The solution to each of the following twenty problems is 12. Focus on finding the most helpful algebraic step to take a reader from the problem as stated to the solution, and be sure you can explain why that step leads to a solution.
Many math students are given strict instructions by their teachers to do all their work in pencil. I disagree.
The advantage of doing work in pencil is that:
- it is easier to erase, so students are less likely to be paralyzed by “I am not sure this is correct, so I don’t dare write it down”
The disadvantages of working in pencil are that: Continue reading Math: Pen vs Pencil
1) Reflect and Summarize
at the end of each class, at the end of each week, at the end of each month. Review your notes and/or think back over the material that has been covered, then decide which skills or ideas you think are most important. Summarize the material you are learning as concisely as you can, because summarizing helps you learn. Identify any skills or ideas that you are not confident about. Write your reflections and summarizations as part of your notes, with reminders about what needs more work.
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!
GeoGebraBook: Exploring Linear Functions, which contains:
Check out this GeoGebraBook of nine applets that will help you explore Unit Circle Symmetries. It contains three applets per type of symmetry on the unit circle, one focusing only on the unit circle, and the other two linking unit circle properties to patterns in the graphs of the sine and cosine functions.
When two angle expressions, such as and , exhibit symmetry on the unit circle, an understanding of unit circle symmetries and reference angles often allow function arguments to be simplified. Mastery of symmetries and reference angles will also be very handy when expanding inverse trigonometric function results to describe all possible answers to a problem.
Suggestions for improvements to these applets, or additional applets, are always welcome via comments on this post.
What Does Absolute Value Mean?
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 Continue reading Absolute Value: Notation, Expressions, Equations
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 Continue reading Combining or Collecting Like Terms
Three concepts help explain the process of simplifying fractions:
- Multiplying a quantity by 1 has no effect
- A fraction whose numerator is exactly the same as its denominator is equal to 1 (unless the denominator equals zero)
- 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? Continue reading Simplifying Fractions
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 Continue reading Negative Differences
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 Continue reading Negative Fractions
Geometric Sequences / Progressions
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 Continue reading Geometric Sequences and Geometric Series
Arithmetic Sequences / Progressions
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 Continue reading Arithmetic Sequences and Arithmetic Series
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 Continue reading Piecewise Functions and Relations
VEX Robots can be more competitive when they have addressed several drive motor control challenges:
- Stopping a motor completely when the joystick is released. Joysticks often do not output a value of “zero” when released, which can cause motors to continue turning slowly instead of stopping.
- Starting to move gradually, not suddenly, after being stopped. When a robot is carrying game objects more than 12 inches or so above the playing field, a sudden start can cause the robot to tip over.
- Having motor speeds be less sensitive to small joystick movements at slow speeds. Divers seeking to position the robot precisely during competition need “finer” control over slow motor speeds than fast motor speeds.
These challenges can be solved using one or more “if” statements in the code controlling the robot, however using a single polynomial function can often solve all of these challenges in one step. A graph can help illustrate the challenges and their solution:
A system of linear equations consists of multiple linear equations. The solution to a linear system, if one exists, is usually the point that all of the equations have in common. Occasionally, the solution will be a set of points.
There are four commonly used tools for solving linear systems: graphing, substitution, linear combination, and matrices. Each has its own advantages and disadvantages in various situations, however I often wondered 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.
Consider the linear system:
which, when graphed, looks like: Continue reading Linear Systems: Why Does Linear Combination Work (Graphically)?