## The Venn Factor

• Lesson
3-5,6-8
2

In this lesson, students use a Venn diagram to sort prime factors of two or more positive integers. Students calculate the greatest common factor by multiplying common prime factors and develop a definition based on their exploration.

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.

### Warm-up Activity

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.]

### Main Activity

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 following questions:

• 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.

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 chooses one.

### References

• 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>.

Assessment Options

1. Use the Venn Factor Assessment form to document observations about students' abilities to complete the lesson activities and activity sheet. Students' answers to questions throughout the activity can also be noted on this chart. This information is useful in determining whether students have met the lesson objectives.
The Venn Factor Assessment Form
2. Give students The Pizza Problem:

The list shows the amounts of money Mr. Sam collected from students for the pizza party. Each student paid the same amount. What is the most the pizza party could cost per student? List: Monday $16, Tuesday$24, Wednesday \$40. Explain how you solved the problem and explain the meaning of the answer.

Students will need to use what they have learned in the lesson to solve a real-world problem involving greatest common factor.

3. Have students write a journal entry telling what they learned, their favorite part of the lesson, and the most difficult part of the lesson.
4. Have students use the blank Venn Factor Cards to make up their own problems in the activity. They can write their own numbers on the cards.

Extensions

1. Show students the Puzzling Products overhead. The prime factors have already been sorted into the Venn diagram. Have students find the two numbers being compared. [60, 140] Multiply all the numbers in one circle to find one of the numbers being compared and repeat the same procedure with the other circle.
2. Organize students into pairs. Have students draw a Venn diagram on their white board or paper. Partner A fills in the Venn diagram with prime factors and partner B finds the two numbers being compared. Then partners switch roles.
3. Students can find the least common multiple of two or more numbers by multiplying all the prime factors in the Venn diagram.
4. Use the Euclidean algorithm to find the greatest common factor.

To find the greatest common factor of 56 and 42, divide 56 by 42. Repeat the process using the divisor as the new dividend and the remainder as the new divisor. Continue this process until the remainder is zero. The last divisor is the greatest common factor. In this case, the last divisor 14 is the greatest common factor of 56 and 42.

56 = 1 × 42 + 14 or 56 ÷ 42 = 1 R 14

42 = 3 × 14 + 0 or 42 ÷ 14 = 3 R 0

5. Have students complete the Ultimate Challenge Activity Sheet. This activity sheet provides students with challenge questions that enrich the lesson.
The Ultimate Challenge Activity Sheet

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.]

Teacher Reflection

• 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?

### Learning Objectives

Students will:

• 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

• CCSS.Math.Content.4.OA.B.4
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.

• CCSS.Math.Content.6.NS.B.4
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

• CCSS.Math.Practice.MP1
Make sense of problems and persevere in solving them.
• CCSS.Math.Practice.MP4
Model with mathematics.
• CCSS.Math.Practice.MP5
Use appropriate tools strategically.
• CCSS.Math.Practice.MP7
Look for and make use of structure.