6.1.1C Factors & Primes / GCF & LCM
Overview
Relationships of equivalence with different forms of rational numbers can be illustrated in a variety of representations. Students use fractions, decimals, and percents to describe equivalent positive rational numbers. Clement (2004) suggests five different kinds of representations for teaching students the concepts of fractions, decimals and percents. These five representations are pictures, manipulatives, spoken language, written symbols, and relevant situations. Conceptual understanding of these equivalencies is developed as students describe these rational numbers in concrete representational forms (fraction strips, Cuisenaire rods, pattern blocks, etc.); visual representational forms (grids, diagrams, pictures, etc.); and abstract symbolic form. Using these same structures, students expand their understanding of equivalence with rational numbers to making comparisons between them. Students' learning experiences with these different forms of representation guide them in identifying and selecting appropriate forms for making comparisons and conversions in a particular situation.
The skills of prime factorization, least common multiple, and greatest common factor become tools for students in their formation of equivalent fractional numbers. Students' understanding of representing whole numbers as a product of factors with exponents is aided by their previous work with whole numbers, multiples, factors and exponents.
All Standard Benchmarks
● 6.1.1.1 Locate positive rational numbers on a number line and plot pairs of positive rational numbers on a coordinate grid.
● 6.1.1.2 Compare positive rational numbers represented in various forms. Use the symbols <, =, and >.
● 6.1.1.3 Understand that percent represents parts out of 100 and ratios to 100.
● 6.1.1.4 Determine equivalences among fractions, decimals, and percents: select among these representations to solve problems.
● 6.1.1.5 Factor whole numbers; express a whole number as a product of prime factors with exponents.
● 6.1.1.6 Determine greatest common factors and least common multiples. Use common factors and common multiples to calculate with fractions and find equivalent fractions.
● 6.1.1.7 Convert between equivalent representations of positive rational numbers.
Benchmark Group C
- 6.1.1.5 Factor whole numbers; express a whole number as a product of prime factors with exponents.
- 6.1.1.6 Determine greatest common factors and least common multiples. Use common factors and common multiples to calculate with fractions and find equivalent fractions.
What students should know and be able to do [at a mastery level] related to these benchmarks:
Use divisibility tests to factor whole numbers;
- Express whole numbers as the product of prime factors, including the use of exponential notation;
- Understand that each whole number has a unique prime factorization;
- Understand relationships among factors, multiples, divisors, products and quotients;
- Use a variety of strategies to determine factors, multiples, least common multiples, and greatest common factors, including prime factorization;
- Use common factors to simplify fractions;
- Understand that dividing the numerator and denominator of a fraction by their greatest common factor results in a fraction in simplest form. Example: the greatest common factor of 12 and 16 is 4. [math]\frac{12 \div 4}{16 \div 4}=\frac{3}{4}[/math], which is [math]\frac{12}{16}[/math] in simplest form;
- Use common multiples as common denominators and find equivalent fractions to perform calculations;
- Understand that the lowest common denominators used to add or subtract fractions are their least common multiples. Example: the least common multiple of 12 and 16 is 48. The lowest common denominator for adding fractions such as [math]\frac{5}{12}[/math] and [math]\frac{5}{16}[/math] is 48;
- Solve a variety of problems involving factors, greatest common factors, multiples, and least common multiples.
Work from previous grades that supports this new learning includes:
Demonstrate fluency with multiplication and division facts;
- Multiply multi-digit numbers, using efficient and generalizable procedures, based on knowledge of place value, including standard algorithms;
- Divide multi-digit numbers, using efficient and generalizable procedures, based on knowledge of place value, and including standard algorithms;
- Apply the commutative, associative and distributive properties and order of operations to generate equivalent numerical expressions and to solve problems involving whole numbers.
NCTM Standards:
Understand numbers, ways of representing numbers, relationships among numbers, and number systems
- work flexibly with fractions, decimals, and percents to solve problems;
- compare and order fractions, decimals, and percents efficiently and find their approximate locations on a number line;
- develop meaning for percents greater than 100 and less than 1;
- understand and use ratios and proportions to represent quantitative relationships;
- use factors, multiples, prime factorization, and relatively prime numbers to solve problems.
Common Core State Standards:
4NF (NUMBER AND OPERATIONS - FRACTIONS) Extend understanding of fraction equivalence and ordering.
- 4NF.7 Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model.
4OA. (OPERATIONS AND ALGEBRAIC THINKING) Gain familiarity with factors and multiples.
- 4OA.4 Find all factor pairs for a whole number in the range 1-100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1-100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1-100 is prime or composite.
5NBT (NUMBER AND OPERATIONS IN BASE TEN) Understand the place value system.
- 5NBT.3b Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.
6NS (NUMBER SYSTEM) Apply and extend previous understandings of multiplication and division to divide fractions by fractions.
- 6NS.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.
- 6NS.6 Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.
- Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., -(-3) = 3, and that 0 is its own opposite.
- Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
- Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.
- 6NS.7 Understand ordering and absolute value of rational numbers.
- 6NS.7.b Write, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write -3^{0}C > -7^{0}C to express the fact that -3^{0}C is warmer than -7^{0}C.
6RP (RATIOS AND PROPORTIONAL RELATIONSHIPS) Understand ratio concepts and use ratio reasoning to solve problems.
- 6RP.3 Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
- 6RP3.c Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.
Misconceptions
Student Misconceptions and Common Errors
- Students believe that 1 is a prime number.
- Students believe that even numbers are composite and odd numbers are prime;
- Students believe that factors of a given number must be smaller than the given number; example: They do not realize that 36 is a factor of 36.
- Students report that some numbers have no common factors; example: They do not recognize 1 as a common factor of 4 and 5.
- Students will sometimes use a decimal as a factor; example: They believe 2.5 is a factor of 10 since 4 x 2.5 = 10.
- Students believe that multiples of a given number must be larger than the given number; example: They do not recognize 36 as a multiple of 36.
- Student will sometimes begin listing the multiples of a number with 1 and then continue accurately with multiples because they are confusing factoring with listing multiples. Example: Multiples of 3 are: 1, 3, 6, 9, 12...
- Students may not understand that each whole number has unique prime factorization.
- Students may incorrectly write the prime factorization for numbers.
Vignette
In the Classroom
In the following vignette, Ms. Phillips assesses her students' understanding of greatest common factor and least common multiple, after instruction, by engaging them to design her flower bed.
Teacher: I need some advice about how to plant the flowers in my rectangular flower. Every year I plant only pink, purple, and yellow flowers, because those are my favorite colors. I usually just purchase one "pack" of each color. Are you familiar with the word "pack," like in "6-pack?"
Student: Yeah, a "6-pack" of soda has 6 cans of soda.
Teacher: Yes, that's correct. In the past, all flowers were sold in 12-packs, meaning there were 12 individual flower plants in each pack; but this year it's different. Only the purple flowers are sold in 12-packs. The yellow flowers are sold in 18-packs, so each pack has 18 individual yellow flower plants, and the pink flowers are sold in 24-packs. I plan to purchase one pack each of purple, yellow, and pink flowers, and then plant each row of the garden with the same color flower. For example, one row might be all pink, another row all yellow, and so on. Here's where I need your advice. I can't figure out how many to put in each row now that the flower packs come in different sizes. What is the largest number of flowers I can plant in a row and use all of the flowers if every row has only one color flower? Remember, I'm planning to buy only one pack of each color flower. Work with your partners to solve my problem, and I look forward to hearing your advice in a few minutes.
A few minutes later....
Teacher: I see several strategies being used to solve my problem. Let's start with the strategy of Student A.
Student A: I drew pictures. First I drew all the possible ways to plant the garden with just the 24 pink flowers.
Student A: Then I drew all the possible ways to plant just the yellow flowers.
Student A: And I did the same thing for the purple flowers.
Teacher: And how did that help you to decide how many flowers I should plant in each row?
Student A: Well, I noticed that I could make rows of either 1, 2, 3, or 6 flowers for each of the colors. Since you asked for the largest number that could be planted in a row, I think that might be 6.
Teacher: How can you be sure?
Student A: I could make another diagram. Let me try it.
Student A: Yup, it works. I've used all the flowers.
Teacher: Nicely done. I saw Student B solving the problem a different way. Let's hear about that.
Student B: I sort of did the same thing, except I didn't draw pictures. I started by making lists of the different ways to make rectangles with each of the flower colors. Like for the 24 pink flowers, I wrote 1 x 24, 2 x 12, 3 x 8, and 4 x 6. I didn't write 6 x 4, 8 x 3, 12 x 2, and 24 x 1 again, because those are really the same rectangles as the first ones I wrote.
Teacher: How do you know that?
Student B: Well, if you draw a rectangle that's 3 x 8 and turn it 90^{०}, it's the same as an 8 x 3 rectangle.
Teacher: Let's use Student A's drawings to help us understand what Student B is saying.
Teacher: Yes, those two rectangles are congruent. Also, what property tells us that 3 x 8 = 8 x 3?
Student C: Is it the commutative property?
Teacher: Yes, the commutative property tells us that you'll get the same result when switching the order of the factors in multiplication. Addition is also commutative, but be careful. It's not true for subtraction or division. OK, back to my flower garden. So what did you do next?
Student B: I noticed that I was really just making lists of pairs of factors. So for 18 I had 1 x 18, 2 x 9, and 3 x 6. For 12 I had 1 x 12, 2 x 6, and 3 x 4. Like Student A, I saw that 1, 2, 3, and 6 were factors that showed up in each of the lists. You asked for the greatest number of flowers that you could plant in each row, and that would be 6.
Teacher: Are you saying that 6 is the greatest common factor of 12, 18, and 24?
Student B: I suppose. I hadn't thought of that.
Student D: I did. May I share my strategy?
Teacher: All right. How did you solve the problem?
Student D: I was thinking that to use all the flowers without having any leftover would mean that 12, 18, and 24 would all need to be divisible by the number of flowers in each row.
Teacher: So what does divisibility have to do with factors?
Student D: Well, if 12 is divisible by 6, that means 6 is a factor of 12.
Student E: Oh, so using divisibility rules really helps you identify factors.
Teacher: Exactly. Now go on with your strategy.
Student D: Since I knew I was looking for a factor, and you wanted it to be the largest factor, I used prime factorization to find the greatest common factor of 12, 18, and 24.
Teacher: What do you mean when you say you used prime factorization?
Student D: I made those trees to find the prime factors of each number. Here, let me show you.
Student D: Then I circled the factors that were in common for all three numbers. Since each shared the factors 2 and 3, I multiplied 2 x 3 to find the GCF.
Teacher: It certainly seems like 6 is the largest number I can have in every row when I plant each row the same color and use all the flowers. But I've been thinking that I really don't want my garden to have more pink flowers than yellow or purple flowers. I want my garden to have an equal number of each color. How many packs of each color should I purchase if I want to have the same number of each color?
Student E: That's a totally different problem, but I think we could use some of the same strategies.
Teacher: Tell me more. How could we use the pictures?
Student A: We can just keep adding "packs" until we have the same number flowers of each color. I'll show you.
My picture shows that 3 packs of 24 has the same number of flowers as 4 packs of 18 and 6 packs of 12.
Teacher: How many flowers of each color is that?
Student A: Uh, 24 x 3 is 72, 18 x 4 is 72, and 12 x 6 is also 72, so 72 flowers of each color.
Student B: Wait a minute. This looks a problem about multiples instead of factors. I wonder if I can use lists of multiples to solve it. I'm gonna try it.
Student B: Sure enough. It works.
Teacher: Is buying 6 packs of 12 flowers, 4 packs of 18 flowers , and 3 packs of 24 flowers, the only way I can get an equal number of each color flower?
Student E: No, I think any multiple of 72 flowers would work. For example, you could get 144 flowers of each color by purchasing 12 packs of 12 flowers, 8 packs of 18 flowers, and 6 packs of 72 flowers.
Teacher: Yes, 144 is a multiple of 72 because 144 is 2 times or double 72. It appears that the number of "packs" needed to get an equal number of flowers also doubled. 72 flowers required 4 packs of 18 flowers, but 144 flowers requires 8 packs of flowers. 8 packs is twice as many as 4.
Student E: If you want the same number of each color flower, the total number of flowers has to be a multiple of 72 - like 144, 216, 288, 360, and so on. It just depends on how many flowers you want to plant.
Teacher: Good point. I don't enjoy planting that much, so I prefer to plant the minimum.
Student D: Then that's the least common multiple. I should be able to find that using prime factorization. Let me think about that for a minute.
Student D: My trees show me that there are some additional factors that need to be considered to find the least common multiple.
Teacher: Which ones?
Student D: The ones not circled - the 2, 3, 2, and 2.
Teacher: So what do you think should be done about them?
Student D: I need to multiply. But when I multiply them by the GCF, 6 x 2 x 3 x 2 x 2 = 144. That's not right. It should be 72.
Teacher: Anyone have ideas about why that happened?
Student A: 144 is 72 x 2, so your strategy gave you an answer that's 2 times as big as the correct answer. Does it have anything to do with the fact that the numbers 12 and 24 share another common factor of 2?
Teacher: Good reasoning. Because 2 is another common factor shared only by 12 and 24, including it gave an answer 2 times as big.
Student D: I'm going to mark that "2" with a box to help me remember that it's another common factor, but just not shared by all three numbers.
Teacher: Sounds like a plan. Thanks for helping me plan my flower garden. Tomorrow we're going to continue this conversation and explore ratios between flower colors.
Resources
Teacher Notes
- By definition, 1 is not a prime number. However, 1 is always a common factor of two numbers.
- Remind students that although 2 is an even number, it is prime since its only factors are 1 and itself. Using examples such as 9, 15, and 21 will help students understand that not all odd numbers are prime.
- Students may not understand that for any given number, 1 and the number itself are factors. They will benefit from listing factors in order from least to greatest so that students have opportunity to see that 1 is always the smallest factor, and the number itself is always the greatest factor. This strategy also helps students to recognize that 1 is always a common factor.
- In earlier grades, students may have come to believe that any numbers being multiplied are considered to be factors. Remind students that one number is a factor of a second number only if it divides that number with no remainder. Although 4 x 2.5 = 10, 4 is not a factor of 10 because 10[math]\div[/math]4 is 2 remainder 2.
- Students may not understand that any given number is a multiple of itself. Remind students that when making lists of multiples, they need to begin by using the factor of 1. For example, the multiples of 36 are 36 (36 x 1), 72 (36 x 2), 108 (36 x 3)....
- Students will easily discover that each number has a unique prime factorization when asked to list factors in order from smallest to greatest and compare answers with peers.
- It is essential that students have repeated practice using prime factorization to find greatest common factors and develop strategies to find least common multiples.
base (of an exponent): the number used as the factor in exponential notation base^{exponent}. Example: In 6^{4} = 6 x 6 x 6 x 6, 6 is the base used as a factor 4 times.
exponent: in exponential notation (base^{exponent} ), the exponent is the number that tells how many times the base is used as a factor. Example: In 8 x 8 x 8 = 8^{3}, the exponent is 3 with base 8.
greatest common factor (GCF): the largest shared factor of two or more numbers. Example: The factors of 27 are: 1, 3, 9, and 27. The factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, and 36. The GCF of 27 and 36 is 9.
least common multiple (LCM): the smallest number, other than zero that is a multiple of two or more given numbers. The LCM is the lowest common denominator (LCD) for adding and subtracting fractions. Example: Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, and 30; multiples of 8: 8, 16, 24, 32, and 40; the LCM of 3 and 8 is 24.
power: the exponent to which a "base" number is raised in exponential notation (base^{exponent}). Example: In 8^{3}, the base 8 is being raised to the 3rd power.
prime factor: a factor greater than 1 that has exactly two whole-number factors, 1 and itself. Example: 7 is considered a prime factor of 63 since 7 x 9 = 63 and the only factors of 7 are 1 and itself.
prime factorization: a whole number written as a product of prime factors. Every whole number greater than 1 has a unique factorization. Example: The prime factorization of 144 is 2 x 2 x 2 x 2 x 3 x 3 or 2^{4 }x 3^{2}.
Professional Learning Communities
Reflection - Critical Questions regarding the teaching and learning of these benchmarks:
- What strategies are students using to find greatest common factors? least common multiples? Are those strategies efficient and generalizable to algebra?
- How will the skills of finding greatest common factors and least common multiples be built on in subsequent years?
- How are the greatest common factor and least common multiple related?
- What are some real-life examples of greatest common factors and least common multiples that can be used to motivate students?
- What evidence do I have that students can transfer their understanding of [math]3^5[/math]^{ }as [math]3 \times 3 \times 3 \times 3 \times 3[/math] to [math]x^5[/math] as [math]x \cdot x \cdot x \cdot x \cdot x[/math]?
Materials - (suggest articles and books for individual or group study with PLC)
- Burkhart, J. (2009). "Building Numbers from Primes." Mathematics Teaching in the Middle School, 15(3),156-167. Print.
- Clarke, D.M., Roche, A., & Mitchell, A. Ten Practical Tips for Making Fractions Come Alive and Make Sense. Article discusses key ideas and concepts involved in understanding fractions and offers a range of hints to the classroom teacher on how to support students to develop a connected understanding of this important topic. Includes a fractions activity sheet.
References
Clement, L. L. (2004). A model for understanding, using, and connecting representations. Teaching Children Mathematics, 11 (2), 97 - 102.
Keeley, P., & Rose, C. (2006). Mathematics curriculum topic study. Thousand Oaks, CA: Corwin Press.
Kilpatrick, J., Martin, W., & Schifter, D. (Eds.). (2003). A research companion to principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.
Mathematics Curriculum Framework. (2000). Malden, MA: Massachusetts Department of Education.
Minnesota's K-12 Mathematics Frameworks. (1998). St. Paul, MN: SciMathMN.
Mathematics Framework for the 2009 National Assessment of Educational Progress. (2009). Washington, D.C.: National Assessment Governing Board U.S. Department of Education.
Mathematics 6-8 GaDOE. Georgia Department of Education, n.d. Web. 29 Mar. 2011.
Cramer, K., Behr, M., Post T., & Lesh, R., (2009). The Rational Number Project (choose RNP: Initial Fraction Ideas) The Rational Number Project (RNP) advocates teaching fractions using a model that emphasizes multiple representations and connections among different representations.
Behr, M. & Post, T. (1992). Teaching rational number and decimal concepts. In T. Post (Ed.), Teaching mathematics in grades K-8: Research-based methods (2nd ed.) (pp. 201-248). Boston: Allyn and Bacon.
Assessment
DOK: (Level 1)
- Which expression represents the prime factorization of 96?
a) 2^{2} x 3^{3} b) 2^{3} x 3^{2} c) 2^{5} x 3 d) 2^{5} x 3^{2}
Answer: c
DOK: (Level 2)
- Write the prime factorization of 108. Use exponents to show repeated factors.
Answer: 2^{2} x 3^{3}
^{ }
(DOK Level 2)
- Which is the greatest common factor of 4 and 15?
a) 60 b) 30 c) 4 d) 1
Answer: d
(DOK Level 2)
- Boxes that are 24 inches tall are being stacked next to boxes that are 18 inches tall. What is the shortest height at which the two stacks will be the same height?
Answer: 72 inches
(DOK Level 3)
- Brigette has guitar lessons every fifth day and singing lessons every third day. If she had a guitar lesson and a singing lesson on May 5, when will be the next date on which she has both guitar and singing lessons?
Answer: May 20.
(DOK Level 3)
- Which is the least common denominator that can be used to add [math]\frac{5}{8}[/math] and [math]\frac{5}{14}[/math]?
Answer: 56
(DOK Level 4)
- Name two numbers whose GCF is 60 and LCM is 600.
Answer: 120 and 300
Differentiation
Struggling Students
- Provide a chart of divisibility tests and examples of their use to help students determine factors.
- Use a multiplication chart and/or a calculator to help students determine multiples.
- Use Venn diagrams to find common factors or common multiples.
- Factor Trees
This website allows students to practice using one or two factor trees. If two factor trees are chosen, it asks students to identify both the LCM and GCF.
This website offers a variety of fraction topics. It gives definitions, explanations, and allows students to practice various fraction concepts.
- A laminated hundreds chart can be used for students to mark multiples of numbers and identify the LCM; erase, and begin again
- Cuisenaire rods of varying lengths or number lines are helpful to show the LCM.
English Language Learners
- Be aware of the use of acronyms, such as GCF and LCM, and ensure that students understand their meanings.
- Words with multiple meanings need special attention; e.g. base, power, prime.
- Post examples that show factors, factorization, and multiples for the same number; example:
Factors of 20: 1, 2, 4, 5, 10, 20;
Factorization of 20: 2^{2} x 5
Multiples of 20: 20, 40, 60, 80...
- Post examples that contrast factorization of numbers that are prime and numbers that are not. Example: Factorization of 47: 1 x 47 (Prime); Factorization of 48: 2^{4} x 3 (Not Prime).
- Pair students with strong English speaking partners for activities.
- Use graphic organizers such as the Frayer model shown below, for vocabulary development.
Extending the Learning
- Ask students to find multiples of algebraic expressions that result from multiplying by whole numbers. Example: 4(x + 3) = 4x + 12
- Ask students to factor simple algebraic expressions and determine the greatest common factor. Example: 2x + 6 = 2(x + 3) and 4x + 12 = 4(x + 3) = 2[math]\cdot[/math]2(x + 3). The GCF of 2x + 6 and 4x + 12 is 2(x + 3).
Parents/Admin
Administrative/Peer Classroom Observation
Students are: (descriptive list) | Teachers are: (descriptive list) |
using a variety of strategies to factor positive integers, discover common factors, and find their greatest common factor. | modeling a variety of strategies for factoring and finding great common factors. |
simplifying fractions by using common factors of numerators and denominators. | posing questions that require simplification of fractions. |
understanding that a fraction is in simplest form when the greatest common factor between the numerator and denominator is 1. | asking students how they know when a fraction is in simplest form. |
expressing positive integers as products of prime factors, including the use of exponential notation. | asking students to use exponential notation to find common factors, find greatest common factor, and least common multiple. |
discovering that each positive integer has a unique prime factorization. | asking students to compare their prime factorizations. |
using a variety of strategies to find common multiples of whole numbers and find their least common multiple. | modeling a variety of strategies, including prime factorization, to find common multiples and the least common multiple. |
using common multiples as common denominators to calculate with fractions. | posing real-world questions that require addition or subtraction of fractions with unlike denominators. |
discovering that the lowest common denominator of fractions is the least common multiple of the denominators. | allowing students to use any common denominator and facilitating a conversation about efficiency. |
solving real-world problems that involve factors and multiples. | encouraging students to think about their world and uses for lcm or gcf both in and out of school. |
Parent Resources
Multiples, Factors, Primes and Composites
http://www.learnalberta.ca/content/me5l/html/math5.html
This lesson shares the basics in an interactive approach that is very concrete and sequential.
Parent Resources
Factors, Multiples, and Prime Factorization
This lesson uses a number line and interactive activities to demonstrate the connections among factors, multiples, and prime factorization
This website allows a student to make one or two factor trees at the same time. It then allows the student to interactively place factors in a math Venn diagram. It further allows students to find the LCM and GCF.
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