Posted in Algebra, What is mathematics

Love and Tensor Algebra

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Come,  let us hasten to a higher plane

Where dyads tread the fairy fields of Venn,

Their indices bedecked from one to n

Commingled in an endless Markov chain!

Come, every frustrum longs to be a cone

And every vector dreams of matrices.

Hark to the gentle gradient of the breeze:

It whispers of a more ergodic zone.

In Riemann, Hilbert or in Banach space

Let superscripts and subscripts go their ways.

Our asymptotes no longer out of phase,

We shall encounter, counting, face to face.

I’ll grant thee random access to my heart,

Thou’lt tell me all the constants of thy love;

And so we two shall all love’s lemmas prove,

And in our bound partition never part.

For what did Cauchy know, or Christoffel,

Or Fourier, or any Bools or Euler,

Wielding their compasses, their pens and rulers,

Of thy supernal sinusoidal spell?

Cancel me not – for what then shall remain?

Abscissas some mantissas, modules, modes,

A root or two, a torus and a node:

The inverse of my verse, a null domain.

Ellipse of bliss, converge, O lips divine!

the product o four scalars is defines!

Cyberiad draws nigh, and the skew mind

Cuts capers like a happy haversine.

I see the eigenvalue in thine eye,

I hear the tender tensor in thy sigh.

Bernoulli would have been content to die,

Had he but known such a2 cos 2 phi!

Love and Tensor Algebra is from the book The Cyberiadwritten by  Stanislaw Lem. Stanis?aw Lem was a Polish writer of science fiction, philosophy and satire. The Cyberiad is one of his best work.

The Cyberiad (Polish: Cyberiada) is a series of short stories. The Polish version was first published in 1965, with an English translation appearing in 1974. The main protagonists of the series are Trurl and Klapaucius, the “constructors”. The vast majority of characters are either robots, or intelligent machines. The stories focus on problems of the individual and society, as well as on the vain search for human happiness through technological means. The poem Love and Tensor Algebra found its way to this blog because of its mathematical flavor. And I love reciting it.

Posted in Algebra, Geogebra

Solving systems of linear equations by elimination method

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This short investigation  about the graphs of the sum and difference of two or more linear equations may be used as an introductory activity to the lesson on solving systems of linear equations by elimination. It will provide a visual explanation why the method of elimination works, why it’s ok to add and subtract the equations.

The  investigation may be introduced using the GeoGebra applet below.

1. Check the box to show the graph when equations b and c are added.

2. Where do you think will the graph of b – c pass? Check box to verify prediction.

3. Check the box to show graphs of the sum or difference of two equations. What do you notice about the lines? Can you explain this?
[iframe https://math4teaching.com/wp-content/uploads/2011/07/solving_systems_by_elimination.html 700 400]

When equations b and c intersect at A. The graph of their sum will also intersect point A.

b:  x + 2y =1

c:  x – y =-5

a:  2x+y=-4

After this you can then ask the students to think of a pair of equation that intersect at a point and then investigate graph of the sum and difference of these equations. It would be great if they have a graphing calculator or better a computer where they can use GeoGebra or similar software. In this investigation, the students will discover that the graphs of the sum and difference of two linear equations intersecting at (p,r) also pass through (p,r). Challenge the students to prove it algebraically.

If ax+by=c and dx+ey=f intersect at (p,r),

show that (a+d)x+(b+e)y=f +c also intersect the two lines at (p,r).

The proof is straightforward so my advise is not to give in to the temptation of doing it for the students. After all they’re the ones who should be learning how to prove. Just make sure that they understand that if a line passes through a point, then the coordinates of that points satisfies the equation of the line. That is if ax+by=c passes through (p,r), then ap+br=c.


The investigation should be extended to see the effect of multiplying the linear equation by a constant to the graph of the equation or to start with systems of equations which have no solution. Don’t forget to relate the results of these investigations when you introduce the method of solving systems of equation by elimination. Of course the ideal scenario is for students to come up with the method of solving systems by elimination after doing the investigations.

You can give Adding Equations for assessment.

Posted in Algebra, Math blogs

Math and Multimedia Blog Carnival #12

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Welcome to the 12th edition of Math and Multimedia blog carnival. Yes, you get a dozen posts this time.  Before we do that let’s look at some trivia about the number 12.

The number 12 is strongly associated with the heavens – 12 months, the 12 signs of the zodiac, the 12 stations of the Sun and the Moon. The ancients recognized 12 main northern stars and 12 main southern stars. There are 24=12×2 hours in the day, of which 12 are in daytime and 12 in nighttime.

In mathematics, twelve as we all know is a composite number and the smallest number with exactly six divisors, its proper divisors being 1, 2, 3, 4, 6 and 12. Twelve is also a highly composite number, the next one being24. It is the first composite number of the form p2q; a square-prime, and also the first member of the (p2) family in this form. 12 has an aliquot sum of 16 (133% in abundance). Continue reading “Math and Multimedia Blog Carnival #12” →

Posted in Algebra, Curriculum Reform

Teaching algebra – it pays to start early

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I believe in early algebraization. I have posted a few articles in this blog on ways it can be taught in the early grades. Check out for example  Teaching  Algebraic Thinking Without the x’s. All the lessons in fact that I post here whether it is a number or geometry or pre-algebra lesson always aim at developing students’ algebraic thinking. What do research say about early algebraization? How do can we integrate it in the grades without necessarily adding new mathematics content?

“Traditionally, most school mathematics curricula separate the study of arithmetic and algebra—arithmetic being the primary focus of elementary school mathematics and algebra the primary focus of middle and high school mathematics. There is a growing consensus, however, that this separation makes it more difficult for students to learn algebra in the later grades (Kieran 2007). Moreover, based on recent research on learning, there are many obvious and widely accepted reasons for developing algebraic ideas in the earlier grades (Cai and Knuth 2005). The field has gradually reached consensus that students can learn and should be exposed to algebraic ideas as they develop the computational proficiency emphasized in arithmetic. In addition, it is agreed that the means for developing algebraic ideas in earlier grades is not to simply push the traditional secondary school algebra curriculum down into the elementary school mathematics curriculum. Rather, developing algebraic ideas in the earlier grades requires fundamentally reforming how arithmetic should be viewed and taught as well as a better understanding of the various factors that make the transition from arithmetic to algebra difficult for students.

The transition from arithmetic to algebra is difficult for many students, even for those students who are quite proficient in arithmetic, as it often requires them to think in very different ways (Kieran 2007; Kilpatrick et al. 2001). Kieran, for example, suggested the following shifts from thinking arithmetically to thinking algebraically:

  1. A focus on relations and not merely on the calculation of a numerical answer;
  2. A focus on operations as well as their inverses, and on the related idea of doing/undoing;
  3. A focus on both representing and solving a problem rather than on merely solving it;
  4. A focus on both numbers and letters, rather than on numbers alone; and
  5. A refocusing of the meaning of the equal sign from a signifier to calculate to a symbol that denotes an equivalence relationship between quantities.
These five shifts certainly fall within the domain of arithmetic, yet, they also represent a movement toward developing ideas fundamental to the study of algebra. Thus, in this view, the boundary between arithmetic and algebra is not as distinct as often is believed to be the case.
What is algebraic thinking in earlier grades then? Algebraic thinking in earlier grades should go beyond mastery of arithmetic and computational fluency to attend to the deeper underlying structure of mathematics. The development of algebraic thinking in the earlier grades requires the development of particular ways of thinking, including analyzing relationships between quantities, noticing structure, studying change, generalizing, problem solving, modeling, justifying, proving, and predicting. That is, early algebra learning develops not only new tools to understand mathematical relationships, but also new habits of mind.”

The foregoing paragraphs were from the book Early Algebraization edited by Jinfa Cai and Eric Knuth. The book is a must read for all those doing or intending to do research about teaching algebra in the elementary grades. Educators and textbook writers should also find a wealth of ideas on how algebra can be taught and integrated in the early years. Of course it would be a great read for teachers.  The book is rather expensive but if you have the money, why not? Here are some section titles:
  • Functional thinking as a route in algebra in the elementary grades
  • Developing algebraic thinking in the early grades: Lessons from China and Singapore
  • Developing algebraic thinking in the context of arithmetic
  • Algebraic thinking with and without algebraic representation: A pathway to learning
  • Year 2 to 6 students’ ability to generalize: Models, representations, and theory for teaching and learning
  • Middle school students’ understanding of core algebraic concepts: equivalence & variable”

Check out the table of contents for more.

The following books also provide excellent materials for developing algebraic thinking.

 

 

 

 

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