Prime factor tiles can be used to introduce prime factorization (the idea that 30 can be uniquely expressed as the product of the prime numbers 2, 3 and 5). The attached pdf includes a printable template for making tiles students can use to explore primes, composites, factors and multiples.

There are numerous questions that can be explored with prime factor tiles:

• Some numbers have their own tiles (2, 3, 5, 7, etc.). Can you make the missing numbers using tiles and multiplication (2×2=4, 2×3=6, etc.)? Are there any numbers you can’t make? (Worksheet: Counting with Factor Tiles)

• After finding the prime factorization of a number (42, for example, can be thought of as 2 x 3 x 7), is there any other combination of tiles that you could multiply to get that number?
• How many total factors does 42 have? Can you use the prime factors 2, 3 and 7 to show all the factors of 42? You might group the tiles in one of the next 3 ways.
  • (2×3)x7 = 6 x 7
  • (3×7)x2 = 21 x 2
  • (2×7)x3 = 14 x 3
  • You could also group all three tiles one side and multiply by 1: (2x3x7)x1 = 42 x 1
• So, 42 has 8 factors: 1, 2, 3, 6, 7, 14, 21, 42. How about a different number, like 50? Before doing anything, do you think it will have more or fewer factors than 42? Take a vote. Now, find the prime factor tiles for 42 and see how many factors it has.

The vocabulary might explored in the following manner:

1. A lot of vocabulary has come up as we have been working with these tiles. Let’s take a few minutes to make sure we understand some important words and have definitions in our notes. There is some important mathematics vocabulary that is helpful when we talk about what is going on with these tiles. Some of these terms may have come up already. It is important that we are able to use precise mathematical words. It will help us understand each other and will help prepare us for other mathematical ideas. It is also helpful to learn math vocabulary in preparation for the HSE exam. The test writers will often use mathematical vocabulary, expecting you to understand what it is meant.
2. Hand out Factor, Multiple, Prime, Composite vocabulary worksheet. With a partner, discuss and fill in the blanks.
3. Possible conversation for each vocab word:

FACTOR

• Each of these numbers are factors of 30. Here are a few examples of what I mean by the word factor:
  • 5 is a factor of 30 because 5 times 6 equals 30.
  • 4 is a factor of 12 because 4 times 3 equals 12.
  • Is 6 a factor of 18? Is 4 a factor of 30? Why or why not?
• What do I mean by the word factor? Give students a definition: Factors are numbers we can multiply together to get another number.
• When we use the word factor in mathematics, we usually say a factor of, as in 5 is a factor of 15 or 3 is a factor of 27. If I told you 7 is a factor, you would probably ask, “7 is a factor of what?”

MULTIPLE

• 3, 6, 9, and 12 are multiples of 3. What are some other multiples of 3?
• Is 15 a multiple of 5? Is 30 a multiple of 7? Why or why not?
• What do I mean by the word multiple? Give students a definition: A multiple is a number that can be divided by another number evenly, with no remainder.
• When we use the word multiple in mathematics, we usually say a multiple of, as in 15 is a multiple of 5 or 27 is a multiple of 3. If I told you 42 is a multiple, you would probably ask, “a multiple of what?”

PRIME NUMBER

• Mathematicians call numbers like 2 and 5 prime numbers. 6 and 15 are not prime numbers.
• What do I mean by the word prime?
  • The only way to get a prime number with multiplication is to multiple the prime number by 1. There is no other way to multiply two whole numbers to get a prime number. For example, 1 x 3 is the only way to get 3 by multiplying whole numbers. 1 x 7 is the only way to get 7. So, 3 and 7 are prime numbers.
  • Give students a definition: A prime number has exactly two factors (1 and itself).
• Can you give some examples of prime numbers that are lesson than 30?
• Please find an example of a prime number from the tiles in front of you. How can you be sure it’s prime? Maybe I’m trying to trick you. What if you could use a calculator? There’s no way to divide them evenly by another number.

COMPOSITE

• Mathematicians call 30 a composite number.
• Has anyone heard the word composite before? What does composite mean outside of mathematics? It means something that is made up different parts.
• What do you think a composite number is? Give students a definition: A composite is a number that has more than two factors.
• What are some other composite numbers? 4, 6, 8, 9, 10, 12… How do you know?
• Is 13 a composite number? Is 14 a composite number? Why or why not?
• Are there any composite numbers in the blue tiles in front of you?
• Can you give some examples of composite numbers that are less than 30? Look at the examples created by students: 12 (2x2x3), 16 (2x2x2x2), 14 (2×7), etc.

Prep

• Cut out prime factor tiles (student and teacher versions) and place in ziploc bags. One student packet for every 2-3 students.