Math Monday: Exponential growth (for young kids)


We went camping recently and on the way back home we stopped at the top of the mountain pass to see some amazing mountain ranges and catch a glimpse of our favorite volcano. (Lol, it’s not erupting — thankfully! — but that’s what our 4 year old always calls it.)


N (the 4 year old) has a delightful habit of picking up rocks everywhere he goes for his “rock collection.” (question: what does he think this is? Does he envision a huge pile of rocks in our house that he collects from his various trips? Is he secretly squirreling them all away somewhere? He’s pretty sure this is a thing, and I’m pretty sure it isn’t. … Time will tell, I suppose.)

So we were at the top of the mountain and N picked up a handful of rocks, per usual, and asked if he could throw them down the mountain. He and lil Dude love throwing rocks in water, so this seemed like a reasonable request — he was quite surprised when i said no. Actually it came out more like “NowayNEVERDOTHATwhatdoyouthinkyou’redoing?!” as terrible visions of rock slides swarmed up in my ever-anxious/protective mama brain.

He stopped, mid wind-up, and looked confused. I tried to explain the basic idea of a rock slide and it turned into a pretty neat math conversation, (although I have reason to fear the actual topic).


On the ride back home we talked about how if he throws one rock down the mountain it might knock loose two more rocks, which isn’t really a problem — except that each of those rocks might knock two more rocks (to keep the math simple) and each of those rocks might knock two more rocks each, and pretty quickly all the rocks are falling down the mountain. N looked properly terrified.

It’s kind of a fun tongue-twister too:

If one rock knocks two rocks, and two rocks knock four, and four rock-knocking rocks knock loose eight rocks, how many rocks will be knocked next?

or something like that.

He counted up to 16 and was impressed enough by the grandeur of the situation so we stopped there. But for the record, the first 10 numbers in the series would be:

1, 2, 4, 8, 16, 32, 64, 128, 256, 512 …

and you get the idea how fast things can get out of hand when a number is doubled every time to create the next number in a series.

Later I was still thinking about this and wondering if this was an example of exponential growth. I drew some diagrams to show N the difference between simply adding 2 to each term in a series (every second after the first rock falls, two additional rocks fall down … or there might be a better way to explain that?) — this would be an arithmetic series — and saying that every rock that falls knocks 2 rocks down (essentially doubling the amount of rocks falling each time) — this is a geometric series. This is admittedly pretty fuzzy math, because you wouldn’t actually have clear intervals to measure the different “groups” of falling rocks, and every single rock wouldn’t actually knock exactly two more, and technically exponential growth that occurs naturally is based on “e,” a constant value that is related to the inverse of the natural logarithmic function (might be missing something here) … etc. But I figured it was a good enough approximation to a complex mathematical idea … and hopefully the exposure to the thought of increasing patterns of numbers in a series (sequence?) is a valuable learning experience. At the very least, I think we’ve all learned not to throw rocks down the mountain! 🙂

And I realized how long it’s been since I learned about all this and how quickly it dissipates. Also, you can learn pretty much anything you want to know on the internet, so really … why go to school? lol … kind of …




Somewhere in there I remembered a song I had run across when I was doing my math major years ago — it’s a song that a college professor (Alan AtKisson) wrote to explain the concept of exponential growth to his students and for some silly reason I loved it and remembered it all these years later. This is the link to the youtube video of him singing it in front of his class. (He has several other songs about other economic principles as well.) So for the past week or so, we all (even lil Dude) have been breaking into song randomly with the refrain “Exponential growth, oh, it’s exponential growth; yes it’s creeping up behind you, yes, it’s exponential growth!”


As I was writing this post, I remembered that we have a book about this concept that does a great job explaining it in a fun, easy-to-understand way for kids with cute illustrations. It’s called 512 Ants on Sullivan Street (by Carol Losi (Author), Patrick Merrell (Illustrator), and Marilyn Burns (Collaborator)). The skill level of this book is definitely beyond where my kids are at, but I think reading books like this with young kids is still a valuable activity because it helps lay a foundation of understanding mathematical principles and, more importantly, helps them to see math as an interesting and fun mental exercise, not a fearful and overwhelming test-driven torture device. Ha, there’s my 2 cents and a little opinion on top. 



Note: None of these are affiliate links. I’m just providing links for things that I’ve found interesting or helpful or entertaining, in case any of it is relevant to you as well. 🙂


Signing off with another pretty picture of the river by our campsite (very safe for rock-throwing exercises!)



Other Math Monday posts:


Math Resources for Teachers, Tutors, and Homeschooling Families:

If you’d like to see my own growing library of educational resources (mostly focused on math) for pre-K through high school students, you can visit my online store at: You can read more about my new venture in this blog post here.

Finding value in screen time: our favorite 15 netflix shows, dvds, and apps for young kids

screen timeIn the complex world of twenty-first century parenting, Screen Time can be the subject of much debate and anxiety.

Every family has different ideas and expectations about this subject. I certainly do not have it figured out. Also, my personal philosophy is in flux regarding the advantages and pitfalls of screen-bound entertainment and education. But …

we have managed to curate a list of several kid’s shows that both I and the kids like. We’ll try out new shows on Netflix every once in a while, but we have our go-to list of shows that (fortunately!) the boys don’t mind watching on repeat.

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Math Monday: RightStart curriculum

We went to our state’s homeschooling convention recently. After surveying the math scene, we ended up buying a beginning math set from the RightStart curriculum. We had been to a homeschooling convention a few years ago and I had been so impressed with this curriculum that I had purchased the Geometry set, thinking I could incorporate it into my classroom since I was teaching high school Geometry that year. Well, as it turns out, there was no time or energy to be found that year, so the entire set sits invitingly on my bookshelf, waiting for new homeschooling adventures instead. 🙂 I decided this year to pick up the first set in the series and start it slowly with my son. (This is not an affiliate post — just stories of our experience with it.) Continue reading

Newly released: fresh-off-the-press resources to help students understand and enjoy Math

logo v3I have a little bit of big news over here in our corner of the world — I’ve hit LAUNCH status!

Over the past few months, I’ve been drafting up and dreaming up plans to start a little company of some sort based on homeschooling/teaching products. I started playing around with the idea of turning some of the projects that I do with my own kids at home into larger lesson pack units. I began with some math games for young kids based on ten frames, and over the course of the next few weeks, my simple explanation of our home-made learning experiment turned into a 20-plus page resource with activities, teaching theory, and full explanations … and I remembered that I love doing this stuff :).

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Math Monday: Tessellations

This week in our spontaneous preschool* math adventures we’ve encountered tessellations, patterns, observations, arts and crafts, and dragons. (*Almost everything explained here could easily be duplicated with students of all ages; in fact, older students would be able to better understand and experiment with these ideas.)

tips fortidyingyourcloset Continue reading

Silly Stories for kids: the Zoo Train Parade

Copy of Copy of autumn art fair

My 4 year old son and I sat down to brainstorm some more silly words (rhyming words or alliteration phrases or just plain silly sounds) and we (ok, mostly I) came up with this story. It’s fun to get him involved in the creative writing process so he can see how to get ideas flowing onto paper, then mash them all up and jumble them back together until they fall into place and you get a silly story. 🙂 That’s our official process at the moment.  Continue reading

Math Monday: Cotton Ball Algebra

Cotton Ball Algebra

If you can lay a strong foundation in math for young children and lead them to discover how things work, keeping the learning environment stress-free and enjoyable, I believe this will give kids the best chance to entirely by-pass math anxiety and succeed in higher levels of math and science as they grow. Continue reading

Creative Writing: Story-telling with preschoolers

Copy of autumn art fair

The tale of the biker man has evolved into a series of stories (The Biker Man Chronicles, perhaps?). We’ve been on a silly-story-telling spree lately, and I had fun working with these particular ones to make them into fun tongue-twister stories.

So here’s the prequel to the story of the Biker man from my last post. Continue reading

On-the-go Storytime

Besides answering All the Questions, another fun activity we like to do on long car rides is make up silly stories. N and I got rolling (umm, no pun intended) on a pretty silly story the other day and he helped me recall it tonight so we could be authors and write our very own book.

autumn art fair (1) Continue reading

Math Monday: 10-frame math with seashells


nothing better than doing math with seashells in paw patrol pajamas!

I’ll randomly do some 10-frame math activities with N (now 4 y.o.). We’ve used egg cartons, sketches on paper, “official” ten frame plastic trains, and now a large plastic floor mat with seashells and expo markers.

I was a math teacher for several years, so sometimes it’s a bit hard to contain my excitement about numbers and patterns and logic and anything else that’s even tangentially math related. (ha! pun intended.) Continue reading