So anyway I thought I'd throw this out there and see if anyone's got anything better. My irrational and complex numbers unit is pretty anemic. The way we have it calendared, we also kind of have to race through it. This intro lesson is my effort to give them something to grab onto. I'm open to suggestions.

The first three are new. The last two are what I used last year (I tried it as a puzzle - see if you can figure out the patterns and determine the missing values - some of them really liked it, and some of them sat and freaked out for 20 minutes because they couldn't find the cube root function in their calculator.) I think I'm going to give them all of it this year, and instruction will be a combo of encouraging them to think/play/struggle and good old d.i.

*I said Greek first. Oops.

Thanks for the Justin link. In 23 years in the classroom, I've never seen his geometric explanation of why sqrt(12) = 2sqrt(3). Very cool. Also Thanks for the handout. How'd you get to be so smart after only teaching a few years :-)

ReplyDeleteI'm not so smart, I just have lots of smart friends from the Internet.

ReplyDeleteHi Kate,

ReplyDelete1) Thanks for the shout out!

2) Radix is Latin, not Greek.

3) radix=radish!

4) And most importantly: Those are some sweet problem sets you've put together there. I especially like the way you've written them on grid paper--it totally beefs up the context factor. I'm really digging the last problems on the second page with the pairs of squares abutted. What a great way to make "like terms" more clear. (Totally stealing that.) A possible addition (thought it might be too far afield) would be asking students to build squares of different areas with their vertices on grid points. Some of these are easy (16), some of them are tricky and on the slant (13--with slope 2/3), and some of them are impossible (11). Finally, I'm super curious to hear what your students do with the root 3 by root 6 rectangle.

Rock on.

Here's another factoid I like... the top bar of the radical symbol is called a vinculum - the original grouping symbol that predates parentheses. It's the same bar that is used over a group of repeating digits in a decimal, and maybe it's even related to the fraction bar that creates two separate groups. The original symbol for root was a capital R, and it eventually became the checkmark thing (which looked like a lower case r for radix), followed by the vinculum.

ReplyDeleteIn terms of ideas, one thing I like is helping students understand how an irrational number is still considered a real number. Most students come out of algebra 1 with the misconception that "2 doesn't have a square root". I do similar things with the sides of squares and cubes as you posted. But what is also nice is to draw a right triangle with 1-unit legs on a number line, with the height at the 1 unit mark and the corner at the origin. Use pythagorean theorem to find that the hypotenuse is root 2 (of course this only works if they believe the pythagorean theorem!). Draw a circle centered at the origin, with radius equal to the hypotenuse. This shows exactly where root 2 would fall on a number line. I ask the kids to imagine cutting a piece of wood in the shape of the triangle. Is this possible? If so, they can cut a piece of wood exactly root 2 units, even though it can't be expressed as a terminating decimal.

ReplyDeleteI had trouble with "squares covered" until I realized that you just meant "area". I was trying to figure out whether partially-covered squares counted as 1 or 0.

ReplyDeleteIf you mean "area", say "area". Putting in confusing substitutes like "squares covered" does not simplify things for the kids.

Yes radish is totally from the same radix! (Check out how you can totally interpret that

ReplyDeletetwo different wayswhich are magicallyboth what I mean!)For some reason this is reminding me when a year and a half ago I was taking an algebra class and we were doing commutative algebra and studying all these theorems about radical ideals in rings. I was so deep in the linguistic milieu of the class that it was about 2 weeks before I double-took and immediately called my social-historian-mom and labor-organizer-oldest-friend to tell them I was studying "radical ideals" in math class.

kate, Yes it is related to radishes, and other "roots"... here is the etymology link from my mathwords page "Root and Radical The Indo-European root werad was used for the branches or roots of plants. Later it was generalized to mean the origins, or beginnings of something whether it was physical or mental. In arithmetic the root of a number is the number that is used to build up another number by repeated multiplication. Since 8 = 2*2*2 we say that 2 is the third root of eight. The word root is also used in the mathematics of functions to indicate the value that will produce a zero (a ground level number) for the function. If f(x)= x^2-9 then x=3 is one of the roots or zeros of the function. The word was used by al-Khowarizmi in his writings and was translated as radix in the Latin translations of his algebra.

ReplyDeleteThe same root word also gave us the word radical which is used for the symbol indicating a root. Students often call this the square root sign. The symbol actually has two parts, the radical and the horizontal bar. The symbol is written so that the two run together and appear as a single symbol, but the actual radical looks more like a check mark. The horizontal bar is a vinculum , just like the bar over a repeating fraction. According to Jeff Miller's web site, "The radical symbol first appeared in 1525 in Die Coss by Christoff Rudolff (1499-1545). "

Other English words related to the Indo-European root werad are rutabaga, radish, race, radix, and eradicate. From the Greek equivalent we get rhizome and licorice (honest). Strangely, I can find no written record of a relation between this root and the word radius, which seems so very much alike in form and usage.?

Wow, you guys are up late. :)

ReplyDeleteThanks for the good ideas. I need to read them more carefully when I've had some more coffee.

@gasstation... I find that dropping the word "area" on them shuts down their thinking. They go into recall-formula-they-don't-understand mode instead of thinking-about-the-physical-model-in-context mode. Of course we'll pay attention to how "area" is our usual word for "squares covered" when the question comes up about what to do with parts of squares. I don't consider avoiding confusion at all costs to be one of my jobs. I like to use it for the good of the cause.

These are really great resources. I like that they incorporate visuals as well. In the class that I'm assisting in, we had the students build their own squares and cubes, and I think this would have been a great extension to give to them.

ReplyDeleteKate --- I got really excited when I read this post because I'm currently in the "uh-oh the first unit is over and the next unit isn't ready" mode.

ReplyDeleteNext unit = irrational and complex numbers... would you be willing to share?

Joe

Kate - would you mind sharing this notebook file (yet again)? I am looking for a way to inject some meaning or interest into this very procedural topic, and this looks great. Thanks! - Wendy

ReplyDeleteTry this link.

ReplyDeleteThanks - you're super!

ReplyDelete