73 – 84 - Prime Factorization, GCF, and LCM

Lesson Focus / The student will understand numbers, ways to represent numbers, relationships among numbers and number systems by using factors, multiples, prime factorization, and relatively prime numbers to solve problems.
Materials / TI-73 Explorer™or TI-84 Plus graphing calculator
TI-Navigator™
Student Activity Sheet
Grouping / Pairs for activity
Whole class discussions
Prerequisite Knowledge and Skills / Know and understand definitions of prime numbers, composite numbers, factors, and multiples. Understand the connection between division and finding factors of a number.
Overview of the lesson / TI-Navigator will be used to review the concept of multiples and factors. Students will develop strategies for finding common factors, GCF, common multiples, and LCM of 2 or more numbers.
The class will discuss criteria and develop strategies to help determine when finding the GCF and/or LCM can be used to solve word problems.Students will use LCM and/or GCF to solve word problems.
Time / Introduction and Activity 1: 50-55 minutes
Activity 2:

Procedure:

Introduction

TI-Navigator™

The teacher will need to determine the students’ understanding of prime numbers, factors, and multiples.

  1. Open TI-Navigator and Start Class
  1. Have students log in to NavNet.
  1. Begin the class usingQuick Poll.

Send each question to students, and do not display the answers. After answering questions, go back to each question and display the Poll Summary. Facilitate a discussion about their answers, asking for processes and decisions about why and how they answered each question.

4. Review the definition of multiples and factors.

Procedure:

Activity 1

TI-Navigator™

  1. Use Quick Poll to ask the following questions. After the students have answered the questions, go back to each question in Quick Poll and show the Poll Summary to facilitate a discussion about their answers; ask what strategy they used to find the answer.

2.Display Venn Diagram transparency. Write the number 6 above one circle and 8 above the other circle.Venn Diagramsare one strategy they can use to help find the GCF and LCM of two or more numbers. On the left side of the first circle, review using division (aka birthday cake method… see explanation below)to find the prime factorization of the number 6. Repeat this process again on the right side of the second circle to find the prime factorization of the number 8.

3.Instruct the students to complete the Venn Diagram using the prime factors for each number. Be sure to use only prime numbers as shown below.

4.The LCM is the product of all factors written on the Venn diagram.The GCF is the product of the factors both numbers have in common (overlapping circles).

  1. Students will work with a partner to complete the lesson activity.

Pose the task:

  • Work with your partner to find the LCM and GCF of two numbers generated by the random integer generator of the calculator. From the calculator home screen press, ,  to PRB,2:randInt(1,100,2). This will randomly select two numbers between 1 and 100 for the students.
  • You may use any strategy to find the LCM and GCF but you must show all work and be prepared to share your strategy/results with the class.

6.Each pair of students will share their findings with the class.

Extension:

Repeat the task for three randomly selected numbers.From the calculator home screen press, ,  (to PRB), 2:randInt(1,100,3). This will randomly select three numbers between 1 and 100 for the students.

Procedure:

Activity 2

TI-Navigator™

  1. Combine pairs of students to work in pods of four or five.
  1. Distribute Student Activity Sheet,GFC/LCM Word Problems.
  1. Pose the task.
  2. Analyze several word problems.
  3. Decide whether you would use GCF or LCM to solve each problem and be prepared to support your decision.
  4. Solve each problem.
  5. Make a class list of criteria to use when solving possible GCF/LCM word problems.
  1. Suggestions for GCF Word Problems:
  • Questions ask to divide items into smaller units.
  • Questions ask to calculate possible number of items or people to include.
  • Questions ask to arrange or sort into groupsor rows.
  1. Suggestions for LCM Word Problems:
  2. Questions include an event or activity that repeats.
  3. Questions involve acquiring multiple items in order to have enough.
  4. Questions involve two events happening simultaneously at some point in the future.
  1. Record the class list of criteria on chart-sized paper to post as a reminder for students.


Venn Diagram Transparency


Venn Diagram ~ Extension
Student Activity Sheet:Name: ______

GCF/LCM Word Problems

  1. Samantha baked cookies for the holiday to share with her family and friends. She has 18 iced sugar cookies, 24 peanut butter cookies, and 36 brownies. If she divides the cookies equally with none left over, what is the greatest number of people with which she can share the cookies?
  1. Rob has two pieces of rope. One piece is 48 inches in length and the other is 3 feet. He wants to cut both pieces of rope into equal length sections that are as long as possible. How long will each section of rope be?
  1. Skylar has dance practice every 4 days. Her best friend Emma has dance practice every 6 days. If Skylar and Emma both had practice today, how many days will it be until they both have dance practice on the same day again?
  1. Mrs. Smith wants to encourage good citizenship in her classroom. She wants to give some students prize bags with candy and stickers in them. If she has 15 pieces of candy and 10 stickers to divide equally among the bags, what is the greatest number of prize bags Mrs. Smith can make?
  1. Frank is buying pencils and erasers at the store. The store sells pencils in packages of two and erasers in packages of 10. If Frank wishes to buy the same number of pencils as erasers, what is the least number of pencils that he needs to buy?
  1. Madison and Max are running the school’s track. Max completes one lap around in 12 minutes. Madison completes one lap around in 18 minutes. If they both started at the same place and time and are running in the same direction, after how many minutes would they meet at the starting point?

KEY ~ Student Activity SheetName: ______

GCF/LCM Word Problems

  1. Samantha baked cookies for the holiday to share with her family and friends. She has 18 iced sugar cookies, 24 peanut butter cookies, and 36 brownies. If she divides the cookies equally with none left over, what is the greatest number of people with which she can share the cookies?

GCF 6 people

  1. Rob has two pieces of rope. One piece is 48 inches in length and the other is 3 feet. He wants to cut both pieces of rope into equal length sections that are as long as possible. How long should each section of rope be?

GCF 12 inches or 1 foot

  1. Skylar has dance practice every 4 days. Her best friend Emma has dance practice every 6 days. If Skylar and Emma both had practice today, how many days will it be until they both have dance practice on the same day again?

LCM 12 days

  1. Mrs. Smith wants to encourage good citizenship in her classroom. She wants to give some students prize bags with candy and stickers in them. If she has 15 pieces of candy and 10 stickers to divide equally among the bags, what is the greatest number of prize bags Mrs. Smith can make?

GCF 5 prize bags

  1. Frank is buying pencils and erasers at the store. The store sells pencils in packages of two and erasers in packages of 10. If Frank wishes to buy the same number of pencils as erasers, what is the least number of pencils that he needs to buy?

LCM 10 pencils

  1. Madison and Max are running the school’s track. Max completes one lap around in 12 minutes. Madison completes one lap around in 18 minutes. If they both started at the same place and time and are running in the same direction, after how many minutes would they meet at the starting point?

LCM 36 min

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