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:

  1.  t chart
  2.  equation
  3.  ordered pairs
  4.  graft
  5. 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.

An example:

 

Week 4 – Math 10 – Bearings and Angles of Elevations/Depression

It can be hard to solve problems using trigonometry if we don’t have a diagram to compare and look at. So when working to solve word problems knowing what bearings, the angle of elevation, and the angle of depression are helpful to create an illustration that will aid in the solving process.

What are bearings?

A bearing is an angle that is measured clockwise from the North direction. Planes and buses use bearings to determine a direction of travel based on their compass (as South from due North).

When solving many word problems, the angle of elevation or depression is determined by the eye level of the person and not from the ground. This is the line of sight, a sight line from the observer to some point of interest. To know what these angle mean, you also need to know what is something when it is vertical or horizontal.

Horizontal: is going side-to-side, for example, the horizon, a shadow, the ground

Vertical: is an up-down direction or position, for example upright, someone’s height, a tree, a building

The angle of elevation if the angle goes “upwards” from the line of sight. Therefore, the angle of depression is the “downwards” angle from the horizotal to a line of sight from the observer to some point of interest.

Week 3 – Math 10 – Finding Side Lengths Using Trig

What is trigonometry (trig)?

Trigonometry is the study of triangles and the relationship of angles and sides in a triangle. Before even finding side lengths using trig, you need to label all the sides of the triangle. There are three special names for each side; there is hypotenuse, opposite, and the adjacent. The hypotenuse is the longest side and is always across from the right angle (90-degree angle). The opposite side is across from the reference angle. And the adjacent is the side beside the reference angle.

What are the ratios?

SINE RATIO = opposite (divide by) hypotenuse

COSINE RATIO = adjacent (divide by) hypotenuse

TANGENT RATIO = opposite (divide by) adjacent

An easy way to memorize these ratios is with the acronym SOH CAH TOA. Therefore is each of the ratios first letters into a big word.

For every trig equation : Ratio (REFERENCE ANGLE) = side 1/side 2

In this example, we need to see what ratio we will use, and from the process of elimination the ratio needed is SINE. Next, we use algebra to find the answer.

 

 

Week 2 – Math 10 – Division Law

When simplifying an expression that includes exponents there are exponents laws that one could use to answer the question easier. The laws of exponents can be thought of just “tricks” or shortcuts that help us work with exponents in equations. Some of these laws for exponents are the Power Law, Multiplication Law, and Division Law. However, we are going to focus on just the division law.

The first thing to do when using the division law is to identify the same bases with exponents. If the bases are not the same, you can’t use the division law and won’t be able to simplify it any more. Next, you will subtract the exponents leaving you with one number with one exponent. To divide exponents (or powers) with the same base, subtract the exponents. The coefficients (if any) won’t be affected by the division law. As division is the opposite of multiplication, so it makes sense that because you add exponents when multiplying numbers with the same base, you subtract the exponents when dividing numbers with the same base.

My example:

 

If when you are subtracting the exponents and the result is a negative exponent then you will use the Negative law to further simplify the expression.