Some may have had trouble using my GeoGebra applets in their browsers. I have moved all of them to GeoGebraTube, which will hopefully fix the problem. You may search for them by typing “MathMaine” into the GeoGebraTube search box.
Links to all updated interactive graph applets are below. Comments and suggestions are always welcome!
The term “Absolute Value” refers to the magnitude of a quantity without regard to sign. In other words, its distance from zero expressed as a positive number.
The notation used to indicate absolute value is a pair of vertical bars surrounding the quantity, sort of like a straight set of parentheses. These bars mean: evaluate what is inside and, if the final result (once the entire expression inside the absolute value signs has been evaluated) is negative, change its sign to make it positive and drop the bars; if the final result inside the bars is zero or positive, you may drop the bars without making any changes:
Another example is:
Note that absolute value signs do not instruct you to make “all” quantities inside them positive. Only the final result, after evaluating the entire expression inside the absolute value signs, should be made positive.
Absolute Value expressions that contain variables
Just as with parentheses, absolute value symbols serve as grouping symbols: the expression inside the bars must be evaluated and expressed as either zero or a positive quantity before the bars may be dropped. While this is fairly straightforward when working with constant values, as shown above, what happens when a pair of absolute value signs contains a variable?
The phrases “combine like terms” or “collect like terms” are used a lot in algebra, and for good reason. The process they describe is used a lot in solving algebra problems. Two approaches, one intuitive and the other algebraic, can help in understanding why some terms are “like” terms, and others are not.
Quantities With Units
Suppose you are sitting in front of a table that holds three piles of fruit: – five apples – three oranges – four apples If someone asks you “What do you see on the table?”, how would you answer the question?
Chances are you answered “nine apples and three oranges”. Why did you combine the two piles of apples with one another, but not with the oranges? How did you know that you could do that?
The quantities of apples may be combined because addition or subtraction only work with quantities that have the same units, such as apples in the example above. If you were asked to add three apples and two oranges, the only answer you could give is “three apples and two oranges”, since neither addition nor subtraction make sense with quantities that have differing units. The concepts of addition and subtraction are based on combining or separating quantities with the same units.
Three concepts help explain the process of simplifying fractions:
1. Multiplying a quantity by 1 has no effect
2. A fraction whose numerator is exactly the same as its denominator is equal to 1 (unless the denominator equals zero)
3. A product of two fractions can be rewritten as a fraction of two products (and vice versa)
To simplify a fraction:
Rewrite both numerator and denominator as products of factors (if they are not already factored)
Examine both numerator and denominator to see if they share any factors
If they do share factors, use concept (3) above to move the shared factors into a separate fraction
That separate fraction should now have a numerator that is exactly the same as its denominator, which by concept (2) above means that it must equal 1, therefore by concept (1) above we can drop it from the expression
Consider the following fraction… can it be simplified?
Algebra is a set of rules that allow us to change the appearance of an expression without changing the quantitative relationship that it represents. Sometimes the changes in appearance are greater than expected, causing us to doubt whether two expressions really do represent the same quantitative relationship. The ways in which negative differences can be rewritten seem to surprise people until they become accustomed to them.
Consider a difference that is being subtracted:
If we wish to eventually drop the parentheses, we’ll have to distribute the negative sign that is in front of them first. Leaving the parentheses in place while distributing the negative produces:
Question: Where should I put the negative sign when I am writing a fraction like negative two thirds?
Answer: As long as you write only one negative sign, it does not matter where you put it.
Two ideas are useful to keep in mind during the explanation that follows: – Subtraction is the same thing as the addition of a negative. – The negative of a number can be created by multiplying the number by negative one.
These principles apply to fractions as well, so:
Placing the negative sign before the entire fraction (subtracting the fraction) is equivalent to adding the same fraction, but with a negative numerator. But this is not the only option…
The terms “sequence” and “progression” are interchangeable. A “geometric sequence” is the same thing as a “geometric progression”. This post uses the term “sequence”… but if you live in a place that tends to use the word “progression” instead, it means exactly the same thing. So, let’s investigate how to create a geometric sequence (also known as a geometric progression).
Pick a number, any number, and write it down. For example:
Now pick a second number, any number (I’ll choose 3), which we will call the common ratio. Now multiply the first number by the common ratio, then write their product down to the right of the first number:
Now, continue multiplying each product by the common ratio (3 in my example) and writing the result down… over, and over, and over:
By following this process, you have created a “Geometric Sequence”, a sequence of numbers in which the ratio of every two successive terms is the same.
Vocabulary and Notation
In the example above, 5 is the first term (also called the starting term) of the sequence or progression. To refer to the first term of a sequence in a generic way that applies to any sequence, mathematicians use the notation
This notation is read as “A sub one” and means: the 1st value in the sequence or progression represented by “a”. The one is a “subscript” (value written slightly below the line of text), and indicates the position of the term within the sequence. So represents the value of the first term in the sequence (5 in the example above), and represents the value of the fifth term in the sequence (405 in the example above).
The terms “sequence” and “progression” are interchangeable. An “arithmetic sequence” is the same thing as an “arithmetic progression”. This post uses the term “sequence”… but if you live in a place that tends to use the word “progression” instead, it means exactly the same thing. So, let’s investigate how to create an arithmetic sequence (also known as an arithmetic progression).
Pick a number, any number, and write it down. For example:
Now pick a second number, any number (I’ll choose 3), which we will call the common difference. Now add the common difference to the first number, then write their sum down to the right of the first number:
Now, continue adding the common difference to the sum and writing the result down… over, and over, and over:
By following this process, you have created an “Arithmetic Sequence”, a sequence of numbers that are all the same distance apart when graphed on a number line:
Vocabulary and Notation
In the example above 5 is the first term, or starting term, of the sequence. To refer to the starting term of a sequence in a generic way that applies to any sequence, mathematicians use the notation
This notation is read as “A sub one” and means: the 1st value of the sequence represented by “a”. The one is a “subscript” (value written slightly below the line of text), and indicates the position of the term within the sequence. So represents the value of the first term of the sequence (5 in the example above), and represents the value of the seventh term of the sequence (23 in the example above).
While many relationships in our world can be described using a single mathematical function or relation, there are also many that require either more or less than what one equation describes. The behavior being described might start at a specific time, or its nature changes at one or more points in time. Two examples of such situations could be:
Acceleration up to a speed limit
Free fall then controlled descent
In the graph on the left, note that the blue line starts at the origin. It does not appear to the left of the origin at all. Furthermore, when x = 3 the blue line stops and the green line begins – but with a different slope.
In the graph on the right, note that the blue curve starts at x = 0. It does not appear of the left of the vertical axis at all. And when x = 3 the blue parabola turns into a green line with a very different slope. And the green line stops at x = 5.5, just as it reaches the horizontal axis.
These graphs do not seem to follow all the rules you were taught for graphing lines or parabolas. Instead of being defined over all Real values of x, they start and stop at specific values. The graphs also show (in this case) two very different functions, but in a way that makes them look as though they are meant to represent a single, more complex function. Both of these graphs are examples of “Piecewise Functions”, functions defined in a way that lets you mix and match as many separate components (pieces) as you wish.
A system of linear equations consists of multiple linear equations. You can think of this as multiple lines graphed on one coordinate plane. Three situations can arise when looking at such a graph. Either: 1) No point(s) are shared by all lines shown 2) There is one point that all lines cross through 3) The lines lie on top of one another, so there are many points that the lines have in common.
There are four commonly used tools for solving linear systems: – graphing – substitution – linear combination – matrices Each has its own advantages and disadvantages in various situations, however I often used to wonder about why the linear combination approach works. My earlier post explains why it works from an algebraic perspective. This post will try to explain why it works from a graphical perspective.
7/17/11 Update: There can be great value in work that contains mistakes. Learning to catch your own mistakes is a critical life skill, as is learning to review other people’s work while seeking to understand it fully (the best way to do this is by looking for mistakes).
Along these lines, I came across an interesting blog posting by Kelly O’Shea. She came up with the idea of insisting that each group who is presenting try to sneak a mistake in their work past their peers. Brilliant!
This post begins a series intended to help introduce or re-introduce some of the core concepts of Algebra. It is often very helpful to re-visit these concepts with students who may have memorized their way through previous math courses without slowing down to contemplate some of the concepts behind Algebra.
Numbers are used in the English language as both nouns and adjectives. However, the only physical instance of a “number” in our world is a symbol or group of symbols such as “23”. Most of the time, we seem to use numbers as adjectives: “Look at the two trees.” Continue reading Algebra Intro 1: Numbers and Variables
Mathematical thinking probably started with addition. Someone may have combined two piles of bricks, and wondered how many were in the single large pile. Addition is the mathematical term that describes “joining quantities together”.
Properties of Addition
Looking at the patterns that can occur when quantities are joined together, you might have noticed that it does not matter whether 2 bricks are added to a pile of 3, or 3 bricks are added to a pile of 2… either way, we still end up with a pile of 5. So, the order in which we add two quantities does not change the result. This probably makes intuitive sense to you when you visualize the situation above and the final pile that results. Continue reading Algebra Intro 2: Addition