Week 17 – Arithmetic Series

What is an Arithmetic series?

While building onto last week’s topic I did, arithmetic is in the same field of ideas when it comes to sequences. A sequence that ends on a certain number (not infinite) is an arithmetic series. We use arithmetic series to find the sum of all of the numbers in the named sequence. Which sounds easy when it comes to a series that is only five numbers and go up by 2. So you can just do simple adding and it is not likely that you would make any mistakes. With sequences that are much bigging or the difference between each number is huge, it doesn’t make any sense to add up each number in that sequence. That would take ages and one small error would ruin all of your work.

So how can you find the sum of all of the numbers in a sequence?

WHILE it is all about the patterns! If you are to take the sequence 1,2,3 … 98, 99, 1oo and add the first and last term together, a really cool trick and a pattern is able to be seen. When you are to add 1 and 100, you get the number 101. If you are to do this again the next smallest and next largest term together, you will also get 101. This means in any arithmetic sequence the smallest number and the biggest added together will equal the next smallest and biggest and so on. How can we use this information to find the sum of all of the numbers in a series? You can take the sum of the largest and the smallest number, then multiply that number by the amount of numbers divided by 2 in the series together.

An example of finding the sum:

Week 16 – Sequences

What are sequences?

It is usually a list of numbers but can also be a list of other things. Each of the objects or numbers in the list are called terms, elements, member meaning all the same thing. If the sequence is to go on forever then it is called an infinite sequence. For sequences that end, they are called finite sequence. Sequences are very similar to a set, where there is an order to each set (does not matter what order) and it sets the same value that appears more than once if it is following the patter. Most sequences have a rule that you must follow to find out the next term in the sequence.

For example, the sequence {3,5,7, 9…} starts at 3 and jumps 3 every time. Even if you are to say that this sequence “starts at 3 and jumps 3 every time” doesn’t help to find out further terms like the 10th, 100th, or the 1000th term in this sequence. So, the rule that you can make is for this sequence is something like 2 times n (the n is the number of the term). However, that won’t work as it is off by one each time. So, if we were to try 2n +1 it would work as a rule.

Here are some examples of patterns that are also known as sequences.

Week 15 – Solving Systems

What is a system?

In math, a system is a two or more linear equations involving the same set of variables. Meaning each system must have x and y in the equation or what other variables you are using for the equation. One example of a system is this:

How to solve a system?

There are many different ways to solve a system such as grafting, substitution and elimination. However, I am going to focus only one of the ways to solve systems substitution. What is the method of substitution? The substitution method is used to eliminate one of the variables by replacement when solving a system of equations. You can think of it as “grabbing” what one variable equal from one equation and “plugging/substituting” it into the other equation. This method works best when one of the linear equations have a variable that does not have a coefficient (so it is all by itself). You can still use it with variables that all have coefficients but it will probably end up in fractions.

The first step is you are going to isolate one of the variables in one of the linear equations, try to make sure it is the best choice (the easiest). Next, with the isolated equations, you are going to “plug” this equation (“substitute it”) for the variable you choose in the other equation, and solve for the other variable. This will leave you with an equation with only one variable. After you are done solving and you have an answer. You are going to take the answer and plug this variable value back into either equation and solve for the other variable that you started with. Lastly, you can verify your answer by inputting the variables that you have come up with into the original equations.

For Example:

Week 11 – Finding Distances on a Coordinate Plane

Before we start to find the distances of lines, there are three different types of lines that you can find. The first one is a horizontal line, a horizontal line is a line that is a straight line that goes from left to right. And on a coordinate plane, a horizontal line runs parallel to the x-axis. The next one is a vertical line that is a straight line that goes up and down. On a coordinate plane, a vertical line will run parallel to the y-axis. The final type of line is an oblique line. It is lines that are drawn at an angle (other than 900) to the horizon. They are neither vertical nor horizontal.

There are two ways to find different types of distances on a coordinate plane. If the coordinate plane you are using has small or easy numbers to count and the line is either horizontal or vertical you can count the points to find the distances between the two points. However, as it gets harder when the numbers get bigger or the line is an oblique line, you need to use another method.

When the line is either horizontal or vertical the way to find the distances is to take the coordinates of the to points that you want to find the distances between. Then the points that are different (the x-axis ones for horizontal lines, and the y-axis ones for the vertical lines) and subtract them. Make sure that the number you get is positive and if it is not, just change it to be positive.

However, is the line is oblique, there is another method to find the distances of the points. When you look at this line on a coordinate plane it looks like a hypotenuse line which gives you a big clue on how to solve for the distance. You can make a triangle and find the distances of the vertical and horizontal line of the triangle. Finally, you can use Pythagoras’s theory ( a squared + b squared = c squared) to find the distances of the last line.

Some examples:

Week 10 – Function Notation

What is a function again?

It is a special group that is a part of relations. This group only has one output for every input. This means that all functions are relations but not all relations are functions. An example of a function is how each person only has one biological father.

So what is function notation?

Well, it can be considered as the way we write functions. It is meant to be a precise way of giving information about a function. Instead of writing out the whole situation and the chance of people getting confused from the words presented. Also, this is to make the function simpler for the reader. To not mix up different functions, each one is given a name, and they are referred to a single letter.

How to write function notation?

While to start you are going to give the function a name. For my example, I am going to use the letter f. However, it can be any letter you want. The name of the function will look like this: f(x). The f(x) notation is another way of representing the y-value in a function, y = f(x). You can also label the y-axis as the f(x) axis when you are graphing. Next, you add the formal to the end.

Week 9 – Relations and Functions

What is a relation?

A “relation” is just a relationship between sets of information. For example; think of all the people in one of your classes, and think of their heights. The pairing of names and heights is a relation. The set of all the starting points (input) is called “the domain” and the set of all the ending points (output) is called “the range.”

What is a function?

One way to think of a function is as a machine that you put something into and will get a new result. The thing that you put into the machine is called the input, domain or x. While the result is called the output, range, or y. And the output is related somehow to the input. When it comes to functions we can think about them in many parts but there will always be three different parts; the input, the relationship, and the output.

Even though a function can be thought of like a machine, it really does not have any real life machine parts and nothing that we put into the machine is being destroyed. A function relates an input to an output. But a function has special rules: It must work for every possible input value and it has only one relationship for each input value. A function relates each element (number) of a set with exactly one element (number) of another set. (it can be possibly the same set).

For example:

Let’s say there is a tree that grows 20cm every year, so this means that the height of the tree is related to its age using the function h: h(age) = age x 20.

What is the difference between a function and a relation?

Functions are a sub-classification of relations. This means that while all functions can also be called a relation as they pair information, not all relations are functions. Compared to a relation, in a function; given a starting point, we know exactly where to go; given an x, we get only and exactly one y. For a relation to be a function, there must be only and exactly one y that corresponds to a given x.

The “Vertical Line Test”

When you are just looking a set of numbers or a tchart it can be quite easy to spot the numbers from the input that repeat and have a different output. However, when it comes to graphs, it can be more difficult to find out if it is a function or just a relation. A way to find if it is a function or relation is by looking at the given the graph of a relation if you can draw a vertical line that crosses the graph in more than one place, then the relation is not a function.

Week 8 – Math 10 – Relations and Variables

To start off with, what are relations?

Well, a relation is a comparison between two sets of elements (also know as numbers) For example the cost of bagged candy is related to the weight. In this example, the cost can also be called the dependent variable, output, y, and the range. When the weight is the independent variable, input, x, and the domain.

What is the independent and the dependent variable?

In any relation or comparison, the independent variable is the factor that determines the dependent variable. In the way my example, how the amount (weight) of bagged candy will determine how much you pay. The dependent variable will always be depending on the number of the independent variable.

5 Ways to Show a Relation:

t chart
equation
ordered pairs
graft
mapping diagram

What is the domain?

The domain is the set of all numbers of the independent variable (input – x) in the relation.

What is the range?

The range is the set of all the numbers of the dependent variable (output – y)

Week 7 – Math 10 – Factoring Ugly Polynomials

When factoring is already difficult to do at times if you don’t know where to start, factoring ugly polynomials are twice as hard. So here is an easy and fun way to get the answer quickly. Firstly, what are ugly polynomials? Are they ugly by the way they look? No! Ugly polynomials are polynomials that can’t factor as easily as one where you can use a simple pattern to solve for. So where to start, while let us go back to multiplying polynomials and how we have certain ways to solve the expression easier.

For Example, Area Model can be also be used for factoring more difficult or ugly polynomials. Same like with using it for multiplying, you are going to draw out of square with the amount of tiny square need to fill in the numbers from the polynomial. First, put the number with the largest degree in the top left square and the constant in the bottom cube on the right. Then take the middle number and find the factor of it. Then find the factor of the numbers in the big square across from each other. Next, doing the same thing to the other number across from themselves. Finally, to check if you did it right, multiply the polynomial.

An Example:

Week 6 – Math 10 – Factoring Polynomials

First of all what is factoring? Well, it is the opposite of multiplication and is parallel to division. So when we are factoring polynomials it is basically doing the opposite to what we did in multiplication.

Here are three easy steps to factoring :

1) Take out the Greatest Common Factor (GCF) if possible.

One way to take out the GCF is by using prime factorization. A short reminder that prime factorization is in number theory, integer factorization is the decomposition of a composite number into a product of smaller integers that are prime numbers.

2) Identify the number of terms.
3) Check by multiplying.

What is the GFC :

It is short for Greatest Common Monomial, you multiply this (the GCM) by the sum/difference of the remaining monomials.

Example using the three easy steps :

After you have the answer to check if it is right, take the factor and multiply it.

Week 5 – Math 10 – Multiplication of Polynomials

Compared to last year where we only did the multiplication of monomials and there was one way that we could evaluate the expression. However, with the multiplication of polynomials, there are a couple of ways. Before moving on to two of the ways of evaluating the polynomial, here is a short review of them.

What are polynomials?

It is an expression that has more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s). There are four types of polynomials; a monomial, binomial, trinomial, and a polynomial. A monomial means that there is only one term when in a binomial there are two terms. So that means that a trinomial has three terms and polynomials have anywhere more than four terms. Polynomials can’t have an exponent of the negative integer in the numerator. As well as the variable can’t be in the denominator.

The first way of evaluating a problem with multiplication: VISUALLY

Like if you were going to use algebra tiles to look at the problem on a table, you can draw out the algebra tiles on a sheet of paper. The first step to doing this is by taking the expression and putting on the side on top of a rectangle and the other side on the right side of the rectangle. Then by lining up the shapes, you will get the simplified answer. After writing out the answer in its numeric form.

An example:

Another way to evaluate a polynomial expression: AREA MODEL

When we were learning about these in math class, it made me remember in science when we were learning about Punnett Square and how they looked and worked similarly. In an area model, you will divide up the expression into different terms. Then make a square and put the terms on the sides of the square. The two terms that meet in the middle to make another square get multiply together. In the end, you add all of the sums from the numbers.