### 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

$a_1$

### 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:

### 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

$a_1$

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 Read more…

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:

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…

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:

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…

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

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

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…

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…

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…

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

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.

### What is a “system” of linear equations?

A “system of linear equations” means two or more linear equations that must all be true at the same time.

When represented symbolically, a system of equations will usually have some sort of grouping symbol to one side of them, such as the curly brace below, which is intended to convey that the set of equations should be considered all at once. For example:

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

When graphed, all of the equations in a system will be shown on the same set of axes, so that they can be compared to one another easily:

### What is “a solution” to a linear system?

A solution to a system of linear equations is Read more…

The phrase “Flipped Classroom” is appearing with increasing frequency in publications and blog postings. Yet, it seems to mean different things to different people. Many of the references I see to flipped classrooms are made by people or organizations who have a vested interest in selling goods or services, which probably affects their view of the issues.

As proposed by Salman Khan in his TED Lecture, flipping the classroom involves using internet-based video to move “lecture” out of the classroom to some other place and time of a student’s choosing. Class time can then be used for student problem solving and group work. Dan Meyer and others have critiqued aspects of Salman Khan’s approach, with some such as Michael Pershan offering constructive ideas for improvements.

Eric Mazur, a physics professor at Harvard, has also been advocating a “flipped” approach  – and for considerably longer than Salman Khan. His conception of “flipping” focuses on getting students to Read more…

What is the difference between a Problem and a Project? While it is difficult to draw a definitive line that separates one from the other, the attributes of each and their differences as I see them are:

### Problems

• Require less student time to complete (usually less than an hour)
• Focus on a single task, with fewer than 10 questions relating to it
• Can involve open-ended questions, but more often does not
• Are often one of a series of problems relating to a topic
• Look similar to many exam questions
• Can be used to introduce new concepts (Exeter Math)
• Can be used as practice on previously introduced concepts (most math texts)

### Projects

• Require more student time to complete (hours to weeks)
• Focus on a theme, but with many tasks and questions to complete
• Provide an opportunity to acquire and demonstrate mastery
• Ask students to demonstrate a greater depth of understanding
• Ask students to reach and defend a conclusion, to connect ideas or procedures
• Can introduce new ideas or situations in a more scaffolded manner

### Why Use Problems?

• Convenience
– Short time to completion makes it easier to fit into a class plan
– Multiple problems allow Read more…