## TEACHER GUIDE TO CLARIFICATION

###### Compute Fluently and Find Common Factors and Multiples

This is the first grade level in which students will be formally taught about LCM and GCF. For the most part, strategies, will remain the same but using and applying the distributive property is a major focus as well.

Students will find the greatest common factor of two whole numbers less than or equal to 100. For example, the greatest common factor of 40 and 16 can be found by

1) listing the factors of 40 (1, 2, 4, 5, 8, 10, 20, 40) and 16 (1, 2, 4, 8, 16), then taking the greatest common factor (8). Eight (8) is also the largest number such that the other factors are relatively prime (two numbers with no common factors other than one). For example, 8 would be multiplied by 5 to get 40; 8 would be

multiplied by 2 to get 16. Since the 5 and 2 are relatively prime, then 8 is the greatest common factor. If

students think 4 is the greatest, then show that 4 would be multiplied by 10 to get 40, while 16 would be 4

times 4. Since the 10 and 4 are not relatively prime (have 2 in common), the 4 cannot be the greatest

common factor.

2) listing the prime factors of 40 (2 • 2 • 2 • 5) and 16 (2 • 2 • 2 • 2) and then multiplying the common factors

(2 • 2 • 2 = 8).

Students also understand that the greatest common factor of two prime numbers will be 1.

Students use the greatest common factor and the distributive property to find the sum of two whole numbers. For example, 36 + 8 can be expressed as

4 (9 + 2) = 4 (11).

Students will need practice using the distributive property to find the sum of two whole numbers.

Students find the least common multiple of two whole numbers less than or equal to twelve. For example, the least common multiple of 6 and 8 can be found by

1) listing the multiplies of 6 (6, 12, 18, 24, 30, …) and 8 (8, 26, 24, 32, 40…), then taking the least in common

from the list (24); or

2) using the prime factorization.

Step 1: find the prime factors of 6 and 8.

6 = 2 • 3

8 = 2 • 2 • 2

Step 2: Find the common factors between 6 and 8. In this example, the common factor is 2

Step 3: Multiply the common factors and any extra factors: 2 • 2 • 2 • 3 or 24 (one of the twos is in common; the other twos and the three are the extra factors.

Examples:

* What is the greatest common factor (GCF) of 24 and 36? How can you use factor lists or the prime factorizations to

find the GCF?

Solution: 22•3 = 12. Students should be able to explain that both 24 and 36 have 2 factors of 2 and one factor of 3, thus 2 x 2 x 3 is the greatest common factor.)

* What is the least common multiple (LCM) of 12 and 8? How can you use multiple lists or the prime factorizations

to find the LCM?

Solution: 23• 3 = 24. Students should be able to explain that the least common multiple is the smallest number that is a multiple of 12 and a multiple of 8. To be a multiple of 12, a number must have 2 factors of 2 and one factor of 3 (2 x 2 x 3). To be a multiple of 8, a number must have 3 factors of 2 (2 x 2 x 2). Thus the least common multiple of 12 and 8 must have 3 factors of 2 and one factor of 3 ( 2 x 2 x 2 x 3).

* Rewrite 84 + 28 by using the distributive property. Have you divided by the largest common factor? How do you

know?

* Given various pairs of addends using whole numbers from 1-100, students should be able to identify if the two

numbers have a common factor. If they do, they identify the common factor and use the distributive property to rewrite the expression. They prove that they are correct by simplifying both expressions.

27 + 36 = 9 (3 + 4)

63 = 9 x 7

63 = 63

31 + 80

There are no common factors. I know that because 31 is a prime number, it only has 2 factors, 1 and 31. I know that 31 is not a factor of 80 because 2 x 31 is 62 and 3 x 31 is 93.

Instructional Strategies (NS. 2 through 4)

As students study whole numbers in the elementary grades, a foundation is laid in the conceptual understanding of each operation. Discovering and applying multiple strategies for computing creates connections which evolve into the proficient use of standard algorithms. Fluency with an algorithm denotes an ability that is efficient, accurate, appropriate and flexible. Division was introduced in Grade 3 conceptually, as the inverse of multiplication. In Grade 4, division continues using place-value strategies, properties of operations, and the relationship with multiplication, area models, and rectangular arrays to solve problems with one digit divisors. In Grade 6, fluency with the algorithms for division and all operations with decimals is developed.

Fluency

Definition

Fluency is something that develops over time; practice should be given over the course of the year as students solve problems related to other mathematical studies. Opportunities to determine when to use paper pencil algorithms, mental math or a computing tool is also a necessary skill and should be provided in problem solving situations.

Greatest common factor and least common multiple are usually taught as a means of combining fractions with unlike denominators. This cluster builds upon the previous learning of the multiplicative structure of whole numbers, as well as prime and composite numbers in Grade 4. Although the process is the same, the point is to become aware of the relationships between numbers and their multiples. For example, consider answering the question: “If two numbers are multiples of four, will the sum of the two numbers also be a multiple of four?” Being able to see and write the relationships between numbers will be beneficial as further algebraic understandings are developed. Another focus is to be able to see how the GCF is useful in expressing the numbers using the distributive property, (36 + 24) = 12(3+2), where 12 is the GCF of 36 and 24. This concept will be extended in Expressions and Equations as work progresses from understanding the number system and solving equations to simplifying and solving algebraic equations in Grade 7.

Kansas Association of Teachers of Mathematics (KATM) Flipbooks. Questions or to send feedback: melisa@ksu.edu.

Retrieved from Math Flipbooks.

Grade 6 PARCC End Of Year Computer-Based Practice Test

http://parcc.pearson.com/practice-tests/math/

What is the greatest common factor of 16 and 48?

Enter your answer in the box.

###### 6.NS.4

Compute fluently with multi-digit numbers and find common factors and multiples.

6.NS.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).

###### Coherence and Connections: Need to Know

Grade Below

Grade Level

Grade Above

5.G.2

6.NS.4

6.EE.3

6.EE.4

7.EE.1

Examples of Opportunities for Connections among Standards, Clusters and Domains: Students use their skill in recognizing common factors (6.NS.4) to rewrite expressions (6.EE.3).

PARCC Model Content Frameworks: Mathematics Grades 3-11 (version 3). (2012, November 1).

Retrieved June 3, 2014, from http://www.parcconline.org/resources/educator-resources/model-content-frameworks/mathematics-model-content-framework

In many traditional treatments of fractions greatest common factors occur in reducing a fraction to lowest terms, and least common multiples occur in adding fractions. As explained in the Fraction Progression, neither of these activities is treated as a separate topic in the standards. Indeed, insisting that finding a least common multiple is an essential part of adding fractions can get in the way of understanding the operation, and the excursion into prime factorization and factor trees that is often entailed in these topics can be time-consuming and distract from the focus of K-5. In Grade 6, however, students experience a modest introduction to the concepts (6.NS.4) and put the idea of greatest common factor to use in a rehearsal for algebra, where they will need to see, for example, that

Common Core Standards Writing Team. (2013, July 4).

Progressions for the Common Core State Standards in Mathematics (draft). Grades 6–8, The Number System; High School, Number. Tucson, AZ: Institute for Mathematics and Education, University of Arizona.

PARCC Evidence Statements: http://www.parcconline.org/assessments/test-design/mathematics/math-test-specifications-documents

Evidence

Statement Key

Evidence Statement Text

Clarifications

**MP**

6.NS.4-1

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.

i) Tasks do not have a context

6.NS.4-2

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).

i) Tasks do not have a context.

ii) Tasks require writing or finding the equivalent expression with the greatest common factor.

7

no

Calculator

no

###### HOT Questions

1. Write an equivalent expression using the greatest common factor. How do you know the two expressions are

equivalent?

50 + 25

Example answer 50 + 25 = 25 (2 + 1) They both equal 75

2. Given the two addends, identify a common factor and use the distributive property to rewrite the expression.

44 + 12

Example answer 4 (11 + 3)

3. If two numbers are multiples of four, will the sum of the two numbers also be multiples of four? Explain the

relationship between these numbers and their multiples.

Yes, because essentially you are counting by 4’s when you add two numbers that are multiples of 4. For example, 8 + 16 = 24 the addends and sum are multiples of 4

4. Aylin is making a scrapbook using 18 photos and 20 newspaper clippings. She wants all the pages to be set up in the

same way, with the same combination of photos and newspaper clippings on every page. She also wants to make sure

that no items are left over. What is the greatest number of scrapbook pages that Aylin can create?

She can have nine pages with 2 photos and 2 newspaper clippings.

Additional GCF and LCM Word Problems http://www.summit.k12.co.us/cms/lib04/CO01001195/Centricity/Domain/445/GCF%20and%20LCM%20WORD%20PROBLEMS.pdf

###### Additional Resources

Illustrative Mathematics

https://www.illustrativemathematics.org/illustrations/255

https://www.illustrativemathematics.org/illustrations/256

https://www.illustrativemathematics.org/illustrations/257

https://www.illustrativemathematics.org/illustrations/259

https://www.illustrativemathematics.org/illustrations/258

Inside Mathematics – Problem of the Month

http://www.insidemathematics.org/assets/problems-of-the-month/what's%20your%20angle.pdf

LearnZillion

https://learnzillion.com/lessons/2721-find-the-gcf-of-two-numbers-using-the-distributive-property

OER Commons

https://www.oercommons.org/browse/alignment/CCSS.Math.Content.6.NS.B.4