What I learned in Grade 9 Linear Relations

What is a Linear Relation

A linear relation is a pattern that increases or decreases by the same amount every time.

What is a T-Chart and How to fill it

When finding the rule for a linear relation, you can put the numbers in a t-chart. A t-chart helps you record your data in one table. This chart helps you find the relationship between x and y. On the left side of the chart is x, and the right is y.

How to find the Rule for a Pattern

The first step is to know how much the pattern changes each time. Once you know how much your pattern goes up/down by you put the number next to x. Next you insert 1 instead of x as it is the first number in the pattern. Example: 6(1)=y. You know that 6(1) does not equal 3. The next step is to find how you get from 6to 3. To get from 6 to 3 you subtract 3. Now add -3 to your rule. Your rule now looks like 6x-3=y. To see if this is correct, try it on the next number.

How to Plot a Point

On a graph, the horizontal line is the x axis and the vertical line is the y axis. The first number on your coordinates is the x number and the second number is the y. Example: 2,3 the 2 is x and 3 is y. The first number tells you which way you are going to go on the x axis. If the number is positive you go to the right and if the number is negative you go to the left. The second number tells you which way you are going on the y axis. If the number is positive you are going up the y axis and if the number is negative you are going down the y axis. If are coordinate is 2,3 you will go right 2 times, and then up 3 times. Once you have done this, you place your point where the coordinates lead.

How to Graph a Linear Relation

To graph a linear relation, put your equation into a t-chart. Once you have a couple pairs of numbers you can insert the points onto the graph. Example, the rule 6x-3=y, the first three numbers are 3,9,15. Now your t-chart has 1,2,3 in the x column, and 3,9,15 in the y column. When you put the two together you get your coordinates. For example, 1,3  2,69 3,15. Once you have these coordinates you can plot them on the graph.

How to Graph Vertical and Horizontal Lines

A vertical line goes across the x axis. This means that the formula you are going to use x = a number. For example x=3. A horizontal line goes across the y axis. This means that the formula you are going to use is y = a number. For example y=-3. This will create a horizontal line to the right and left of -23 on the y axis.

Vocabulary:

X Axis – The horizontal line on a graph.

Y Axis – The vertical line on a graph.

T-Chart – A chart that has two columns.

Quadrant – A graph that is split in 4 parts.

Origin- The middle of the graph.

Plotting – Where you place your point on a graph.

Linear Pattern – A pattern that increases or decreases by the same amount each time.

Increasing Pattern – A pattern that increases each time.

Decreasing Pattern – A pattern that decreases each time.

Horizontal Line – A line that runs from left to right.

Vertical Line – A line that runs top to bottom.

Something Else I learned

Something else that I learned in this unit is restricting your line to between two points. This is useful because without the formula, you will have a long line running from one side of the graph to the other.

What is an Inequality

An inequality compares two values, showing if one is less than, greater than, or simply not equal to another value. Example: 2 > 1.

What the symbols mean

An inequality has 1 out of 4 different signs. The first sign is >, which is the greater than sign. This sign is used when the number on the left is bigger than the on the right.  The second sign is <, which is the less than sign. A less than sign is used when the number on the left is smaller than the number on the right. The third sign is ≥, which is the greater than or equal to sign. This sign is used when the number on the left can be greater or equal to the number on the right. The fourth sign is ≤, which is the less than or equal to sign. This sign is placed when the number on the left is less than or equal to the number on the right. Examples: 3 > 2,  7 < 8,  x ≥ 13,  9 ≤ x.

How to solve

Solving inequalities is almost exactly like solving an equation; legal moves. The only difference is when you are dividing by a negative, you need to flip the sign. Example, if you have -3x < 21, after you divide and get x < -7 you need to flip the sign, so your answer becomes x > -7

How to graph an inequality

To graph an inequality on a number line, you need to know how to identify what the sign is. If the sign is < or > than you will put an open dot on your graph and if the sign is ≤, or ≥, you will place a closed dot on your graph. Once you know what kind of dot you need, you need to find the number that is the inequality with x. For example x > 10. You would find 10 on your number line. Once you have found 4 draw the dot on top of the 4. The direction your arrow goes depends on the sign. For our example, x > 4 the arrow would go to the right of 4. This is saying that x can be any number to the right of 4.

How to check an inequality

Since solving inequalities are very similar to solving equations, checking is also mainly the same. To check your solution, you replace the variable with your answer. Example: 2x+4 > 8. When you do all your legal moves you end up with x > 2. From here, your answer should be 2. Now you replace the variable with 2 and solve. 2(2) + 4 > 8 turns into 4 + 4 > 8, then 8 > 8.Now you know that your number is right. There is also a second to check and inequality. The second way to check is to make sure the sign is going the right way. To do this you need to put  any  number that is greater than 2 into the question. Example: 2(10) + 4 > 8 turns into 24 > 8 which is a correct statement.

What I learned in Grade 9 Solving Equations

What is an equation?
An equation is a mathematic statement showing that on both sides of the equal sign that the two statements are equal to each other.

What are equivalent equations?

Equivalent equations are two or more algebraic equations that have the same solution as each other.

How to solve equations (find what x = ?)

Visually with algebra tiles:

First, model the equation with the tiles. After that, to find x, you need to make legal moves. Using legal moves helps you isolate the variable to help you find the solution. A legal move is doing the same move on both sides of the equation. A move you can do is adding tiles or taking away tiles. With legal moves, you need to create zero pairs. A zero pair is when you make a legal move to make one term to cancel itself out. But, if one term cancels itself out, its like term will get the same move that its like term got. Eventually, you will get to the point where you cannot do anymore moves because you have the smallest possible set of x’s on one side and the “ones” on the other. The final step is to divide. You have to divide the amount of x’s by the amount of ones.

Algebraically:

An easy way to solve an algebraic equation on paper is to visualize the equation in your mind with algebra tiles. Creating zero pairs with legal moves is your next step. A good way to keep track and to show your work with zero pairs is to cross out the zero pair with a large letter “z”. Keep making legal moves until you get to your final divide step.

BFSD (brackets, fractions, sort, divide)

This acronym is pretty self explanatory. It is a lot like BEDMAS or PEMDAS. In an equation, you might counter brackets or fractions. The first thing to do is distribute. This is the exact same distributing as any other equation. With the fractions, you need to find the lowest common denominator and multiply everything in the equation by the LCD. With the fractions being multiplied, you need to simplify by changing the fraction to a whole number. The next two steps are easy and self explanatory.

How to verify (Check) a solution (answer) is correct

Vocabulary (Definitions):

Equation: a statement that the values of two mathematical expressions are equal (indicated by the sign =).

Equivalent: equal in value, amount, function, meaning, etc.

Solution: a means of solving a problem or dealing with a difficult situation.

Coefficient: a numerical or constant quantity placed before and multiplying the variable in an algebraic expression (e.g. 4 in 4x y).

Zero pairs: a pair of numbers whose sum is zero, e.g. +1, -1. • used to illustrate addition and subtraction problems. with positive and negative integers.

Variable: A variable is a quantity that may change within the context of a mathematical problem or experiment. Typically, we use a single letter to represent a variable

Constant: A fixed value. In Algebra, a constant is a number on its own, or sometimes a letter such as a, b or c to stand for a fixed number.

Common denominator: a shared multiple of the denominators of several fractions.

Distribute: Distributing items is an act of spreading them out equally. Algebraic distribution means to multiply each of the terms within the parentheses by another term that is outside the parentheses.

Vocabulary:

Degree: The largest exponent attached to a variable in an expression.

Constant: A term without a variable.

Coefficient: The whole or large number in a term.

Leading coefficient: The coefficient at the start of the expression (most left).

Binomial: When the number of terms in an expression is 3.

Trinomial: When the number of terms in an expression is 2.

Monomial: Is equivalent to one term.

How to use algebra tiles:

You will have 3 types of algebra tiles. 1, x and x². Expressions will usually have the largest amount to least amount. So the order will be in x², x then 1/whole numbers. With x² tiles, since they are the largest, they will also be physically the largest pieces. You will follow the expressions instructions. If it tells you to place 4 x’s, then place 4 x’s. But when an negative number is involved, all you need to do is flip the tiles backwards.

When starting out learning polynomials, it is good to sort the alike terms with each other. Example: 2x² + 3x + 3x². The first thing you should do is sorting the terms with the other ones that are alike. This step is not mandatory, but will help you when starting out learning to simplify polynomials. After sorted by amount, the expression should turn into 2x² + 3x² + 3x. After sorted, add the coefficients that have the same variable and exponent to each other. It should look like 5x² + 3x. Starting with small expressions is easier to learn than trying to simplify larger ones.

Subtracting polynomials:

Just like simplifying every other polynomial expression, it’s good to sort the terms. Example: 7x + 5 + 4x² – (2x² – 2 + 3x). In subtracting, all of the terms inside the bracket that is after the subtracting symbol gets flipped to its opposite number. 7x + 5 + 4x² + (-2x² + 2 – 3x). After doing this, just add and follow what the expression is. 4x² + (-2x²). 7x + (-3x). 5 – 2. The simplified version of the expression should be 2x² + 4x + 3.

Multiplying polynomials:

2(4x + 2). Multiplying polynomial expressions is almost like multiplying normally. The only difference is the variable.

Using algebra tiles for learning large multiplying expressions is very helpful. When you use algebra tiles, model the expression in a shape in a rectangle. In the example, you would take 2 small tiles and place them on the left side. On the other side, you would take 2 long tiles and 2 smaller ones and make a line with them. To simplify the expression, you must fill the space with more tiles. But, using algebra tiles and multiplying polynomials is mainly used for large expressions.

Without algebra tiles, you need to multiply normally. When there are variables, it is different. When the number that is multiplying has no variable attached, you follow whatever the variable is. In the example above, the simplified version is 8x + 4.

Some expressions might have you multiply by 2 numbers. Example: 3x + 2(4x + 2)

First, you multiply the number on the far left of the expression by the numbers inside the brackets. After that, you do the same the the number on the right; multiply them by the numbers inside the brackets. Once you have both of these numbers, you add them. The simplified version would be 12x² + 14x + 4.

Dividing polynomials:

4x + 2

_______

2

Using algebra tiles for dividing is almost pointless. Dividing polynomials is very simple if you can divide whole numbers. You would divide these numbers normally except for the variables. Just like multiplying, the same rules apply except of adding the variables, you subtract. So in the expression above, the simplified version is 2x + 1.

What is an exponent?

Official definition: A quantity representing the power to which a given number or expression is to be raised, usually expressed as a raised symbol beside the number or expression.

An exponent has 2 parts to it; the base and copies. The base is a large number and the copy is a smaller number raised right beside the right of the number. The copies shows how many times the base is multiplies itself by. Mathematicians made the exponent to keep multiplication expressions shorter, I think. Instead of saying 4x4x4x4, you say 4 to the power of 4. When a negative sign is introduced in the question, depending if it has brackets or not, the answers can be different. I if negative symbol is outside with no brackets, the answer will always be negative. I there are brackets in the question, it depends the amount of copies there are. If the copies are an even number, the answer will be positive, if the copies is an odd number, the answer is negative.

What is the difference between evaluating and simplifying?

Evaluating an exponent means for you to find the answer to the exponent. When a question asks you to simplify, it means to write the question in more simple terms. Usually when a question asks you to simplify a exponent, there will probably be two exponents. Sometimes you might need to use the multiplication law or division law if the exponents are doing that to each other. Example: Evaluate 5 to the power of 3. To solve the question, you need to understand and break down how to solve this. The expanded from is 5x5x5 because the base is 5 and has 3 copies of itself. 5×5 is 25, 25×5 is 125. So 125 is 5 to the power of 3 evaluated. Example: If a question says to simplify 3⁴x3², the simplified version 3⁶.

Multiplication law and why it works.

The multiplication law is when two exponents are being multiplied to each other and the bases are the same and the copies are added to each other. This law is not for solving the problem, it is just for simplifying and so are the other laws. Example: 5³x5⁴= 5⁷. 3 and 4 were the copies and added together is 7 making that the new number of copies.

Division law and why it works

The division law is the opposite of the multiplication law; instead of adding the copies, you subtract them. But, only when the question asks you to divide the exponents provided. Example: 5 to the power of 5 divided by 5⁵÷5³=5². By taking the two copies (5-3) and subtracting them equals 2.

Power of a power law and why it works

The power of a power law is when there are two copies copying the same base. To simplify, you need to multiply the two copies to each other. Example: (5⁸)⁷ turns into 5⁵⁶. 8×7=56.

One more thing you learned about exponents

Before this unit, I didn’t know how to solve a fraction with an exponent, but now I know how to solve these problems. Example: 1/3² becomes 1/9. Because 1×1=1 and 3×3=9 so the solution is 1/9

Fractions and Number Lines: I already had lots of previous knowledge with number lines, but I learned more and improved on this especially when using not only fractions but also decimals on number lines. Learning negative numbers was not a problem for me. To prove this, the negative fractions stay on the left side of the zero and the positives are on the right side. If a fraction is proper and positive, it is less than one. Meaning that the numerator is smaller than the denominator.

Comparing Fractions: Most of the time I can just eye ball a fraction to know if it is larger than the other, but in this class, I need to show all off my work which has helped me complete this further. A good way is to make the denominators the same and then multiply the same number to the numerator.

Adding/Subtracting Fractions: I also had a lot of previous knowledge on this, but the trick of making the denominator the same number was stuck in my head making me understand this more. Including the negative sign caused me a little bit of trouble, but I kept practicing and later became a skill of mine. Making the denominator the same is the main and most efficient way and then doing the same to the numerator.

Multiplying/Dividing Fractions: I am very experienced in multiplying numbers in general, so multiplying fractions was very easy for me to learn and understand. Before we started dividing fractions, I forgot how to answer these types of questions, but after learning about it and how multiplying is the main tool and solution to solving the problem, I understood this rule and now I feel very confident solving this types of problems. By adding the negative sign next to a number, did not cause me any confusion on how to solve the problem. When multiplying the fractions, you multiply both of the numerators to each other and the same for the denominator. When dividing, if the denominators are not the same, then you need to reciprocate. This means that on the fraction on the right, you flip the numerator and the denominator. After this step, you just multiply the new numerators and the new denominators. Example: 3/4 ÷ 5/16 turns into 3/4 ÷ 16/5.

Square roots: I was taught how to find the square root last year, but we never elaborated on this. So practicing this made me have a better understanding on how square roots work. But one thing I still have a little of trouble with is finding the square root to a decimal, but I am still practicing this skill. To find the square root of a number, you find a number and multiply it by itself to become the square root. Example: You want to find the square root of 36. Now you need to identify what number times itself will get 36. The solution is 6*6=36. So 6 is the square root. Square root means that the root of the number is a multiple of the number you want to find the square root of. Once found the root, you need to square it.