The Sieve of Eratosthenes as an Aid to the Memorization of Prime Numbers

Just as memorizing sight words can help a kid read better and more confidently, there are tons of math facts that, if memorized, will make a kid's calculation work quicker and more confident. 

Our culture is well used to having kids memorize the multiplication tables through at least 10 (through 12 is better!), and certain formulas like the quadratic formula or the Pythagorean theorem, but it's so helpful to just know, when you're busy doing your algebra, say, if a number is a perfect square or a Pythagorean triple, etc. It builds confidence when a student is learning advanced math concepts, and it increases their speed and fluency, which they will VERY much appreciate whether they're working through a page-long proof or an SAT problem set!

When my kids were pretty little, we dedicated the first ten minutes of the first car trip of the day to memory work, and they memorized a lot of advanced concepts by rote then (most famously, the first 25 digits of Pi, a party trick that they still both often pull out over a decade later, lol!), but it's a better aid to learning and to memorization to have them, whenever possible, create for themselves the anchor chart that contains the information I want them to memorize.

So when I realized recently that my teenager has lost most of the prime numbers to 100, I pulled back out the same activity that she used to create her Prime number chart back when the kids memorized the primes to 100 the first time around back in 2016.

It's the Sieve of Eratosthenes!

Creating the Sieve of Eratosthenes is simple. All you need are a hundred chart and some colored pencils or crayons. This hundred chart has the numbers by rows, and this hundred chart has the numbers by columns. This hundred chart is blank, for some sneaky real-world handwriting practice writing the numbers to 100. 

To create the sieve, you simply start with the first Prime number, 2. Don't color it, but color all of its multiples. Bonus points if you unlock the pattern and color it that way! The next uncolored number is your next Prime number, 3, so leave it blank but color all its multiples. It makes a pretty pattern, too!

Carry on through 7, and by the time you've colored the last multiple of 7, you'll have colored every composite number through 100, and every uncolored number is a Prime. Your grid will look like this:

photo credit: Wikipedia

I think the patterns that it makes are beautiful and fascinating!

While you're working, it's best if you have a Ginger Gentleman supervise you:

While you're working, you also might notice that you have a sudden, inexplicable swarm of Asian lady beetles inside your home. Would the Ginger Gentleman like to meet one?

He very much would!

Please note: no Asian lady beetles were harmed in *that* particular encounter. When Matt got home and found the swarm and went for the vacuum cleaner, though, well...

The Sieve of Eratosthenes is a quick, enjoyable, non-rigorous enrichment activity for an older kid, best used for a review of Prime numbers or to construct a memory aid/anchor chart. However, you can actually also do this activity with quite young kids, since multiplication is the only skill required. It's fun and hands-on, the patterns are pleasing, and it gives kids a really interesting math concept to explore.

Here are some good books to use with younger kids in partnership with this activity:

To extend the fun, younger kids can play Prime Number Slapjack or color in a Prime path maze. If kids are a bit older and are ready to properly learn about Primes, composites, factor trees, and the factorization of Primes, this lesson and this lesson are excellent jumping-off points. 

We have a lot of wall space in our home, and my kids have always enjoyed making large-format posters, maps, and charts to put on our walls. A large-format hundred chart mounted on a wall lets kids have a different experience coloring it in mural-style, and would also allow room for kids to write each composite number's factors into those squares. Alternately, extend the hundred chart to 1,000 and keep sieving, although I wouldn't blame you for eventually pulling out the calculator!

Here are some books that older kids and adults would enjoy; completing a reading assignment (and perhaps even a response essay!) builds context and adds rigor to an otherwise simple activity, and is a good way to facilitate different ages/abilities working on the same project:

Here are some other math facts that a student could aid fluency by memorizing:

• fraction/decimal conversions
• PEMDAS
• Quadratic formula
• squares
• square roots (perfect square factors and simplified square roots to 100)
• Pi to several digits
• Pythagorean theorem
• Pythagorean triples
• triangle identities
• SOH CAH TOA

P.S. Want to follow along with my craft projects, books I'm reading, dog-walking mishaps, encounters with Chainsaw Helicopters, and other various adventures on the daily? Find me on my Craft Knife Facebook page!

Celebrate the 100th Day of the Year with Me

Every year, the teenagers and I volunteer with the Children's Museum of Indianapolis' 100 Days of School celebration. Area schools bring their kindergartners and first graders to the museum, and in between visiting the exhibits, the volunteers help the kids do fun activities relating to the number 100.

This year, my teenager and I had hoped to be assigned to the 100-bean maraca station again, but I actually loved the station we ended up at even more!

When kids came to our station, we helped them measure their height and their arm span, and helped them record the information in inches and centimeters:

If the kids had a little extra time because they were waiting on a few kids to finish up, I'd sometimes also have them stand on their tiptoes and then measure their height so we could compare how much taller they were on tiptoes (usually something like three inches!), or we'd see how long they could step or how high they could reach, etc. There's a lot you can do with a horizontal ruler and a vertical ruler!

It's always fun to me to see the range of kids we encounter, and the differences and similarities--we run through something like 250 kids in two hours, so those differences and similarities are really noticeable! Most of the kids, for instance, were around 143-147 centimeters tall, and their arm spans were always a little shorter. BUT they were all wearing shoes, so I told them that if they hadn't been, their arm span would be the same as their height! It was especially fun for kids to measure themselves against the horizontal ruler and then step back to visualize their arm span, so now I'm on a whole kick about how early ed classrooms ought to have those rulers set out the same way that most of them probably have height charts, so kids can visualize lengths in two planes. 

Many of the kids could not write numbers bigger than 100, but many could, and nearly all could write the numbers below 100. Several kids did a cute thing in which if I said, for instance, that they were 145 centimeters tall, they would write "100 45" on the line. I'd then show them what it looked like to combine it into 145, but I thought their solution was so clever, especially coming from different kids from different schools!

The 100th Day of School wasn't a thing when Matt or I were in school, so it wasn't on my radar when the kids were little enough to have fun with it, and I'm actually really sad about that, because we LOVED random little holidays and celebrations like that, and it would have been as super cute as our yearly celebration of Pi Day and everyone's half birthdays and May the Fourth. The celebration is a fun excuse for kids not just to practice the one-to-one correspondence of counting and the fine-motor skills of writing the numbers, but also to build context and meaning for the concept of 100, and explore the way that larger numbers work.

It turns out that this year, the 100th day of the year is Monday, April 10, which is quite a respectable homeschool day to celebrate a holiday! Here are some activities that I think would be fun--and educational!--to do with younger kids to celebrate the 100th day of the year:

paper chain. We made SO MANY paper chains when the kids were younger! Syd, especially was all in on paper chains, and we used them a lot to count down to various big events. Here's the paper chain birthday countdown that we made in anticipation of her fourth birthday, including the discovery that tearing a link off of a paper chain? OMG, such horror. Such despair.

That's why I actually think a paper chain counting UP to the 100th day of the year would be so great. Every day you don't tear a link off--you ADD one!!! Much less distressing to those tender, tiny hearts. 

For bonus points, make these laminated index cards with the numbers and number words on them for kids to match, trace, and add to their collection each day.

hundred grid fraction art. This reminds me a little of the mathematical map coloring that the kids loved just a few years ago! Kids color a pixel design onto a hundred grid, then can play with rearranging the colors and recording the fractional or decimal representation of each color.

roll to a hundred (or roll to zero). This is a fast-paced game that both of my kids loved long after they'd mastered their numbers, addition, and subtraction to 100. And it incorporates coloring, which is ALSO super fun (and utilizes those fine-motor skills, ahem):

This would make a fun "party game" for the hundredth day of the year, and you could even possibly convince a kid to fill out a blank 100 grid in preparation.

build with 100 things. This works well if you've got sets of more than 100 of various building toys, like LEGOs or blocks, but it would also probably be even more fun and creative if you chose seemingly random things. Can you build something or create a design using 100 pennies? Can you build a structure using 100 books? Who can build the biggest pyramid out of 100 rocks? Tiny little things in bulk also make fabulous math manipulatives, so it wouldn't be a terrible idea to splash out and buy your kid a 100-set of something small and cute as a 100th Day of the Year gift.

write a googol. That time that we read a book about googols, and then I asked each of my small children to write one, turned into a bit of a wacky adventure.

I scrolled Pinterest to look at other ideas, and while a lot of projects made me cringe or looked super corny, there was also tons of non-cringe, non-corny ideas to build a kid's numeracy and inspire them to love larger numbers and help them feel festive and celebratory. Many of the printables displayed would also work well for us homeschoolers to celebrate the 100th day of the year. I know that *I*, for one, will be coloring and wearing a giant cardboard 100-shaped hat on April 10!

How to Homeschool Math: Our Curricula, Resources, and Activities for Elementary

Yes, you CAN homeschool math.

Do not give me that sulky face, and don't you dare tell me that you clearly cannot homeschool your child because you obviously don't remember how to divide fractions.

Of course you don't remember some math stuff! You learned all that a zillion years ago! But you know what?

You are a grown-up adult. You can pick it back up. It's really not that big of a deal.

Also, remember that time that you tried to learn some math thing back in school and it was too hard and you got mad at yourself and said that you hated math and you're not good at it?

You were NOT a grown-up adult back then. You were a child.

How fun, then, not to pass that same negative mentality on to your own kids?

Elementary math can be so much fun for kids! Elementary math is fun manipulatives. It's fun games. It's jumping and screaming and singing. It's drawing and painting. It's measuring and pouring. It's reading books together, and doing puzzles, and figuring out interesting problems. It's some of my happiest memories with my kids.

ELEMENTARY MATH

Both my kiddos studied math informally until second grade, then began in Math Mammoth's Light Blue Series at either Level 2 or Level 4, depending on the kid. Math Mammoth took them both all the way through elementary and middle school math.

For part of elementary and middle school, we also used this text as a combined math/science/history study:



Here are some other textbook, reading, and viewing resources that we used:

Almost all of these came from the library, and most consist of fun picture books that I found for us to read together--even older elementary kids (and even high schoolers!) love great picture books!

And here are our most important manipulatives:



There are tons more manipulatives that I created myself or Googled, downloaded, and printed. I'm a BIG proponent of DIYing as much as possible!

Below, I've listed tons of the enrichment activities, extra reading/viewing resources, and fun stuff, that I did when my kiddos were elementary ages. They're not listed in any particular order, because they don't have to be done in any particular order! I liked to have the kids doing some kind of hands-on math activity every day, and if it wasn't inspired by them wanting to bake or build or draw or read together, then I invented a fun invitation or set up a game or suggested a project or offered up a puzzle.

I also set up enrichment and hands-on activities to support skills that the kids were struggling with, or to expand on skills that they were blazing through. Seriously, ask me how many hands-on activities I know for teaching rounding!

So scroll through my list of fun enrichment activities, and definitely set them up to support something specific, but also feel free to engage in them at random. I mean, most of what we did in elementary was totally random, and those were some of the best years of my life!

• Building Big Numbers with Base Ten Blocks. This is the BEST math activity that my younger elementary kids ever did. It's endlessly repeatable when they're little, and builds that crucial number sense that they need to make everything else make sense!
• Clock Cake. Clock reading is a skill taught in elementary math, but it's ever more abstract these days, as every device has a digital clock and analogue clocks are becoming more of a rarity. The solution? Bake and decorate your own clock!
• Roll the Time. To get even more clock-reading practice, write minutes onto a blank die, combine it with a twelve-sided die, and have a kid roll the time. She can tell you the answer and draw it onto a blank clock face.

• Demonstrating the Commutative Property of Addition using Cuisenaire Rods. Cuisenaire rods are superb at demonstrating equivalent equations.
• Pascal's Triangle. You can find the blanks for these online, and seconds later, you've got an in-depth addition review!

• Roll to a Hundred. I'd completely forgotten how many of those printable hundred charts we used in the early elementary years! There are endless amounts of games you can make using them, all of which reinforce numeracy and computational skills.

• Add Multi-Digit Numbers using an Abacus. Just as when the kids were learning to read I wanted them to see all kinds of letters in all kinds of fonts and all kinds of sizes and made of all kinds of textures, I liked the kids to practice counting and adding and subtracting in all kinds of ways, with all kinds of manipulatives. If you've never used an abacus, it's well worth learning how to work it properly!
• Write a Googol. Big numbers are so fun for little kids!
• Build Decimal Numbers using Base Ten Blocks. Just as you can build whole numbers with Base Ten blocks, you can also build decimal numbers--it's all Base Ten!
• Subtraction with Borrowing across Zeroes using Base Ten Blocks. If you can do it in Base Ten, you can generally model it with Base Ten blocks.

• Metric Conversions in Grams using Rice Models. If you have a balance scale and some rice, kids can make their own metric conversion models, using a LOT of problem-solving and practical life skills in the process. And even in high school, we STILL pull out those rice models whenever a kid needs a metric conversion refresher!

• Metric Conversions in Liters using Water. You can't keep this metric model forever, but you can use it to cool you off on a hot day!

• Hands-on Rounding. For some reason, Syd really struggled to master this concept, and so now I have a zillion concrete, physical, hands-on ways to teach rounding.
• Decanomial Square. You're going to see me refer to this a lot, as it's one of our most-used math manipulatives. It's well worth DIYing or buying one, because your kids will still be using it in high school!

• Skip Count with Coins on the Hundred Grid. It's good to practice skip counting a lot, and in a lot of different formats. This version also reinforces coin values.
• Multiplication Touch. It's quite the project to build the roll-up hundred mat and the number tiles from scratch, but Multiplication Touch is well worth it! It's surprisingly fun to play, so works great not just for memorizing the facts, but also for reviewing them for years.

• Multi-Digit Multiplication using Area Models. Centimeter graph paper, Base Ten blocks, and Cuisenaire rods are your best friends in elementary!

• Model Long Division with Base Ten Blocks and Cuisenaire Rods. You can cover a LOT of mathematical ground with just those two types of math manipulatives! By the time a kid hits long division, she's already starting to get math instruction that's more focused on teaching her how to plug numbers into an algorithm than it is on understanding what she's doing, and that's not cool with me. When I teach long division, I like to drag out all the blocks and show a kid exactly--and I mean EXACTLY!--what is physically happening to the literal numbers that she's dividing.
• Literally Walk through Long Division. The beauty of homeschooling is that nobody--not you, and definitely not the kids!--are trapped at a desk. Get the wiggles out and bring whole body learning into your math study by drawing a GIANT long-division problem on your driveway and then having your kids solve it. Who knew there was so much walking, crawling, and skipping in math?

• Compose Fractions with Rice Krispy Treat Models. Because number sense is delicious!
• Model Converting Mixed Fractions to Improper Fractions. You need circle fraction manipulatives for this activity, but the good news is that it makes converting between mixed fractions and improper fractions as clear as a bell and as easy as pie! Here are the circle fraction manipulatives that we own:

• Level these fraction models up to middle school and high school by bringing them out whenever kids need a review. They WILL randomly forget how to convert between mixed and improper fractions, but going back to the manipulatives will quickly remind them, and every time they re-learn it, they'll cement it into their brains even better!

• Montessori Pink Tower. Playing with the Pink Tower can be soooooo satisfying for the pattern-loving child, and it's great for their hand-eye coordination and number sense.
• Montessori Pink Tower and Cuisenaire Rod Patterns. If you're loving the idea of open-ended exploration of math manipulatives for young learners, then you should really check out Montessori math! These kid-built, elaborate patterns are satisfying to create and look at, and since they're built on a Base Ten system at a centimeter scale, they're terrific for building a kid's number sense. Here's the Montessori Pink Tower:

• Level pattern-building using the Montessori Pink Tower and Cuisenaire rods to middle school by having kids create their own diagrams or pattern cards from their creations. Perhaps they want to share them with a younger learner, or even publish them!

• Blow Bubbles with Geometric Figures. What happens inside a prism when you dunk it into bubble solution? More fascinating geometrical models! Here are our Zometools:

• Find the Area of Irregular Polygons using Graph Paper. Everything is more fun when it's on graph paper! For even more reinforcement, use centimeter graph paper and fill in the polygons with Base Ten hundreds, tens, and ones, and Cuisenaire rods.

• Make Mandalas with a Compass and Protractor. Here's a fun art activity that happens to teach compass and protractor use while also offering a lot of scope for creativity and process-oriented art. 
• Level this activity up for middle and high school by encouraging even more elaborate, thoughtful creations.
• Origami. This is one of the best geometry activities that your kids won't even recognize as math work! Start with SUPER simple builds so that they don't get frustrated with the fine motor requirements, but no matter what they make, they're building an intrinsic knowledge of angles and shapes and how they work together in two- and three-dimensional spaces.
• Level this up to middle school and high school by working through ever-more-challenging how-to books.

• Pentominoes. Allowing kids free exploration of mathematical models and manipulatives and interesting shape puzzles is one of the best things that you can do with them in elementary, whether or not you're officially homeschooling. Pentominoes are especially fun for older elementary kids who enjoy stretching themselves with challenging puzzles.
• Level pentominoes up to middle school and high school by working through ever-more-challenging puzzles, and calculating the number of possible solutions to puzzles.
• Ink Blot Prints that Show Bilateral Symmetry. Get really messy in order to get really symmetrical!

• Symmetry and Similar Figures with Pattern Blocks. This is a more challenging activity for older elementary, but you get to drag out ALL the pattern blocks and it makes the kinda abstract concept of similar figures crystal clear!
• Tangrams. This is another math manipulative that's endlessly entertaining, even onto middle school and high school. Heck, *I* still love to play with tangrams!

• Zometool Stellations. To stellate a polygon is to extend its line segments. It always makes something beautiful, and it encourages kids to stretch their imaginations and become more mathematically creative thinkers.
• To level this to middle school, have the kids measure the angles both inside and outside their polygons.

I don't blog about every single thing that I do with the kids, so here are some of the Pinboards where I collect even more elementary-level math resources. Some I've done, some I haven't, but I think they all look pretty cool!

• Homeschool Math: Addition and Subtraction
• Homeschool Math: Base Ten
• Homeschool Math: Fractions
• Homeschool Math: Geometry
• Homeschool Math: Measurement
• Homeschool Math: Money Math
• Homeschool Math: Multiplication and Division
• Homeschool Math: Telling Time

I LOVED doing math with the kids in elementary. I loved exploring open-ended activities with them--activities that I, too, thought were fun! I loved reading math poems and picture book biographies of mathematicians together. I loved every single time a kid was happy while doing math, every single time she felt confident, every single time she took a risk, every single time she struggled and struggled and finally understood. 

It's exactly what learning should be, and it was my privilege to give that to my kids.