We played Pig again.

1. In turn, each player is handed four dice. He can choose to roll 1, 2, 3 or all 4 dice. He can roll as many times as he likes.
2. Each time he rolls, he computes the sum of all the dice he rolled. That is his score.
3. If he rolls one “1″ at any point, his turn ends, and he loses all the points he’s scored in that round.
4. If he rolls two “1″s — his turn ends, and he loses all the points he’s scored so far in the whole game.
5. The first player to reach 100 points wins.

• with 1 die there are 6 (6 to the 1st power) possibilities
• with 2 dice there are 6×6 (6 to the 2nd power) possibilities or 36
• with 3 dice there are 6×6x6 (6 to the 3rd power) possibilities or 216

This time I asked the kids this question: “What are the odds of getting a “1″ if you roll one die?” (That’s easy: 1:6) “What are the odds of rolling at least one “1″ if you roll 2 dice?” Again, I showed them how to construct a tree diagram to show that there are 36 possible outcomes when you roll 2 dice. The odds of rolling at least one “1″ are 11:36. So when you choose two dice rather than one — in hopes of getting more points — you almost double your odds of getting a “1,” which would cause you to lose your points for that round. Then I asked “What are the odds of rolling at least one “1″ when you roll 3 dice, and tried to guide them through using the tree diagram to discover that there are 216 possible outcomes with 3 dice. Another way of looking at it:

This was too difficult for both kids. I tried a different approach, using grids instead of tree diagrams to illustrate the possible outcomes. I’ve never seen it done this way, but I thought it would visually spotlight the multiplication & exponents involved.

with one die: 6 possibilities (1 chance out of 6 of getting a “1″):

with two dice: 36 possibilities (11 chances out of 36 of getting at least one “1″):

James and I looked at several patterns on the last chart, such as the odds of rolling doubles.

The kids “got” these tables, but they didn’t see the pattern or understand how this information could be used to work out the odds of rolling at least one “1″ when three dice are rolled. I didn’t want to explain it; I want them to discover it for themselves. So I set it aside; we’ll come back to this at some point in the future. (math: computation (addition) & probability) ]]> http://tribeofautodidacts.homeschooljournal.net/2008/09/05/more-probability/feed/ http://tribeofautodidacts.homeschooljournal.net/2008/09/05/horseback-riding/ http://tribeofautodidacts.homeschooljournal.net/2008/09/05/horseback-riding/#comments Fri, 05 Sep 2008 19:08:23 +0000 tribeofautodidacts http://tribeofautodidacts.homeschooljournal.net/2008/09/05/horseback-riding/ What a great program! Today was Sarah’s first day of lessons. She chose the horse she will ride. They are teaching her to groom and care for “her” horse, Tiny, as well as to ride.

I thought she might be anxious at first, since it’s her first time on a horse (and — needless to say — Tiny is not aptly named). But she looked very confident and had fun.

Of course one of our goals here in remediation for challenges related to her “mild” Asperger’s. Horseback riding is a great experience for developing sensory integration and motor planning, which helps with self-regulation.

More importantly, it’s something she’s wanted to do for years! And it’s a great way to get some exercise and fresh air.

• We reviewed the six kingdoms of life and made a new chart — kind of a “mind map” — which the kids decorated with magazine clippings.

• We talked a lot — again — about how the animal kingdom is divided into phyla.

• We talked about how scientists divide animals into kingdoms (by the type of cells, how it is nourished, etc.)

Lab:
John led us in a biology lab — studying specimens of various animal phyla. We used a biology textbook for reference. Since my kids already have a basic understanding of natural selection and evolution, I wove that into the discussion. We emphasized the way animals are believed to have evolved from protists (which we studied in our last lab).
We started with a sponge — which is basically a mass of unspecialized cells, then moved on to cnidarians (like jellyfish), which have simple specialized parts (basically a tube-like body and tentacles, both adapted for catching, eating and digesting food). Then we moved on to annelids — an earthworm and a leech — which show another evolutionary milestone: the development of a segmented body. We continued through mollusks (a snail), arthropods (such as insects and arachnids), and so forth. We discussed the physical features of each, and the kids studied these features under the stereoscope.

The specimens studied included:

• PORIFERANS — Commercial Sponge (Euspongia)
• COELENTERATES — Common Jellyfish (Aurelia), Portuguese Man-of-War (Physalia) & Sea Anemone (Metridium)
• NEMATODES — Roundworm (Ascaris)
• ANNELIDS — Earthworm (Lumbricus), Leech
• CRUSTACEANS –Crayfish (Cambarus)
• ARACHNIDS –Garden Spider (Argiope)
• INSECTS — Orthopterans, Grasshopper (Romalea), Odonates — Dragonfly Adult, Hemipterans — Giant Water Bug (Lethocerus), Lepidopterans — Butterfly, Dipterans — Horsefly (Tabanus), Hymenopterans, Wasp (Vespa)
• MOLLUSKS –Land Snail (Helix)
• ECHINODERMS — Starfish, Sea Urchin
• CHORDATES — Lancelet (Amphioxus)
• AMPHIBIANS — Frog
• REPTILES — Turtle
• AVES — Chick
• MAMMALS — Mouse, Pig Embryo

I was ambivalent about buying these specimens. I have always been disturbed by the way animals are killed to be used as “specimens.” The sight of the dead mouse in a jar made me cringe, and I didn’t even look at the chick or the pig fetus. (Yeesh! What a brutal species we are. And all in the name of the study of LIFE)

But the kids thought it was kind of cool, and it did make an excellent lab. Sarah is keen to start dissecting stuff. :-O

Anybody want the position of lab instructor around here for a while? Must bring scalpels.

We played “Pig,” a simple probability game that has become a regular for us. This game provides practice with addition up to sums of 24 along with intuitive experience with probability. (math: computation & probability)

1. In turn, each player is handed four dice. He can choose to roll 1, 2, 3 or all 4 dice. He can roll as many times as he likes.
2. Each time he rolls, he computes the sum of all the dice he rolled. That is his score.
3. If he rolls one “1″ at any point, his turn ends, and he loses all the points he’s scored in that round.
4. If he rolls two “1″s — his turn ends, and he loses all the points he’s scored so far in the whole game.
5. The first player to reach 100 points wins.

Since I’m focusing a bit on adding without counting, I modeled logical ways of adding the numbers on the dice. For example, if you roll 5, 6, 5 & 4 … group the 6 & 4 together and the 2 5’s together. Two 10s is 20.

We had a round when Sarah and I both chose to roll only two dice and rolled two “1″s. (VERY bad luck!) After the game, I asked Sarah and James what they thought were the odds of that happening. Since the chances of rolling a “1″ when you roll one die is 1:6, they thought the odds of rolling 2 “1″s when 2 dice are rolled would be 2:12. I pointed out that 2:12 is the same as 1:6. Since James hasn’t studied fractions yet, I drew a diagram to show him why this was so. Then I asked, “So you think the chances of rolling a “1″ with one die are the same as the odds of rolling two “1″s when you roll 2 dice? They confirmed that this was their prediction. Even though we’ve done the tree diagram thing before, it hasn’t sunk in yet.

So I showed them, using a simple tree diagram, why the odds of rolling two dice and getting two “1″s are actually much lower (1:36)

(Poking around online, I just found a link that explains this.

It seems the more I plan and the more money I spend on a unit or project, the worse I flub it up. Our Biology unit is unfolding kind of like a big meal, with pots and pans on all the burners, with nothing being ready at the same time. My big order from Wards science company never arrived, and they haven’t responded to my e-mails. Meanwhile, Carolina Science sent the living specimens for the lesson we were SUPPOSED to be on this week (on protists). So we just had to move forward with what we had. I am feeling lost, confused, and totally inept — but I am determined to keep going and keep learning.

We had our first “class” yesterday. The first thing we did was review and explore the six kingdoms of life a bit. My kids have learned a bit about classification — through nature books, discussions, and activities like this. But they don’t see the “big picture.” So my first step was to sketch out the six kingdoms a large roll of newsprint. Then they helped me add the 21 phyla in the animal kingdom (I used this source), the subphyla of chordates (vertebrates are one), then the five main classes of vertebrates. Then they could see where the orders of mammals, which they learned last year, fit in — on a small corner of the map.

We also included different kinds of protists, and talked a bit about how they are sorted, such as by how they move or how they nourish themselves.

My goal was NOT to encourage them to memorize this stuff (why memorize something that can be easily looked up?? I certainly don’t plan to try to commit it all to memory) My goal was just to create a kind of “mind map” that would offer a mental picture of how scientists think, when it comes to classification, and how all the diverse forms of life fit together. We’re just getting our feet wet now, but I hope that — in time — they will hold this broad picture in their minds.

I’m not sure I am explaining it right. Does this make sense? Does anyone have any thoughts on all this?

While we were sketching out the six kingdoms, we talked a bit about how this was like a “family tree,” and we touched on evolutionary theory. I mentioned that archebacteria seemed to have evolved from eubacteria, when they adapted to live in less than ideal environments, then eukaryotic cells developed (more complex cells with nuclei and other specialized parts), bringing the protists. Organisms became more complex with multicellularity and sexual reproduction. Protists, in turn, are believed to have evolved into animals, plants and fungi. (Sarah and James already understand the basics of natural selection and evolution)

Another thing I highlighted was how complex words, like flagella and pseudopod, can be deciphered by looking at the word roots and other ways they are used (flagellate, pseudonym).

We also talked a bit about what protists are (eukaryotes that aren’t plants, animals, or fungi) and — again — we briefly explored six basic types.

Paramecia from Encyclopedia Britannica online

Euglena from cartage.org

Then our whole family got to look at some protists through a stereoscope. We observed paramecia, euglena, and blepharisma, and looked at diagrams of them in a textbook.

By the way, I am grateful to my favorite biologist turned homeschooling mom and unschooler for helping me select this microscope and steroscope (and you’ve got to check her recent nature posts).

We’re just doing observations at this point, not attempting experiments or involving a whole lot of critical thinking. That will come. John did point out his observation that the critters moved more quickly when under the microscope, and guided Sarah and James to make an inference (that they were more active in the light).

A Few Questions — raised by Sarah — that we discussed (Yes I know I can — and should — look these up myself, but if you have answers, please jump in )

1.  How are these little critters attracted to the light? Instinct?

2. If so, they must have some sort of brain or a primitive neurological system (I didn’t see it noted in the diagrams I looked at, though I did see a primitive sort of eye on some). Is this accurate? If so, what sort of neurological system do they have?)

One last thing I wanted to mention is that I would like — eventually — to talk about the origins of life, scientific theories on it, my and John’s belief in theistic evolution and similar topics. Delving into religious and philosophical ideas about the beginning of life — strictly speaking — is not science, but since when do home educators keep things in tidy boxes? I ordered an interesting book that Willa reviewed but I have not started it yet. It looks like it will offer a thoughtful discussion of these topics.

There are a few more thoughts about this here.

After an unsuccessful effort to get the older kids motivated to help, I settled into working on the gardens with Trishy. Little ones LOVE to participate in whatever mom or dad is doing, and they love to help do stuff. Once they get old enough to really do a good job, they lose interest and getting them involved is like pulling teeth. *LOL*

While planting flowers, Trishy and I watched bees and butterflies visit flowers. We talked about how these insects and flowers “help each other.” The insects pollinate flowers and the flowers give them nectar. We also looked at Trishy’s wilted sunflowers and talked again about how birds and sunflowers “help” each other. Birds eat the seeds, and in the process they strew some which will help create new plants.

We also looked at rocks, bugs, and the exoskeleton of an eviscerated insect tangled in a spider web.

And we discussed other things — like Trishy’s imaginary friends — Friend Horse and numerous baby dogs, who all seem to be named “Sandy.” Except for one who she just christened “Vinnie” in honor of a friend’s dog. *Laugh* She used to have 2 baby dogs named “Ashlyn” and “Tori” — but apparently Tori was hit by a car and killed and Ashlyn ran away. Yikes! There is a lot of cold realism in her fantasies.

This post needs a picture. I’ll add one later. First … off to do some errands.

• (very easy) Flip a coin. If it lands on heads I get a point; if it lands on tails, you get a point. (Fair, each person has a 1:2 — or 50% — chance of scoring)
• (very easy) Spin a spinner equally divided among 4 colors: red, blue, yellow & green. If we spin red, I get a point; if we spin blue, you get a point. (Fair, each person has a 2:4 — or 1:2 — or 50% — chance of scoring)
• (easy) Spin the same 4-color spinner. If we spin a “cool” color, I score a point; if we spin a “hot” color, you get a point. (Fair — there are 2 cool colors (blue & green) & 2 hot colors (red & orange): each person has a 1:2 — or 50% — chance of scoring)
• (easy) Spin the same 4-color spinner. If we spin a primary color, I score a point; if we spin a non-primary color, you get a point. (Not Fair — there are 3 primary colors (red, blue & yellow) & 1 other color (green): 1 have a 3:4 — or 75% — chance of scoring)
• (harder) Spin 2 4-color spinners. If the 2 colors match, Sarah gets a point. If there is a mismatch, James gets a point. They felt intuitively that this wasn’t fair, but they didn’t know why. I showed them how to make a simple tree diagram. There is a 4:16 (or 1:4 or 25%) chance of getting a match and a 75% chance of a mismatch.
• (harder) Roll two dice and multiply the numbers shown. If the product is an even number, James gets a point. If it’s an odd number, Sarah gets a point. Since an even/odd game with ONE die is clearly fair –you’d have a 3:6 chance of rolling a even number (2, 4, or 6) and a 3:6 chance of rolling an odd (1, 3 or 5) — the kids felt intuitively that this was a fair game. After they started playing they quickly changed their minds.

I asked them to fill in a square from a multiplication table showing all possible rolls in this last game, coloring even and odd numbers in different colors. They quickly saw WHY it was so unfair and immediately saw the pattern. The odds are 4:16 … or 1:4 … or 25%.

Then we played around with Tootsie Roll pops. (One day it was ice cream; the next day it was lollipops. There is a pattern here).

• If you put 4 lollipops in a bag — grape, cherry, pomegranate & chocolate — and you pull one out a random what are the chances of getting cherry? (easy — 1:4)
• If you pull out TWO at random, what are the odds of getting a cherry AND a pomegranate? (We used a tree diagram to solve that one. The odds are 2:12 or 1:6)

We bought six painted lady caterpillars. Two are still in cocoons. Three have become butterflies, and have been released. Before we set them free, Sarah helped me research what kind of plants we needed to create an appropriate habitat for them, and we are starting a butterfly garden.

Trishy particularly enjoyed this.

Trishy, James and I read:

Butterfly Story by Anca Hariton

and Trishy and I read:

From Caterpillar to Butterfly (Stage 1 Let’s-Read-and-Find-Out Science) by Deborah Heligman, illustrated by Bari Weissman

And the kids got to see baby bunnies at a good friend’s house.

James has really never done any writing except for occasional captions for his pictures and comics. He is a prolific artist (mostly monsters and dark creatures), a competent reader, an enthusiastic storyteller, and a kid who’s been able to give detailed, articulate narrations since about he was about 6. But he likes scratching out even a few sentences about as much as he would enjoy having sharp nails driven into his feet.

We just started doing “Breakfast School,” which includes — in addition to board games and a few other things — Five-Minute Freewrites. Here is a good resource on freewriting.

Last week, James wrote a few lines. I was pleasantly surprised that he did that without too much anguish. Despite the fact that I’ve assured him that I am not worried about spelling or mechanics, he wrote his sentences out carefully, choosing simple words he thought he could spell (Even so, he misspelled many words. *LOL*)

Here’s what he wrote (I corrected spelling and punctuation):

Aengus and Me
Playing YuGiOh!
We both got good hands. He got a Dark Hole and Witch of the Black Forest and Heavy Storm and Hammer Shot and Armed Dragon N7! And I got these!!

Then he went visual — which is his normal learning mode — and sketched 5 YuGiOh cards.

We read the freewrite together, and I thanked him for his hard work. We didn’t say much else about it.

This week, his freewrite was much different. It was a few sentences long, and the spelling and mechanics were thoroughly atrocious (I corrected them here). But I LOVED it. He risked bringing his imagination into the process and doing a bit of vivid, original descriptive writing:

Helicopters were flying above as the dark howling creature began to screech. People fled from the giant form waving its arms like a three-year-old having a tantrum.

I got excited about these lines, and I certainly didn’t comment on the mechanics or urge him to “finish” it. James read his work to everyone in the family. His sister and dad encouraged him to continue the story, and he seemed interested in doing that. We may do this every morning, encouraging him to write a line or two each day.

I am interested to see what the coming year brings. He is going to be doing a bit more grammar and spelling (at his request), starting with the basics. It’ll be a long process.

I guess if he were in public school (at almost 10), they’d be insisting that he write compositions and even research papers by now. They’d need a truckload of Ritalin to get him to stay still that long. But I am loving learning about his energetic, creative “right brained” learning style. I’m in no rush. (It’s not as if he needs to write essays for his SATs next year.) And I saw a spark of something today, and there’s no way I am going to suffocate it.

*******************************************************************************************

We have started doing brief freewrites together, and I am really liking the direction we’re going with this. Freewrites — and the role they play in writing development — are discussed in depth by Julie at Brave Writer. She has a good article here. (By the way I have invested in both The Writers Jungle and Help For High School from Brave Writer, but I’ve never quite gotten around to reading them. They do look excellent.)

In short — family freewriting accomplishes many of the goals you’d try to achieve in a writing “curriculum,” but is it very compatible with unschooling and eclectic learning. And it seems to “fit” everybody — my 14-year-old daughter, who is an articulate and fairly prolific writer, my 9-year-old son, who is one of the most reluctant writers in the Western World, and even my preschooler.

Last week I wrote about a spark I saw in James — the quintessential reluctant writer. (”Reluctant” is putting it delicately. Even getting down a few words is about as much fun for him as shoveling a ton of cow manure.)

The key here seems to be lots of patience and faith (Which almost requires more wisdom than I have! Believe me.) James is a good reader. He’s listened to plenty of good books. He’s a terrific artist and a prolific storyteller. For years, I have been printing cartooning templates from donnayoung.org and he’s been creating comics. Occasionally he includes captions. All the seeds are there waiting to germinate. It’s all there for this “right brained” learner — except the writing.

One obstacle seems to be that his spelling and grammar are very weak (which at his age, as far as I’m concerned, is O.K. Though he will be doing more formal spelling and grammar this year, at HIS request!) He is a perfectionist. No matter how many times I assure him that he needn’t worry about these things yet — they’ll come in time — he is reluctant to put “incorrect” words and sentences on paper.

Still, I keep reassuring him. I would much rather correct sloppy habits in mechanics and spelling later than have him be afraid to express himself or twaddlefy his ideas to fit a narrow range of skills. Have fun with it. Use your imagination. The rest will come. He seems to finally be going along with this a bit.

It’s kind of liberating, really, to just write for a set period of time (for us, 5-10 minutes) without any purpose or product in mind, and without needing it to be “good.”

Today James continued the story he started last time. I corrected errors so it’s easier to read.

James’s Freewrite:

Helicopters were flying above as the dark howling creature began to stretch. People fled from the giant form waving its arms like a three-year-old having a tantrum. Its scales glistened in the light of fire and moonlight. Its snout was as big as a helicopter itself. Its glowing red eyes were fixed on jet scooters and police below. It had teeth the size of golf clubs. Its huge needle-like claws were as big as cars. It stormed at anything that moved.

I love it!

He decided to draw a picture too. Notice that the monster is anatomically correct. *LOL*

Sarah has been writing her freewrites in a special notebook, and she has opted to keep them private.

Mom’s Freewrite:

I like this time of year — late summer — when sultry afternoons melt into gloriously cool evenings, nights, and mornings. Crickets softly chirp throughout the day.

I liked working in the butterfly garden with Trishy. With small shovels, we slowly chipped away at the hard, dry, rocky earth. We slowly collected the rocks we unearthed, noticing their colors. We saw the exoskeleton of an eviscerated insect in a spider web and wondered where the spider had gone. We noticed crawling and flying bugs, a tiny bee landing on a flower and a speckled white butterfly fluttering by. We planted phlox, bee balm, and yarrow. I told Trishy about the rhythms of nature, like how bees pollinate flowers as they gather nectar, and she told me about her imaginary friends — Friend Horse and her baby dogs — who always seem to be christened “Sandy.”

I gave Trishy a pencil and paper, just to keep her occupied while we wrote. She drew several pictures, chattering non-stop about them as she went. Then I realized, when it was time to share our freewrites (if we chose to), she would want to be included. And it would be interesting to record some of the words that went with the quickly scribbled images. So I asked her to tell me about each picture and transcribed her words.

Trishy’s Freewrites:

It’s a dog, and it has sharp teeth, and it growed new fur. It didn’t grow a new tail. It didn’t grow a new ear. It is under the ground, not in a cave. It’s under a cave. Somewhere it’s looking for monsters to eat.

It’s a guy and it has a dress on. He’s in a house. There’s stairs, too.

I am interested to see how her ideas and skill with words develop throughout the year. And I am excited to see what direction all this is going to take us.

In high school, we used square and cubic numbers. It was something I understood two-dimensionally (literally) as problems on the page of a textbook. Much later, as a 30-something-year-old home schooling mom, I was playing around with some of our 1-centimeter cubes (you know — math manipulatives — those things we home schoolers have all over our carpets, between the cushions, and under couches. *LOL*) And I finally got WHY they’re called square and cubic numbers. I will concede that my “AHA” moment would be a “duh” moment for most. Everybody else probably already knew this. But when I saw that square numbers, constructed with cubes, actually make squares and cubic numbers make cubes I thought it was VERY cool. Later I learned about triangular and pyramidal numbers. Cool stuff.

I will never be a math expert, but — through the things I’ve learned as a home schooler — I can see glimmers of what makes it profound and beautiful. And I’m having fun with it. Once in a while, my kids even have a little fun too.

This week, we are continuing our simple unit on probability and statistics. Today we did the “31 Flavors” problem.

I got this problem out of an old edition of About Teaching Mathematics by Marilyn Burns, and I played with it a bit.

If an ice cream parlor only served one flavor of ice cream, how many kinds of double dip ice cream cones could you make? One (2 scoops of vanilla) What if there are 2 flavors? Three (2 scoops of vanilla, 2 scoops of chocolate, and a scoop of chocolate with a scoop of vanilla). Following this pattern, how many possible combinations would there be with 31 flavors?

When I first read the problem, I thought it was confusing. Then I realized it was just a series of triangular numbers.

To help Sarah and James solve the problem, I set up a table on the white board, and I showed them how to use centimeter cubes (white cubes for scoops of vanilla, brown cubes for scoops of chocolate, and so forth — cute, huh? :-P) For instance, 2 white cubes were stacked up to make a double dip vanilla cone.

They worked on the problem for a while with a little help from me.

It goes like this –

1 flavor — 1 possible combination

1. 2 scoops of vanilla

2 flavors — 3 combinations

1. 2 scoops of vanilla
2. 2 scoops of chocolate
3. 1 scoop of vanilla & 1 scoop of chocolate

3 flavors — 6 combinations

1. 2 scoops of vanilla
2. 2 scoops of chocolate
3. 2 scoops of strawberry
4. 1 scoop of vanilla & 1 scoop of chocolate
5. 1 scoop of vanilla & 1 scoop of strawberry
6. 1 scoop of strawberry & 1 scoop of chocolate

4 flavors — 10 combinations

1. 2 scoops of vanilla
2. 2 scoops of chocolate
3. 2 scoops of strawberry
4. 2 scoops of mint chocolate chip
5. 1 scoop of vanilla & 1 scoop of chocolate
6. 1 scoop of vanilla & 1 scoop of strawberry
7. 1 scoop of vanilla & 1 scoop of mint chocolate chip
8. 1 scoop of chocolate & 1 scoop of strawberry
9. 1 scoop of chocolate & 1 scoop of mint chocolate chip
10. 1 scoop of strawberry & 1 scoop of mint chocolate chip

It’s easiest for me if I do it systematically. For example:

• 4 flavors means 4 possible cones with 2 of the same flavor.
• The 1st flavor, vanilla, can be paired with 3 other flavors.
• The 2nd flavor, chocolate, can be paired with 2 other flavors (besides vanilla, which has already been covered).
• The 3rd flavor, strawberry, can be paired with 1 other flavor ( besides vanilla & chocolate, which have already been covered)
• We’ve already exhausted the possibilities with mint chocolate chip

Then the kids figured out the pattern:

1 (+2) … 3 (+3) … 6 (+4) … 10 (+5) … 15 and so forth

When I asked them to work out how many possible combinations there were with 31 flavors, they got stuck. I tried to “scaffold” the task by helping them write a sequence of numbers across the board:

Hmmm … 496 different kinds of double dip ice cream cones to choose from? I guess we’d better get busy.

Sarah, James, Trishy and I went to Baskin Robbins to get … what else? Double dip ice cream cones. A scoop of mint chocolate chip ice cream is a lot more interesting than a green centimeter cube.

By the way, I just found a variation of this problem online.

On the way, I asked how many different kinds of double dip ice cream cones could we choose from — with 31 flavors — without getting 2 scoops of the same kind. (496 - 31 = 465, since there are 31 options if you get 2 of the same kind).

Sarah gave me the Quotable Kid Moment of the Day — “Did you just get us in the car so you could ask us another math question. ‘Cause you’re torturing me.”

Yup … these kids are really suffering. I got to spend part of my “math period” in the ice cream shop when I was in school. Didn’t you?