The Shapes of Numbers — Exploring Prime Numbers through Eratosthenes’ Sieve

The sieve of Eratosthenes is one of my favorite math exercises to do with kids, and it’s a great intro into prime numbers. It eliminates numbers through the multiples, leaving primes on the 100 chart. We usually use block crayons, because there is a definite pattern on the 100 chart for multiples, and block crayons allow us to cover multiples quickly. They also have some transparency, so the overlay is more obvious. You can find these exercises to see all the multiple patterns for multiplication in my curriculum: Multiplication and Division Year 2

Like introducing multiplication, we begin with the multiples of 10 (shaded yellow,) in contrast to the 2’s. This starts with the easiest multiple pattern, but also allows us to see where the 2’s fall on the chart in relationship to the 10’s. Then we move onto the 5’s (also shaded yellow.) We can see that the 5’s also hit all the 10’s every other time.

From here we travel back to the 2’s, which move in verticle lines down the page (shaded purple). Then we slide over to do the 3’s (shaded blue). This diagonal pattern for the 3’s is not only fun, but if you explore the digit-sums of the 3’s, we discover it’s rule for divisibility. (This is covered in my Number Sums Curriculum.) The 4’s, 6’s, and 8’s are covered by the 2’s since 2 is a factor for each, and that leaves us with 7’s. (If finding the multiple patterns in a 100-chart is not something that you’ve done before, I suggest also working through the 4’s, 6’s, and 8’s, so you can see the patterns, understand why the 2’s also cover all even numbers, and see where numbers overlay showing common multiples. Ideally, each number would be done on a 100-chart on it’s own. This activity is done in my Quality of Numbers and repeated with new pattern discovery of the multiples in my Multiplication and Division.)

After 7’s are shaded, we are left with the *primes* — the numbers of only two factors – one and themselves. I find primes so interesting, and they are the favored numbers in this house at the moment. Each year my children decide how to take turns opening our Activity Advent Calendar. It’s usually even and odd numbers; this year my youngest child took prime and the older took composite.

There is so much to notice about primes, and people (mathematicians) have looked for years. All the primes except the number 2 are odd. This is because all the other even numbers are multiples of 2. There are more primes that end with 7 than any other digit within the 100 chart. Interestingly it is the 7’s was the last that multiple in our sequence (and if you sequenced numerically) of elimination. There are prime gaps, the distance between two subsequent primes, prime twins, primes separated by exactly two numbers, and prime theorems and conjectures, ideas proven to be true and those thought to be true. One conjecture, by Goldbach, states all even numbers can be expressed by the sum of two primes. We are exploring some of those relationships in the current block we are doing: The Shapes of Numbers. I hope to eventually write a curriculum for this exploration. If you want to learn more about why primes are so fascinating, the site, Frontier for Young Minds, has some interesting information. Additionally, if you haven’t see Data Pointed’s Dancing Dots that illustrate composite and prime numbers, it’s also worth exploring. This is a fabulous animation with which to do a ‘ notice and wonder’.

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