Just talkin’ Algebra!

For one to articulate an understanding of algebra, one needs to be able to speak the vocabulary of algebra. Knowing how to correctly reference algebraic terms helps one to ask better questions and to describe methods in a more meaningful manner.

In this series of short video clips, Virtual Nerd revises the meaning of ConstantVariableCoefficientLike Terms and Simplest Form. Review these video clips as the vocabulary represents the building blocks of algebraic expressions. Thank you Virtual Nerd!

 

Carolin’s Notation for Solving Equations

Note: This entry is a non-technology post. Carolin is a visiting student from Germany incarolin200 my 5th Year Maths class.

To solve a basic algebraic equation, we isolate the unknown to one side of the equation. In the case of  x  as the unknown, we say, “we get the x term by itself.”

The steps we go through in order to get  x  by itself often requires us to either (i) add or subtract the same term to/from BOTH sides of the equation or (ii) multiply or divide the same factor to BOTH sides of the equation. By performing the same operation to both sides of the equation, we keep the equation balanced.

As discussed in class, we DO NOT use some strange type of VOODOO maths logic that allows us to move a term from one side of an equation to the other side and then magically “change” its respective sign.

As an example of documenting every step involved in solving a specific linear equation, consider the worked out solution below:

Notation1_350

The highlighted terms emphasize how I’ve worked to keep the equation balanced every step of the way.

Not unreasonably so,  many teachers and students find documenting the operations on both sides of the equation laborious – overkill to be precise.

The problem with not writing out the steps on each side, however, is that it can lead to silly mistakes and, if the original attempt to solve the equation doesn’t work out, the process of finding where you made your mistake can be made more difficult than if you left a clear trail of your steps taken.

In steps Carolin!

Carolin is a visiting student from Germany who is in my 5th year mathematics class. In the process of marking one of her tests, I noted she had a “different method” for documenting her steps. Below is how she would document the solution to the above linear equation:

Carolin's Notation for Solving Equations

Carolin's Notation for Solving Equations

Carolin documents each step she is  “about” to take by way of a single reference in the right-hand margin. So as to avoid confusing her next step with the actual equation, she separates her “next step” from the equation with a small vertical line. It is implied that the operation in the margins is to be performed on BOTH sides of the equation.

Carolin’s notation for solving equations provides us with an easy to understand, concise method for documenting our steps. Along with assisting us in re-tracing our steps (in case of an error), it also provides the person checking our work with clarity on how we worked through our problem.

carolin200Carolin tells me she learned the notation as part of her maths studies in Germany. I’ve begun using Carolin’s notation with my second year maths students and it has proved very helpful. Thank you Carolin and thank you Germany!

Be a Happy Grapher!

Throughout post primary maths you will encounter the need to sketch a number of different types of functions (graphs).  At minimum, you will be sketching linear, quadratic and cubic functions.  Unfortunately, many textbooks don’t provide students with graphed solutions to check the accuracy of their work. Additionally, if you are student in Ireland, it is unlikely you have a handheld graphing calculator.

This online “graphing calculator” webpage will allow you to check the accuracy of your graphs. Along with displaying  the graphed function, it also shows the corresponding x, y couples in table format. 

Online Graphing Calculator

When checking your homework, if your graph doesn’t match the displayed graph, investigate if the reason is due to (i) a mistake in calculating your x, y couples or (ii) an error in plotting your points on your graph paper.

Another use for this tool is checking your solutions to simultaneous equations. When presented with a set of simultaneous equations and asked to solve algebraically, you can check your solution(s) by typing the equations into this graphing calculator and have it check for points of intersection. There is a dedicated tab on the graphing calculator for finding the points of intersection.

Access the Online Graphing Calculator here!  Thank you to Holt Online Learning for providing this application. Happy Graphing!

Simultaneous Eq’s: One Linear, One Quadratic

To date you have learned how to solve simultaneous equations using the elimination method –  for example, getting either the x term to cancel or the y term to cancel and following on from there.

When presented with one linear equation and one quadratic equation and asked to solve for simultaneous solutions, the substitution method is advised. The video clip below by David Handley will introduce you to the substitution method.

Simplifying Roots that include Surds

In using the quadratic formula and Theorem of Pythagoras, you will sometimes arrive at a solution or part of a solution that seeks to take the square root of a number that is not a perfect square, i.e.

3

or

15

These type numbers are generally called surds. When written in decimal format, they are non-repeating and go on forever!

The examples above are called entire surds, while the examples below are called mixed surds – they are made up of a surd multiplied by a rational number. The standard format is to write the rational number in front of the surd


2√ 3


means ” 2 times √ 3


8√ 7

7

This short video clip titled “Simplifying Roots” by ehow.com explains how to simplify roots when surds are involved.


Simplifying Roots — powered by eHow.com

The Quadratic Formula to the Rescue!

In this video clip, Patrick of www.justmathtutoring.com walks us through solving a quadratic equation using the Quadratic Formula.

Keep in mind, we typically turn to the quadratic formula after first trying to solve the given quadratic equation by the factoring method. This is because the factoring method is often quicker. If you can’t factor the quadratic, however, then turn to the quadratic formula to find the solutions.

On some exams, the quadratic problem will be presented to you in such a way that the wording of the question tips you off that you’ll need to use the quadratic formula. If the question requests your answer “in surd form” or correct to so many significant digits, then you will definitely have to use the quadratic formula.

Don’t forget,
* all quadratic equations have two solutions
* another name for the solutions of a quadratic equation is roots
* some quadratic equations have a double root – two identical solutions.

Thank you Patrick and www.justmathtutoring.com !

Algebraic Division

Have you ever gone into maths class and said to your teacher something along the lines of, “I understand how to do the problems when you do them on the board, but when I get home, I can’t do them.”

There is something about algebraic division that brings about this situation with students. Perhaps it is because some students were never comfortable with numerical long division or the fact that algebraic division requires one to be very careful in managing +/- signs.

If you find yourself in need of a “walk through” once you get home, I recommend this video clip by Greg Twitt.  Greg presents the simplest type of problem, but once you understand this standard example, the other problems are only slightly more difficult.

I trust this video clip will make you happy as Larry!

The Quadratic Formula Song

…and all across America and beyond, teenagers have been creating music videos dedicated to The Quadratic Formula, a.k.a. the -b formula!

That’s right, so determined to solve the most stubborn type of  quadratic equations – those that refuse to be solved by the factoring method – students are turning to The Quadratic Formula to ease their solution woes.

Come on, listen to it one more time, you know you want to! Extra marks for those students who transfer this clip to their MP3 player!

Quadratic Equation Summary

Lawrence Spector’s Quadratic Equations web page revise’s the standard form of the quadratic equation, the definition of a root and provides interactive examples of the types of quadratic factoring you’ll come across in the Junior Certificate (H) and Leaving Certificate course.

Quadratic Equations Revision

P.S.  Teachers –  when projected, this page works well in facilitating end-of-chapter discussion.

Factorising Quadratic Trinomials

There is no escaping a degree of trial-and-error when factorising quadratic trinomials. That said, the structured step-wise approach introduced in this YouTube clip will minimise the angst these type problems may cause you – and many other students. I call this gentleman’s approach the “No Drama” method and it is the same method we use in class.

Note:  This video was made for older maths students or teachers, but that doesn’t mean you won’t be able to understand the key points. Don’t try to take in all the speaker is saying, simply roll with his lecture until he gets his example problem to the stage you are are familiar with. You’ll be glad you did.