6.NS.4 - About the Math, Learning Targets, and Increasing Rigor
Grade 5AGL The Number System
6.NS.4
Full Standard
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).
Learning Targets (I can)
- Find factor pairs for any number between 1 and 100.
- Find multiples of a number (multiples up to 100).
- Explain what the greatest common factor is.
- Find the greatest common factor of two numbers less than or equal to 100.
- Explain what a least common multiple is.
- Find the least common multiple of two numbers less than or equal to 12.
- Express two addends with a common multiple using the distributive property (i.e. 36 + 8 is the same 4(9 + 2), note: this standard is explored during daily number routines).
About the Math
- Find factor pairs for any number between 1 and 100.
Factor pairs can be found in a variety of ways. Students should have many opportunities to find factor pairs and look for patterns within factor pairs. Finding and using factor pairs is more than creating "rainbows." Instead, students should reason about why a halfway number is useful for finding factor pairs and how knowing factor pairs of some numbers can help with others. For example, knowing the factors of 40 can help with finding the factors of 80. Explicit instruction of divisibility rules is not appropriate, however student discovery of these rules could be an enrichment within this standard for students who show proficiency with factors, multiples, and prime/composite numbers.
- Find multiples of a number (multiples up to 100).
0 is often overlooked as a multiple. 0 is in fact a multiple of any whole number. However, it is often ignored as it isn't useful or practical. However, 1 is not a multiple yet many students often believe that it is.
- Explain what the greatest common factor is.
- Find the greatest common factor of two numbers less than or equal to 100.
In the earlier grades, students identified primes, composites, and factor pairs. In 6th grade 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:
Listing 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 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 10 and 4 are not relatively prime (have 2 in common), the 4 cannot be the greatest common factor.
Listing the prime factors of 40 (2 x 2 x 2 x 5) and 16 (2 x 2 x 2 x 2) and then multiplying the common factors (2 x 2 x 2 = 8).
For this standard, students should also understand that the greatest common factor of two prime numbers is 1.
- Explain what a least common multiple is.
- Find the least common multiple of two numbers less than or equal to 12.
Common multiples can be used for various purposes. Here they are useful for finding common denominators. Often, the least common multiple is used so that a fraction is in simplest form. However, students can find common denominators without using LCMs. Often, students need to find common denominators or common multiples before then finding a least common multiple.
- Express two addends with a common multiple using the distributive property (i.e. 36 + 8 is the same 4(9 + 2), note: this standard is explored during daily number routines).
Common multiples can be used to simplify equations and expressions among other things in later grades. As noted with this learning target, students can use common multiples, even greatest common multiples to find express addends.
Essential vocabulary for this standard includes greatest common factor, least common multiple, distributive property, prime numbers, composite numbers, relatively prime, factors, multiples, and prime factorization.
Progression of Standard within Grade 6 (5 AGL)
This progression informs how to develop the standard within the grade level. This progression is provided by HCPSS Elementary Mathematics.
| Quarter 1 | Quarter 2 | Quarter 3 | Quarter 4 |
|---|---|---|---|
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Progression of this Standard Across Grades
This progression is informed by the Achieve the Core Coherence Map Links to an external site.. Information is not the complete standard.
| Grade 5 | Grade 7 |
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TASKS
These tasks can be used with small group or whole group instruction.
- 54 + 9 is the same as 3 (18 + 3). What is another way you could use the distributive property to represent 54 + 9?
- What is the smallest number that has four and six as factors? What about six and nine?
- The weather is reported every 18 minutes on WBAL and every 12 minutes on WJZ. Both stations broadcast the weather at 1:30. What is the next time the stations will broadcast the weather at the same time?
- How many common factors do 36 and 84 have?
- How do you know when you have found all of the possible factors for a given number?
- A set of 36 red checkers and 60 black checkers is to be arranged into piles. Each pile may contain only red or black checkers. All piles, both red and black, must contain the same number of checkers. What is the greatest number of checkers that each pile can have? (12)
- Christian has 48 chocolate chip cookies and 64 gingerbread cookies. What is the largest number of bags he can make with an equal number of chocolate chip cookies and an equal number of gingerbread cookies in each bag (assuming he uses them all)? Explain your reasoning.
- The following expressions (in red) are possible ways to make 36 + 12. Are they all correct? Explain your reasoning. 3(12 + 4), 9(4 + 3), 2(18 +6), and 4(9 + 3)
SLIDE-BASED TASKS
These links are HCPSS created instructional tasks. These tasks are provided in Google slides. These tasks should be used for inspiration and resources, but instruction should start with students having the opportunity to engage with the math first (often involving physical and/or visual models) followed by discussion and explicit instruction to ensure student understanding.
Module 5 • Expressions
Express Two Addends with a Common Multiple Using the Distributive Property:
Module 11 • Problem Solving with GCF/LCM and Fractions
Find Greatest Common Factor (GCF) and Least Common Multiple (LCM) and Solve Problems with GCF and LCM:
Additional Tasks
These links provide instructional ideas connected to this standard.
- Adding Multiples (Links to an external site.) Links to an external site.(Illustrative Math)
- Factors and Common Factors Links to an external site. (Links to an external site.) (Illustrative Math)
- Multiples and Common Multiples (Links to an external site.) Links to an external site. (Illustrative Math)
- Common Factors Links to an external site. (Illustrative Math Lesson 7.16)
- Common Multiples Links to an external site. (Illustrative Math Lesson 7.17)
- Using Common Multiples and Common Factors Links to an external site. (Illustrative Math Lesson 7.18)
- The Florist Shop (Links to an external site.) Links to an external site.(Illustrative Math)
- Bake Sale (Links to an external site.) Links to an external site.(Illustrative Math)
- Geared Up Links to an external site. (3-Act Task, G. Fletcher)
- It All Adds Up Download It All Adds Up (Lesson Collection)
- The Venn Factor Links to an external site. (NCTM Illuminations Lesson)
- Greatest Common Factor Links to an external site. (MSDE Lesson)
- GCF and LCM Links to an external site. (MSDE Lesson Seed)
- Shipping Routes Links to an external site.(101 Questions task)
Tasks From Print Resources
These publications have been provided for each school. They are typically stored in team closets or the media center. Check with your team leader if you cannot find them.
| Book Thumbnail | Book Title | Grade | Pages |
|---|---|---|---|
| Teaching Student-Centered Mathematics | 6-8 | Common Multiple Flash Cards, Activity 8.17, Page 128 | |
|
Nimble With Numbers
|
6-7 |
28-29 32-33 35 |
|
|
Hands-On Standards |
5-6 | 36 (Factors, Primes, and Prime Factorization) | |
|
Fundamentals |
4-5 | 28 (Factor Find) | |
|
Math Intervention: Building Number Power |
3-5 | 147-152 | |
|
Brain-Compatible Activities for |
4-5 |
61-62 66-68 |
|
|
Problem-Driven Math |
5 |
53 (Animals in the Classroom) 56 (Celia's Cookout) 32 (Bowling for Fish) |
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Centers
These print resources can be used during independent or center time. These resources could also be used as lesson seeds.
INDEPENDENT PRACTICE/HOMEWORK/ASSESSMENT
These resource sheets can be used for independent practice, homework, or assessment. They are intended to reinforce procedures and concepts. They should not be used as a source of direct instruction or whole-group practice.
- Factors Links to an external site.
- Greatest Common Factor Links to an external site.
- Multiples Links to an external site.
- Least Common Multiple Links to an external site.
- Find and Describe LCM Links to an external site.
- Explain GCF and LCM Links to an external site.
- Find GCF and LCM Links to an external site.
- Apply GCF to Solve Word Problems Links to an external site.
- Apply GCF to Solve Word Problems_2 Links to an external site.
- Apply LCM to Solve Word Problems Links to an external site.
- Represent Addition Expression Using Distributive Property Links to an external site.
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Learning Targets
- Find factor pairs for any number between 1 and 100.
- Find multiples of a number (multiples up to 100).
- Explain what the greatest common factor is.
- Find the greatest common factor of two numbers less than or equal to 100.
- Explain what a least common multiple is.
- Find the least common multiple of two numbers less than or equal to 12.
- Express two addends with a common multiple using the distributive property (i.e. 36 + 8 is the same 4(9 + 2), note: this standard is explored during daily number routines).
Learning targets identify what students should be able to do. This rubric can be applied to tasks and observations for assessment and/or grading.
Rubric for Tasks Links to an external site.
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Links to an external site. license. Content in this course can be considered under this license unless otherwise noted.