Category: Math 9

What I Learned About Grade 9 Linear Relations

What is a linear relation?

A linear relation is simply a pattern of some sort. There are many ways to explain a Linear Relation but the most common one is usually something to do with patterns and relations. You can graph a Linear Relation, you can make a t-chart and you can even make it nonlinear! You may not not what that means yet but here’s a pattern that is simple and you can tell how much its going up by.

This is a pattern! And you can tell how much its going up by. Every time its going up by 3, and it get bigger and bigger if we add more to it.

 

 

 

How to find the rule of the pattern?

Finding the rule is super easy when you have a t- chart! You plot numbers in that t- chart and you will always end up with #x= y or it can be a bigger relation #x+#=y, the hashtags represents which number goes in. Sometimes the t- chart is some what filled up, but some times it not and you would have to find out what the “rule” is. There are two sides in a t- chart side one has all the x values and the other side has all the y values. here is an example of a t- chart.

 

As you can tell there is some sort of pattern going on inside this t- chart. Any linear relation has a pattern and this one is going up by 2 on the y side and going up by one on the x side. We need to add 2 because when we times 1(2) we get 2 but we need to end up at 4! So that’s where we add 2 more to get 4.

 

 

 

T-chart – what is it? how to fill it:

A t- chart is the easiest way to find the rule of a pattern. You are just putting the coordinate numbers on the chart and finding out how much it goes up by and how much do we need to add, or subtract to get the final answer.

The coordinates can be any number as long as we can find the pattern and solve the equation. The x’s are usually 1,2,3,4 but it can be any number you choose. The y’s are different form the x’s because they are the ones that actually have a pattern to it.

 

 

 

 

How to plot a point (coordination):

In plotting/ coordinating you have to be very careful of where your putting your plot because since the plots are so small you cant tell where the plot is unless you look very carefully and go over it with your finger or pencil. Everything starts with the x value so you should look at the x value and go up to the y. Once you have all the coordinates and you are confident that those are it, you would put them into a t- chart like i said above and find the rule.

 

 

 

 

How to graph a linear relation:

linear relations are just straight lines and everything is connected and has a relation for example,

linear relations are simple because they are lines that have a pattern to it.

 

 

How to graph vertical and horizontal lines:

for horizontal and vertical lines you can tell if they are on the x- axis or on the y- axis. for the vertical line you can see that the number three is on the x axis so it would be x=3. And for the horizontal line you can see that its not on the x- axis, its on the y- axis so for that one it would be y=2.

 

 

 

Vocabulary: X and Y axis , t- charts, coordination, quadrants (1-4), origin, plotting, linear patterns, increasing and decreasing patterns

 

X axis is on the bottom line also known as horizontal

Y axis is on the upper line and also known as vertical

We first look at the x axis for everything then we go up to the y axes. It can be negative or positive but always starts with x.

 

T- charts:

The charts are used to find the rule or just to plot your coordinates.

Coordination:

Coordination is where you put your plotting. May look like this (-2,3)

Plotting: 

Plotting is where your coordination is located and where to put it on the graph.

Origin:

the origin is in the middle of the graph so that means the plotting for it would be (0,0)

Linear pattern:

It is where the x and y have a pattern going any way but, both are connected by a certain rule. Most linear pattern are in a straight line

4 Quadrants: 

Increasing pattern:

This is when your pattern is just increasing and going up wards, it will always be positive. For example +3x

Decreasing pattern:

This is the opposite of the increasing pattern so the pattern goes down and it will always be negative. For example -3x

Horizontal line:

The x- axis is horizontal so we would always be on the x axis (flat) never going up or down.

Vertical line:

The y- axis is vertical so we would be going straight up or down we wouldn’t be flat.

 

 

Include a screen shot of the desmos (initials NDA)

 

 

 

 

Core Competencies:

 

Math 9 cc

 

 

 

sources:

https://encrypted-tbn0.gstatic.com/images?q=tbn%3AANd9GcSnh_p1jiEkQOAZ1E0kl_PoSXmRvYahWslRSouTjLALfOOub7o4&usqp=CAU

 

What I Have Learned About Grade 9 Inequalities

What is an inequalities?

Inequalities can mean a lot of things due to what the symbols mean. Some of the symbols mean less than, equal to or even more than/ equal to! Inequalities don’t have equal signs they only have their own signs. some inequalities actually don’t equal to each other and that’s when you do the equal sign with a line in the middle

 

What does the symbols mean?

there are a lot of meaning behind each symbol and what they mean and how to make sense of it. Some people think of it as the PAC man symbol. So, if the PAC man symbol is open to the number than it is smaller because its eating the number so becomes small. But if the PAC man symbol is facing where the point is then the number is bigger because its not eating it. this is what i mean about the *eating it* and *pointing it*

 

 

How to solve – Divide by a negative?

if you are solving by a negative number there is always this one step you will have to take. And it is just simply flipping the symbol the other way of where it was already at. You do everything the same when you divide by a negative its just at the end when your dividing you would have to flip EVERYTHING over! its literally flipping everything over.

 

 

Graphing Inequalities ————-3———-

There are two main points when you are graphing Inequalities. there is an open whole and also there is a closed whole it is pretty simple to tell which is which when you get the hang of it. And you would put it where ever it says to. If it says 3 is less than x, you would put three on the number line and then figure out if its on the right side or the left (left side is smaller than the right side).

 

How to check solutions

when your done solving the inequalities, to check if its right you would have to actually do the equation and put the number you solved for *x* or any variable (mostly its x). So, you would input the x with the number for example 7 so you would input the number 7 with brackets (7) then solve both sides and they should be equal to each other!

What I Have Learned About Grade 9 Solving Equations

What is an equation?

an equation can be anything that is with an equal sign, for example this would be and equation 4x-7= 24

since there is an equal sign this question can be answered (= equal sign means solve!)

 

What are equivalent equations?

when you think of equivalent you think of equal and that’s exactly what this question is asking, equivalent equations are equal both sides and have the same value.

 

How to solve equations (find what x=?)

1. visually with algebra tiles

there are so many ways to use algebra times and it helps o much with any equation you might be stuck on. Alebra tiles are super easy to use and a fun way to learn, you can slide the tiles over *depending on your question* and or using the legal moves and add the algebra tiles.

 

Algebraically

For algebraically its really important to know all the steps you are taking and doing because, if you mess up then the whole equation messes up and then you would have to re start! and that’s not good. so its important to the the first step and after your done you take the second step. Doing it algebraically is actually keeping things organized and you will be happy with what you end up in the end. This is a chance of not getting confused and messing up.

 

 

BFSD (brackets, fractions, sort, divide)

B– Brackets

When you get brackets on your equation you always want to get rid of that first. That’s step one

F– Fractions

If you have fractions or any ugly questions you want to deal with that second because then you will find out what the ugly equation actually meant.

S– Sorting

For the sorting stage would be step 3 we are almost there but just one more step after this one. When you got everything all figured out and there is nothing else left to do then you would use the legal moves and adding on this or subtracting

D– Divide

This last part is where you divide the two numbers and find what “X” is. Dividing plays a huge part in this solving equations unit because it comes in handy from time to time.

 

How to verify and check your answers?

when you got your answer and you found out what the “x” was then what ever equation you had before you insert the answer to where ever the “x” places are. And both sides should be equal to one an other. If they are not equal you may have done something wrong maybe miss placed the negative sign or positive (that’s some common mistakes us students do) so when you placed the “x” with the number you got, you would solve to get the right answers

 

Vocabulary 

Equation: Has to have an equal sign

Equivalent: When two numbers are equal to each other

Solution: The end of the equation (answer)

Coefficient: Its the number before the letter/ variable “2x” and the number 2 would be the coefficient

 

Zero Pairs: Zero pairs is where you add or subtract numbers or variables from both sides

 

Variable: A variable is the letter in the equations, there can actually be multiple variables depending on the question. For example 2n+5=13 the variable in this equation would be *n*

 

Constant: Constant is the number that has nothing attached to it. it usually at the end of the equation and sometimes can be the middle

 

Common Denominator: Common denominator is usually used in fractions, you would use the common denominator when the bottom numbers are the same. Easiest way to use them is to multiply both the bottom numbers together to get your common denominator.

 

Distribute: This is used in multiplication you would use distributive property when you have brackets and one number outside the brackets, this way it can make the number smaller

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

what are legal moves and zero pairs?

legal moves and zero pairs only happen when what ever you do to one side you will have to do to the other. If you add a number to one side then the other side has to be the same thing, so you would add a number on the other side. That’s the big golden rule!

 

 

 

What I Have Learned about Grade 9 Polynomials

What is a polynomial?

A polynomial is  where you get all the algebraic terms like subtraction, addition, multiplication, division. There is also a key part we were missing and that’s the variables ex. 2x(x+5) this is one example of a polynomial. So many terms fit into a polynomial (fractions, coefficient, constant) and we will be talking about them in the next section.

 

Vocabulary :

Degree: A degree is the number of an exponent on the variable for example with the number 5x to the 4 power, and 5x to the 2

the degree for that one would be 4 (which ever one is higher exponent that would be the degree)

 

Constant: The constant in a polynomial is where the number is all by it self, it had no variables in it nor a power. Its usually at the end of the equations.

 

Coefficient: Is where there is a variable, its similar to a constant but just one thing changed. for example you would have to put the number in front, ex. 7x this would be a coefficient and the answer would be 7.

 

Leading coefficient: The leading coefficient is where its leading the equation, to it would start you off. in that case the leading coefficient would be the first number in the equation.

 

Monomial: is where there’s one term and only one term ex. -2x

Binomial: Is where there’s two terms in one equation ex. -2x+ 3

Trinomial: “Tri” stands for three so trinomial is where there’s three terms in one equation ex. -2x+ 3- 5x(squared)

 

 

Add polynomials/ Subtract polynomials:

Adding and subtracting polynomials is really nice.Because, you get to use the algebra tiles and when using the algebra tiles the equation get easier. There is also another way to make it simple, you would just group them together (all the like terms) and you would do the equation when you group them for example the equation here

ex.

When you are subtracting polynomials the sings flip, it doesn’t flip in the beginning it flips after the minus sign. Like what I have done in this photo

as you can see you could add the algebra tiles to show your work.

 

 

 

Multiply polynomials:

for multiplying polynomials there is really one way that works for me and its the distributive property. The distributive property is where you take the number outside of the brackets, and you multiply EACH number inside the brackets. for example this equation here

ex.

 

 

 

 

 

Divide polynomials: 

The high school way of dividing polynomials is really like the other unit we have done with the fractions, so you would just make it into a huge fraction and divide it. Their also different steps for the exponents as well, you don’t divide the exponents you subtract them in stead.

ex.

 

 

Make connections to previous units:

in the previous units we have done a huge fraction where you take everything and divide it with the number from the bottom, so that connects to the fractions unit. But also for the rational numbers unit we learned about negative and positive numbers and how we subtract, add, multiply, and divide them. Know in this unit we added exponents with variables!

What I Have Learned About Grade 9 Exponents

What is an exponent?

Exponents are lazy that is the first thing about them! but the real explanation of an exponent its the number of times its being multiplied by itself, for example 2 is the base and its to the 3 power= and what this means is just 2 x 2 x 2= 8. And it usually contains two numbers first one is base and the second one is the copier (it copes the base), they all wont be positive numbers there will be some that are negatives like this

= -25

= 25

these are not the same but may look  alike

Since it has a negative sign the answer would not be the same as the second one so negative 5 to the power of 2 would be -25. there are actually two different types of negative numbers, one was the one in the example and the second one is where its in the parentheses.

$latex (-5)^2= 25

Even though you may think it should be the same thing as the one in the top but its really not! It can get confusing at times but in brackets its gonna be a positive when its not in brackets it is a negative. Its because exponents are LAZY! It knows its there but wont take it if it doesn’t have to so when there is a negative with no parentheses then it wont take the negative fraction. But, when there is parentheses then it knows it has to take it but doesn’t want to. It was confusing at first but then when you get used to it you will get it. so really exponents are the number of times it being multiplied by itself, it has the base the power those are the two main things in a exponent

 

 

The difference between evaluating and simplifying?

When you simplify you use the order of operations and get it the smallest number possible. It means to simplify and do the question but leaving it in a exponent

To evaluate an expression sometimes variables are given to you and you evaluate the variable.

you really just have to do it and simplify as much as you can at the end

 

 

Division+ multiplication  law and why it works?

 

The division law you would have to subtract the equation to get the answer I know you thing you would have to divide because it says to divide but it’s actually subtract

 

The multiplication law is different just like the other ones you add the exponents together when the bases are the same as you can see in my example I have the 3 + 4 = 7 so the answer would be  its that simple you just add the bases together to get your answer!

 

 

Power of power law and why it works?

Power laws is when you have an exponent on the out side and it gets multiplied together like the example

So for power law you just times the exponents together even thought it does not say anything to do with multiplication but you have to times the 2 and the 4 to get your answer which is 6 or if its a negative question it would be a -6. I like power law because all you have to do is multiply quickly and you have your answer

 

 

 

Applications of exponents

Exponents are used in many ways multiplying a number by itself

it has the power law that we just talked about

it has the multiple law

and the division law

 

One more thing you learned about exponents?

exponents are LAZY! I didn’t know what that meant first but then when they explained it I knew what and why it was super easy and fun to do. So when there is no brackets it only like the positive number even when its negative it *pretends* its a positive. Another thing that I really like and was really cool was anything to the power of 0 = 1!! Even if its a big and huge number and if you see a zero up there then you already know what the answer will be!

= 1

But you can only do it with the zero you cant do it with the other ones zero is the magical number and can get the question answer quickly.

those were the things that were very cool!

What I Have Learned About Grade 9 Fractions

Fractions On a Number Line

Fractions on a number line was confusing at first but when I was trying a couple of examples I was doing fine and grate. The negatives on a number line are really the same thing as the positives.

 

 

 

 

 

Comparing Fractions

For comparing fractions I just wrote it down on paper but ill just say it here as well. When its a negative fraction vs. a positive fraction, the negative fraction will win because negatives are more bigger and powerful than the positive ones, the more number it goes down the powerful it gets. In this picture I have and the would be bigger because of how small the numbers are and plus its also negative.

 

 

 

 

Adding+ Subtracting Fractions 

For adding and subtracting fractions the key thing you will need is a common denominator! Common denominators are really important when dealing with equations like this. But when you have the same denominator you add or subtract what ever the sign is. So much better when you have the same denominator.

 

Multiplying And Dividing 

Multiplying fractions are super easy! there are many ways to get your answer for example cross- cancelling it. You could also just multiply top by top bottom by bottom, this is how i see multiplying fractions.

 

Dividing!

Dividing is the same thing as multiplying because you change it into a multiplication question, you change it by flipping (reciprocal) the second fraction. then you can multiply top by top and bottom by bottom.

 

The other thing that I learned in math this unit is identifying rational numbers, and rational numbers have to do with positive and negative numbers it also means that it can be expressed as a quotient/ fraction. For example these are rational numbers -5, 9, 3/4, 0.36 (decimals are also rational numbers), 1.4, 5/8….. and so on. To me rational numbers just mean numbers in all shapes of form, that’s how I picture it. Comparing rational numbers was not hard at the beginning because the first thing you had to do was look for the bigger number and that’s when you know its bigger than the other.