Cheesecake crisis

Jon turned 50 recently.

Okay, fine. He turned 50_8. [That should read 50 subscript 8.] In base 10, he turned 40.

Now, some men might panic at this “over the hill” marker. Some men might rush out and purchase a red sports car. Some men might even trade in a current spouse for a trophy wife.

Providentially, Jon is too practical to buy a Ferrari--and he already has a trophy wife. ;)

Further, as life expectancy lengthens, we can be grateful that we all have an extra decade or so. He’s barely hit middle age! (Wait, if 50 really is the new 40, does that mean my assertion in the first line was right?)

Nonetheless, it occurred to me that my sweetie might be worried about going senile. I wanted to cheer him up, just in case he was concerned.

I decided to give him a mental mountain to surmount. What could be a better birthday present than the euphoric endorphins that come from meeting and mastering a challenge?

My natural inclination was to write rhyming riddles, but part of marriage is putting the other person first. While Jon could no doubt solve my Tolkienesque wordplay, wouldn’t he much rather do something involving math and encryption?

I wrote—okay, fine, I plagiarized—a bunch of hard math problems, instead. Thanks to Tim Sanders, the original author, for letting me hack his questions. It would have taken me ages to create them from scratch.

Oh, and I also sacrificed my natural inclinations by ministering to Jon’s domestic needs. (In other words, I actually planned, prepped, and cooked a real meal.)

For homeschool that day, I took the kids grocery shopping. Each boy picked out one chocolate bar for daddy and helped purchase ingredients to make a special dinner. (I also threw in a mini lecture about choosing grades of hamburger based upon the fat/protein ratio.)

After we got home, they made a special cheesecake for their dad. Daniel made the crust, Eric made the filling, and I scorched it in the oven. It was a group effort.

Then I assigned Eric and Daniel to design a treasure hunt for Jon. The hidden treasure was the chocolate bars.

Meanwhile, I worked on hacking math problems.

Jon normally gets home around 7 p.m., so it was a nasty shock when he called around 4 and said he was leaving work early.

*Ahem* Sorry. I mean, it was a delightful surprise that he was coming home early--but aaaaaaaack! Only halfway through the math, and I hadn’t even started the second cheesecake!

Jon’s brother, who was in town visiting friends, came over. Poor Ryan showed up and found the house in chaos, dinner unstarted, and me scribbling frantically, muttering mysterious “mod” math to myself. He very bravely took over my messy kitchen and made the pasta. Thanks, Ryan! You’re a lifesaver.
 
When Jon got home from work that day, the kids ambushed him with a treasure hunt. Solutions are at the very bottom of this post.

First clue: Your electronic display
Second clue: Zuchst in dein Zimmer nah die Tur.
Third clue: guy ebbz bs rzretrapl fgbentr
Fourth clue: fvyirejner
Fifth clue: “A chair that changes pitch conceals a thing you wish.”
Sixth clue: A map.

While Jon worked on those clues, I finished up writing mine and then provided reinforcements for Ryan.

After dinner (rigatoni noodles with kielbasa sausage and extra cheese),  I handed Jon his math clues with an apology. “The treasure they point to isn’t ready yet,” I confessed. “But you can solve them anyway and collect tomorrow.”



Here they are:

[Standard moaning about blogger's idiotic formatting. It looked cooler as printed in Word.]
1. Four friends are sitting around one afternoon thinking up large numbers.  If Olive chooses 100^200^300, Lily chooses 300^200^100, Heather chooses 200^300^100, and Peony chooses 200^100^300, which person has chosen the largest number?

2. The 26 letters of the alphabet are arranged in alphabetical order clockwise around a wheel with a counter.  On each trial, the counter starts at 0 on A and is moved clockwise around the wheel.

On Trial 1, the counter moves 2222222 letters.
On Trial 2, it moves 33333333 letters.

On Trial 3, it moves 4444444 letters.

On Trial 4, it moves 23456 letters.

On Trial 5, it moves 5678912 letters.

On Trial 6, it takes a break and moves only 212 letters.

On Trial 7, it moves 78912 letters.

On Trial 8, it moves 202020 letters.

On Trial 9, it moves 404440 letters.

Finally, on Trial 10, the poor thing moves 108 letters and then collapses in exhaustion.

In order, what are the letters the counter ends on?

3. Assume a base-36 numbering system where the digits 0 through 9 represent 0 through 9, as usual, and the letters A through Z represent 10 through 35, respectively; what is the base-36 value of the base-10 number 625700?

**Spoiler Alert. The answers are given four paragraphs below. Quit reading now if you want to solve them yourself.**

Poor Ryan. I should have waited, because Jon spent the next 30 minutes obsessively working on math problems instead of capitalizing on his limited minutes with his brother. That was an error on my part.

As Jon finished his first clue, I said, “Doh! I should have changed the question to ask ‘Whose number is the largest’ so that the answer had an s on the end.”

Jon shrugged that off. How forgiving he is of these glitches! I appreciated his attitude: why obsess about minor issues when there is more math to be done?

Once he emerged from his hyper focus, I confirmed his decryption. (“Olive(‘s), cheesecake, desk.”)

I had intended to hide olives and cheesecake, Jon’s two favorite foods (the rigatoni is a distant third) on my desk, behind my computer. (Actually, I had briefly considered hiding them in the dryer, but didn’t want to deal with the extra computation. And while “van” was a non-obvious spot and only had three letters, it was also disgusting. Not quite as disgusting as olives, but close.)

After dinner, Jon spent some time talking to Ryan and I put the kids in bed.

The next day, I made the second cheesecake--with a hitch.

I had lied to my husband. Lied. To his face. (I figure there should be a grace period on these things. As long as I ‘fess up within 24 hours, we’re fine, right?)

You see, the “mistake” on problem one wasn’t actually a mistake. I intended the clue to read “olive cheesecake, desk.” I didn’t want to admit that, though, so I tried to confuse the issue, such that Jon would think I had planned to serve him olives and cheesecake separately.

Instead, I served them together.

I was trying to make a point about the nastiness of olives. “If he loves cheesecake and olives so much,” I reasoned, with false sophistry, “Then he should love them both even more if they are put together. Synergy!”

Behold, pictures of my efforts:

When Jon got home, I presented the treat to him.

I hoped that Jon would take a bite and gag in revulsion.

Instead, he spotted the olives in advance. (Ah well. At least this proves dementia hasn't yet set in.)

"Did you think I wouldn't notice?"

And then he ate them anyway!

Olives. And cheesecake. And strawberries! Together.

EWWWWWWWW!!!!!


(No, beloved, you’re not senile. Instead, you’re a sick, sick man.)

I had spent the last 12.43 years believing that I was “the crazy one” in our marriage, but now Jon--calm, steady, sensible Jon—had just exhibited the most behemoth-sized case of "the bonkers” I’d ever observed. He was a batty basket case. And I had never noticed this before???

I fear the shock of this seismic paradigm shift will send ME into a mid-life crisis.

Right. It’s not all about me. Sorry. Focus on Jon, instead.

Dearest, I want to assure you that I still love and respect you. I will devote my energies to helping you find the highest quality of medical care available. I will gladly sacrifice all the money you earn toward your cure.

Or we could compromise: I promise never to make olive cheesecake again, and you promise never to have a midlife crisis. A cheesecake crisis is bad enough. Or, even better: You promise never to eat olive cheesecake again, and I promise never to turn 40.

Deal?

Solutions to the treasure hunt:

1: Jon’s computer monitor. (“Do you know how many electronic displays we have in this house?” asked Jon, rhetorically, but still got it on the first try.)

2: Search in your room near the door. (It was written in German. Eric spent quite some time with a German/English dictionary. I gave him some grammatical advice, although I fear the cases still aren't quite right.)

3. Storage room. (Jon thought it was the garage, based upon how often I throw things in there when company approacheth.)
4. Silverware
5. Rocking chair
6. The map, once interpreted, placed the treasure (chocolate bars) behind some encyclopedias in the library.

2013 Retrospective

My new year's goals for 2013 were to write more and exercise more. I succeeded at the former and failed at the latter. (I didn't exercise more because I was writing so much.)

I'm going to focus on the positive, though. Let's look back at the year's accomplishments:

* Daniel's DI team went to state.
* Eric did well at a state math competition.
* Sam transitioned from "talking like a toddler" to "talking in recognizable, complete sentences."
* Sam is now potty trained. One more to go. I can do this. I mean, Jeff can do this.
* Jeff actually decided to start talking. He is now babbling meaningful phrases, like "Oh, thanks, mom."
* Jon continued to be wonderful and awesome. He says he can't think of a specific accomplishment, but then, he has so little room for improvement. It's so much easier to move from the 48th to the 49th percentile than to move from the 98th to the 99th. In a world where too many people abrogate responsibility, a calm, steady, smart, honest, hardworking, unselfish, quietly funny dad who unobtrusively always does the right thing should be cherished. Thanks, sweetheart.
* Eric and Daniel delivered "The Summer of Harry Potter," a fantastic two months in which they read and re-read the books, analyzed characters, predicted plot points, acted out wand duels, sent "owl post" messages to each other, and taught their little brothers Latin words. It was a glorious moment when the boys both finished book 5 and were then allowed to watch the movie for the first time. In the opening scene, as the Dementors swooped in to attack, Sam, age 3, pointed a chop stick "wand" at the TV and shouted "Expecto Patronum!"--before Daniel Radcliffe said it. What reflexes!

It has been a good year.

For myself, here is a list of what I'm most proud of:

* I stopped whining about "nobody ever giving me a chance to teach" and just decided to make my own opportunities. I started a writing camp for kids over the summer. It was successful.
* Afterward, a chance to lead an elementary math group fell into my lap, and I seized it. So much fun! Isn't it interesting how decisions drive destiny?
* I stopped thinking of myself as a "wanna-be/prospective/someday/future" writer, and started thinking of myself as a "real" writer, now. This happened at the same time I pitched my services to mathleague.org and said "You should hire me to do some technical writing." Something about getting paid to write--even if it is just "soulless procedures"--made it all seem official.
* It wasn't just the paycheck, though. I effected an internal shift. "Adulthood isn't an award they'll give you for being a good child," wrote Lois McMaster Bujold. "You can waste...years, trying to get someone to give that respect to you, as though it were a sort of promotion or raise in pay. If only you do enough, if only you are good enough. No. You have to just... take it. Give it to yourself, I suppose." I decided to start thinking of myself as a writer. Period. Even if I never get published formally, writing is part of my identity. I can't not write.
* I have forced myself to be more outgoing. Invite people over more. This has met with generally positive results.


I feel like everyone in the family has grown in positive ways. I know I feel like a "bigger" person than I was last year.

Let's see what growth 2014 brings.

Although...actually, what I'd like for 2014 is to see some shrinkage--of my waistline. [Wink]

Pickets and Picnics and Pigs, Oh My! (Also Bears, My Bane.)

I enjoyed playing legos with my boys today, but people are accusing me of sneaking in a homeschool lesson on a Sunday. You be the judge.

Daniel assigned me to set up a "store" and stock it with merchandise in legoland.

Bear added that he refused to be sold as meat. I suggested that I could sell him as a bear skin rug, instead. He protested. What about bear claws? He still objected. Then I suggested selling bear fat. He began to protest, but I pulled him aside and whispered a business deal: I would help him lose weight, and he would submit to liposuction. He countered that he got to keep 33% of all my proceeds, not my profits. I sighed that he struck a hard bargain, but we shook on it.

Our jars of bear fat were pretty successful. I began marketing them as a cooking ingredient, or a way to waterproof one's boots, or a healing salve for chapped winter skin, or even a method for slicking back one's hair, if desired. I emphasized that this was an animal-friendly, fair trade process, and that no animals were being exploited or harmed.

Soon, however, Bear began demanding a bigger take of the proceeds. I declined, and began hunting for Checkov (Eric's Russian bear) as an alternate supplier. Bear thwarted this idea by organizing a union, though they did sign at the 33% figure.

When the contract was expiring, Bear attempted to use his collective bargaining power to push greedily for a 40% take. I refused.

Next thing I knew, there was a line bears picketing outside my factory. I brought in some pigs as scabs and started mixing bear and pig fat together, changing the labels from "Mommy's 100% Bear Fat" to "Mommy's Bear Fat: 100% Organic!"

The bears shouted some insulting things (and oinks) at the scabs. The pigs retorted "We may be working for only 25%, but at least this way she'll have an incentive to keep us alive; otherwise we'd be murdered this winter.") Bear claimed this demonstrated my unfair leverage over them and made the exploitation worse.

Then Ursa Major crossed the picket line. ("But I have a baby to support!") Bear was furious.

Bear launched a PR campaign to uncover my duplicitous labeling. It turned out my new jars were only about 10% bear fat. (But my labels weren't LYING, darn it!) Still, sales fell off.

After some unhappy financial consequences for both sides, we sat down at the negotiating table again. I agreed to 35%, plus free food to help the bears fatten up again. (They had already been low on fat storage, and the long strike hadn't helped.)

Things have now calmed down considerably, although Ursa Major is still not  being invited to the union picnics. ("No picket, no picnic!")


Later, Eric finished his emporium and announced it was "a bricks and mortar store." I couldn't help laughing. They asked for clarification, and I gave a brief--brief!--explanation of the dot-com bubble in the late 90s.


The whole thing was unplanned and unscripted. Really, it felt more like an impromptu bedtime story than a sneaky homeschool lesson. After surviving the dispute with the bears, the last thing I need now is a dispute with my own children.

So, what do my loyal readers think? I'll admit things got out of hand, but Bear totally started it.

I'm still not sure what happened to the pigs...

In the Manner of the Adverb--Hilariously, Compassionately, Contentedly

Today after homeschool and chores were done, we played "In the manner of the adverb." My thanks to Renae Kingsley for teaching me this game!

I could explain the rules, but you'll pick them up better in context. Let me just describe a few rounds and you'll get it.

I left the room while Eric and Daniel conferred. When they were ready, I came back in. "Eric," I said. "I want you to tap dance in the manner of the adverb."

It was comical to see him trying to tap dance. It was hard, though, to tell how much of the wild gesticulation was due to his mysterious adverb, and how much was due to his inexperience.

Next I asked Daniel to pretend to cook dinner in the manner of the adverb. He walked into the kitchen, picked up various ingredients lying around, and mimed dumping them into a pot. He made a point of mentioning that the pot was dirty, then "threw in" stovetop stuffing, cheesecake mix, various spices, empty jars formerly containing alfredo sauce, pieces of trash...

Finally, I asked both boys to pretend to dig holes in the manner of the adverb. Eric's "hole" was more of a "trench" that careened wildly all over the living room. Daniel comically mimed tossing dirt high into the air, and then flailing about with his shovel when the dirt landed in his face. (I'm pretty sure he "hit" Eric with his imaginary shovel a few times, but Eric didn't react appropriately. Tsk.)

After those three prompts, I had to guess the adverb. I tried "crazily," "randomly," and "wildly" to no avail. If I'd gotten a fourth try, I would have tried "haphazardly." Nope. They stumped me. It turned out their adverb was "untraditionally."

I laughed.

Next Daniel left the room while Eric and I conferred. When Daniel re-entered, he asked me to "build a redstone circuit computer in minecraft in the manner of the adverb." I stood sharply, saluted, and marched over to an imaginary computer screen. "Hmm," I boomed. "Yes, I can see the benefits to having a computer internal to minecraft. We could use it to calculate trajectories. This will require a lot of contract work. It will cost a million dollars and it will be three years late, but the end result will work, mostly. Remember it's vital to national security!"

(I was proud of that improvisation, by the way.)

Next, Daniel turned to Eric. "Eric, tell me what the adverb is, in the manner of the adverb." Eric froze. "You're not allowed to ask me that--" he began, but I pulled him aside. After a whispered consultation in the library in which I coached him, he returned to Daniel, saluted, and said, "Sir. That was an illegal order, sir!" (I wanted him to say "According to the UCMJ", but I figured he wouldn't remember that, alas.)

Finally, Daniel asked both of us to spar in the manner of the adverb. Eric and I began pretending to trade punches, and I started yelling. "You call that a block? You think the enemy will be impressed with your puny efforts, private? Drop and give me twenty push-ups, you miserable excuse for a recruit!"

Daniel said, "I know what you're doing, but I don't know the word for it. It's like you're in the army."

I gave it to him. "The word," I beamed at him, "was 'militaristically.'"

It was fun. Sam ran around, imitating everyone else and getting in the way. Jeff was asleep in the office.

I enjoyed playing with my boys.


Yesterday was hard. It was concurrently Marian's sixth birthday and the yearly kid's program at church.

As I looked at all the cute little girls singing "I am a Child of God" and "Families Can Be Together Forever," I thought "She ought to be up there." I reminded myself that every person has problems in life. Everyone has some heartache, often privately held. He might be facing divorce, addiction, guilt, cancer, infertility, debt, abuse, mental illness. She might be facing a floundering business, a dying parent, an unhappy marriage, a spiritual crisis. If Christ's infinite suffering gave him perfect understanding, surely my lesser griefs can give me expanded empathy for others.

One more reason to be kind to everyone; you never know who is suffering secretly.


Today was better. "You do not betray your sorrow to set it aside for an hour....Remember to take joy." [*1] I have four healthy, brilliant, creative, hilarious boys. Today I played with them. I enjoyed the moment, and then I recorded it for posterity. I chose to be thankful for my blessings, and happy in my circumstances.

In my head, I'm replaying the footage of Eric trying to tap dance "untraditionally." (It looked vaguely like Dick Van Dyke doing "Step in Time" in Mary Poppins.)

Priceless.


[*1 From The Sharing Knife by Lois McMaster Bujold.]

Sinister, Subversive Silliness

"Oh, frabjous day! Callooh! Callay!" She chortled in her joy.

In Daniel's defense, I volunteered him for this primary talk. When word went out that senior primary needed a substitute, I turned to him, asked if he would like to do it, and volunteered to assist with the writing thereof.

In my defense, Daniel agreed promptly. Like me, he generally enjoys giving talks.

In Daniel's defense, he helped outline the talk in great detail. His ideas and examples were excellent and well-organized.

In my defense, he let me draft the paragraph-form version.

In his defense, he didn't have much time to read the thing. I typed it up during second hour of church, printed it from the library computer, and handed it to him right before third hour started.

In my defense, he still had several minutes. And, really, you'd think he'd be more suspicious of his dear, sweet mom by now. I mean, if I handed you a piece of paper and said, "Here, read this out loud in front of fifty people," wouldn't YOU glance through it first for subversive content? Of COURSE you would!!! But then, you are more experienced than a nine-year-old. Poor naif, trusting, innocent, boy...

Here is a picture of the talk:




I added the highlighter in after the fact. It reads, "I want to serve God with my might, so I have decided to do all my chores at home cheerfully, and even volunteer for extras." (Ain't no way I was going to call attention to that language in advance.)

I was curious to see if Daniel would read the talk through before giving it. And if he would notice that wording. And if he would edit out that sentence.

He didn't. He delivered it verbatim.

I have witnesses.

Heck, I have VIDEO. It will join the the burgeoning ranks of mama's blackmail folder.

BWAHAHAHAHA...!

It is a glorious day.

Tomorrow, when I remind Daniel that he is going to help me clean the entire house in advance of his grandparents' arrival, that will be--(hopefully, oh, please, let it be so)--that will be an even better day.

My house is very messy. It's a combination of still working the kinks out of a new chore system, saving money by dismissing the cleaning help I've had the last two years, and starting a new math  group. Plus doing some more freelance stuff.

"If seven maids with seven mops
swept it for half a year,
Do you suppose," the Walrus said,
"That they could get it clear?"
"I doubt it," sighed the mama, stressed,
and shed a bitter tear...

But here's the good news: I weep no more. You can't see my cheshire-cat-sized grin right now, but even if Daniel balks tomorrow, I have this moment. The glories of my victory will carry me through the next several hours of Sabbath-violating cleaning.

It is, indeed, the most frabjous of days.


--I reference, above, "Jabberwocky" and "The Walrus and the Carpenter", both by Lewis Carroll, whose skill with silly poetry rivals that of Dr. Seuss.

Number Line Protocols

This year I'm helping to start an elementary math group for homeschooled kids, similar to the groups Eric does but on the fourth- or fifth-grade level.

On Tuesday, I tried an experimental "number line" activity. I wasn't sure how well it would work, but the theory seemed sound.

First, I took four different clotheslines, each 50' long. Then I took clothespins, wrote on them in permanent marker, and made four different number lines. One was simple, going from 0 to 50, with each clothespin spaced a foot apart. Another two were harder, with knots or clothespins only 6 inches apart. They covered -50 to +50.

The fourth and final number line dealt with fractions and decimals. Not very many kids wanted to attempt it, but the few who did seemed to "get" the idea pretty quickly that adding .4 + .7 wasn't any harder than adding 4 + 7. There were just dots involved.

The decimal number line. It ran from 0 through 10. Spread out you see 4.4 through 5.0; halfway through I ran out of clothespins and had to get more before finishing. I face constant tangling and can't wait to upgrade to something better, like large beads. Empty ribbon or thread spools might be even better. Feel free to donate!

(You can argue that I was oversimplifying, and that's a fair criticism. My reply lies in Lockhart's Lament, which argues that formal arithmetic, taught only on paper, kills a kid's intuition for how numbers actually work. The essayist argues that this approach is like teaching a kid "music" only by giving him mind-numbing, abstract theory exercises ("transpose this up three steps") without encouraging him to hum the tune, pick it out by ear on the piano, or compose a variation.)

Several parent volunteers and I laid out the lines in the grass at Girlstart, our fabulous host venue. Here's a picture of some kids attempting a few problems:

Why are people walking, not hopping? Ah well, give it time.

There are many things I would like to tweak now that the debut is done. I want to make a new number line where the numbers are spaced further apart and the markers are larger. We were all, kids and parents, inexperienced, but I expect we'll all improve with a little practice. Hopefully by March, children will be bounding up and down like professional bunnies, blithely solving "8 times negative three [flip!] equals...[hop, hop, hop]...negative 24!"

It's a work in progress. I welcome feedback and suggestions. In particular, I wish there were some way to handle multiplying and dividing by negative numbers better. It seems like 8 x (-3) should involve hopping backwards, somehow...

I also have this idea for a color-coded number line, with dots of paint or pipe cleaners representing different ideas. Example: 20, normal; 21, red (multiple of 3), orange (multiple of 7), and blue (Fibonacci number); 22, normal; 23, yellow (prime); 24, red (multiple of 3); 25, green (square number)..."

I was very proud of the kids who were introduced to negative numbers and didn't flinch. I believe in exposing them to big ideas, even if they aren't completely grasped.

This girl did great with it, though, and picked up on the idea immediately:

For some reason, the girl sandals at -12 made me suddenly want to do a number line decorated with big, cheerful foam flowers. And not just because I was sick of the cheap clothespins snagging and tangling all the time.

Some of the parents asked for a copy of my number line rules so they could try something similar at home.

I'm happy to share! Anyone may borrow this for non-commercial, educational purposes. Once again--I get tired of saying this--forgive blogger's idiotic formatting issues. Thanks to my brother, Ronald, for his help working out some kinks, and to Eric and Daniel for helping to alpha-test.

---

Number Line Protocols

by Gail Berry

“Facing positive” means that you are facing the direction in which numbers get bigger. Start at 0. If you see 1, 2, 3, 4… stretching out ahead of you, then you are facing positive.

“Facing negative” means that you are facing the opposite direction. Start at 0. If you see -1, -2, -3, -4… stretching out ahead of you, then you are facing negative.

“Flipping” means changing your orientation. If you are facing negative and then “flip,” you are now facing positive.

Note: Always start by facing positive. “Flip” from positive to negative only as indicated below.

ADDITION/SUBTRACTION:

·         Stand [first number], facing positive.

·         Face [operation].This means you face “forward/positive” for addition, “backward/negative” for subtraction.

·         Move [second value].

Examples:

·         3 + 5, you start at three, face the positive direction (for addition), and move forward 5, ending at 8.

·         For 3 - 5, you start at 3, face the negative direction (for subtraction), and walk 5, ending at -2.

·         For 3 + (-5), you start at 3, face the positive direction (for addition), and walk BACKWARD 5 (for the negative second value), ending at -2.

Note: 3-5 and 3 + (-5) yield the same result. The only difference is the direction the kid is facing. Feel free to invite the kid to think about why this is or to suggest a better protocol.

·         For -3 - (-5), you start at -3, face negative (for subtraction), and then move BACKWARD 5, ending at +2.

       

                                                     
MULTIPLICATION:

·         Start at zero, facing positive.

·         Listen to the entire problem.

·         “Flip” once for each negative sign you hear.

·         Jump [first number], [second number] times.

 Examples:

·          For 2 x 4, you start at zero, face positive, and then hop forward by 2s, 4 times. End at 8.

·          For 2 x (-4), you start at zero facing positive, "flip" so once you're facing negative, and then hop by 2s, 4 times, ending at -8.

·          For (-2) x (-4), you start at zero facing positive, "flip" TWICE so you end up facing positive, and then hop forward in increments of 2, 4 times. End at positive 8.

DIVISION:

·         Start at zero, facing positive.

·         Listen to the entire problem.

·         "Flip" once for each negative sign.

·         Move in increments of the second number.

·         Count your increments.

Examples:

·         For 15 / 3, you start at zero, face positive, move forward by 3s, and count how many iterations you move. (“Zero to three, that’s the first one; three to six, that’s the second; six to nine, I’ve taken 3 steps, nine to twelve, that’s four, twelve to fifteen, I took five steps.”) It’s fine to use fingers to keep track of which step/iteration you’re on. When you reach 15, you have moved 5 iterations, so you say "5." 15 / 3 = 5.

·         For 20 / 3, you start at zero, face positive, move forward by 3s, and count iterations. You get 6 iterations, arriving at 18, with a remainder of 2. So 20 / 3 = 6, remainder 2.

·         For (-15)/3, you start at zero, “flip” negative, move forward by 3s, and reach -15 after 5 iterations.

·         Note: at the moment, I am avoiding negative division problems with remainders. If you want to try them, I suggest using decimals. Or having an argument about why it’s “negative six, remainder two,” not “negative six, remainder negative two.”

VARIATIONS:

·         Crawl, slither, dance like ballerinas, gallop, “galumph,” saunter, stroll, or imitate any animal of your choice. For addition or subtraction which involve moving backward, you could try a crabwalk. For multiplication and division, try “hop like a frog” (or kangaroo, or bunny, or astronaut on the moon in 1/6^th gravity) or "hop on one foot."

·         Let the kids assign problems to adults, or to each other.

·         Ask what happens when you run out of number line? What if the problem is 7 x 8 but the markers truncate at 50?

·         Try "skip counting" by a number, starting at somewhere that is not a multiple of that number. For instance, start at 2 and skip-count by 3s. (2, 5, 8, 11...). Or start at 1 and skip-count by 3s. (1, 4, 7, 10...) See if you can find any patterns.

Make your own number line. What if it only counts by 7s or 12s? How would you use your special number line to solve a problem like 15 + 9?


QUOTE: “Negative numbers aren’t scary. They’re just in the opposite direction.”