Given a mathematical or real-world context, find the greatest common factor and least common multiple of two whole numbers.
Examples
Example: Middleton Middle School’s band has an upcoming winter concert which will have several performances. The bandleader would like to divide the students into concert groups with the same number of flute players, the same number of clarinet players and the same number of violin players in each group. There are a total of 15 students who play the flute, 27 students who play the clarinet and 12 students who play the violin. How many separate groups can be formed?
Example: Adam works out every 8 days and Susan works out every 12 days. If both Adam and Susan work out today, how many days until they work out on the same day again?
Clarifications
Clarification 1: Within this benchmark, expectations include finding greatest common factor within 1,000 and least common multiple with factors to 25.
Clarification 2: Instruction includes finding the greatest common factor of the numerator and denominator of a fraction to simplify a fraction.
Subject Area: Mathematics (B.E.S.T.)
Grade: 6
Strand: Number Sense and Operations
Date Adopted or Revised: 08/20
Status: State Board Approved
Benchmark Instructional Guide
Connecting Benchmarks/Horizontal Alignment
Terms from the K-12 Glossary
- Area Model
- Composite Number
- Dividend
- Divisor
- Factor
- Greatest Common Factor (GCF)
- Least Common Multiple (LCM)
- Prime Factorization
- Prime Number
- Rectangular Array
Vertical Alignment
Previous Benchmarks
http://flbt5.floridaearlylearning.com/standards.html
Next Benchmarks
Purpose and Instructional Strategies
In grade 4, students determined factor pairs for whole numbers from 0 to 144 and determined if numbers are prime, composite, or neither. In grade 6, students will determine the least common multiple and greatest common factor of two whole numbers. In future grades, students will use their understanding of factors, factorization and the distributive property to generate equivalent expressions and solve equations.
- Instruction includes a variety of methods and strategies to determine the least common multiple (MTR.2.1, MTR.3.1, MTR.5.1).
- Instruction includes a variety of methods and strategies to determine the greatest common factor (MTR.2.1, MTR.3.1, MTR.5.1).
- T-charts
The greatest common factor of 96 and 138 is 6.
- Factorization
GCF is represented as 3.
- The greatest common factor can be determined by dividing the numbers multiple times by common factors and then multiplying the common factors that were used as divisors.
The greatest common factor of 75 and 150 can be found by multiplying 5×3×5 = 75.
÷ =
- Instruction includes the use of technology to explore, interpret and define greatest common factors and least common factors.
Common Misconceptions or Errors
- Greatest common factor and least common multiple are often mixed up or confused by students. Stressing the academic vocabulary of what is a factor versus what is a multiple can help students keep the two straight in their minds.
- Students may incorrectly think that the greatest common factor of two numbers cannot be 1. However, if given two prime numbers, the greatest common factor will always be 1.
- Students may incorrectly think that if one of the numbers is prime, the greatest common factor must be 1. However, the greatest common factor is not 1 if one of the numbers is a prime number and that prime number is a factor of the second number.
- Students may incorrectly think the least common multiple has to be larger than both of the given numbers. Provide examples to showcase the least common multiple being the same as one of the given whole numbers.
- For example, the LCM of 12 and 24 is 24.
- Students may incorrectly think the least common multiple can be found by multiplying the two numbers. Provide examples where this is not the case, such as the LCM of 4 and 6 is 12 not 24.
Strategies to Support Tiered Instruction
- Teacher shares their thinking when determining if a given situation requires finding the greatest common factor or the least common multiple. Steps include reading the given context out loud and thinking out loud about if the context requires the final answer to be greater than or equal to the given numbers (least common multiple) or less than the given numbers (greatest common factor).
- Teacher creates and posts an anchor chart with visual representations of factors and multiples and encourages students to utilize the anchor chart to assist in utilizing correct academic vocabulary when referring to factors and multiples.
- Teacher provides students with flash cards to practice and reinforce academic vocabulary.
- Teacher provides instruction on the definitions of greatest common factor and least common multiple then co-creates a graphic organizer using student created definitions, lists of key characteristics, examples, and non-examples of each term.
- A non-example for greatest common factor might include finding a least common factor, which would result in 1.
- A non-example for least common multiple might include attempting to find the greatest common multiple, which cannot be determined because multiples are infinite.
- Teacher provides examples to showcase the least common multiple being the same as the larger of the given whole numbers or the same as the product of the two numbers.
- For example, the least common multiple of 12 and 24 is 24.
- For example, the least common multiple of 14 and 5 is 70.
- Teacher provides examples where this is not the case, such as the least common multiple of 4 and 6 is 12 not 6 or 24.
- Instruction includes the use of a graphic organizer to compare the multiples or factors of two numbers to help recognize common multiples and factors.
Instructional Tasks
Instructional Task 1 (MTR.3.1)
Parker reads a book every 12 days and Leah reads a book every 8 days.
- Part A. If today is Wednesday and they both started a book today, how many days will it be when they both start a new book on the same day?
- Part B. What day of the week will it be when they both start a new book on the same day?
Instructional Items
Instructional Item 1
What is the greatest common factor between 636 and 132?
Instructional Item 2
What is the least common multiple between 17 and 12?
*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.
Related Courses
| Course Number1111 |
Course Title222 |
| 1205010: | M/J Grade 6 Mathematics (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current)) |
| 1205020: | M/J Accelerated Mathematics Grade 6 (Specifically in versions: 2014 - 2015, 2015 - 2020, 2020 - 2022, 2022 and beyond (current)) |
| 1204000: | M/J Foundational Skills in Mathematics 6-8 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current)) |
| 7812015: | Access M/J Grade 6 Mathematics (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2022, 2022 and beyond (current)) |
Related Access Points
| Access Point Number |
Access Point Title |
| MA.6.NSO.3.AP.1 | Use tools to find the greatest common factor and least common multiple of two whole numbers 50 or less. |
Related Resources
Formative Assessments
| Name |
Description |
| Least Common Multiples | Students are asked to find the least common multiple of 8 and 12 and to explain how they found their answers. |
| Greatest Common Factors | Students are given two whole numbers less than or equal to 100 and asked to find the greatest common factor. |
Lesson Plans
| Name |
Description |
| Can You Find the Relationship? | In this lesson students will first define in their own words what the greatest common factor (GCF) and least common multiple (LCM) mean. They will take this understanding and apply it to solving GCF and LCM word problems. Students will then illustrate their understanding by creating posters based on their word problems. There are examples of different types of methods, online games, a rubric, and a power point to summarize this two-day lesson. |
| Can you say that another way? | Students will model how to express an addition problem using the distributive property. |
| Digesting the Distributive Property | This lesson will show the student how to use the distributive property to express a sum of two whole numbers 1-100. |
| You Can Never Have Too Many Shoes! | This lesson teaches Least Common Multiples. |
| Finding the Greatest Crush Factor | This lesson uses a real-life approach to exploring the use of Greatest Common Factors (GCF). The students will utilize math practice standards as they analyze math solutions and explain their own solutions. |
| Factoring out the Greatest | This lesson teaches students how to find the GCF and LCM by factoring. This is a different method than is normally seen in textbooks. This method easily leads to solving GCF word problems and using the distributive property to express a sum of two whole numbers. |
Original Student Tutorials
| Name |
Description |
| Homework Help: Least Common Multiple Part 2 | Use the least common multiple to solve real-life problems with Brady and Natalia in this interactive tutorial.
This is part 2 of 2-part series, click HERE to view part 1. |
| Homework Help: Least Common Multiple Part 1 | Learn how to find the least common multiple by helping Brady and Natalia work through some homework questions in this interactive student tutorial.
This is part 1 of 2-part series, click HERE to view part 2. |
Problem-Solving Tasks
| Name |
Description |
| Bake Sale | The purpose of this task requires students to apply the concepts of factors and common factors in a context. A version of this task could be adapted into a teaching task to help motivate the need for the concept of a common factor. |
| Multiples and Common Multiples | This problem uses the same numbers and asks similar mathematical questions as "The Florist Shop" file, but that task requires students to apply the concepts of multiples and common multiples in a context. |
| The Florist Shop | Students are asked to solve a real-world problem involving common multiples. |
| Factors and Common Factors | This problem uses the same numbers and asks essentially the same mathematical questions as "Bake Sale," but that task requires students to apply the concepts of factors and common factors in a context. |
Tutorials
| Name |
Description |
| Least Common Multiple | This video demonstrates the prime factorization method to find the lcm (least common multiple). |
| Least Common Denominators | In this tutorial, students will be exposed to the strategy of finding the least common denominator for certain cases. Elementary teachers should note this is not a requirement for elementary standards and consider whether this video will further student knowledge or create confusion. This chapter explains how to find the smallest possible common denominator. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. |
Video/Audio/Animation
| Name |
Description |
| LCM and GCF Word Problems | Here we have a couple of word problems searching for the least common multiple and the other for the greatest common factor. |
Student Resources
Original Student Tutorials
| Name |
Description |
| Homework Help: Least Common Multiple Part 2: | Use the least common multiple to solve real-life problems with Brady and Natalia in this interactive tutorial.
This is part 2 of 2-part series, click HERE to view part 1. |
| Homework Help: Least Common Multiple Part 1: | Learn how to find the least common multiple by helping Brady and Natalia work through some homework questions in this interactive student tutorial.
This is part 1 of 2-part series, click HERE to view part 2. |
Problem-Solving Task
| Name |
Description |
| The Florist Shop: | Students are asked to solve a real-world problem involving common multiples. |
Tutorials
| Name |
Description |
| Least Common Multiple: | This video demonstrates the prime factorization method to find the lcm (least common multiple). |
| Least Common Denominators: | In this tutorial, students will be exposed to the strategy of finding the least common denominator for certain cases. Elementary teachers should note this is not a requirement for elementary standards and consider whether this video will further student knowledge or create confusion. This chapter explains how to find the smallest possible common denominator. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. |
Parent Resources
Problem-Solving Tasks
| Name |
Description |
| Bake Sale: | The purpose of this task requires students to apply the concepts of factors and common factors in a context. A version of this task could be adapted into a teaching task to help motivate the need for the concept of a common factor. |
| The Florist Shop: | Students are asked to solve a real-world problem involving common multiples. |
The greatest common factor of 96 and 138 is 6.
