A Puzzle Equivalent to Factoring a Quadratic

In response to The Number Warrior’s post, I would like to propose an alternative, since the choice to do the sum or product inheres more to the nodes than the connections.  In the following, the style of line tells you what to do with the numbers from the lines coming into the node:

  • solid black = multiply the incoming numbers (so that the node is the product)
  • dashed red = add the incoming numbers (so that the node is the sum)
A puzzle equivalent to factoring a quadratic

A puzzle equivalent to factoring a quadratic

factorpuzzle (PDF)

N.B.: I did this more as an exercise in using Asymptote (and its flowchart package) than to make any real contribution.  For the interested, here is the source code:

size(6cm,0);
import flowchart;
// for lines coming into a node, solid lines mean multiply, dashed lines mean add

int a = 6;
int b = 17;
int c = 12;

real step = .2;
real radius = 18;
pen number = blue+linewidth(2)+fontsize(20pt)+Palatino();
pen sum = red+linetype("2 2")+linewidth(2);
pen product = black+linewidth(2);
pen connection = black+linewidth(2);

block A=circle(Label(string(a),number),(0,step),drawpen=product,mindiameter=2*radius);
block B=circle(Label(string(b),number),(0,0),drawpen=sum,mindiameter=2*radius);
block C=circle(Label(string(c),number),(0,-step),drawpen=product,mindiameter=2*radius);

block d1=circle("",(-step,step),drawpen=product,mindiameter=2*radius);
block d2=circle("",(step,step),drawpen=product,mindiameter=2*radius);
block e2=circle("",(-step,-step),drawpen=product,mindiameter=2*radius);
block e1=circle("",(step,-step),drawpen=product,mindiameter=2*radius);
block cross12=circle("",(-step,0),drawpen=product,mindiameter=2*radius);
block cross21=circle("",(step,0),drawpen=product,mindiameter=2*radius);

add(new void(picture pic, transform t) {
    draw(pic,path(new pair[]{d1.right(t),A.left(t)}),MidArrow);
    draw(pic,path(new pair[]{d2.left(t),A.right(t)}),MidArrow);
    draw(pic,path(new pair[]{e2.right(t),C.left(t)}),MidArrow);
    draw(pic,path(new pair[]{e1.left(t),C.right(t)}),MidArrow);
    draw(pic,path(new pair[]{cross12.right(t),B.left(t)}),MidArrow);
    draw(pic,path(new pair[]{cross21.left(t),B.right(t)}),MidArrow);
    draw(pic,path(new pair[]{d1.bottom(t),cross12.top(t)}),MidArrow);
    draw(pic,path(new pair[]{e2.top(t),cross12.bottom(t)}),MidArrow);
    draw(pic,path(new pair[]{d2.bottom(t),cross21.top(t)}),MidArrow);
    draw(pic,path(new pair[]{e1.top(t),cross21.bottom(t)}),MidArrow);
  });

draw(A);
draw(B);
draw(C);
draw(d1);
draw(d2);
draw(e1);
draw(e2);
draw(cross12);
draw(cross21);
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Published in: on 2011.06.02 at 14:51  Leave a Comment  
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Geometric Algebra

Another teacher and I have been exploring Geometric Algebra in application to geometry and physics together on most weekends using the online collaboration suite Elluminate (through LearnCentral, which allows one up to three collaborators for free).  I love David Hestenes’ Oersted Medal Lecture, but it was a bit deep for someone not steeped in the culture of physics.  So, I was delighted when I found a series of more introductory lectures by Chris Doran and Anthony Lasenby.  Occasionally they get into deep physics, but the bulk of the treatment is easy to follow for the mathematician, and I enjoy their style of content presentation.

Using the tools of geometric algebra I was able to prove (easily) a theorem that I have never before seen in physics.  Concerning a constantly accelerating model

\begin{array}{rcl} a &=& \text{constant} \\ v &=& v_0 + a t \\ x &=& x_0 + v_0 t + \frac{1}{2} a t^2 \end{array}

one can (and usually does) prove the following intermediate steps

\begin{array}{rcl} v-v_0 &=& a t \\ v+v_0 &=& \frac{2}{t}(x-x_0) \end{array}

on the way to proving

v^2-v_0^2 = 2 a\cdot(x-x_0)

but, by taking the wedge product instead of the dot product, one gets

v\wedge v_0=a\wedge(x-x_0).

In other words, the parallelogram formed by final and initial velocity has the same area as the parallelogram formed by acceleration and displacement. Not only this, but on the way to proving it, one finds that (x-x_0)=\frac{v+v_0}{2}t, so that the displacement lies in the same direction as the average velocity, which we proved intermediately. It makes for a great GeoGebra applet, but I can’t embed it with WordPress. It’s also the quickest way to prove (also using the first relation that v^2=v_0^2) that range is \frac{v_0^2\sin(2\theta_0)}{|a|}.

Published in: on 2011.01.09 at 11:44  Leave a Comment  
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Reflections after the first week

I just finished Reluctant Disciplinarian by by Gary Rubinstein, which sums up my own inclinations toward being a “softy”.  The things I take away from this book include:

  • Start the first few weeks of class with lessons that have clear expectations for students and, in my case, may be similar to what students are used to.  This sets up the students for a minimum of culture shock.  To this baseline I can slowly add more pizazz.  The time that I have spent “substituting” for my CT have been a convenient baseline.
  • All the standard advice about not arguing with students, etc.
  • Be mindful about over-talking and over-sharing.
  • Criticize one’s own lessons with a list of “cons”.

My tendency is to become so absorbed by the content and zany lesson that I forget to respond to student needs.  Any misbehavior is then so shocking that I deal with it too publicly, thereby alienating the rest of the students.  I have seen firsthand how a “softy” becomes the “mean teacher”.  I have also seen that it is possible to improve, and so I hold that hope going into the next week, the beginning of which my CT will again be gone.  I will have to prepare the students for an upcoming test.  Reviews are always hard for me to do well because every student has different needs.  I know that the students will be expecting a review game on the last day before the test, so I hope that my CT was able to put something together.  If not I will have to invent something.

This Monday is also the first day of official classes for my University.  I wouldn’t be so worried by this, but I am also studying to take the Praxis Secondary PLT this Saturday, and it has been quite some time since I took Educational Psychology.  I’m finding that the questions are IMHO so poorly written that the right answers are flawed, which leads me to reject them and try to find truth in another answer.  As a result, I’m not doing so well on the practice questions, so I will need to put in quite a bit of studying all week.  If anyone from ETS reads this, I suggest that rather than rewarding the top 15% (for Excellence!), you reward the top 5% by refunding their testing fees.  If you offer to refund even other materials (like your lousy online review course with terrible questions), you may even encourage more people to buy those.

Published in: on 2011.01.09 at 11:09  Leave a Comment  
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Teaching already?

Due to an unexpected event, my cooperating teacher had to leave for a few days, putting a substitute in charge of the class.  Thus begins my student teaching!

On the plus side (opposite the minus of being unprepared), my CT has already prepared lessons, which will give me the chance to teach in a similar manner as I get started.  This should make the classroom management situation a little easier.

Published in: on 2011.01.05 at 21:10  Leave a Comment  

First non-professional-development day

Yesterday was a semi-productive inservice day.  None of the inservice topics were new to me, but we were tasked with improving on a previously written lesson using differentiated instruction.  Seeing as I had no previously written lesson plan, I assisted one of my CTs with improving one.  I plan to try to write one later using differentiated instruction methods to satisfy the principal’s call, but I haven’t yet selected a topic.  Although I agree with differentiated instruction in principle, I find myself getting anxious over a tendency to cater to each student’s preferred modality rather than build on weaker ones.  Surely some balance in this dialectic exists?  Perhaps even synthesized into a harnessing of group diversity to advance individual learning à la Japanese mathematics?  I like the idea of improving stale lessons, but the process lacks the accountability and collaboration of lesson study.

Today was the first day with students.  I assisted only in a minor capacity while trying to learn student names, something at which I am exceedingly bad.  The day, however, passed without incident, so I shall use the remainder of my evening to make some flash cards to help me learn student names (and for later when asking questions).  The one thing I will note, which carries increased salience now that I have read The Teaching Gap and its comparisons with Japan, is that there were an aweful lot of interruptions to class today.  The teaching could not help but be fragmented, though the teachers did the best they could.

Lesson Study

I just finished reading the updated (2009) version of The Teaching Gap by James W. Stigler and James Hiebert.  I know it’s a bit old, but I really like the idea of lesson study.  Moreover, I think that in the absence of superintendent and principal support that it might be possible to subvert this power structure by using the Internet to conduct distributed lesson study.  It might go something like this:

  1. Work collaboratively with other teachers interested in the same lesson topic.  Tag it with any state, CC, NCTM, ISTE, or any other standards that are relevant.
  2. Create assessments that can test the effectiveness of the lesson.
  3. Give the lesson.  Gather any in-class paper that the students used (to scan), video record the lesson, and otherwise obtain any data you can.
  4. Repeat steps 1-3 until the lesson is as good as it can be.
  5. Collect the materials on a website on which other teachers can comment, rate, and view the materials.  Do not publish it until at least three teachers have been involved in the development of the lesson and at least two have tried it out in a live classroom.

Blogs already provide a partial mechanism for Step 1, helping teachers to gather a wider range of ideas to use in lessons, but this alone will not help to create better lessons, especially if individual practitioners misinterpret the point of the materials (see the author’s notion of teaching as a cultural institution).

Technorati couldn’t find much on “lesson study” or “teaching gap”, so I wonder how much these ideas have circulated.  The University of Wisconsin La Crosse has incorporated lesson study into a project for college faculty.  There is also NSF- and IES-funded research into lesson study, but it is still conducted by researchers!  The Education Development Center also has an NSF-funded project going.

Published in: on 2011.01.01 at 17:06  Leave a Comment  
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Student Teaching

I start student teaching this coming semester, starting January 3, 2010.  I remain anxious about classroom management issues and whether I can translate my inner light into thoughtful and fun learning experiences for high school students.

I’ve started thinking about the curriculum for the Geometry and Algebra 2 classes that I will be teaching.  I’ve found many good resources on the blogosphere, including stunning treatments of logarithms by Kate Nowak (Log Laws , Introducing Logs) and Dan Greene (Intro to LogarithmsBig L).  I’ve integrated this into my own treatment using the idea of relations/relationships.

I’ve also pondered over the last few months the many ways to teach factoring.  With a general quadratic trinomial ax^2+bx+c, the ac-method usually results in numbers that are unnecessarily big.  Instead, I start from separate prime factorizations of a and c.  This builds on the idea of natural numbers being “bags” of prime numbers (the “Bag Model”).  Just like the guess-and-check method, this one takes practice.  It’s similar to the X-method but different.  I haven’t settled on one form yet and have developed a filtering-based approach (only using prime factors indirectly to generate factor pairs and much like the X-method) and a “spin” approach that is visually appealing but not helpful.  This got me into a digression on the combinatorics of “possible” factorizations based on the first and last terms, which turned out to be more complicated than I thought.

I’ve also been working on ways to make geometry relational (filled with relations between objects) to give students cognitive tools to attach geometry problems and proofs.  On top of this, I am making Glenn Doman-style flashcards in reading and math for my young daughter and preparing to take the Praxis PLT (blech!).

Modeling

If you haven’t heard of Modeling and you teach math or science, you should check it out.  It’s an awesome curriculum and allows teachers to focus on planning for their students rather than inventing effective lesson plans.  http://modeling.asu.edu/ To access the full curriculum, one has to attend a workshop.

Published in: on 2009.07.15 at 08:33  Leave a Comment  
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The Hot Tub

Found at: http://math.rice.edu/~lanius/Algebra/hottub.html

The technologies used in this lesson are a word processor, a blog, a microphone, and sound editing software (such as Audacity).

Objectives of the technologies:

  • Word processor – Students will use a word processor to compose a brief (less than one page) story following the idea given in the “Another Story…” section of the lesson plan.
  • Blog – Students will publish their stories on a class or public blog. This follows the meme of “explain this picture mathematically”. They will publish both their printed and their reading of the story. (This motivates a conversation on the pros and cons of different representations of the story.)
  • Microphone – The microphone will be used to capture each student telling their story in front of the class.
  • Sound editing software – The students will use Audacity to prepare their story for inclusion on their blog as a podcast.

Software and Hardware requirements:

  • computer with enough storage space for a hour or so of uncompressed audio
  • web browser
  • sound editing software
  • microphone
  • Internet/Intranet access to the blog

Skills required:

  • Teacher – The teacher must be able to troubleshoot audio problems and levels with the student to record the student’s presentation, should be familiar with sound editing, and should know how to upload audio to a blog.
  • Students – Students should be familiar with a word processor and know how to use their class/individual blog account.

Reflections on the technology:

The comprehensiveness of the storytelling might be overkill, but it does illustrate an edit cycle of preparing a word processing document for something, rather than an end in itself. The underlying assumption here is that students will enjoy hearing the stories that their peers concoct. The process of creating a story emphasizes thinking deeply about how mathematics is connected to life. To shorten the process, an alternative to formal presentations is to set up a sound booth station in a closet or a corner, but then one cannot ensure that other students will hear the stories.

Published in: on 2009.06.30 at 01:29  Leave a Comment  
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Multiplying binomials with F.O.I.L.

Found at: http://www.lessonplanspage.com/MathMultiplyingBinomialsWithFOIL812.htm

The technologies used in this lesson are a Wiki, a graphics program (such as Inkscape), and the World Wide Web.

Objectives of the technologies:

  • Wiki – The Wiki will be used to present and share the group’s solution to a way to multiply binomials in the expanded discovery phase of the lesson; it should explain in detail to an audience including their classmates how to perform their method. At the end of the creation phase, students will use the Wiki to compare their solutions. Each student will individually choose a method and add to that Wiki page an example of how to use it.
  • Graphics program – Inkscape or another program will be used to create illustrations for the Wiki.
  • World Wide Web – After completing their project, students will use the World Wide Web to explore other solutions to the project and compare other student’s solutions to existing solutions.

Software and Hardware requirements:

  • A class-wide Wiki should be selected—one that can handle mathematics is preferable.
  • A graphics program
  • Internet access and web browser

Skills required:

  • Teacher – The teacher must be facile with the software and hardware above, including how to export images for use on the wiki and how to represent mathematics on the wiki.
  • Students – Familiarity with searching and conducting research is essential, as is knowing how to evaluate Internet sources. Familiarity with graphics software is helpful.

Reflections on the technology:

The students will eventually use the Web to evaluate methods for multiplying binomials, but they will first struggle on their own to find their own representations. The culminations of the struggle is their group page on a class Wiki, a tangible form of effort spent. The hardest part for students will probably be not using computers in the discovery phase, as they will simply want to look up an answer and consider it the right answer. Hopefully, the emphasis on comparing their solution to the solutions of other groups in the class and those found on the Internet will require them to think critically and realize that there is no one right answer. Some students might not want or need to use the graphics program, but those students who think visually should have some outlet for their creativity. An alternative for those students who like to draw is using a scanner to digitize a paper drawing.

Published in: on 2009.06.30 at 00:55  Comments (1)  
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