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

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A Snarky Dracula Literature Study Appropriate for Snarky High Schoolers

Out of all of the hundreds of audiobooks that my kids and I have listened to together, I think Dracula might be our favorite. It's got so much to offer, from its status as a classic of literature to its Gothic horror vibes to its vampire creation mythology to its depiction of some of the best characters ever to grace the pages of a novel, particularly my personal badass heroine, Mina Harker, a woman who can write shorthand, has memorized London's entire train schedule, is a loyal friend and wife, and is the only one of Dracula's victims who lived to tell the tale.

Here are some of the things that my teenagers and I did during our brief Dracula unit study. This was part of their high school English credits, although the cross-curricular art activities were also recorded as part of their Studio Art credits.

Audiobook Narrated by Gildart Jackson

The kids and I loved Gildart Jackson's narration, so much so that we've specifically looked for him when picking other audiobooks (he also does an awesome Frankenstein!). 

We also loved Dracula!

The trick when you read Dracula with kids, or really probably any iconic work of literature, is to remind them that once upon a time, Dracula wasn't iconic--it was just Stoker's newly-published book! There weren't a ton of vampire tropes or cliches to tap into, because he'd mostly just invented them. You literally start the novel thinking that you're going along on one of Jonathan Harker's boring work trips. Probably, you're just going to be reading an epistolary travel novel about the wonders of verdant Transylvania. I love imagining it, how the Victorian reader must have grown ever more suspicious of the mysterious Dracula, how they'd gasp in shock at the big reveal: Dracula is a VAMPYRE!!!

Other semiotically rich elements worth much discussion include everyone's shitty, misogynistic treatment of Mina, how even with her misogynistic depiction Mina is still the most badass of all the Dracula characters (how/why did Stoker write her to be completely disregarded but also the most efficient, practical, and intelligent?!? Did he do it on purpose or by accident?), and the absolutely hilariously bonkers depiction of Quincy, the "American." Like, he literally carries a Bowie knife--had Stoker ever seen one of those to know how impractical that is? The kids and I howled with laughter every time he had a line of dialogue.

To make some comparative literature analyses, pair with any other vampire novel ever written. Excerpts from the Twilight novels are fun because they're just so corny, or you could do something well-written like Interview with the Vampire or 'Salem's Lot. I'm currently reading The Historian, and I am VERY into it! For a quicker project, go for a Goosebumps book or even a picture book, or do a sub-genre like paranormal romance with Dead Until Dark. So many semiotically rich vampires!

1931 Dracula Film

This first (and best!) Dracula film has a lot to add to the study, and everything about it is ICONIC. 

I mean, come on: Bela Lugosi? Iconic!

This 1931 movie introduced conceits that you don't even realize are conceits, they're so ingrained. The "Transylvanian" dialect that Dracula speaks in? That's from here! Turning into a bat that talks in squeaks, and the vampire's subjects can understand those squeaks? Yep, that's from here, too. It's super funny to watch this film, then turn on just, like, the first ten minutes of Hotel Transylvania. 

For a deeper analysis, pair with any vampire movie that came afterwards. We watched The Hunger for a recent Family Movie Night, because David Bowie, and it had some interesting things to say about age, sexuality, and the power dynamics of romantic love. Funnily enough, Dracula ALSO has interesting things to say about age, sexuality, and the power dynamics of romantic love!

Art Experiences

I used to get the Dover Publications free samples weekly, and the kids LOVED them! These Dracula paper dolls from Dover are a good inspiration for artistic response, because kids can then create the costumes for them. They can go old-school vampire tropes, or Victorian fashions, or make modern reimaginings of the novel's characters. I want to see Mina Harker as a steampunk heroine!

My younger teenager is very into the art of the book, so I've been encouraging her to create book cover images and design book jackets for the books we read together. You can actually get those printed to fit your favorite books, you know!

Artistic kids who want to use the book for more cross-curricular explorations can also illustrate memorable scenes or quotations. If they need some inspiration, check out my favorite edition of Dracula, this one illustrated by Edward Gorey (there's also an Edward Gorey Dracula toy theater that I covet). Even kids who don't see themselves as super artistic can create memes; memes are one of my favorite student creations, because kids love them, they get to show off how witty and sarcastic they are, and the most successful memes show natural evidence of a sophisticated understanding of the book.

Place-based Studies

I'm not very excited about Romania or Bran Castle when it comes to a Dracula study, because I'm not convinced that Bram Stoker even knew of those places, or of Vlad the Impaler. He liked the name "Dracula" when he read it somewhere, and his inspiration for the novel was Whitby, England. We didn't visit there when we went to England this summer, but we DID make a point of collecting ourselves some Gothic vibes. Next time I visit England, which I hope will be soon, I want to go north and see Whitby!

But I mean, you know, if I was IN Romania I'd definitely go to all the Vlad the Impaler sites.

To add rigor and more composition practice to any of these activities, just tack on a writing assignment! Write an opinion piece about Stoker's depiction of one of these now-iconic characters. Write a compare/contrast essay with another vampire book or film. Write Dracula fanfiction that changes the ending or the setting or a key plot detail. Write a research paper on the history of the vampire myth, or Gothic horror, or Whitby Abbey. Write an essay explaining your artistic process and decision-making when creating the book cover or illustrations. Create a travel brochure for Dracula's Transylvania. There are so many ways to make high schoolers suffer!

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The Cavetown Concert Made Me Cry

Matt and my older kid both seem to lack the music appreciation gene, so when my younger kid, around the ages of 12-13, started to get really into music, omg I was thrilled. Finally, someone to listen to music with! Someone to make mixtapes for! 

Someone to go to concerts with!!!

Cavetown and Ricky Montgomery are a couple of the kid's OG loves, at the forefront of that first wave of music that she discovered for herself. I love these guys, too, in a complicated mix of nostalgia for that awesome little tween I used to have and genuine appreciation for how genuinely good they both are. Cavetown, especially, is always going to be, for me, the experience of driving to and from ballet daily with my newly-minted teenager, listening to those earnest teen folk-pop anthems on endless repeat. 

I was VERY excited, then, for the teenager and I to join the rest of Central Indiana's emo teens at our first Cavetown/Ricky Montgomery concert. Imagine: mother/daughter bonding to the dulcet crooning of Ricky Montgomery! 

This is an accurate depiction of my level of happiness for the entire evening:

Fun fact: this is the photo I now text the kids whenever I want to enthusiastically agree with something they've written. They both have steadfastly refused to give me any positive reinforcement in any manner for this, even though it is objectively awesome.

In the interests of Family Bonding, we dragged a couple of less-enthusiastic companions with us, but then immediately ditched them when they weren't willing to stand up front with us for two hours before the concert started. Vibe killers, the both of them!

I'm pretty sure I had an out-of-body experience when Ricky Montgomery came onstage, I was so happy and excited. Like, my god. I fell in love with his music during the pandemic lockdown, when I thought that we were all going to die. But we didn't die, and now here I am, at an outdoor concert with the people I love the most, listening to this music that I listened to back when I thought I might never go to another concert again.

I tried taking some videos of my favorite songs, like I saw all the other cool kids doing. But, um, nobody told me that you're not supposed to be loudly singing along when you take your videos, ahem. So please join me in my own combined horror/amusement at the following clip, in which you can clearly hear me both sobbing with happiness and loudly singing off-key to my favorite Ricky Montgomery song:

Bless. My. Heart.

Here are some of my other favorite Ricky Montgomery songs:

 

Most of them are from his pre-pandemic album that I played on repeat all during lockdown. Lockdown Julie would have loved to know this day was coming!

Check out this gross guy, though:

That's a photo of him in between sets, as we're all just standing there, putting his foot directly in front of the teenager's body, then leaning his ass back into her. The teenager immediately got my attention, because GROSS, but we weren't giving up our standing spots for nothing, so I nudged my foot in right beside hers, then, like Indiana Jones switching a golden idol for a bag of sand, I sidled into her exact spot while she took mine.

I waited a beat so that I could experience that, indeed, this absolute nasty asshole of a man was, indeed, pushing his nasty ass against me, and then I said VERY loudly, "Can you stop pressing your butt against me!?!?" 

The nasty asshole jumped away like he'd been slapped when he realized at this moment that he was assaulting not a teenaged girl, but a shouty late-40s woman, and everyone turned to look at us, including his female concert companion, and I gave him SO much stink-eye while he lied about "saving someone a spot," and he left us both well alone after that.

Seriously, though, imagine a world in which a teen girl could go to a public place and not be assaulted in some way by a gross man. If she can't avoid it at a Cavetown concert, of all places, then where in the world IS she safe?

 Fortunately, no other incidents occurred, and eventually, there came Cavetown!

When it comes to Cavetown, my emotions are deep. I literally love him the way that I love my kids' friends. He's easily young enough to be one of my own kids, and I've been listening to him with my own kid since before his voice changed. To me, Cavetown's songs feel like it felt to have that young teen--witnessing a new social maturity, a new and uncomfortable level of self-awareness, those first signs of the anxiety and depression that it feels like all teenagers these days suffer through, the accompanying respect and acceptance of a diversity of people that these same kids are also graced with--how can it have been so few years ago, because it seems so far away?

But look at us now.

Here's "Juliet," with just a little off-key singing on my part:

And here's where Ricky Montgomery came back out and I lost my mind with happiness again:

I really love that they're all just wearing their comfy clothes to perform in:

They're already working hard--I *want* them to be comfortable!

Also, apparently that song was a popular moment to make into a Tiktok! Here's another I found:

Here's where I realized that I was going to cry during this set, too, because all these kids here, and in particular my own kid, have worked so hard and been so brave and look where they get to be tonight:

Thank goodness for concerts. Thank goodness for Cavetown. Thank goodness for a beautiful summer night, and a crowd of young people screaming the lyrics to the song that was your kid's favorite when she was thirteen.

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Heaven Will Be an Eternal Game of Bananagrams

 

There's this unschooling tactic called "strewing" that I started to follow when the kids were little. Basically, if there's a material you'd like a kid to explore, you don't hand it to the kid, or direct the kid to it--instead, you just leave it temptingly out on a shelf or a table for them to discover for themselves. And it's supposed to be okay if it takes days, weeks, even months for them to pick up that material, because when they do, it will be self-directed and self-motivated, and following their interests is how kids best learn.

Until I really discovered the true dimensions of our local university's library (Hint: filtering for Three-Dimensional, Mixed-Media gets you all the puzzles and toys and manipulatives!), this was mostly just a way for me to waste my homeschool budget on stuff the kids didn't touch. I mean, sure, they became obsessed with the Geomags and the Kapla blocks, eventually they spent entire weeks with the Perler beads and the Sculpey clay, listened to so many books on cassette tapes that I worried they would never do anything else, but the 3D pen? Nope. Nope to the Zometools for free play. Nope to the crystal growing set and the build-your-own maze set and the Snap Circuits and the Turing Tumble and the balloon animals kit and probably over a thousand dollars more of awesome kid crap over the years that I would have given a year off my life to have played with when I was their ages, ahem.

Anyway, became much cheaper to strew when what I strewed came from my local university's library, and since then, over the years, we've had logic puzzles and board games and STEM toys and math manipulatives and scientific instruments all enticingly placed on our family room bookshelves ready to be explored and easy to return.

Thanks to the library's infinite renewals, I can't even tell you how long we've had Bananagrams. I think we got it around the time we also got the giant poster of the Greek mythology family tree and the giant map of the Moon, and those have both gone back to the library, but Senet and the leaf identification kit are also still here, so maybe it hasn't been too ridiculously long. Although I'm pretty sure several sets of tessellation puzzles have come and gone in that time, as have the Proofs of Pythagoras kit and the French vocabulary flash cards, sooo...

Now that the kids are grown or nearly grown, though, it turns out that the person I'm maybe actually strewing for is myself. I was wandering around the family room the other day, aimlessly tidying while the teenager and I listened to The Haunting of Hill House (it's not translating as well as I'd hoped to audiobook; we're going to finish it, but we don't love it, whereas I LOVED this book when I read it a couple of Halloweens ago), when I noticed, for the first time in ages, the Bananagrams game sitting on the shelves where we keep our library materials, and thought, "Huh. I should play that and see if it's fun."

So I rallied the teenager, and we did play it:

And it IS fun!

You know how Scrabble is generally really fun, but it's also boring waiting for other people to take their turns, and it's terrible when you have a terrific plan for an awesome word to play but before you can do it another person takes your spot?

Bananagrams solves ALL of that. There is never any downtime. You make your own crossword grid that's all for you, so nobody can ever mess it up, and when you see a better play you can rearrange your own crossword however you like:

Teenager peeled an "I" and decided to turn "DOPE" and "DAMN" into "DOPAMINE." 

So, everyone draws the same number of letter tiles (every time I've played it's been with 2 or 3 players, and we always draw 21 tiles), and you each work on your own individual crossword grid.

Matt, our college student home for Fall Break, and I are playing at the kitchen table on a Saturday night, listening to the drive-in's broadcast of the Taylor Swift concert like a good old-timey family.

When you've all of your own letters in your own crossword, you say "Peel," and everyone takes a new tile from the pile and continues playing.

Sometimes you get a new tile and it's like an S or something, so you can just pop it onto the end of a noun. But sometimes you get a Q and you realize that your only U is already busy, so you have to disassemble half your grid to get it back and then figure out how to rebuild while continuing to take a new tile every time someone else says "Peel."

The game continues that way, with occasional breaks to look weird words up as a family or neg someone else's word choice or lore dump about Scrabble games of old, etc., until there are fewer tiles left in the pile than there are players. At that point, we declare that the next person to use up all their letters wins, and then we get in everyone else's business to "help" them finish their own crosswords, but you could also go by Scrabble points.

I think you could also play Bananagrams as a solitaire game, going by how much fun the teenager and I had one time simply turning all our tiles face-up and using them all to make one giant grid. We started off just trying to build the most emotionally unhealthy words we could, as a "joke," so maybe it's also a little bit therapeutic, as well!

Or you could just build words representing the biggest thing on your mind these days...

I'm now officially on thrift store/garage sale lookout for a set of my very own, although I'm also toying with the idea of DIYing a set. They're literally just letter tiles in a zippered bag, and the only requirements are that the letter tiles have enough chonk to be able to pick them up easily, and that they have two sides for facing them up or down.

I think it would be fun to take a set of blank wooden tiles and handpaint each one, maybe with little background decorations like an illuminated manuscript. Think how pretty your crossword grid would be!

I Read The Chalice of the Gods, Because the Scariest Adventure in a Teenager's Life is Applying to College

The Chalice of the Gods by Rick Riordan
My rating: 5 of 5 stars

When my kid and I read The Lightning Thief, she was ten, and Percy was twelve.

Last week, we each read this latest book in the series. Now, my kid is seventeen, and so is Percy.

Reading this latest adventure of my kid’s all-time favorite book character, the character she’s stayed loyal to even as she’s grown up, is bittersweet as hell, just like everything else about this last year at home before college. What happened to that little girl who wore an orange Camp Half-Blood hoodie (with "Daughter of Apollo" printed on the back) and wielded a wooden sword and a cardboard Spartan shield that she'd painted a Pegasus on? Where is the kid who made endless batches of slime one summer while listening to The Demigod Files on repeat on my old CD player? Could I please have another day with that small daughter who traipsed around Greece with her American Girl doll, petting feral cats and lighting Olympic flames?

Just like Percy, my own little demigod is almost all grown up.

I am not mad, then, that this particular Percy adventure is lighter fare than the usual magical mayhem he gets himself into. In this book, Percy's big adventure is obtaining one of the three godly college recommendations that he needs to be accepted to his dream school of New Rome University. In order to obtain a recommendation, he has to go on a quest. It's demanding and complicated, there are snakes, but it's no Battle of the Labyrinth, thank goodness. My gods, can you even imagine if my kid’s favorite book character who happens to be her exact same age in this book and is also going through college application bureaucratic nonsense just like her had DIED (RIP Jason)?!? Or was betrayed by a loved one or ended the book on the verge of an apocalyptic war or got turned into a tree or was the subject of an awful prophecy, etc.?

Ugh, just parenting/being a kid in their Senior year of high school is hard enough, thank you.

So, the plotline of this Percy Jackson book, and the next two, may feel trivial to many readers, but I, for one, find it comforting and validating that the mythology is barely metaphorical here, and instead, Percy’s difficult adventures are quite like the difficult Senior year adventures of many American kids. Asking for recommendations IS hard! Coming to terms with growing old(er) is hard! Preparing to leave your best friend is hard! Realizing that you’re going to miss a lot of big moments with your family in the coming years is hard! Moving away from young siblings is hard!

But to tangentially speak about the actual book for a moment, instead of just my own personal feelings ABOUT the book, I am thrilled to report that, unlike the hot mess that was The Sun and the Star: A Nico di Angelo Adventure, The Chalice of the Gods is, happily, good! It felt like a way quicker read than what I remember from my other Percy Jackson adventures, but maybe that’s just because it was fast-paced throughout. I learned a couple of new gods, which is always fun, and enjoyed the deep cuts and callbacks to other Percy adventures.

The only bummer to the book is that I feel like we’re all just meant to ignore the fact that Ganymede is an honest-to-Zeus human trafficking victim. It’s very disturbing, and I didn’t really know what to make of the tone there. Lighthearted-esque, but clearly Ganymede is being held against his will. Zeus has one line of dialogue about Ganymede that’s not sexually explicit but you know what he means, and what he means is gross. Also, Ganymede is canonically a CHILD! He’s Percy’s age, if he’s even that old! Which means that he’s my kid’s age! I’m glad he got his cup back, but he’s still enslaved to his abuser, sooo...

Deeply troubling Zeus sex crimes aside, The Chalice of the Gods is beautiful and sweet, and I loved it. I’d have no idea what was happening or why if I came in cold, but if you read and loved the previous Percy Jackson books, surely you’ll love this one, too.

As for the rest of Senior year… well, my own kid is planning a completely unironic Percy Jackson-themed set of Senior photos, complete with her Camp Half-Blood hoodie, her sister’s Ren Faire sword, and a post-production chimera or two created by her graphic designer father. Over Christmas, she and I plan to buy ourselves a month of Disney+ so we can unironically watch the Lightning Thief reboot. And she has GOT to get her own set of college recommendations completed, even though at the moment she’s acting like this is a feat much more challenging than wrestling the god of old age.

When she and I read the next book in the series, she will, for the first time, be older than Percy. Stay tuned for my whole other set of tears on that day!  

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