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Thursday, April 1, 2010

Formulas? What Formulas?

I banished the textbook slope, distance, and midpoint formulas from Geometry class this year. I feel like such a rebel.
First of all, kids have trouble with subscripts. They write them like exponents and then madness ensues. Second, that big long radical sign freaks them out. Third, if they learn and remember it conceptually, they won't try to memorize a formula they don't understand and dink it up later.
So here's what they did learn, and even though I'd like a more organic way for them to get there, the outcome was not bad at all.
Really Just the Pythagorean Theorem. Distance2 = (subtract the x's)2 + (subtract the y's)2. (To be followed next week by Standard Form of a Circle: Really Just the Pythagorean Theorem.) This is awesome because they heart them some pythagorean theorem.
First I asked how far away are these places as the crow flies. (We had to discuss what "as the crow flies" means.) (Thanks Visalia, CA, for laying out your streets on a square grid.)
Then we practiced a little treating the segment whose length we wanted as the hypotenuse of a right triangle, drawing the right triangle, using PT:
Then...dun dun duuuuuun:

We go back to the ones we can graph, and figure out how we can get the legs of the right triangle from the coordinates:

Then we are basically home free:

(average the x's, average the y's)
Same basic plan. Practice a few times, throw one in that doesn't fit on the graph, backtrack.
Subtract the y's, subtract the x's, divide. Yep we played the song. (We sang the song. We claimed we wanted to write our own song about the distance formula and film it and put it on youtube. We are probably not that ambitious.)

Which part of the roof would you rather stand on in a flood?

How do we compare their steepnesses numerically? They've seen slope before.

Do the y's or the x's go on top? Well...which of these slopes do you think should come out to be the bigger number?

Put it together:

I was a little uneasy about not making them copy down and memorize the formulas as they appear in I setting low expectations? Will they get to college and fail because they are unfamiliar with subscripts? But then we were settling in for a quiz this afternoon, and someone said all panic-stricken "What's the midpoint formula?!" And someone else said, "Calm down, you just average the x's and average the y's." And I think that's all right.


  1. I hate when students don't think. I have found that often times they just want the formulas because they think they will solve all of their problems. Take away the formulas and they're useless. Not only have you eliminated some of the anxiety of needing to remember formulas you've also eliminated them as a crutch. You've forced your students to think.

    I like your image with the superimposed grid.

  2. My only suggestion would be to horizontally flip the barn picture. Negative slopes may be a bit less intuitive? And "bigger" is an odd word when talking about negatives, too. Otherwise, I really like this lesson!

  3. We've had this conversation? Maybe not. I routinely deny kids these formulas until they've been doing them your way for a while.

    Especially distance.

    And I can justify never giving kids a formula for midpoint. But the justification wouldn't fit in this tiny space. (actually, it would. Can't generalize the special case. Find the point a third of the way between (c,d) and (f,g) and uh-oh)


  4. I've been going with a similar approach the last few years (heck, my wording of the midpoint 'formula' is pretty much identical to yours), though I am not yet so bold as to outright banish the textbook formula.

    On a related note, one of the high points of my past month was hearing one of my AP Calculus kids (who I also taught in Geometry) say something along the lines of "I finally get it -- you don't have to worry about memorizing a bunch of formulas if you just understand where they come can always figure them out again."

    (At some point in this year's geometry class I started referring to subscripts as being like the numbers on athletes' jerseys -- they're used to tell different players on the same team (or different values of the same variable) apart.)

    Anyway, my original point in commenting was to give a big thumbs up and say Awesome!

  5. Thanks. It actually took three 45-minute period, if anyone was wondering.

    @Dave Good point. When the brains shut off, bad things happen. I think much of what I do differently is driven by trying to develop their intuition more than anything else.

    @CalcDave I agree about the negative slopes and the "bigger" wording being weird - that was noticed and objected to by some kids, and good on them. We amended it to "the absolute value is bigger." But you are right, reflecting the image would avoid that, since positive/negative wasn't the point of the discussion anyway.

  6. The *only* reason I show kids the formulas at all is because on our state test they are provided with the most ridiculous formula chart ever, and I know many kids will try to use a formula if they have one in front of them. So I would hate for them to not know how to use them.

    But, as a geometry teacher myself, I HATE formulas. I am only 10 years out of high school and I don't remember learning as many formulas as are printed in the textbook. Why on earth is there a formula for the perimeter of a rectangle on the formula chart?

    The second you put a formula in front of a kid, they stop thinking about the problem. The ones we're dealing with now are for surface area of prisms and pyramids, all of which have faces that they already know how to find the area of. Ugh. My rants about this are legend around my school. I could go on, but the point is, I agree with you, and will man up next year and just say to heck with all the formulas.

  7. I got rid of these formulas awhile ago too, especially since here kids learn the Pythagorean theorem in grade 5 or something so they NEVER forget it. I really like how you introduced it on the map though, that's way better than my "how long is this line" intro...

  8. This is off topic, but what are you using to write your notes? I have a smartboard, but could never write as small or neatly on it as your writing is. Just curious what you're using . . .

  9. @jd - probably! (have had this conversation)

    @doug - When I hear things like that, I know I'm doing something right. Good work. Also I love the jersey analogy - very apt.

    @Stacy - One reason I felt more free to dispense with these formulas is that they are not on the formula sheet for the state exam. But, the volume/SA ones are - and oy, what a mess that is. Especially because they have volume of a cylinder, for example, as Bh, and kids are just substituting radius for B, because it's the only other number in the problem. (Hello?! McFlyyyy?!)

    @meandthedoor - thanks!

    @MrsA - The notes are written on a Smartboard. One tthing you can do is change your default pens to a thinner line. They come pre-set with a line I find too thick. Pick up a pen. Click the tools settings (the tab with the 4-colors and the "A), click Line Style and select a thinner line. Then click the 'Save Tool Properties' button at the bottom.

  10. Love it! I am about to start our conics unit as well and was thinking of doing away with formulas as much as possible and trying to teach with shifts on the graph. We will see. I do basically the same thing as you for distance and midpiont to come up with the formula, but this year I will try to use words like you instead of the letters and subscripts. Hopefully it works!!!

  11. I always teach these concepts this way. I would much rather the students remember how to do a problem then to memorize a formula they will forget. I show them the subscripts and then tell them it is similar to having two Bobs in class and calling one of them Bob1 and the other Bob2.

  12. >Thanks Visalia, CA, for laying out your streets on a square grid.

    How did you find that treasure?!

  13. I just started looking at maps of all the places I've lived that were very, very flat. :-) Central CA in particular had very grid-like streets. I actually lived in Hanford but couldn't find a section that was regular enough.

  14. And I was trying to imagine a google query that would find it. ;^)

    Most blocks are long rectangles, even when the streets are grids, so that was quite a find.

  15. The old parts of town in most Utah cities are perfect grids too. Pick a Utah city that's been around for more than 100 years, zoom into the zero spot (they also number their streets going both ways--100W, 100E, 100S, 100N) and you almost always have a nice square grid! (Having single person in charge of organizing the settling of a state has a few nice side effects).

  16. gorgeous work here kate;
    you've done it again. wow.
    (great lesson, too!)

  17. Find the point a third of the way between (c,d) and (f,g) and uh-oh)

    c+ A[f-c]
    d+ [1-A][g-d]
    naturally. let A=1/3
    for the special case
    of interest (and take
    A= 1/2 for the mid-
    point formula).

    this all makes pretty good sense
    with the drawings and handwaving.
    it's great fun presenting it, too.
    vectors without scary notations.
    ("formulas" are another case
    of scary notation... with the
    added mystique of supposedly
    being the whole point (for some)
    or the "easy" part (for others)
    or a prerequisite (g-d forbid)
    or what you will; it's a mess

  18. This comment has been removed by the author.

  19. Love using the Pythagorean Theorem instead of teaching the Distance Formula. I HATE the Distance Formula. The kids do mix up subscripts powers. I hope to use the map next year when I teach distance. Thanks for the post!

  20. A third of the way...

    kids (and adults) have no problem finding the street a third of the way from 14th to 20th.

    (actually, many of them do... but that's not my audience).

    I can ask helpful/annoying questions: "Why is the answer not 2nd street? Isn't 18 also a third of the way?"

    And then refer them back to 14th to 20th when they are working it out for points.

    Not wonderful, but gives them something to hang onto.

    And I'm hoping it's clear that none of this generalizes from the midpoint formula.


  21. I love all the stuff you have done above. I think you are doing exactly the right thing, and I think that there is a fair amount of research to back it up. You might try to pick up "How Students Learn: Mathematics in the Classroom" by the National Research Council. I once left the book in the middle of the street but it was there when I came back 3 hours later. So if a local library has it, it probably isn't checked out.

    I am interested in the subscripts part. I do think that it is better to start the way that you have with intuitive contextualized problems, but now that your students have some meaning to what they are doing maybe now is the time to show why subscripts are useful and good. The lingo I would use for this is that as we were setting up these problems using variables (a, b) and (c, d) for points I might point out it would be helpful if I could just use x and y, since that is how I think about them, the problem is that there are two x's and two y's. How could we handle this? With something like last names. If we had two Erics in the class we might call one Eric C. and one Eric M. this is the meaning of subscripts. They are last names that help us differentiate things that have the same first name. They don't do anything mathematically they are just names. Exposing your students to this lingo is probably worth something, and now you have a way to motivate it.

  22. I don't know, Matt. I see too many kids trying to cram a teacher-given formula and apply it badly, even if I've done my due diligence in showing why the formula is what it is.

    On the other hand, I noticed a few kids writing x_1 - x_2 themselves. Whether they came up with it themselves, or they applied their knowledge of subscripts from another class, or a parent or tutor showed them, I think that's fine. But I still think me presenting it to the whole class has too many negative consequences in these very heterogeneous-ability regents classes.

  23. Wow! Great job - I love your work!
    So glad I stumbled across your blog.
    I will definitely be back!!

  24. Love this :) Agree that it's especially powerful for students and their developing conceptual understanding when they can reason through the formula and see how/why it makes sense.

    In my Geometry class, my students had often already memorized these formulas (often poorly) in previous courses, and I had to do some work explaining why they actually worked-- and why we should think of distance/midpoint this way rather than as a string of x's and y's that won't be provided to you on a test.

    Another teacher recently asked me though-- at what point do you, if ever, give (struggling) students the formula? A week before the state test if they still haven't made sense of the concept? I wasn't sure I had a good answer and would be curious to hear your thoughts.

  25. Very creative! I've never heard that slope song before so that was fun too! Thanks for sharing so many ideas for making the math relate to real life!

  26. I really like this approach to these formulas. Have them create their own formula once they have the ideas down...fantastic!

    Appreciate the shout out to the Central Valley. About 30 miles south east and you could've used my street for your lesson. Find a way to incorporate the manure smells, would ya?

  27. @grace Today, after three days off, the first thing I did was to ask everyone to take a blank sheet of paper and write down the distance formula.

    I was really pleased to see nobody (that I noticed) forgot the squares or the addition sign.

    But many kids instead of writing "subtract the x's" were writing "x - x". I had one of them dictate the formula while I wrote it on the board, and I said we had the basic idea down and could probably use this to calculate a distance, but the x - x bothered me, because x - x should always come out to be zero! So then we talked about how we could distinguish between the two x-coordinates, and I introduced the idea of subscripts. Some of them like x_A and x_B better than x_1 and x_2 - and that's fine - but I feel like they have some ownership and understanding behind applying a formula now.

  28. Whenever I teach The Pythagorean theorem and the slope of a line I don't teach "the formula". In my classroom the slope of a line is the change in vertical divided by the change in the horizontal. Sometimes I let them get away with change in y over change in x. Pythagoras theorem in the cartesian plain is then "the square root of the change in x squared plus the change in y squared".

  29. Not to be too snarky, jason, but do you also teach them that it's the Cartesian "plain", instead of "plane"?

  30. Can't a man make an innocent typo? Dag.

  31. If only it were just a typo. I didn't even mention:

    "Pythagoras [sic] theorem in the cartesian plain [sic] is then 'the square root of the change in x squared plus the change in y squared'."

    This is a formula -- the distance formula he claims not to teach, in fact -- but it's not a theorem. It asserts no true-or-false statement. It's completely missing a phrase to the effect of "equals the length of the hypotenuse". Further: the Pythagorean theorem now applies only to right triangles whose legs are parallel to some God-given coordinate axes?

    No wonder freshmen in college don't know which end of an equation is up.

  32. Groovy stuff. I like it.

    1) When I was a kid, I spent the first five or ten minutes of every test looking over all the questions and deriving all the formulas I would need. I couldn't remember formulas for the life of me, but I could derive the heck out of them. I like helping kids who happen to learn the way I do.

    2) I love the phrase "...they heart them some pythagorean theorem."

    3) One thing to add to the Midpoint discussion is that this is just a two-dimensional version of the midpoint formula in one dimension. When they get that, it can make more sense and going part ways (e.g., a third of the way) can be easier as well.

    4) Have you thought about introducing a tool like GeoGebra to this discussion? I think it could be pretty handy.


  33. Definitely in agreement that teaching formulas often encourages students to short-circuit their own thinking. One question - most of my students seem to have seen the formulas before. Does it make sense to references the formulas just so students can make connections and don't feel like math class is just years and years of random trivia?

  34. @KH: I love this statement: "math class is just years and years of random trivia". I hadn't thought of it that way but imagine there are many students who see things that way.

  35. My sister also teaches Geometry in central VA, and asked me to write a song to help her kids remember the formulas. I realize this goes somewhat against the grain of your original post, but thought it might be helpful as an additional aid for learning/memorization.

  36. This is hysterical, David. There should be like a Geometry Slow Jams compilation. :) Thanks!


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