Launch
The following problem uses Venn diagrams to organize information
about numbers. A Venn diagram uses circles to show sets of numbers or other
objects with the same attributes. Your students may have used string circles in
the elementary grades to look at relationships or attributes. The important
thing in this problem is for students to look for relationships and
characteristics of numbers and to determine what numbers belong to a descriptor
and what numbers belong to more than one descriptor.
You might start the lesson
by drawing two overlapping circles, labeled "Factors of 30" and "Factors of 36,"
on the board.
Give me
some examples of numbers that go in each
circle.
Record a few numbers that students give, always
asking in which area of the diagram the number goes.
Are there any numbers that
belong in both circles? Why or why not?
We put the numbers that belong in both
circles in the intersection, or overlap, of the circles.
Write some numbers that are factors of both 30 and
36, for example 2 and 6, in the intersection of the circles.
Are there any numbers that do not belong
in either circle?
Help students see that 7, 8, 11, and many other
numbers are not factors of 30 or 36, and therefore do not belong in either
circle. Draw a rectangle that encloses the circles.
We can show these numbers by putting them
in the area outside of the circles.
Write these numbers inside the rectangle and outside
of the circles. When you finish, your diagram will look something like
this:
Explore
Students should work in pairs to complete the Classifying Numbers activity sheet.
Summarize
One way to summarize is to have each group draw their
Venn diagram on a blank transparency. The groups can then present their work at
the overhead and explain their thinking. Or, select three pairs of students to
put their work on the board. Have one pair draw their Venn diagram from the activity sheet
and the other two pairs give their answers to the questions that follow.
If any
group wants to add or disagree with anything on the board, come up and write
your comments in another color.
In the discussion, have students share the strategies
that they used to solve the problems. Be sure to ask what is special about the
numbers in the intersection of the circles. Students should realize that these
numbers are divisible by both 2 and 3, which means they are also divisible by 6.
In other words, they are all multiples of 6.
