The general idea comes from generalizations of yesterday's Side-Splitting Theorem. If a trio of parallel lines splits a transversal proportionally, then it splits any transversal proportionally. In particular, if the parallel lines are equidistant, then any transversal is split equally. This extends to any number of equidistant parallel lines.
So all we need are a bunch of equidistant parallel lines. And many students just happen to be carrying not just one, but several pages filled almost completely with equidistant parallel lines.
I'm speaking, of course, about lined notebook paper.
Wu's activity is simple. He begins by picking a point A near the middle of the top line of the notebook paper, and then two points B and C on a certain line below the top -- Wu chooses the fifth. Then he finds the points where rays AB and AC intersect another line -- in this case the seventh -- and labels these points B' and C', respectively. Then it can be proved using the Side-Splitting Theorem that
This activity can be made more dramatic by presenting a segment on one of the lines and asking the student to draw another segment that is exactly 7/5 as long as the first, without measuring. That's the tricky part, since the straightedges that the students use to draw the lines are likely to be rulers that allow the students to measure them.
So far, I haven't mentioned the U of Chicago text. Notice that so far in Chapter 12, I haven't been giving the Exploration Questions at the end of each lesson as a bonus. But notice that, right at the end of yesterday's Section 12-10, Exploration Question 22 is similar to the Wu activity.
Here the student is asked to take a segment
The best way to do the U of Chicago activity is to have the students take something small, such as an index card, and divide it into something like thirds or fifths. (Halves and quarters are trivial.) To divide the index card into fifths, the student places one corner of the card on the top line of the notebook paper and another corner on the fifth line down from the top. Then the four lines in between touch the paper at exactly the one-fifth marks of the card, so the student labels these points. Finally, the card is lifted from the paper and folded. If these are done exactly (but this is difficult), then the card has been successfully divided into fifths.
Meanwhile, I've still been thinking about the Common Core lesson that I taught earlier this week, the one on slope that confused many of the students. As I mentioned before, giving a word problem to get the students thinking about slope does occur in pre-Common Core texts.
For example, Section 3-5 of the Glencoe Algebra I text -- the book I used for student teaching -- is on "Proportional and Nonproportional Relationships." The lesson begins with a chart showing the number of miles driven for each hour of driving, and students have to figure out that every time the number of hours increases by one, the number of miles increases by 50. And the very next lesson, Section 4-1, "Rate of Change and Slope," the students calculate the speed -- and therefore the slope -- by dividing the change in distance by the change in time.
But when actually teaching Chapter 3, my master teacher had me teach Sections 3-1 and 3-2 on relations and functions, then skip directly to Chapter 4 -- there was no Chapter 3 test. And I don't remember how much of Sections 4-1 and 4-2 we actually taught -- we might have just given the students a separate worksheet on slope before diving right in to Section 4-3, on graphing equations in slope-intercept form. I believe that this practice was common for most Algebra I classes before Common Core forced us to emphasize these sections that we used to skip.
In the Common Core debates, those who oppose the standards due to traditionalism will argue that students shouldn't have a "guide on the side" discovery lesson like Wednesday's. The students discovered nothing because few of them were calculating the rates correctly, so none of them realized that the rate was always the same no matter which points they chose. Instead, according to the traditionalists, the "sage on the stage" should just tell them the slope formula and have the students practice plugging in points and finding the slope.
But then again, the students in my student teaching class still struggled with the slope lesson even when taught traditionally. Chapter 4 was a turning point for many students in that class, as many students who performed well on previous chapters struggled from this point on. Many students simply did not remember the slope formula, no matter how many times I gave the formula -- and some simply chose not to listen to me when I told them the formula.
Of course, students will struggle no matter how slope is taught if they are generally weak on operations with integers. As we already know, many students simply don't remember how to add, subtract, multiply, and divide integers from year to year -- that is, a student can be taught to excel on an integer test one year, yet won't remember the integer rules the following year. It is one of the two main topics from grades 4-7 that students can't remember -- the other, of course, is fractions. But unfortunately, slope is heavy on both integers and fractions. To calculate slope, one must add -- actually subtract -- integers for both the rise and the run, and when one divides these, the answer may be a fraction.
Is it possible to compromise between the traditionalist and progressive philosophies here to obtain a slope lesson that will get most of the students to do well? I can't help but notice that the example that I mentioned earlier has a rate, or slope, of 50 mph. First of all, it's positive, unlike the negative slope from Wednesday. Also, this one is easier to calculate as it uses numbers such as 50 and 100, as opposed to the 48 and 96 that appeared on Wednesday. Therefore, students are more likely not to make a mistake and reach the Aha! moment that the rate is the same no matter which times they happen to choose.
And so we, as teachers, must be careful when designing the sort of lesson that the Common Core Standards encourage. The traditionalists are correct that discovery lessons won't work unless the students get the basic calculations correct, so let's design "Opening Activities" and "Anticipatory Sets" so that the students are likely to get the answers correct. This means delaying problems with negatives and fractions until after the opening activity, by which time the formula is given.
Then again, my slope lesson in a couple of weeks is from a geometric, not algebraic, perspective. I won't be focusing on how a constant speed of 50 mph is related to the constant slope of the line, but rather how similarity will be used to prove that a line has constant slope. So this will be tricky.
When I was in that classroom, as I said earlier, I found out how all the packets where eighth, not ninth, grade packets. (This means that the other part of the Common Core debate -- how will these freshmen ever reach calculus by senior year -- is still unresolved.) And so I looked ahead for a packet on geometry, specifically congruence and similarity, since that is the current topic on this blog.
In Student Packet 8-14, there are four sections, on congruence, dilations, similarity, and review. The dilations section began with a rubber band experiment to perform the transformation. It sounds as if it could have been a good opening lesson instead of the one I posted Monday, but using the rubber bands correctly might have been tricky. This is followed by another "experiment," except by using graph paper to perform the dilation. So far, I'm avoid graphing because of the temptation to use coordinates to prove the major theorems -- but once again, this is a packet for eighth grade, not high school geometry, so such a proof isn't emphasized. In eighth grade, the relevant standard is:
Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
Notice that eighth grade is the first in which transformations appear.
The activity that I will post contains both the Wu and U of Chicago versions of the activity. It is not typed because I used my own notebook paper for this, and I don't want the notebook paper to jam my new printer. I'm afraid, though, that the notebook lines won't necessarily show up when reprinting.