Wednesday, September 25, 2013

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.

·         Stand [first number], facing positive.
·         Face [operation].This means you face “forward/positive” for addition, “backward/negative” for subtraction.
·         Move [second value].


·         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.
·         Start at zero, facing positive.
·         Listen to the entire problem.
·         “Flip” once for each negative sign you hear.
·         Jump [first number], [second number] times.


·          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.

·         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.


·         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.”


·         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 1/6th 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.”

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