Category: Algebra

Posted in Algebra

Tough Algebra Questions about Equations and Expressions

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Here are some questions your students have been wanting to ask you in your algebra class. Daniel Chazan and Michal Yerushalmy in their article On Appreciating the Cognitive Complexity of School Algebra posed these questions about equivalence of equations , solving equations, and equivalence of expressions for us teachers to ponder upon.

function_notationHow will you answer the following questions? What explanation will you give to the students?

Continue reading “Tough Algebra Questions about Equations and Expressions” →

Posted in Algebra

What is an algebraic function?

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An algebraic function is a function created by applying the operation of addition, subtraction, multiplication, division, and extracting the nth root. Let me give an example. Suppose you have the function f and g where f is a linear function and g is a constant function.  Let f(x)=x and g(x) = -3. We can create another linear function h by multiplying f and g that is h(x) = -3x. We can also create another linear function l where l = f – g, that is l(x) = x-3.

What about quadratic functions? A quadratic function (with real roots) is a product of two linear functions. So we can make a quadratic function n by multiplying f and l for example. That is, n(x) = f(x) x l(x) = x(x-3). And cubic function? A cubic function is a product of three linear functions or of a quadratic function and a linear function. And quartic function? Well, you must have figured it by now. This process of creating function by multiplying linear functions produces a family of functions called polynomial functions so called because its algebraic representation is a polynomial.

functions
Polynomial Function Family

What kind of function is produced when you divide a function by a function in x? Using the function defined earlier, what is g÷f?  g÷l? f÷l? Getting the quotient of two polynomial functions give us a new family of functions: p(x) = -3/x; q(x) = -3/(x-3); and, r(x) = x/(x-3). These expressions defining the functions will not simplify to polynomial expressions so they do not belong to the family of polynomial functions. They belong to what is called the family of rational functions so called because they are defined by rational expressions.

You can also raise a function to a fractional power, that is get the nth root of the function. For example we can have t(x)= x^0.5. That is t(x)=sqrt of x. I don’t know what this family of function is called. Maybe we can call then nth root functions.

These three families — polynomial functions, rational functions, and nth root functions, all belong to the family of algebraic functions. Functions that are not algebraic functions are called transcendental functions.

You may also want to read ideas for teaching functions.

Posted in Algebra

What is algebra?

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Didn’t we say in our algebra class that in the grades they study about numbers and so now they will be studying letters instead? Didn’t we say that in algebra we now use x instead of box (in 3 + ___ = 15, we now write 3 + x = 15)? And isn’t it that since this announcement our algebra class activity has been about finding that 24th letter?

Well, we reap what we sow.

what is algebra

Just  a friendly reminder to take the teaching of variables and unknown quantities with meaning.

For a serious discussion about what algebra is, I  suggest the following articles.

1. What is Algebra? by Prof. Keith Devlin

2. Algebra vs. Arithmetic

Posted in Algebra

Guest Post: Mega Math Challenge

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Only a few days remain to register for Moody’s Mega Math Challenge

math challengeSo what’s the best way to teach mathematics to the fast-paced, multi-tasking young people of the information age? Crunching numbers with pencil and paper and poring over the pages of conventional textbooks just doesn’t cut it with this tech-savvy generation, so used to instant gratification!

Today’s math teachers, for their part, are finding new ways to make the subject relevant and engaging for their pupils, ultimately helping them use mathematics to solve everyday problems. There are many ways to stimulate the children of the digital revolution but perhaps none capture the spirit of applied mathematics like Moody’s Mega Math (M3) Challenge (http://m3challenge.siam.org/about/), which allows students to do rather than just read, memorize, or calculate.

The M3 Challenge is a free applied math competition for high school students that connects textbook and classroom learning to the “real world” by simulating the genuine and practical issues we face as a society and in our daily lives. Teachers who coach M3 Challenge participants realize the contest’s potential to educate students in math modeling. And while teacher involvement is critical – they register and prepare teams of juniors and seniors – the responsibility of developing a viable solution paper by the end of Challenge day (and possibly winning a share of the $115,000 in total scholarships) falls squarely on the shoulders of the thousands of students who participate each year.

Challenge Champion

“You have to let the kids do their thing. I try not to direct, I try not to drive,” Ellen Leblanc, an experienced coach from New Jersey’s High Technology High School, shared. “Initially, and prior to the Challenge weekend, the students and I do a little bit of brainstorming: what could the Challenge problem be this year? If the question were “X,” how would you approach it and what is important? Beyond that, you have to leave it up to the students,” she said.

With so much technology at their fingertips, high schoolers in 2013 are used to doing more than just reading and answering textbook problems. Some have the benefit of being offered math modeling classes at their high school, some experience technology-based lessons in their classrooms, and others use the skills in their math toolboxes for extracurricular activities.

“This is really the only competition in the nation where kids come together and have this day-long charrette in a high-performance work team that is so similar to what we do in industry. To have that experience as a high school junior or senior really opens their eyes to what a career in a math-related field can be like. It is incredibly influential,” explained Mary Redford, team coach from Nashoba Regional High School in Massachusetts.

Registration must be completed by each team’s teacher-coach by Feb 22 at 6:00 p.m. EST. It is both quick and easy and there are no fees whatsoever. Register now at http://m3challenge.siam.org/participate/.