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BODMAS
Posted in Elementary School Math, Math videos, Number Sense

Why PEMDAS is ‘morally’ wrong

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Here’s a video that explains why you need to memorise PEMDAS (or BODMAS, BIDMAS, depending where you are in the world) order of operation and why you don’t need to. Minute Physics who made this video made a mistake in assuming that PEMDAS is taught in schools without emphasising that multiplication and division should be done whichever comes first from left to right. But he does explain the ‘why’ behind the rule plus the importance of knowing fundamental ideas such as the distributive property and the associative property. This is what makes the video worth viewing.

You want to test your PEMDAS skill try this problem.

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 Geometry

Unpacking mathematics – a geometry example

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Engineers, mathematicians, and mathematics teachers all deal with mathematics but it is only the math teacher who talks about math to non-mathspeakers and initiate them to ‘mathspeak’. To do this, the math teachers should be able to ‘unpack’ for the students the mathematics that mathematicians for years have been so busy ‘packing’ (generalising  and abstracting) so that these learners will learn to do the basics of packing by themselves. This is in fact the real job description of a mathematics teacher. I won’t comment about the remuneration as this is not this blog is about. I thought it would be best for me to continue sharing about the ways we can unpack some of the important ideas in mathematics as this is the mission of this blog. Just in case you haven’t read the blog description, this blog is not about making mathematics easy because math is not so stop telling your students that it is because that makes you a big liar. What we should try to do as math teacher is to make math make sense because it does. This means that your lesson should be organised and orchestrated in a way that shows math does makes sense by making your lesson coherent and the concepts connected.

Today I was observing a group of teachers working on a math problem and then examining sample students solutions. The problem is shown below:

congruent triangles

The teachers were in agreement that there is no way that their own students will be able to make the proof even if they know how to prove congruent triangles and know the properties of a parallelogram. They will not think of making the connection between the concepts involved. I thought their concerns are legitimate but I thought the problem is so beautiful (even if the way it is presented is enough to scare the wits out of the learners) that it would be a shame not to give the learners the chance to solve this problem. So what’s my solution to this dilemma? Don’t give that problem right away. You need to unpack it for the learners. How? To prove that AFCE is a parallelogram, learners need to know at least one condition for what makes it a parallelogram. To be able to do that they need to know how to prove triangle congruence hence they need to be revised on it. To be able to see the necessity of triangle congruence in proving the above problem, learners need to see the triangles as part of the parallelogram. So how should the lesson proceed?

Below is an applet I developed that teachers can use to initiate their learners in the business of making proofs where they apply their knowledge of proving triangles and properties of quadrilaterals, specifically to solving problems similar to the above problem.  Explore the applet below. Note the order of the task. You start with Task 1 where the point in the slider is positioned at the left endpoint. Task 2 should have the point positioned at the right end point. You can have several questions in this task. Task 3 should have the point between the endpoints of the slider. Of course you can also present this using static figures but the power of using a dynamic one like the geogebra applet below not only will make it interesting but the learners sees how the tasks are related.

Task 1

  1. What do the markings in the diagram tell you about the figure ABCD? What kind of shape is ABCD? Tell us how you know.
  2. Do you think the two triangles formed by the diagonals are congruent? Can you prove your claim?

Task 2 – Which pairs of triangles are congruent? Prove your claims

Task 3 – What can you say about the shape of AFCE? Prove your claim.

[iframe https://math4teaching.com/wp-content/uploads/2013/05/Parallelogram_Problem.html 550 500]

Here’s the link to the applet  Parallelogram Problem

Note that Task 3 has about 4 different solutions corresponding to the properties of a parallelogram. I will show it in my future post.

More of this type: Convert a Boring Geometry Problem to Exploratory Version