Mathematical definitions are precise, each word chosen with care. A prime example is below:
Prime Number: A number that has exactly two factors.
Composite Numbers have more than two factors.
The first prime number is two. The factors of two are 1 and 2. That makes two total factors, making two prime. Since two divides into all even numbers (why they’re called even), two is the only even prime number. An example would be 20. Factors of twenty include 1 and 20, and 2 and 10. That’s more than two factors, making twenty composite.
One way to determine if a number is prime or composite is to try and divide the number by primes smaller than it. If three, for example, divides evenly into 4,695, then it is composite. Note that since 4,695 ends in a five, five divides into evenly, so we already know it is composite. But there’s a nifty trick, a short cut, to check divisibility by three.
To see if a number is divisible by three, add the digits together. If the sum is divisible by three, then so is the entire number. Our number 4,695 for example adds to 24, which is divisible by 3. Take 24, 2 + 4 = 6, which is also divisible by three.
Perhaps the easiest way to know if a number is prime or composite is to be familiar with the primes less than 100. There are twenty-five and they are listed below:
What About One?
Since one has just one factor, itself, it does not satisfy the definition of prime or composite.
This is not just a minor issue, one being neither prime or composite. The Fundamental Theorem of Algebra says, basically, that every number can be uniquely expressed as the product of prime numbers. For example, 14 is the product of 2 and 7. If one was prime, then there would be another way to write 14 as the product of primes, 1, 2 and 7. This has major consequences. So the math seems to work best if one is not prime or composite.
Prime versus Relative Prime
Prime numbers have just two factors, and you only consider the number itself. When discussing relatively prime numbers you are comparing more than one number to another. If they share more a factor other than one, they are not relatively prime.
Relatively prime numbers will share exactly one factor, the number one.
An example of a relatively prime pair of numbers would be 15 and 77. Neither number is prime, but 15 has factors of 1 and 3 and 5 and 15. Seventy-seven has factors of 1 and 7 (for this case we only need list one of the sevens), and 11. The only common factor they share is one.
An example of numbers that are not relatively prime is 15 and 9. They share a factor of one and of three.
If two numbers are prime are they also relatively prime? How do you know?
A factor is a part of a number. Factors multiply together to make the number at hand. For example, the factors of 24 are 1, 24 and 2, 12, and 3, 8 and 4, 6. This would be a complete factorization of 24.
The prime factors of 24 are: 2, 2, 2, 3.
The purpose of factoring is often to find some property of the number in question. Sometimes you will factor to determine if a perfect square or perfect cube exists as a factor of a number.
Other times you will be looking for a lowest common multiple so that you can find a common denominator, for something like:
The purpose of finding a greatest common factor, in Algebra, will be to reduce an algebraic expression, like:
What About Variables and Factoring?
In Algebra you will be factoring algebraic expressions like 15x2 . The confusion comes in with how to factor the x2 . To factor x2 we would just write x•x. We will discuss this in greater detail when we learn how to find the lowest common multiple and greatest common factors of algebraic expressions in the next section.
Perfect Squares and Perfect Cubes
A perfect square is the product of a number times itself. Examples would include 1, 36,49, 400, x2 and a4. Each of these is the product of a number squared (times itself).
A perfect cube is the product of a number times itself three times (cubed).
There is an interesting pattern in perfect squares. See it below?
How many prime numbers exist? Take a guess. How do you know? In a future section we will introduce mathematical proof and discuss this question.
In summary, a prime number has exactly two factors, one and itself. Composite numbers have more than two factors, and one is neither prime nor composite.
This is a one or two-day lesson. You can trim the PowerPoint to one day, or explore a second day to really encourage student thinking.
Note: The objectives of this teaching this topic include:
- Create mystery and interest in mathematics through questioning
- Promote mathematical thinking
- Introduce the idea that an algebraic expression could be prime
- Knowledge of prime, versus composite, and how the number 1 fits
- Knowledge of prime numbers less than 100
The simple definition of prime is not sufficient to achieve all of these objectives. It will be important for you, the teacher, to be creative in questioning and forcing students to articulate their thinking. This material will provide you some ideas.
Numbers can be built, and deconstructed with multiplication. This comes into play later with LCM and GCF, and with algebraic expressions, it becomes quite confusing because students mix up the meaning of multiple and factor.
- A prime number has exactly two factors, one and itself.
- A composite number has more than two factors.
- One is not prime or composite because it has exactly one factor.
Know the prime numbers less than 100, especially those numbers close to 100. Understand that an Algebraic expression can be prime!
To download the PowerPoint, click the icon below.
Introduce the Big Idea, the take-away, and what you want students to do, how you want them to engage. If they only try to memorize facts instead of applying facts to solve problems and create questions of their own, they’ll struggle.
This material is easy to understand, but the habits that make more difficult mathematics more easily learned need to be developed now.
Have students write down their thinking, without talking in their notes for one minute. Then allow them to share their ideas with a “shoulder partner.”
Poll the room and ask for justification. Pressure (positively) students to be concise and articulate.
15 – 20
Have students use a calculator to verify that this example of why prime numbers are infinite is valid. Question along the way to make sure they really understanding the approach and purpose. Have students explain why we are assuming there are only prime numbers from 2 through 11.
4 – 9
Have students write the prime numbers less than 100 in their notes in a place easily referenced. They will be asked, on tests, questions that require this recall knowledge. Numbers like 91 are always tricky for them!
5 – 10
The fact that all natural numbers can be written as a product of prime numbers is a good chance to explore the difference between factors and multiples.
This is practical knowledge you’re testing here. Seek out students that might not know, in a helpful way. If everybody has this, you’re ready to move on. If not, it is time to bring this knowledge to life so they do know it.
12 – 13
5 – 10
Have a stack of recycle paper, halved, for students to answer questions, closed notes, on slides 20 – 26.
5 – 10
Explore with students what relatively prime means.
15 – 16
Review the homework and introduce the expectations for day 2.
5 – 10
The idea that x and xy and x2 can be treated as prime or composite depending on if they can be factored or not is a new idea. New ideas must be tested by the learner in order to be integrated into their understanding. As such, ask questions and conduct discussion about this idea! Go beyond what is provided in the PPT. The PPT is just structure, it cannot account for everything your students might need.
18 – 19
Have students put away their notes. Hand out the halved recycled paper for their pop-quiz.
20 – 26
Have students put their pencils away and get a highlighter or pen. Then, have them verify their answers by searching in their notes.
Allow students two minutes to discuss answers with other students.
Conduct a full class discussion of the questions. Unless the class fails to come up with correct reasoning, do not offer answers. Elicit answers from the body of students. Have students that agree with what has been said by another student expound upon the idea. Encourage question asking.
Remind students that these skills are what will be required as material gets more difficult.
Closure: Have students reflect on their experiences with the questions on slide 28. If time permits, explore their answers with discussion.
Homework two is printed on slide 29.
Prime Numbers Practice Problems
- For the following consider that n is a number such that 60 < n < 70.
- The two values of n that are prime.
- Could n be a perfect square?
- Could n be a perfect cube?
- Why is one not composite or prime?
- Describe the short cut divisibility check for three.
- Is 5,821 prime or composite?
- The number a is divisible by 7. What could the last digit of the number a be?
- If Relative Prime Numbers are numbers that only share one common factor, the number one, are 12 and 39 relatively prime? How do you know?
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