Before beginning the lesson, cut out the Venn Factor Cards
and place into a clear Ziploc bag. Construct the Venn diagram circles
by twisting together the ends of three pipe cleaners of the same color.
Repeat this procedure two more times using different colors of pipe
cleaners for each circle. This results in three circles of three
different colors with which to use for the Venn diagram. Place all the
materials into a large brown envelope. Create enough envelopes for each
group to get one. Each group should have 2 students so that one can
arrange the factors while the other records the results.
The Venn Factor Cards
Draw a blank Venn diagram with two circles on the board. Organize
the class into pairs. Tell students that partners in each pair will
perform a specific task. First, Partner A records a list of interests
on his/her white board generated by both students. Then, Partner B
draws a blank Venn diagram like the one drawn on the board on his/her
white board. Together, the students sort their list of interests into
the Venn diagram. Circulate to pairs of students asking what criteria
students used to place their interests into the Venn diagram.
After the activity has finished, pose the following questions to students:
- Why are some items placed in only one of the circles? [These are interests of only one of the students.]
- Why are some items placed in the overlapping section of the Venn diagram? [These are interests that we share or have in common.]
Tell students they will be using Venn diagrams to find factors that
various numbers share or have in common. Assess students' prior
knowledge by asking the following questions:
- What is a factor? [They are counting numbers that are multiplied
together to form a product. They are numbers that can evenly be divided
into another number.]
- What is a product? [The result of multiplying two or more factors.]
- What are prime numbers? [Numbers that have exactly two factors, one and the number.]
- What are composite numbers? [Numbers that have more than two factors.]
Pass out the large brown envelope of materials to each pair of
students. Tell students to use the cards to find all factors of 18
and 24 and place the cards on their table. Circulate to groups and
assist students if they are unable to complete this activity. Once
students' cards have been checked, ask students to name factors of 18
[1, 2, 3, 6, 9, 18.] and 24 [1, 2, 3, 4, 6, 8, 12, 24.]. Record these
factors on the board.
Tell students to use two different color circles to make a Venn
diagram. Have them suggest what factors will go in each circle.
Students then complete their Venn diagram with their partner. After
students have finished sorting the cards, pose the following questions
to the class:
- How did you sort the cards? Have a student record his/her answer on
the Venn diagram drawn on the board. [1, 2, 3, and 6 are in the
overlapping circle, 9 and 18 are only in the 18 circle, and 8, 12,
and 24 are only in the 24 circle.]
- Why did you sort the cards this way? [1, 2, 3, and 6 are in the
overlapping circle because those are the factors they share or have in
common, 9 and 18 because those are the factors that only 18 has and
8, 12, and 24 because those are the factors of only 24.]
- What factors do the two numbers have in common? [1, 2, 3, and 6.]
- What is the greatest common factor? [6.]
Students choose two of their own numbers to find the GCF. They may
use the blank Venn Factor Cards and write in their own numbers or they
may use their white boards. Repeat this activity until students feel
confident with this method.
Explain to students that you will be teaching them a new method for
finding the greatest common factor using a factor tree. Ask students
for two factors that have a product of 18. Record their answer on the
board as the beginning of the factor tree. Ask students whether the
factors are prime or composite. Answers will vary depending on factors
chosen. Explain that if a factor is prime they are done factoring that
number. If the factor is composite, they will continue to factor it
using the factor tree until all factors are prime. Repeat this
procedure with 24. Students use individual white boards to record the
answers while the factor tree is being completed on the board. Once
students have found the prime factorization for 18 and 24, they use the
Venn Factor cards and sort them into the Venn diagram. Ask students how
this Venn diagram is different from the one they created in the
previous activity. [This Venn diagram has all prime numbers.]
Allow sufficient time for all students to finish the activity. Draw
a blank Venn diagram on the board and have student volunteers write in
their answers. After completing this section of the activity, ask
students the following questions:
- What is the prime factorization of 18? [2×3×3.]
- How do you know it is the prime factorization? [2 and 3 are both
prime numbers because they only have factors of 1 and itself and if you
multiply all three numbers the product is 18.]
- What is the prime factorization of 24? [2×2×2×3.]
- What prime factors do 18 and 24 have in common? [18 and 24 share one 2 and one 3.]
Tell students to multiply the common prime factors. The result is
six. This is the same greatest common factor they found in the previous
activity. Ask students to identify the greatest common factor of
18 and 24.
Tell students to find the greatest common factor of 14 and 72 using
any strategy. As students are working, circulate to groups and ask the
- How did you find the greatest common factor? [Answers will vary based on the strategy chosen.]
- How do you know it is the greatest common factor? [It is the factor of the two numbers with the largest value.]
Have students use the third circle to make a Venn diagram comparing
three numbers. They repeat the same activity finding the greatest
common factor of 36, 48, and 60. Again, circulate to groups asking the
previous questions. Allow students enough time to complete the activity.
Once students have completed the activity, have them clean up their materials. Pass out The Venn Factor
activity sheet. Students develop their own definition for greatest
common factor. In groups of four, have students share their
definitions. Student 1 reads his/her definition, then student 2,
student 3, and student 4. This allows students having difficulty the
opportunity to hear examples.
The Venn Factor Activity Sheet
Independently, students complete the remainder of the activity
sheet. While students are completing the activity, read their responses
and assess their ability to use the skills learned in the lesson.
Notorious Nutshell Questions
These questions are notorious because they are a well known closure
procedure and nutshell comes from "in a nutshell" which summarizes a
topic. Before the lesson begins, tape the following questions
underneath students' desks:
- What is a factor? [They are the numbers multiplied together to form
a product or numbers that can evenly be divided into another number.]
- What is a prime number? [A number that has exactly two factors, one and the number.]
- What is prime factorization and how do you find it? [A
multiplication problem using only prime factors. It can be found using
a factor tree.]
- What is the greatest common factor? [A factor with the largest value that two or more numbers share.]
- How do you find the greatest common factor of 18 and 24? [Use a
factor tree to find prime factorization. Sort the prime factors into a
Venn diagram and multiply the common prime factors. Or, list all the
factors of the numbers and the largest factor all numbers have in
common is the greatest common factor. Answers will vary depending on
students preferred strategy.]
Tell students to look under their desks for a Notorious Nutshell
question. They read the question aloud and either answer the question
or others who think they can answer raise their hand and the student
The Venn Factor Answer Key
- Charles, Randall I. "Chapter 7: Fraction Concepts." Scott Foresman-Addison Wesley
Mathematics. Glenview, Ill.: Pearson/Scott Foresman, 2008. 414-15. Print.
- Hines. "Math 402-Lesson Plan and Reflection." Math.niu.edu. Northern Illinois University. Web.
2 June 2010. <http://www.math.niu.edu/courses/math402/hines/402-3LessonXPlanXandXReflection.pdf>.
- Megan. "Math Forum - Ask Dr. Math." The Math Forum @ Drexel University. Drexel
University, 27 Oct. 1998. Web. 20 May 2010. <http://mathforum.org/library/drmath/view/51897.html>.
- Middle, Allen J. "Lesson Plan-Greatest Common Factor." GCS Technology Services. Guilford
County Schools, 26 Jan. 2005. Web. 16 Mar. 2010. <http://its.guilford.k12.nc.us/act/grade6/act6.asp?ID=904>.
- Unknown. "Lesson Plan - Adding Fractions Algorithm." UTeach: Home. University of Texas, 15
Feb. 2001. Web. 16 Sept. 2009. <http://uteach.utexas.edu/~portfoliohelp/math_cs/full/Instruction/Adding Fractions Algorithm.html>.
Questions for Students
1. Can the GCF of 12 and 30 be greater than 12? Explain.
[No, the greatest common factor can never be greater than
the smallest number being compared and since 12 isn't a factor of 30
the greatest common factor has to be less.]
2. Give an example of a prime number. How do you know it is prime?
[2, it only has two factors one and itself. Regardless of the number chosen, the reason should remain the same.]
3. Give an example of a composite number. How do you know it is composite?
[4, it has more than two factors. Regardless of the number chosen, the reason should remain the same.]
4. What are the factors of 12?
[1, 2, 3, 4, 6, and 12.]
5. How do you know 3 is a factor of 12?
[It is a number that can be multiplied to get 12 or because it divides evenly into 12.]
- Was students' level of enthusiasm/involvement during the Venn diagram activity high or low? Explain why.
- How effective were you in allowing students to demonstrate their
knowledge or explain their reasoning during questioning and group work?
Explain. Give examples.
- What additional assistance, support, and/or resources would have further enhanced this lesson?
- Did you find it necessary to make adjustments while teaching the
lesson? If so, what adjustments, and were these adjustments effective?
- How effectively did the lesson involve students in reasoning and
problem solving? Explain. What changes could you make to encourage more
student reasoning and problem solving?
- Did the Notorious Nutshell Questions help students to consolidate
their knowledge? Explain. What changes in closure might be indicated?
- Find prime factorizations of various positive integers.
- Organize prime factors into a Venn diagram.
- Calculate the Greatest Common Factor.
- Develop a definition for Greatest Common Factor.
NCTM Standards and Expectations
- Describe classes of numbers according to characteristics such as the nature of their factors.
- Use factors, multiples, prime factorization, and relatively prime numbers to solve problems.
Common Core State Standards – Mathematics
Grade 4, Algebraic Thinking
Find all factor pairs for a whole number in the range 1-100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1-100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1-100 is prime or composite.
Grade 6, The Number System
Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1-100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4 (9 + 2).
Common Core State Standards – Practice
Make sense of problems and persevere in solving them.
Model with mathematics.
Use appropriate tools strategically.
Look for and make use of structure.