Skip to content

Simplifying Fractions

August 14, 2014

Consider the following fraction… can it be simplified? If so, why can it be simplified?

\dfrac{12}{15}

The above fraction can be simplified, because both numerator (the top expression) and denominator (the bottom expression) share a common factor. By factoring both numerator and denominator, then pulling out the factor they each have in common (using the reverse of how fractions are multiplied), we end up with a fraction that equals one being multiplied by a simpler fraction than the original:

\dfrac{12}{15}~~=~~\dfrac{3\cdot 4}{3\cdot 5}~~=~~\dfrac{3}{3}\cdot\dfrac{4}{5}~~=~~1\cdot\dfrac{4}{5}~~=~~\dfrac{4}{5}

Since multiplying by one does not change a quantity, we can drop the multiplication by one from our expression, leaving behind the simplified fraction as our final expression on the right.

This is the only process that allows you to Read more…

Negative Differences

August 3, 2014

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:

b-(a-3)

If we wish to eventually drop the parentheses, we’ll have to distribute the negative sign in front of them first.  Leaving the parentheses in place while Read more…

Negative Fractions

July 31, 2014

Where should I put the negative sign when I am writing a fraction like negative two thirds?  As long as you write only one negative sign, it does not matter where you put it.

Subtraction is the same thing as the addition of a negative. The negative of a number is created by multiplying the number by negative one. These rules apply to fractions as well, so:

-\dfrac{3}{5}\\*~\\*~\\*=(-1)(\dfrac{3}{5})\\*~\\*~\\*=(\dfrac{-1}{~1})(\dfrac{3}{5})\\*~\\*~\\*=\dfrac{-3}{~5}

So, placing the negative sign before the entire fraction (subtracting the fraction) is equivalent to Read more…

Summary: Geometric Sequences and Series

May 8, 2014

Geometric Sequences

Pick a number, any number, and write it down.  For example:

5

Now pick a second number, any number (I’ll choose 3), multiply it by the first number, then write their product down to the right of the first number:

5,~15

Now, continue multiplying the second number by previous product and writing the result down… over, and over, and over:

5,~15,~45,~135,~405,~1,215, ...

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, 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

a_1

This notation is  Read more…

Summary: Arithmetic Sequences and Series

May 5, 2014

Arithmetic Sequences

Pick a number, any number, and write it down.  For example:

5

Now pick a second number, any number (I’ll choose 3), add it to the first number, then write their sum down to the right of the first number:

5,~8

Now, continue adding the second number to the sum and writing the result down… over, and over, and over:

5,~8,~11,~14,~17,~20,~23,~26,~29, ...

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:

ArithSequence

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

a_1

This notation is Read more…

Piecewise Functions and Relations

April 21, 2014

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:

Piecewise1
Acceleration up to a speed limit
Piecewise2
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 Read more…

Polynomials and VEX Drive Motor Control

January 20, 2014

VEX Robots can be more competitive when they have addressed several drive motor control challenges:

  1. 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.
  2. 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.
  3. 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:

Read more…

Linear Systems: Why Does Linear Combination Work (Graphically)?

December 26, 2013

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:

\begin{cases}y=-3x+2\\y=x-6\end{cases}

which, when graphed, looks like: Read more…

Successfully Asking Questions In Class

September 6, 2013

Do you ask questions in class at least once per week? For many students, the answer is probably “no”.  Reasons for such an answer may include one or more of:
– I don’t want to let my peers or the teacher know I don’t understand something
– I am uncertain about what to ask… I just don’t get what the teacher is talking about
– I don’t wish to appear to be the teacher’s “pet”
– I am not being called on when I raise my hand
– Someone else asked a question first, and the teacher needed to move on
– The teacher has not answered my past questions – they just said “see me after class”

Preparation

A number of small preparatory steps may help get your questions answered in class, particularly if your class is a large one.  The need for such steps will vary greatly from one school to another, or one teacher to another, but they will not hurt your efforts to master the subject even if they are not necessary to get your questions answered during class time:

  • Ask questions of your teacher outside of class time on several occasions early in the semester, particularly if you have never had this teacher before.  This will help you get to know your teacher a little better, and will also help your teacher:
    – associate your name with your face
    – gain some Read more…

What A Parent Wants From A School

August 22, 2013

As a parent, I look for two categories of attributes when choosing a school for my child:
– Ones which benefit my child directly
– Ones which benefit my child indirectly, by helping others (teachers, parents) do their jobs more effectively

Schools that satisfy more of the attributes in both categories are likely to have happier parents and more successful students.

The Administration and Teachers Should Help My Child

Directly By:

  • Being aware of history. Before the start of each school year, my child’s current teacher(s) should have reviewed all of
    – last years’ teacher comments for my child
    – my child’s transcript (all courses, all years at the school)
  • Helping my child to both pursue existing  Read more…

Domain, Range, and Co-domain

June 24, 2013

When working with quantitative relationships, three concepts help “set the stage” in your thinking as you seek to understand the relationship’s behavior: domain, range, and co-domain.

Domain

The “domain” of a function or relation is:

  • the set of all values for which it can be evaluated
  • the set of  allowable “input” values
  • the values along the horizontal axis for which a point can be plotted along the vertical axis

For example, the following functions can be evaluated for any value of  “x”:

f(x)=2x+1\\*~\\*g(x)=x^2+5

therefore their domains will be “the set of all real numbers”.

The following functions cannot be evaluated for all values of “x”, leading to restrictions on their Domains – as listed to the right of each one:

h(x)=\dfrac{1}{x}~~~~~~~~~\text{x cannot be zero}\\*~\\*j(x)=\dfrac{1}{(x-2)(x+4)}~~~~~~\text{x cannot be 2 or -4}\\*~\\*k(x)=3x+2,~1<x<10~~~~\text{only values between -1 and 10 may be used for x}

The values for which a function or relation cannot be Read more…

Summary: Roots and Rational Exponents

May 7, 2013

Although addition and multiplication are commutative, exponentiation is not: swapping the value in the base with the value in the exponent will produce a different result (unless, of course, they are the same value):

2^3 \ne 3^2

Therefore, two different inverse functions are needed to solve equations that involve exponential expressions:
– roots, to undo exponents
logarithms, to undo bases

Just as there are many versions of the addition function (one for each number you might wish to add), and many versions of the “logarithm” function (each with a different base), there are many versions of the “root” function: one for each exponent value to be undone.

Notation

The symbol for a root is \sqrt{~~~~}, and is referred to as a “radical“.  It consists of a sort of check mark on the left, followed by a horizontal line, called a “vinculum”, that serves as a grouping symbol (like parentheses) to Read more…

Short Assessment Grading: Add or Average?

May 6, 2013

Long assessments can waste precious class time unless there is much material to be assessed, but shorter assessments (with few questions) can cause small errors to have too big an impact on a student’s grade.

For example, consider the following assessment lengths where each question is worth 4 points, and the student has a total of two points subtracted from their score for errors:

# Questions Points % % Grade
1 2 / 4 50% F / F
2 6 / 8 75% D / C
3 10 / 12 83% C+/ B
4 14 / 16 88% B / B+
5 18 / 20 90% B+/ A-

The “% Grade” in the table above reflects a 7-point / 10-point per letter grade approach. A one question quiz is risky for students: they could get a failing grade for losing two points on the only question. Two question quizzes are only slightly less risky.  Only with three or more questions does this scenario start to minimize the risk of actively discouraging a student who loses several points.

Should quizzes therefore only have three or more questions? What if I don’t want the class to spend that much time on an assessment, or don’t have Read more…

Summary: Logarithms

April 25, 2013

Unlike the two most “friendly” arithmetic operations, addition and multiplication, exponentiation is not commutative. You will get a different result if you swap the value in the base with the one in the exponent (unless, of course, they are the same value):

3^2 \ne 2^3

The most significant impact of this lack of commutativity arises when you need to solve an equation that involves exponentiation: two different inverse functions are needed, one to undo the exponent (a root), and a different one to undo the base (a logarithm).

Just as there are many versions of the addition function (adding 2, adding 5, adding 7.23, etc.), and many versions of the “root” function (square roots, cube roots, etc.),  there are also many versions of the “logarithm” function. Each version has a “base”, which corresponds to the base of its inverse exponential expression.

Inverse Functions: Logarithms & Exponentials

Logarithms are labelled with a number that corresponds to the base of the exponential that they undo. For example, the Read more…

Unintended Consequences of a 0 – 100 Grading System

January 17, 2013

If a student makes four errors in the course of answering ten questions, what is an appropriate grade? Presumably, it would depend on the severity of the errors and the nature of the questions. Consider how your approach to grading might vary if students had been asked to:

- match ten vocabulary words to a word bank, or
– define each of ten words, then use each appropriately in a sentence

- complete ten 2-digit multiplication problems, or
– solve ten multi-step algebra problems, each requiring a unique sequence of steps

- answer ten questions similar to what they have seen for homework or in class, or
– answer ten questions unlike ones they have been asked before

Would you label each answer as right or wrong, then use percentage right as the grade?
Would you assign a number of points to each answer (if so, out of how many points per question)?
Would you assign a letter grade to each answer (whole letters only, or with +/-)?
What would you consider a “D” set of answers?
What would you consider an “A” set of answers?

Would your answers vary depending on whether you had created the assessment yourself, or were using someone else’s questions?

Many math/science teachers seem to use a percentage approach (based on total points earned or number correct) more often than any other, particularly when their school defines its letter grades using a 0 – 100 scale. Teachers of other subjects also use this scale often, but less so for “free-response” questions. While a percentage approach can work well for some assessments, it can have unintended consequences for others.

Similar Right/Wrong Questions

When asking a series of similar questions, such as Read more…