Monthly Archives: November 2011

Room to think …

This post describes what happens when students are allowed to develop their own problem-solving approaches.  Admittedly, this particular situation involves nothing deeper than different forms of symbol manipulation, but I still find it interesting to see how students tackle even simple problems when they aren’t given mandatory methods of solving.

I’m particularly fond of multiple solutions like these.  They encourage students to take approaches that match their thinking and provide spectacular opportunities for classes to compare and contrast different approaches.  Some are elegant, some always work (but require additional algebra), some are just painful–but each offers a different way to think about the problem.  Through my work with CAS, I have always emphasized the importance of recognizing that just because answers look different does not necessarily mean they are different.

Some of our precalculus classes were given the following problem.


Following are 4 different approaches our students used to solve for x, the first three from my classes, and the last from a colleague’s.  (Thanks, Heather!!)

Method I:  One of the most common approaches recognized that logarithm properties can be used to pull down exponents.  These students applied a common logarithm to both sides.

log(5^{x+4}) = log(3^{2x-1})
(x+4)log(5) = (2x-1)log(3)
x log(5)-2x log(3) = -log(3)-4log(5)
x = \frac{-log(3)-4log(5)}{log(5)-2log(3)}

Method II:  A slightly more sophisticated approach applied a logarithm with a base matching one of the given exponentials.  Of course, a base-3 logarithm would work the same way.

log_5(5^{x+4}) = log_5(3^{2x-1})
(x+4) = (2x-1)log_5(3)
x-2x log_5(3) = -log_5(3)-4
x = \frac{-log_5(3)-4}{1-2log_5(3)}

Method III:  Two students decided to employ inverses to change the base of one exponential before equating exponents.

5^{x+4} = (3^{log_3(5)})^{x+4} = 3^{2x-1}
log_3(5)(x+4) = 2x-1
x log_3(5)-2x = -1-4 log_3(5)
x = \frac{-1-4 log_3(5)}{log_3(5)-2}

Method IV:  Yesterday, Heather showed me another approach one of her students used.

5^{x+4} = 3^{2x-1}
5^x 5^4 = 3^{2x} 3^{-1}
5^x (625) = 9^x (\frac{1}{3})
1875 = \frac{9^x}{5^x}
x = log_{\frac{9}{5}}(1875)

This means four very different looking answers were found.  Which were “correct”?  Of course, the question of equality of student responses motivated proof investigations far better than any prompts I could have planned.  It was a good day.

Quadratic Explorations

The standard form of a quadratic is y=ax^2+bx+c.  Most students understand the effects of a and c on the graph of the parabola, but what does b do?  I hinted at this problem earlier, but just created and uploaded a GeoGebraTube Web document to model the problem.

In class, that’s about all I give before releasing my students to explore on their own.  As always, my students are required to do two things:  state their conjecture and prove that it works.

Results from my students coming in a future post.

CAS and Probability

A recent thread on the TI-Nspire Google Group asked about uses of CAS in probability.  There are so many possibilities–one uses CAS for binomial probabilities. For example, what’s the probability of getting exactly 3 heads in 5 tosses of a fair coin?  A CAS approach expands (\frac{1}{2}h+\frac{1}{2}t)^5.  The \frac{1}{2} coefficients of h (heads) and t (tails) are the respective probabilities of each outcome and the exponent is the number of trials. Obviously, there’s lots to unpack here to prevent this from being a black box tool, but note the power of the output.  The three heads event is represented by the \frac{5h^3t^2}{16} term, and the coefficient is the desired probability, \frac{5}{16}. Early in my career, I taught this by expanding (h+t)^5, picking the appropriate term, and substituting for each variable its probability.  The great power of this approach is that the meaning of each fractional term remains by the presence of the variables while you gain the answers simultaneously.  Also note that while the problem asked only for the probability of exactly 3 heads, the CAS output gives the result of every possibility in the entire sample space. Variations 1)  What is the probability of 3 heads in five tosses if the coin was bent in a way that P(h)=0.4?  Adjust the coefficients to get 0.2304. 2)  The technique is not restricted binomial probabilities.  If there are three possible outcomes (a, b, and c) where P(a)=0.4, P(b)=0.35, and P(c)=0.25, then what is the probability of exactly 2 as in 3 trials? Because only 2 outcomes are specified for the 3 trials, the third could be either b or c.  These two outcomes are highlighted above, giving a total probability of 0.288. While these values certainly could be computed without a CAS, the point here is to use technology for computations, freeing users to think.

Calculators and Testing II

As a follow-up to an earlier musing about the future of graphing calculators in the face of emerging technologies, I offer this perspective from Justin Lanier (@J_Lanier)–The College Board and Calculators: Some Thoughts.

I’m shocked to learn that I’ve been teaching lots longer than Justin (I swear I just started teaching yesterday!), but it some ways, I’ve grown up–at least in my teaching–with the same tech-presence he has.  Two days before I was to begin my first year of teaching twenty-one years ago, my department chair walked into my room and unexpected laid a brand new TI-81 on my desk, announcing that we would be introducing that machine in my classes that year.  I loved my unexpected new toy, and in retrospect, realize that Jerry’s move completely reset my entire mindset for teaching before I met even my first student.  Any successes I’ve had as a teacher can be traced in some way to technological enhancements in my understanding or my students’ ability to explore.

Given the stunning and ever-evolving ability to explore that technology grants us, I’m equally frustrated by educationally nonsensical limitations. Justin captures this well on his aforementioned ‘blog post.

As educators, I feel like we too often feel beholden to the College Board and standardized tests and take them as monolithic givens.  We betray this in the way that we talk: “But they have to take the SATs,” and “What about the AP scores?”  But the College Board is just a human institution that can change over time, and we can help that change to happen.  In fact, we are best positioned to do so.

Think about it, and consider signing an open letter to the College Board.

Curvy areas

Just encountered this image on William Emery’s ‘blog (Twitter:  @Maths_Master) who made some comments on a puzzle posted at Nrich Maths.

In addition to other questions, both sites ask students to determine how the image was created–a nice enough problem for students learning to play with areas of circles. This immediately reminded me of the construction of the yin-yang (ignoring the “eyes”, of course).

Nrich maths offered two more images that fill the gap between the yin-yang and the original image:

In my opinion, the single best question on both sites was:  What generalizations can you make about these images?

Personally, I would lead with that and then shut up to let them think.  Students are remarkably insightful when we allow them to be, and too often a teacher’s desire to be “helpful” robs students of key opportunities to be creative and to grow.  (I’ve certainly been guilty of this.)  The reality is that I’ve often learned far more from inspirations gleaned from my students’ musings than the questions I posed for them in advance.  Whatever generalization(s) they determine, my sole insistence is that students prove their claim(s).

Another nice extension of this is the arbelos.  So much cool math in this shape.