venn diagrams and label the loops with class attributes such as boy/girl, owns pet dog/cat, age eight/ nine, likes to ski/skate/bike and so on. Some examples are shown below. Have students come to the board and write their names in the appropriate part of the diagram. Try to make a different one every week to hang on the wall.

Venn number sense (Grades 3-8)
Use Venn diagrams to sort numbers according to different properties. Here's some that work well and create interesting discussions about number properties.

• Numbers 1–15: Odd and prime. 1 is odd but not prime, 2 is prime but not odd.
• Numbers 1–20: Multiples of 2 and 3.
• Numbers 1–25: Even and square numbers.

• Numbers 1–25: Fibonacci and composite (the Fibonacci sequence starts with two ones, then the next number is made by adding the previous two: 1, 1, 2, 3, 5, 8, 13, 21, ...)
• Numbers 1–36: Square and triangular. (Square numbers are n x n: 1, 4, 9, 16, 25, 36…Triangular numbers are 1, 3, 6, 10, 15, 21, 28, 36… and are made by starting with 1 and increasing the adding number +2, +3, +4, +5, etc...)

Quadrilateral relationships (Grades 6-8)
This challenging activity will stretch your students' logic and geometric concepts. First, demonstrate set inclusion to your class with the following three examples.

1. Some sets overlap, like rectangles and rhombi (or rhombuses), because some rectangles are not rhombi and some rhombi are not rectangles. Ask your students for examples that go in each area (the square belongs in both groups).

2. Some sets don't overlap at all, like triangles and trapezoids. Why?
3. Some sets are entirely inside of other sets, like rectangles are inside of parallelograms. It's impossible to make a rectangle that is not also a parallelogram.


Now, ask your students to brainstorm a class list of quadrilateral names. Be sure the list includes these categories:

Polygons Quadrilateral Rhombus Square

Rectangle Parallelogram Trapezoid

(Don't include a kite for this exercise – a rhombus is a special case of a kite and this makes things complicated. You can add a kite to a student's list as an extension.)

Ask your students to use every item on the list to make a single drawing showing the relationships between the sets. They should check their own work by drawing an example of a figure for each region of their diagram – if they can't, they will probably need to revise their drawing.

Many interesting questions will arise as they try to relate all of these groups; be sure to point them towards a dictionary as they struggle to precisely define names and classes of shapes.

The correct relationships are shown below.



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Michael Naylor is a professor of math education at Western Washington University, Bellingham, WA.