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Classifying Numbers

Number and Operations
Location: Unknown

Students use Venn diagrams to represent the relationships between the factors or products of two numbers.


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:1067 venn 


Students should work in pairs to complete the Classifying Numbers Activity Sheet.

pdficon Classifying Numbers Activity Sheet 


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.


Number and Operations

Connections and Extensions

Students make connections and expand on what they have learned in the first three lessons. Students explain the effects of different moves on the game board. Finally, students "Guess My Number" using various clues.

Learning Objectives

Students will:

  • Review multiplication facts.
  • Develop understanding of factors and multiples and the relationships between them.
  • Use Venn Diagrams to demonstrate these relationships.
  • Understand that some products are the result of more than one factor pair.

NCTM Standards and Expectations

  • Develop meaning for integers and represent and compare quantities with them.
  • Develop and analyze algorithms for computing with fractions, decimals, and integers and develop flue
  • Develop and use strategies to estimate the results of rational-number computations and judge the reasonableness of the results.

Common Core State Standards – Practice

  • CCSS.Math.Practice.MP1
    Make sense of problems and persevere in solving them.
  • CCSS.Math.Practice.MP2
    Reason abstractly and quantitatively.
  • CCSS.Math.Practice.MP6
    Attend to precision.