### 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

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).*- Multiple listing method
- FactorizationLCM is represented as 2×3×5.

- Multiple listing method
- Instruction includes a variety of methods and strategies to determine the greatest common factor
*(**MTR.2.1, MTR.3.1, MTR.5.1*).- T-chartsThe 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.

- T-charts

- 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.
- Example:

- Example:
- 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*

Parker reads a book every 12 days and Leah reads a book every 8 days.

**(***MTR.3.1*)- 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?

*What is the least common multiple between 17 and 12?*

Instructional Item 2

Instructional Item 2

**The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.*

## Related Courses

## Related Access Points

## Related Resources

## Formative Assessments

## Lesson Plans

## Original Student Tutorials

## Problem-Solving Tasks

## Tutorials

## Video/Audio/Animation

## MFAS Formative Assessments

Students are given two whole numbers less than or equal to 100 and asked to find the greatest common factor.

Students are asked to find the least common multiple of 8 and 12 and to explain how they found their answers.

## Original Student Tutorials Mathematics - Grades 6-8

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.

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.

## Student Resources

## Original Student Tutorials

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.

Type: Original Student Tutorial

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.

Type: Original Student Tutorial

## Problem-Solving Task

Students are asked to solve a real-world problem involving common multiples.

Type: Problem-Solving Task

## Tutorials

This video demonstrates the prime factorization method to find the lcm (least common multiple).

Type: Tutorial

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. *

Type: Tutorial

## Parent Resources

## Problem-Solving Tasks

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.

Type: Problem-Solving Task

Students are asked to solve a real-world problem involving common multiples.

Type: Problem-Solving Task