Prime Factorization -- From Fingerprints to Factorprints

Summary

This activity provides an introduction to composite numbers and prime numbers through factorization.

Materials

Invitation to Learn

• Ink pads (1 per group)
• 1 1/2" x 2" Post-it® Notes
• Wet wipes
• Poster of main fingerprint patterns

Instructional Procedures

• Overhead color tiles
• Overhead markers
• Centimeter graph paper
• Colored pencils
• Prime Factorization (pdf)
• Centimeter cubes
• Prime Factorization -- Centimeter Cubes (pdf)
• GCF Mat (laminated) (pdf)
• LCM Mat (laminated) (pdf)
• Dry-erase markers
• Paper towels

Additional Resources

Books

Discovering Mathematics with the TI-73: Activities for Grades 5 and 6, by Melissa Nast; ISBN 1-8886309-22-1

Background for Teachers

The number one is a unique number because it only has itself as a factor. A prime number is a counting number larger than one that has exactly two factors. The two factors are one and the number itself. A composite number is a counting number that has more than two factors. Each composite number is divisible by three or more whole numbers.

Each composite number can be renamed as a product of prime numbers. This is known as prime factorization. Understanding prime factorization helps students understand the composition and decomposition of numbers.

Prime factorization is a strategy students may employ to find the Greatest Common Factor (GCF) of two or more numbers. Students may also use prime factorization to find the Least Common Multiple (LCM) of two or more numbers. It may be interesting to note that the product of the LCM and the GCF of two numbers is equal to the product of the two numbers themselves.

Instructional Procedures

Invitation to Learn

Pretend you are a detective. What is one piece of evidence that would help you to identify suspects from a crime scene? Fingerprints would be one type of evidence. Every person has a one-of-a-kind fingerprint. Have students make a fingerprint of their right index finger on a Post-it® note. Have students place their Post-it® note on the line plot, matching their fingerprint with one of the nine main patterns pictured on a teacher-made categorical line plot poster. Even though there are nine fingerprint patterns, allow students time to notice that each individual fingerprint is still one-of-a-kind.

Write the following analogy on the board: human is to fingerprint as number is to "factorprint." Tell students that just as each human has a one-of-a-kind fingerprint, we will learn that each number has a one-of-a-kind "factorprint."

Instructional Procedures

(The activities listed below are intended to be taught sequentially. They will take several lessons/days to complete with students.)

1. Explain to students that you will be creating a pattern with color tiles. They will copy the pattern by coloring the same pattern on their graph paper. Students are to observe, reflect, and predict the pattern after they see the first few representations
2. Have students copy the following pattern demonstrated on the overhead with color tiles:
   • To begin the first row, color one square black near the top left-hand corner. Label underneath "1."
   • Skip two squares horizontally and color one square red. Label underneath "2."
   • Skip two squares horizontally and color one square green. Label underneath "3."
   • Skip two squares horizontally and color two vertical squares red. Label underneath "4."
   • Skip two squares horizontally and color one square yellow. Label underneath "5."
   • Skip two squares horizontally and color one square red and one square green, placed vertically. Label underneath "6."
   • Skip two squares horizontally and color one square blue. Label underneath "7."
3. Have students predict what will be next in the pattern. Have students justify their prediction.
4. Continue the pattern, stopping to predict and justify answers at each number, as follows:
   • Skip two squares horizontally and color three vertical squares red. Label underneath "8."
   • Skip two squares horizontally and color two vertical squares green. Label underneath "9."
   • Skip two squares horizontally and color one square red and one square yellow, placed vertically. Label underneath "10."
5. By this time, students may have discovered that "4" was created by multiplying the value of the red square by itself or "2 x 2." They may have found that "6" was created by multiplying the value of the red square by the value of the green square or "2 x 3." Go back over the first 10 numbers and label the expressions for the composite numbers. Label "4" as "2 x 2," "6" as "2 x 3," "8" as "2 x 2 x 2," "9" as "3 x 3," and "10" as "2 x 5." Write PRIME under each prime number.
6. If a horizontal row is full, start a new row about halfway down the page. Continue the pattern, stopping to predict and justify answers at each number, as follows:
   • Skip two squares horizontally and color one square orange. Label underneath "ll" and write "PRIME."
   • Skip two squares horizontally and color two squares red and one square green, placed vertically. Label underneath "12," and label with the expression "2 x 2 x 3."
   • Skip two squares horizontally and color one square purple. Label underneath "13" and write "PRIME."
   • Skip two squares horizontally and color one square red and one square blue, placed vertically. Label underneath "14," and label with the expression "2 x 7."
   • Skip two squares horizontally and color one square green and one square yellow, placed vertically. Label underneath "15," and label with the expression "3 x 5."
   • Skip two squares horizontally and color four squares red, placed vertically. Label underneath "16," and label with the expression "2 x 2 x 2 x 2."
7. Remind students that each composite number is being formed by the multiplication of prime numbers. As you are labeling the expression for "16," teach students that there is another way to write this expression that would use fewer symbols and would be more efficient. We could use base numbers and exponents. Have students write the expression underneath "16" and "2 x 2 x 2 x 2" as "2^4." Look back over earlier numbers and write the expressions using base numbers and exponents on each composite number.
8. Continue the pattern, stopping to predict and justify answers at each number as follows:
   • Skip two squares horizontally and color one square pink. Label underneath "17" and write "PRIME."
   • Skip two squares horizontally and color one square red and two squares green, placed vertically. Label underneath "18," label with the expression "2 x 3 x 3," and label with the expression "2^1 x 3^2."
   • Skip two squares horizontally and color one square brown. Label underneath "19" and write "PRIME."
   • Skip two squares horizontally and color two squares red and one square yellow, placed vertically. Label underneath "20," label with the expression "2 x 2 x 5," and label with the expression "2^2 x 5^1."
9. Have students complete the patterns to the number 50. This could be done as cooperative teams or as a homework project. Since there is a new color for each prime number, the teacher will need to provide these patterns to create uniformity in correcting:
   • 23 is a black-outlined box with a red dot in the center 29 is a black-outlined box with a green dot in the center
   • 31 is a black-outlined box with a yellow dot in the center
   • 37 is a black-outlined box with a blue dot in the center
   • 41 is a black-outlined box with an orange dot in the center
   • 43 is a black-outlined box with a purple dot in the center
   • 47 is a black-outlined box with a pink dot in the center
10. Have students complete the handout Prime Factorization. Label the number one with the word "UNIQUE." Label each prime number with the word "PRIME." For each composite number, write the prime factorization expressions found in the color tile activity.
11. Have students place centimeter cubes on the handout Prime Factorization -- Centimeter Cubes in the same number and color as the color tile pattern. For example, "1" would have one black cube, "2" would have one red cube, "3" would have one green cube, "4" would have two red cubes, and so on.
12. To find the greatest common factor of two numbers, students must first find what prime factors they have in common. Have students write the numbers 8 and 12 as the two selected numbers on the GCF Mat with dry-erase marker. Have students take the cubes from Prime Factorization -- Centimeter Cubes, for 8 and 12 and place them on the GCF Mat in the proper squares. Help students to see what factors (represented by the colored centimeter cubes) these two numbers share. The number 8 has three red cubes, while the number 12 has two red cubes and one green cube. These two numbers each have two red cubes, so students would place two red cubes in the GCF column. Thus, the GCF of 8 and 12 is 2x2 or 4.
13. Repeat Step 12 with other pairs of numbers such as 9 and 18, 15 and 20, 8 and 24, 10 and 22, and so on.
14. To find the least common multiple of two numbers, students must first find the prime factors of each number. Have students write the numbers 8 and 12 as the two selected numbers on the LCM Mat with dry-erase marker. Have students take the cubes from for 8 and 12 and place them on the LCM Mat in the proper squares. Help students to see what factors (represented by the colored centimeter cubes) these two numbers each has. The number 8 has three red cubes, while the number 12 has two red cubes and one green cube. Place three red cubes in the LCM column to represent 8. Add one green cube (using two of the red cubes already there) to complete the factors of 12. Thus, there will be three red cubes and one green cube or 2x2x2x3 and the LCM of 8 and 12 is 24.
15. Repeat Step 14 with other pairs of numbers such as 3 and 9, 4 and 5, 4 and 7, and so on.

Extensions

• Find the prime factorization of a number using the tree method.
• Find the prime factorization of a number using the cake method.
• Find the Greatest Common Factor of two numbers using the prime factorization of the numbers from the color tile activity.
• Find the Least Common Multiple of two numbers using the prime factorization of the numbers from the color tile activity.

  Family Connections

• Have students share their graph paper patterns of prime factorization with parents.
• Ask students to explain to parents the difference between unique, prime, and composite numbers.
• Have students explain how a composite number may be renamed as a product of prime numbers to their parents.
• Have parents select a composite number under fifty and have students share a strategy for determining the prime factorization of that number.
• Have students teach parents how to find the GCF and LCM of two numbers using prime factorization.

Assessment Plan

• Informal assessment includes observation of students as they complete the color tile activity to the number 50.
• Have a class discussion of answers for the numbers 21 through 50. Model the answers on the overhead projector using color tiles or pictorial representations.
• Correct the handout Prime Factorization with the expressions from the color tile activity. Have students save this in a math journal or portfolio for future reference.
• Make a concentration game with 20 index cards. Put composite numbers on ten different cards, and put the prime factorization of the selected composite numbers on the other ten cards.

Bibliography

Gerlic, I., & Jausovec, N. Multimedia: Differences in cognitive processes observed with EEG. Educational technology research and development, September 1999, Vol. 47, Number 3, p5-14.

This study investigated the cognitive processes involved in learning information presented in three different methods: with text; with text, sound, and picture; and with text, sound, and video. Students' brain activity was measured using an EEG in each format. Less mental activity was found using the text only presentation. The results showed higher mental activity with the video and picture presentations, confirming the assumption that these methods induced visualization strategies on the part of the learners.

Zazkis, R., & Liljedahl, P. Understanding primes: The role of representation. Journal for research in mathematics education, May 2004, Vol. 35 Issue 3, p164-186.

The authors of this article investigated how preservice elementary teachers understood the concept of prime numbers. They attempted to describe the factors that influenced their understanding. The authors suggested that an obstacle to a full conceptual understanding is a lack of a representation for a prime number. The importance of representations in understanding math concepts is examined.