HAPPY THANKSGIVING!
Alright, you're teaching about the rule for slopes of perpendicular lines in Algebra or Geometry.
Here are some of the instructional strategies or approaches you may have used...
(1) State the theorem without explanation followed by 3-4 demo examples of how it's used
(2) Motivate the theorem using the lines y = (3/4)x and y = (-4/3)x, choosing the points (4,3) and (-3,4) to demonstrate why these lines are perpendicular
(3) A more abstract approach using the following diagram
NOTE: Q(-b,a) is the point on line M in quadrant II. The label is too far from the dot!
FROM THE GIVEN INFORMATION IN THE DIAGRAM PROVE THAT ∠QOP IS A RIGHT ANGLE, THAT IS, LINES L AND M ARE PERPENDICULAR.
Comments
(a) If your group was advanced, would you omit the perpendiculars QR and PS?
(b) Would you draw the diagram to scale to prevent confusion for most students?
(c) Would you even consider Option (3) with a regular or weaker group of students? Would Option 2 be more than enough to get at the main idea?
(d) To more strongly suggest the use of slopes and/or similar triangles, would it be better to use the points (4,3) and (-6,8) on the lines? I personally would prefer this (and not give the equations of the lines). What do you imagine most students would do with this problem a few weeks (or even days!) later? Would they make the connection to slopes immediately if they had moved on to another unit or if this appeared on an assessment?
(e) Would some students need more than one example to suggest a generalization? Exactly what questions would you ask to promote a generalization?
(f) What have you done with this topic and/or how would you modify the above ideas??? The floor is open..
By the way, do you believe it is likely or unlikely that some version of this problem might appear on a standardized test like ADP's Algebra 2 End of Course Test or the SATs?
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10 comments:
I teach beginning algebra in a community college. It's pretty much the equivalent of algebra I, but done in a semester, instead of a year.
Every textbook I've seen uses your option one - giving the "formula" for perpendicular lines and then having problems using it. I always tell the students that's not math, and usually ask them to try to see why it might be true. I've never scaffolded with your option two, though.
I usually point to it on one day, and ask them to think about it. Then I walk them through a proof the next day. My students would not have the stamina to work through this proof on their own.
Perhaps this is a topic I could think more about, and figure out how to build scaffolding so they could do it.
Happy Thanksgivong, Sue!
I appreciate your comments. With youngsters who struggle with conceptual understanding, abstraction and generalization, I have sometimes been successful with repetition of concrete numerical examples. This is effective with all learners - the difference is in the number of examples needed.
Your comment about option1 is dead on. It's sad to see this approach still so common. Fortunately there are teachers such as you who care about the integrity of our discipline.
Scaffolding has become the "in vogue" term for what we have always done in the classroom - an incremental approach. We know that some youngsters have difficulty or lack the confidence to make a leap by themselves. Here's a basic example of what I mean:
1) 2 x 3 + 4 x 3 = ??
(2 + 4) x 3 = ??
2) 2 x 5 + 4 x 5 = ??
(2 + 4) x 5 = ??
Some youngsters, particularly those who have learning disabilities, need many reps before being able to state:
Two times a number added to four times the number equals six times the number.
I could say more but I'll stop now!
How many students, do you think, would use congruence to find equalities between the numbered angles, and then use traversals to relate the angle between L and the negative x-axis to one of the numbered angles? This would show exactly what the angle between the two lines is.
The problem, of course, is that when you make up a problem, and specially mark various points and angles, it is hard for a novice to see that it's perfectly legal to add and use unmarked points and angles.
Eric,
You're making a strong case for not labeling the angles and drawing the perpendiculars. This is always a difficult decision for me:
Does "steering" students toward a particular way of thinking actually constrict the thought process of those who see things differently.
I believe that using the points (4,3) and (-6,8) for example would help students focus more on the slope relationships than setting up congruent triangles as I did in this problem. The difficulty lies in working with students who give up quickly when the problem is more unstructured. I show them strategies they should follow in this case:
(1) Label all angles
(2) If it's a coordinate problem,particularly involving lines passing thru the origin, consider slopes early on.
(3) Draw perpendiculars from the points to the x- or y-axes. This creates triangles which may help.
(4) Look for similar as well as congruent triangles.
(5) Look for and LABEL vertical angles
(6) Look for parallel lines and focus on angles formed by a single transversal (redraw as needed!)
Will they remember much of this when faced with problem-solving situations under timed conditions? Not much but I have observed that frequent exposure to higher-order problems OVER TIME does make a difference. If I didn't believe that, why would I have become a teacher.
Finally, "test-smart" kids know that answers to questions like this are almost always one of the special case angles: 30,45,60,90,120,etc...
These youngsters make quick assumptions about the diagram and are confident risk-takers. They are not bothered about the reasoning behind their guess. They trust their instincts. I was the opposite of this kind of student. I usually had to know the "why" before putting an answer down. I was not a smart test-taker...
Me?
1. lose the lines. Segments are enough.
2. lose the coordinate axes? I just ignore them. Usually (habits are strong) I sketch in the second quadrant.
3. Draw a small right triangle, oriented with one leg vertical, one horizontal. Label the vertical leg a and the horizontal b.
4. Sketch a second right triangle, congruent to the first. Start by extending the horizontal, but make it length a, and then let the vertical go up b.
(both triangles point up. If coordinates help you see, try (-10,7), (-10,3) and (-7,3) for the first and (-7,3), (-3,3), and (-3,6) for the second. I DO NOT give kids coordinates)
From this point I've done different things.
I like asking if the triangles are congruent (of course), and supplying the measure of one acute angle (maybe n) and ask them to find the measures of the other angles, then for the angle between the hypotenuses.
But I've also asked for a proof that they are perpendicular, without offering help.
In any case, the kids always have to calculate the slopes (a/b and -b/a, or their opposites depending on the labeling).
But I think you may have already known that I do this...
Jonathan
Thanks for the input, Jonathan. Your approach reminds me of one of the famous proofs of the Pythagorean Thm.
Actually this post was motivated by an example in the College Board's new SAT book.
Do you believe that a majority of algebra students are given a conceptual development of the theorem about slopes of perpendicular lines?
Also, there is a nice vector/matrix approach which is often overlooked. Apply a rotation matrix of 90 degrees to the vector (a,b) and the result is (-b,a).
Nice.
No, I think most kids are just given the rule, and told to go apply it.
I've done that. But not for years.
Jonathan
I don't understand the "just give them the rule" people, philosophically and fundamentally. Does anyone really do that? Really?
Although I still haven't found an ironclad way to approach perpendicular slopes. The past few years I have been giving them grid paper, a ruler, and having them discuss how they would draw perpendicular lines (that are not grid lines). Once they decide on a method (which they don't need much help figuring out a good way to do this), they're asked to impose some coordinates and write equations for the lines. Then we compare all the different equations of all the different lines and look for a pattern.
I don't think the variable coordinates would go over with on-pace 9th graders. Maybe in geometry, with accelerated 9th graders and on-pace 10th graders, it would be do-able, with lots of hints.
Kate,
note, I do not introduce any coordinates - just the lengths of the sides of two (congruent) right triangles.
They are calculating slope by looking at an a-b-c right triangle.
I use this with advanced 9th graders, but also with students in College Algebra.
I agree, the version with coordinates can be overwhelming.
Jonathan
Kate and Jonathan--
Thanks for jumping in on this and sharing your knowledge and expertise. Kate, I love your "hands-on" experimental approach using grid paper. Now I remember why I started this blog!
I'd like to pursue the rotation idea with both of you further. I'm not suggesting that this would replace anything you already do for algebra or geometry students.But it allows students to revisit an important idea through the lens of transformations, rotations in particular, and can lead to more general notions of rotations.
I think all geometry students should understand that two lines are perpendicular iff one line is the rotation image of the other under a 90 degree rotation about their point of intersection. Despite the sophisticated terminology, the visualization is very straightforward. Further, this definition or theorem covers all cases and no exception need be made for the horizontal-vertical line case. Students can first be asked to find the rotation images of various points on the unit circle. R_90 ((1,0) = ??
R_90 ((0,1)) = ??
R_90 ((√2/2, √2/2)) = ??
R_90 ((√3/2, 1/2)) = ??
The 45-45 and 30-60 cases are easy to visualize using right triangles. I have asked students to first draw the rotation image of the triangle. The stronger spatial students see this easily and come to the board to demonstrate it.
After all of this we calculate slopes and, behold, we've rediscovered the opposite reciprocal rule!
Just a thought...
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