Learning Math: Number and Operations
Divisibility Tests and Factors Part C: Factors (35 minutes)
You saw that you could devise tests for such numbers as 6 and 18 based on their relatively prime factors. Let’s explore factors further.
A prime number is a number with exactly two factors. For example, the number 1 is not a prime number because it only has one factor, 1. The number 3 is a prime number because it has exactly two factors, 1 and 3. See Note 4 below.
Some numbers factor into two factors only, while others may have two factors, one or both of which can be factored further. See Note 5 below.
An important distinction can be made between the terms “factor” and “prime factor.” By factors, we mean all the factors of a number. To find all the factors of 12, you can list them as shown below:
1 • 12
2 • 6
3 • 4
You know you can stop here because the next factor on the left would be 4, and you already have it listed on the right.
To find the prime factors of 12, you could use a factor tree:
The numbers on the bottom branch of this tree are the prime factors of 12 — they can’t be factored any further. So we say that 12 has only two prime factors, 2 and 3, and the prime factorization of 12 is 22 • 3. Note that we could have started the factor tree with the factors 3 and 4, and we would have derived the same prime factorization, 22 • 3.
This is the non-interactive version of Problem C1:
Complete the following tables to explore the factors and prime factorization of the numbers from 2 to 36.
a. What are the factors of the numbers from 2 to 36? Enter each number in the appropriate column in the table below. For example, enter 4: 1, 2, 4 in the Three Factors column and 16: 1, 2, 4, 8, 16 in the Five Factors column.
The Fundamental Theorem of Arithmetic states that every positive integer other than 1 has a unique factorization into primes (up to rearrangement of the factors). Now, any negative integer is simply �1 times a positive integer. So we can extend the theorem to all integers in a natural way: each integer (except, of course 1, 0, and �1) can be written uniquely as a product of primes and either +1 or �1.
Here are some unique prime factorizations:
12 = 1 • 22 • 3
(Note that we usually leave off the 1 for positive numbers.)
-12 = -1 • 22 • 3
36 = 32 • 22 will have (2 + 1) • (2 + 1), or 9 factors total.
TAKE IT FURTHER
Look at the prime factorizations of numbers. Do you see any patterns — for example, how many factors in total a number will have based on its prime factorization?
Make a table that allows you to complete the prime factorization and total number of factors. Using numbers that are powers of 2 may be a useful way to see that pattern initially.
Notice that, for our purposes, we’ve defined “factors” as a positive divisor, so prime numbers will also be positive.
If you’re working in a group, you may want to play the Factor Game, which takes about 20 minutes. The Factor Game is a fun activity that requires understanding of factors; players must use some strategy to ensure a win!
Show participants a chart with the numbers 2 through 24:
2 – 24
Explain that they will be divided into two teams, Squares and Circles. Each team in turn chooses one of the numbers. The other team then claims all of the numbers on the chart that are factors of that number (if they’re not already taken). When all the numbers have been chosen, the team with the highest sum wins the game.
For example, say the Squares team chooses 21 and draws a square around it. The Circles team then draws a circle around all of the factors of 21, in this case 3 and 7. Then the Circles team chooses 24 and circles it. The Squares team then puts a square around all its factors, 2, 4, 6, 8, and 12 (they can’t put a square around 3, because it’s already been taken). Teams alternate choosing numbers, continuing until all the numbers are taken.
There are many variations of this game. For example, you could start with more numbers on the board, or you could decide that teams are not allowed to choose numbers with no factors left on the board.
After playing the Factor Game, ask participants to describe a strategy that would help them win the next time they play. Would they rather be the first or second team to choose a number?
Looking at the prime factorization of numbers, you can tell how many factors a number will have in total. For example, the prime factorization of 2 is 21, and 2 has two factors in total, 1 and 2. The prime factorization of 4 is 22, and it has three factors in all: 1, 2, and 4. To further investigate this pattern, let’s look at the following:
36 = 32 • 22 will have (2 + 1) • (2 + 1), or nine factors total.
24 = 23 • 31 will have (3 + 1) • (1 + 1), or eight factors total.
81 = 34 will have (4 + 1), or five factors total.
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.