Learning Math: Number and Operations
Number Theory Part A: Models for Multiples and Factors (75 minutes)
In This Part: The Venn Diagram Model
The numbers 24 and 36 have certain things in common, including many common factors — numbers that divide evenly into both of them. For example 2, 3, and 6 are all common factors. The largest such number is called the “greatest common factor.” In this case, the greatest common factor of 24 and 36 is 12. No number greater than 12 is a factor of each of these numbers. See
Another characteristic numbers can share is a common multiple — a third number that is evenly divisible by both 24 and 36. The smallest such number is called the “least common multiple.” In this case, the least common multiple is 72. No number less than 72 is evenly divisible by each number.
One way to explore the common factors and multiples of the two numbers is to use a Venn diagram:
The circle on the left contains all the prime factors (i.e., counting numbers that have exactly two factors: themselves and 1) of 24, and the circle on the right contains all the prime factors of 36. (The number 1 doesn’t qualify as prime, because it has only one factor.)
The numbers contained in the intersection are those factors that are in both numbers; i.e., their common factors. That means that the 2s and the 3 in the intersection, both separately and multiplied together (2 • 2, 2 • 3, and 2 • 2 • 3 — or 2, 3, 4, 6, and 12), are all common factors.
Note that the largest of these factors is 12. The greatest common factor (GCF) of 24 and 36 is 2 • 2 • 3, or 12, the product of all the numbers in the overlap.
Since the circle on the left contains all the factors of 24, every multiple of 24 must contain all of these factors. Likewise, since the circle on the right contains all the factors of 36, every multiple of 36 must contain all of these factors.
In order to be a multiple of both numbers, a number must contain all the factors of both numbers. The smallest number to do this is 2 • 2 • 2 • 3 • 3, or 72, the product of all the factors in the circles. Thus, the least common multiple (LCM) of 24 and 36 is 72.
See Note 3 below.
Use a Venn diagram to determine the GCF and LCM for 18 and 30.
In This Part: Finding Prime Factors
The numbers we’ve examined so far have been fairly simple to factor. Now let’s look at a general method for finding prime factors. See
One method is to draw a factor tree. To do this, write a number — 24, for example — and then draw an upside-down V under it. This V represents two “branches” of the factor tree. Think of a pair of numbers with the product 24; for example, 4 and 6. Check to see if either of these numbers is prime. In this case, the answer is no.
Draw another V under each number that is not prime, and find two factors for each of these numbers. In this case, we will find the factors 2 and 2 for 4, and 2 and 3 for 6. Now we have four factors, 2, 2, 2, and 3, all of which are prime numbers. This is the prime factorization of 24.
In mathematics, we like to do things via consistent algorithms. So rather than just picking two factors, it’s a good idea to make the process more consistent by first finding the smallest prime factor and its partner and then repeating that process on the partner (since the first number is guaranteed to be prime). For example, you could factor out all of the possible 2s, then all the 3s, then 5s, 7s, and so on, until the number is completely factored. Here is what the diagrams look like for the numbers 24 and 36:
Does the order in which you factor a number matter? Is the product always uniquely that one number? To answer these questions, use a factor tree to find the prime factorization of 60 in the following ways:
a. Start by factoring out 2s.
b. Do another diagram, but this time start by factoring out 10s.
c. Do a third diagram, but this time start by factoring out 6s.
d. What is the same and what is different about your results?
When you factor a number, no matter where you start, you always get the same set of factors; the only difference might be the order in which they occur. This phenomenon is called the fundamental theorem of arithmetic, which states that any integer (other than 0, and 1) can be factored into a product of prime numbers and that this product is unique except for the order of the factors. This is another reason why 1 cannot be considered prime — otherwise, this, and every other result that builds on it, falls apart.
For example, we could factor 6 in an infinite number of ways:
…and so on, for any number of 1s that we cared to use.
Draw a factor tree to find the factors of 231 and 195.
Use a Venn diagram to find the GCF and LCM of 231 and 195.
In This Part: The Area Model
The area model makes the process of finding GCFs and LCMs visual. See
Greatest Common Factor
If we think of the numbers 24 and 36 as the dimensions of a rectangle, then it follows that any common factor could be the dimensions of a square that would tile that entire rectangle.
For example, a 1-by-1 square would tile the 24-by-36 rectangle without any gaps or overlaps. So would a 2-by-2 or a 3-by-3 square. Notice that these numbers are all common factors of 24 and 36.
To determine the GCF, we want to find the dimensions of the largest square that could tile the entire rectangle without gaps or overlap. Here’s one quick method.
Start with the 36-by-24 rectangle:
The largest square tile that fits inside this rectangle and is flush against one side is 24 by 24. Only one tile of this size will fit:
The largest square tile that fits inside the remaining rectangle and is flush against one side is 12 by 12. Two tiles of this size will fit. The original rectangle is now completely filled:
Least Common Multiple
Note that the 24-by-24 square could also be filled with the 12-by-12 tiles, so 12 by 12 is the largest tile that could fill the original 24-by-36 rectangle; therefore, 12 is the GCF of 24 and 36.
Conversely, if we think of 24 and 36 as the dimensions of a rectangle that could tile a square, then it follows that any common multiple could be the dimensions of a square that could be tiled by this rectangle.
For example, since 24 • 36 = 864, a square that is 864 by 864 could be tiled by the 24-by-36 rectangle. The LCM of 24 and 36 would be the dimensions of the smallest square that could be tiled by the 24-by-36 rectangle.
Here’s a quick method for determining the LCM.
Start with the 24-by-36 rectangle. Your goal is to make a square tiled with rectangles of these dimensions:
Since the width (24) is less than the height (36), add a column of tiles to the right of the rectangle (in this case, one tile). This makes a 48-by-36 rectangle:
The width (48) is now greater than the height (36), so add a row of tiles under the existing rectangle (in this case, two tiles). This makes a 48-by-72 rectangle:
The width (48) is now less than the height (72), so add another column (two tiles) to the right of the existing rectangles. The dimensions are now 72 by 72 — and you’ve made a square!
Use the following Interactive Activity to answer Problem A5. The 72-by-72 square is the smallest square that can be tiled with a 24-by-36 rectangle. Therefore, the LCM of 24 and 36 is 72.
For a non-interactive version of this activity, use graph paper when drawing the squares and rectangles you wish to represent to ensure that the dimensions of the shapes are precise.
Use the area model to find the GCF and LCM of the following:
a. 30 and 42
b. 18 and 30
In this video segment, Ben and Doug use the area model to find the GCF for two numbers, following the analogy of tiling a rectangle with the biggest square they can fit. Watch this segment after you’ve completed Problem A5.
Notice that the teachers omitted one step and didn’t use the square with the dimensions of 12-by-12 to tile the 12-by-30 rectangle. Think about why going through all the steps will ensure that the result will be the largest common factor rather than just any common factor.
You can find this segment on the session video approximately 6 minutes and 4 seconds after the Annenberg Media logo.
Can you explain in your own words why the area model works?
Here, Ben and Doug use the area model to find the LCM for two numbers, following the analogy of finding the biggest square that can be tiled with a rectangle whose dimensions are the two original numbers. Notice the connection they make between the area model and Venn diagrams.
You can find this segment on the session video approximately 8 minutes and 32 seconds after the Annenberg Media logo.
The greatest common factor is equivalent to the greatest common divisor. The greatest counting number that evenly divides a and b is both the greatest common factor and the greatest common divisor of both a and b.
The LCM and GCF can be difficult concepts to understand because we hear the words in the opposite order of their importance: For example, for the LCM, first we hear “least,” then we hear “common,” and last we hear “multiple.” However, the most important word of the three is “multiple.” The multiples of 24 are 1 • 24, 2 • 24, 3 • 24, and so forth, and the multiples of 36 are 1 • 36, 2 • 36, 3 • 36, and so forth. The next-most important word is “common.” We are looking for a number that is common to both numbers. The third most important word is the one we hear first, “least.”
So the number we want is a multiple, common to both numbers, and the least of all such numbers. This would be the same case for the GCF, for which we want a number that is a factor, common to both numbers, and the greatest of all such numbers. It would be worth taking the time to have a class discussion of the three words when introducing the LCM and GCF.
When you’re asked to find factors, pay careful attention to the specific question that is posed to you. Were you told to find the prime factors, the prime factorization, the number of factors, or all the factors? These are four very different questions.
For example, for the number 36, the following statements are all true:
• The prime factors are 2 and 3.
• The prime factorization is 22 • 32.
• The number of factors is nine.
• Those nine factors are 1, 2, 3, 4, 6, 9, 12, 18, and 36 — 1 • 36, 2 • 18, 3 • 12, 4 • 9, and 6 • 6. (Note that we only count the 6 once. This is why squares have only odd number of factors.)
The area model shows how to fill a rectangle with squares (to find the GCF) or make a square with rectangles (to find the LCM). It can be a useful method for visual learners.
18 = 2 • 3 • 3, and 30 = 2 • 3 • 5. The factors they have in common are 2 and 3. In the left circle is 3, and in the right circle is 5:
The GCF is the product of all the numbers in the intersection: 2 • 3 = 6. The GCF is 6.
The LCM is the product of all the numbers in the Venn diagram: 3 • 2 • 3 • 5 = 90. The LCM is 90.
a. b. . c.
d. All three methods yield the same prime numbers (two factors of 2, one factor of 3, and one factor of 5).
The intersection contains the only common factor, which is 3. On the left are the factors of 231 that are not factors of 195; i.e., 7 and 11. On the right are the factors of 195 that are not factors of 231; i.e., 5 and 13.
The GCF is the product of all the numbers in the intersection (in this case, just the 3). The GCF is 3.
The LCM is the product of all the numbers in the Venn diagram: 7 • 11 • 3 • 5 • 13 = 15,015. The LCM is 15,015.
a. The largest square that can tile this entire rectangle without any gaps or overlap is 6 by 6. Therefore, the GCF of 30 and 42 is 6.
The smallest square that could be tiled by this rectangle is 210 by 210. Therefore, the LCM of 30 and 42 is 210.
The largest square that can tile this entire rectangle without any gaps or overlap is 6 by 6. Therefore, the GCF of 18 and 30 is 6.
The smallest square that could be tiled by this rectangle is 90 by 90. Therefore, the LCM of 18 and 30 is 90.
Answers will vary.
Session 1 What Is a Number System?
Understand the nature of the real number system, the elements and operations that make up the system, and some of the rules that govern the operations. Examine a finite number system that follows some (but not all) of the same rules, and then compare this system to the real number system. Use a number line to classify the numbers we use, and examine how the numbers and operations relate to one another.
Session 2 Number Sets, Infinity, and Zero
Continue examining the number line and the relationships among sets of numbers that make up the real number system. Explore which operations and properties hold true for each of the sets. Consider the magnitude of these infinite sets and discover that infinity comes in more than one size. Examine place value and the significance of zero in a place value system.
Session 3 Place Value
Look at place value systems based on numbers other than 10. Examine the base two numbers and learn uses for base two numbers in computers. Explore exponents and relate them to logarithms. Examine the use of scientific notation to represent numbers with very large or very small magnitude. Interpret whole numbers, common fractions, and decimals in base four.
Session 4 Meanings and Models for Operations
Examine the operations of addition, subtraction, multiplication, and division and their relationships to whole numbers. Work with area models for multiplication and division. Explore the use of two-color chips to model operations with positive and negative numbers.
Session 5 Divisibility Tests and Factors
Explore number theory topics. Analyze Alpha math problems and discuss how they help with the conceptual understanding of operations. Examine various divisibility tests to see how and why they work. Begin examining factors and multiples.
Session 6 Number Theory
Examine visual methods for finding least common multiples and greatest common factors, including Venn diagram models and area models. Explore prime numbers. Learn to locate prime numbers on a number grid and to determine whether very large numbers are prime.
Session 7 Fractions and Decimals
Extend your understanding of fractions and decimals. Examine terminating and non-terminating decimals. Explore ways to predict the number of decimal places in a terminating decimal and the period of a non-terminating decimal. Examine which fractions terminate and which repeat as decimals, and why all rational numbers must fall into one of these categories. Explore methods to convert decimals to fractions and vice versa. Use benchmarks and intuitive methods to order fractions.
Session 8 Rational Numbers and Proportional Reasoning
Begin examining rational numbers. Explore a model for computations with fractions. Analyze proportional reasoning and the difference between absolute and relative thinking. Explore ways to represent proportional relationships and the resulting operations with ratios. Examine how ratios can represent either part-part or part-whole comparisons, depending on how you define the unit, and explore how this affects their behavior in computations.
Session 9 Fractions, Percents, and Ratios
Continue exploring rational numbers, working with an area model for multiplication and division with fractions, and examining operations with decimals. Explore percents and the relationships among representations using fractions, decimals, and percents. Examine benchmarks for understanding percents, especially percents less than 10 and greater than 100. Consider ways to use an elastic model, an area model, and other models to discuss percents. Explore some ratios that occur in nature.
Session 10 Classroom Case Studies, K-2
Watch this program in the 10th session for K-2 teachers. Explore how the concepts developed in this course can be applied through case studies of K-2 teachers (former course participants) who have adapted their new knowledge to their classrooms.
Session 11 Classroom Case Studies, 3-5
Watch this program in the 10th session for grade 3-5 teachers. Explore how the concepts developed in this course can be applied through case studies of grade 3-5 teachers (former course participants) who have adapted their new knowledge to their classrooms.