Here are a couple of songs I made up about prime and composite numbers:
"Prime and Composite" by Mrs. Wippich (Sung to the tune of the chorus in "Cool for the Summer" by Demi Lovato) Count each factor up one at a time Only two (one and itself) it's priiiiiiiiiiiime If of three or more it is comprised You'll realize It's a composite number "Prime or Composite?" by Mrs. Wippich (Sung to the tune of "Uma Thurman" by Fall Out Boy) I can take numbers and then factor them out, factor them out Wo oah Count up the factors tell whether it's prime or composite You want to know if it's prime or composite Factor it-DON'T JUST GUESS! You want to know if it's prime or composite Factor it- You'll pass the test Just factor numbers with the rainbow method or t-chart, it's your choice, just factor it divide it down to the factors that can be multiplied together and then you will see I can take numbers and then factor them out, factor them out Wo oah Count up the factors tell whether it's prime or composite You want to know if it's prime or composite Factor it-DON'T JUST GUESS! You want to know if it's prime or composite Factor it- You'll pass the test A number's prime if it only has two factors (one/itself), only two But what happens if a number's factors do count up to more than 2 (composite!) I can take numbers and then factor them out, factor them out Wo oah Count up the factors tell whether it's prime or composite Factor it out, and then count up the factors only one and itself, you'll know that you've found a prime Factor it out, and then count up the factors more than two factors, it's a composite number this time You want to know if it's prime or composite Factor it-DON'T JUST GUESS! You want to know if it's prime or composite Factor it- You'll pass the test I can take numbers and then factor them out, factor them out Wo oah Count up the factors tell whether it's prime or composite I can take numbers and then factor them out, factor them out Wo oah Count up the factors tell whether it's prime or composite
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Yesterday we learned some divisibility rules and played a game to help us with our prime and composite numbers.
The kids have learned what prime and composite numbers are: Prime numbers- have only two factors, one and itself EXAMPLE) 2- the only two factors that can be multiplied together to make 2 are 1 and 2 (itself) Composite numbers- have 3 or more unique factors (can also stated as "more than 2 unique factors") I am so impressed by how many kids know their math facts this year, however, since we have to learn about prime and composite numbers between 1 and 100, these facts don't always help. One way to figure out how many factors a number has is to just do the division, which could take forever... the student would have to do 89 divided by 2, 89 divided by 3...and so on until they got to 89 divided by 9. Ain't nobody got time for that! So, in order to make it easier to determine a number's factors, we started talking about divisibility rules yesterday. What are divisibility rules? They are rules that tell you if a number is evenly divisible by (can be divided by) a certain number. Here are the rules we follow: 2- A number is divisible by 2 if the number ends in an EVEN number: 0, 2, 4, 6, 8 Examples) 482 is divisible by 2 because the digit in the ones place is an even number. 891 is NOT divisible by 2 because the digit in the ones place is an odd number. 3- A number is divisible by 3 if, after adding up the digits in the number you end up with a sum that is evenly divisible by 3 Examples) 482: 4 + 8 + 2= 14. 14 is not divisible by 3 (when you divide 14 by 3 you have a remainder), so neither is 482. 891 is divisible by 3 because 8 + 9 + 1 = 18 and 18 is evenly divisible by 3 (when you divide 18 into 3 groups, you don't have a remainder). 5- If a number ends in 0 or 5 in the ones place, then the number is divisible by 5. Example) 615 is divisible by 5 because it ends in a 5. 892 is not divisible by 5 because it does not end in a 0 or 5, it ends in 2. 6- If a number is divisible by BOTH 2 and 3, it is also divisible by 6. Example) 624 is divisible by 2 because it ends in an even number in the ones place. 624 is also divisible by 3 because 6 + 2 + 4 = 12 and 12 is divisible by 3 with no remainder. Therefore, 624 is also divisible by 6 because it is divisible by both 2 and 3 (which are factors of 6). 7- Sorry, the rule takes longer to figure out than to just divide whatever number you have by 7 to see if it is a factor. Example) 57 divided by 7 is 8 remainder 1, so it is not divisible by 7. 91 divided by 7 is 13 remainder 0, so it IS divisible by 7, meaning 7 is a factor of 91. 9- Just like the threes rule, if you add up the digits of the number and it is divisible by 9, then the whole number is divisible by 9. Example) 711 added together is 7 + 1 + 1 = 9, and 9 is evenly divisible by 9, so 711 is as well. To find out how many factors a number has, start applying the rules from 2-9 until you find one that works. Once you've figured out even just one additional factor for a number other than 1 and itself, you can automatically stop and call it composite. Example: 93 We know that two factors for 93 are 1 x 93 because all numbers have the factors 1 and itself. Let's test the 2s rule: It is NOT divisible by 2 because it ends in an odd number. So we still only have 2 factors so far: 1 and 93. Let's try the 3s rule: 9 + 3 = 12. 12 is divisible by 3 with no remainder, so that means 93 is as well. Guess what! We don't even need to figure out what the other factor is that has to be multiplied times 3 to get 93- we now know that 3 is a factor because the rule worked, 93 has at least 3 factors- 1,3 and 93, so we know it is composite! We don't even need to go to the 4s rule because we know three factors already- 1, 3, and 93. Of course, it is a lot less work on paper to memorize the 25 prime numbers from 1-100, however, if you don't do this, you can just follow these rules or do all of the math. For those of you who want to memorize the primes from 1-100, here they are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97 Always remember that 0 and 1 are neither prime nor composite This week we did an activity to demonstrate the difference between even and odd numbers. Each student received a handful of plastic tiles/chips and then made two rows of their pieces. Some students noticed that all of their pieces had a partner (that is, they were able to put two chips side-by-side with none leftover) and some noticed that they had one leftover at the end (all but one tile had a partner). We then made a chart of the total number of chips each child had in two columns- one with numbers in which all tiles had a partner (in the "partner" column) and one with numbers that had a leftover (the "leftover" column). After we'd completed the chart, we looked for patterns in the numbers and students discussed that all of evens ended in a 0, 2, 4, 6, or 8 and all odds ended with 1, 3, 5, 7, or 9. We also reviewed that if all of the chips had "a dancing partner," they were considered even. The group that had one "lonely" tile with "no dancing partner" were called odd. See example below: This was review for most students, however, it helped students to conceptually understand even and odd numbers better than just memorizing what even and odd numbers end in (or that all even numbers are divisible by two evenly and odd numbers always have a remainder of 1 if divided by two). The conceptual understanding of even and odd numbers will help students greatly as we move on to prime and composite numbers as one of the biggest misconceptions about prime/composite numbers is that all odd numbers are prime and all evens are composite- not true!!! In order for students to see the difference between these two types of number classifications, a conceptual understanding is extremely important. Prime/Composite Numbers This week we will be working to discover the difference between prime and composite numbers and then will compare them to even and odd numbers. Through a class activity, students will discover that prime numbers only have two factors (factors are numbers you can multiply together to get another number)- one and itself, that composite numbers have three or more factors, and the number 1 only has one factor (1) so it is neither prime nor composite. In order to understand what this looks like, students will make arrays to figure out all of the different factors a number has. Those numbers that only have one array (1 x itself) are prime. Here are a couple examples. 3- In order to figure out if this number is prime or composite, we think of all the numbers we can multiply times each other to get 3. We then make arrays to model the multiplication problems, like so: The only two numbers (factors) that can be multiplied times each other to make 3 are 1 and 3, because 1x3=3. Therefore, I know that 3 is prime because it has exactly two factors- 1 and itself (3). 6- In order to figure out if this number is prime or composite, we think of all the numbers we can multiply times each other to get 6. We then make arrays to model the multiplication problems, like so. I start with the factor 1. Next, I think of the next number that is a factor of 6 after 1 and know that 2 is a factor that can be multiplied times 3 to get 6 (so both 2 and 3 are factors). After 3, the next factor is 6, which I've already included, so I know that I can stop. Once you reach a factor you've already found, it means that you don't have to keep going because you've found them all. Therefore, the number 6 has 4 factors: 1, 2, 3, and 6. Since it has three or more factors, it is composite. 1- In order to figure out if this number is prime or composite, we think of all the numbers we can multiply times each other to get 1. We then make arrays to model the multiplication problems, like so. The only factor that can be multiplied times 1 to get 1 IS 1, so I really only have 1 factor for 1. Therefore, 1 is NEITHER prime nor composite because it doesn't even have two factors (which are needed to be prime-remember, prime numbers have exactly two factors and composite numbers have three or more factors). Last week, students colored in a hundreds chart, coloring in each multiple of 2, 3, 5, and 7. In the end, all of the composite numbers were colored in and the primes (those with only 2 factors) were left! Here is a video that talks about what we did: |
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