1. Exponent Laws

• For me, exponent laws were big essentials that helped me out in math 9. They helped me remember when to add, subtract, multiply and divide the exponents in equations and gave me little tricks and tips that really came to my advantage while completing this unit. Exponent work was a little tricky for me so to learn these laws really made my learning experience a lot easier (Like how Ms. Burton says “work smarter not harder”.)
• First, there is the Multiplication Law: If the question is a multiplication question and the bases are the same, you can add the exponents together and keep the base the way it was. Example:  =
• Then, there was the Division Law: If the question is a division question and the bases are the same, you can subtract the exponents and keep the base the way it was, similar to the multiplication law but instead dealing with subtraction. Example:  =
• After that, there was the Power of a Power Law: If the question looks like  you can use this law. All you have to do is multiply the exponents and keep the base the way it was. Example:  =
• And last but not least, there is the Exponent of Zero: If the question has an exponent of zero, you automatically know that the answer will equal to one. Example: = 1

2. Similar Triangles

• I thought this was very useful when it came to this unit. Not only did this teach us how to find the missing side of a triangle but this also taught us about the cross multiplication technique which in my preference came in handy immensely. This was something that I believe is very good to comfortably know how to do for the years in math to come.
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• To find the missing side length (x), you have to create ratios by using the information in the given similar triangles above. This gives us the ratios 5/10, 10/20 and 15/x
• Now this is where the cross multiplication comes into play. You will take one of completed ratios and place it beside the ratio that contains the variable. From there, you will multiply the information in a cross-like formation. Example:   = x=30

3.     Adding and Subtracting Polynomials

• This was a huge part of math 9 so I feel that this was definitely very important for me. It is used a lot and will be used even more as math excels so this is why it is good to understand this concept and are able to execute it with ease. For me, sometimes I have a hard time remembering some of the steps in this concept so I feel that this is not only important but is also something that I can work and improve on. For me, practicing this makes all the difference in the world so that is why I feel so strongly about the importance of adding and subtracting polynomials.
• When adding and subtracting polynomials without brackets in the question, all you need to remember to do is group your like terms together and simplify. Like terms are all the terms that contain the same variables and exponents. You cannot group unlike terms together or your equation will not work. Once you have grouped all your like terms together, you can start you add and subtract your like terms.
• When adding and subtraction polynomials with brackets in the question, you do the exact same thing but you have to remember and essential step before you group and simplify. If there is a subtraction sign before a bracketed polynomial, you must remember to switch the term’s sign from negative to positive or vise versa. After completing that step, then you can move on to grouping the like terms and simplifying the equation.

4. Dividing and Multiplying Rational Numbers

•     This is one of the most used parts in math and it is a good idea to be able to comfortably understand how to properly do this for almost every single math unit. The tricks and tips that make dividing and multiplying fractions easier really seem to work with me so I feel that this was a very important unit for me.
• When multiplying fractions, you need to “just do it” as Ms. Burton says. All you do is multiply across ans simplify if possible. Example: x = $latex \frac{3}{8}
• When dividing fractions, all you need to do is flip the reciprocal and multiply the two fractions together like a multiplication question. Example: =  x   =

5. Graphing Linear Relations

• For me, this is the hardest thing that we learned. For the longest time I struggled with graphing, but now that I understand it more, I feel that it is really important. It is something that took me a while to fully understand but I feel more confident with it now and realize the impact it has in math 9.
• When graphing, x is vertical and y is horizontal
• Let’s say that step is represented by x and number is represented by y.
• Here is the graph for the information above:
• 

This blog post represents what I have learned in this similarity unit.

What is an Enlargement and a Reduction?

• An enlargement is to make an original object bigger and a reduction is to make an original object smaller.
• We use multiplication to create an enlargement or reduction for an object.
• Examples of enlargements: Any numbers over 1, impropre fractions, percentages over 100%
• Examples of reductions:  Any numbers under 1, propre fractions, percentages under 100%
• If the scale is 1 or 100%, that means that is not an enlargement nor a reduction and actually stays the same.

What is a Scale Factor?

• It is a ration that demonstrates two corresponding lengths in two figures.
• For the scale factor, the original length goes on the bottom and the length in the original goes on the top.

Equations with Scale Factors:

• Here is an example of a question including a scale factor.
• What is the actual length of an object if the scale is 1:10 and the length of the object in the diagram in 4?

x=40

Similar Triangles:

• Similar triangles are triangles with equal corresponding angles and proportionate sides.
• To figure out if two triangles are similar, you have to create ratios that correspond to the sides. If they equal the same number, they are similar but if they do not, then they are not similar.
• To figure out a missing side length from a similar triangle, you have to use the butterfly technique (multiplying the information in a cross like formation).

Indirect Similarity:

• To figure out how to measure and object is taller than you are, you can use this technique by using a mirror.
• You place a mirror on the ground and measure how tall you are from your eyes, the distance between where you are standing and the mirror and the distance between the mirror and the object.

I chose to measure one of the walls from the outside of my house.

Person’s Height: My height is roughly 5’4.5, which the same as saying 163.83cm

The Distance Between the Mirror and I: The distance between where I was standing and where the mirror was placed was 35cm.

The Distance Between the Mirror and the Wall: The distance between where the mirror was placed and the wall was 105cm.

Here is a diagram I created to show my information..

If I look at the middle table, my enlargement is a scale factor of 2. To enlarge my image, I multiplied the given information by two.

My reduction was a scale factor of 0.5. I reduced the given information my multiplying by 0.5.

What is a Linear Inequality?

• A linear inequality is an equation that involves linear functions. An equation also includes one symbol of inequality:
• < is less than
• > is greater than
• ≤ is less than or equal to
• ≥ is greater than or equal to
• ≠ is not equal to
• = is equal to
• A linear inequality ressembles just like the linear equations from the last unit, except with an inequality symbol to replace the equal sign.

What do these Equations Mean?

• If we were to write “x<4”, that would be that a number is less than four. Same goes for “x>4”, which would mean that a number is greater than four.
• If we write “x≤4” that would mean that a number is less or equal to four. Same goes for “x≥4”, which would mean that a number is greater or equal to four.

How do you Graph a Linear Inequality?

• When graphing linear inequalities on number lines, we use different types of dots to identify the inequality sign. if it’s an open dot, we use those for equations that contain less or greater signs. If it’s a closed dot, we use those for equations that contain less or equal to, and greater or equal to signs.
• 

Here are some examples of linear inequalities plotted onto graphs:

(Graph 1: x<-2, Graph 2: x≤-2, Graph 3: x>-2, Graph 4: x≥-2)

How do you Solve Linear Inequalities?

• Solving linear inequality equations is just like solving normal linear equations,
• One thing that always helps me is to remember is

Best: Brackets

Friends: Fractions

Share: Sort

Desserts: Divide

These will help you remember what to do first when you are solving your equation.

• Here is an example of the steps of solving the linear inequality equation “2x+4>3x+1”..
• 
• If an answer to an inequality equation is negative, you have to switch the sign to the opposite sign.

What is a Linear Equation?

• A linear equation is an equation between two variables that gives a straight line when plotted on a graph.
• Here are 2 simple examples of linear equations:   5x=6+3y       or     y=2x+1
• Here is an example of a linear equation that has been plotted on a graph:

•  Linear equations can contain variables that are whole numbers, integers, decimals and fractions. When dealing with linear equations that have fractions, the best way to solve it is to find a common denominator.

How can Equations be Modelled Using Algebra Tiles?

If you don’t have algebra tiles you can always draw out and solve the equation by drawing the tiles on paper. I did not have tiles so below I visually represented how I solved the linear equation.

• When using algebra tiles, coloured tiles are positive integers and non-coloured tiles (white tiles) are negative integers.
• Larger coloured rectangular tiles are used as 1x and the smaller coloured squares are used for 1.
• Larger non-coloured rectangular tiles are used as negative 1x and the smaller non-coloured squares are used for negative 1.
• See photo below:

This is an example of how I visually solved the equation below. I drew it out using algebra tiles.

How to Solve Equations Algebraically?

To solve the following simple algebraic equation I have get x all by itself. We call that isolating the variable.

If you have the equation:      2x+3=7

• I want to get x alone. The first step would be to get rid of the +3 by subtracting 3 (this cancels each other out). I remember that what I do to one side of the equation, I have to do to other so I subtract 3 from 7. Now, I’m left with 2x= 4
• to isolate the x I divide 2 by 2 to cancel each other out (which leaves me with x on its’ own). What I do to one side, I do to the other so I divide 4 by 2 and I get the answer x=2

Patterns:

• A pattern is a repeated design that can increase or decrease the quantity of objects in each image.

• This is an example of a linear pattern. You can see how the quantity of squares increases in each step.
• A linear pattern is that if you plotted the information of the pattern onto a graph and the plotted points make a pattern, then the coordinates of each point may have the same relationship between the x and y values.
•  Here is a T-chart that is made from the information included in the pattern above. If you look at the T-chart, you will notice that the numbers in the white column increase by four each time. This is a good thing to notice, as it will help you with the questions to come.

Linear Relation Rules:

• A linear relation is a relationship between two products, that if plotted on a graph, it would make a straight line.
• A linear rule is a algebraic expression that is used to demonstrate how to get the output Y-values for a given set of input X-values. This rule will help us determine what the rest of the pattern would be without even looking at the image! It would also help us know where to successfully plot the information onto a graph.
• Going back to the information on the given T-chart above, we know that the number of squares goes up by four each time.
• Let’s think that the grey column is also named “X”, and the white column is also named “Y”.  We would write “4x” to represent the X-value increasing my four, each step.
• (4)(1) would not bring us to 6. Now it is time for you to figure out whether you should add of subtract to find the answer given in the “Y” column.
• Now we know that the rule would be 4x+2

Plotting Points on a Graph:

• Plotting points on a graph is simple. A graph is basically a horizontal number line (x-axis), crossing through a vertical number line (y-axis).

  

  
  
• Let’s say that we had a T-chart that represented the rule “2x+1”.
• We will use this information to plot our dots successfully onto our graph.
• If you remember when I said that the grey column is named “X” and the white column is named “Y”, you will place the dot where 1 on the x-axis and 3 on the y-axis meets in the middle.
• You will continue that by using the rest of the information in the T-chart and you will begin to notice that plotted dots create a straight line that will run through your graph.

In this term, we have learned about polynomials and how to add, subtract, multiply and divide them.

Definitions: 

• Term: An expression formed from the product of numbers and/or variables.  An example of an expression with 3 terms would be + 3x – 5.  This example would be called a Trinomial because it is a expression that consists of three terms. The terms can be separated by addition, subtraction, multiplication or division.
• Coefficient: When a term consists of a number before a variable, the number is thought of as a coefficient. An example would be 6z which means 6 times z, and “z” is a variable, so 6 is a coefficient.
• Constant: A number on its own would be a constant. In the expression 4x-7 = 5, the numbers 5 &7 would be constants because they are on their own, without and exponents or variables after them.
• Degree of a Term: The sum of the exponents on the variables in a single term (monomial). An example of the degree of a monomial would be The degree would be 3.
• Degree of a Polynomial: The degree of the highest-degree term in a polynomial. In the example -3a the degree of the first term is 2 and the degree of the second term is 1, so the highest degree is 2, so the degree of the polynomial is 2.

Adding and Subtracting Polynomials Without Brackets:

• When you are adding and subtracting polynomials without brackets, all you have to do is group and simplify.
• Once you have the question, the first thing you need to do is to group your like terms together (Like terms are terms that all consist of the same variables and exponents. Unlike terms are terms that do not).
• Your second step is to add or subtract your like terms together. You may not add or subtract unlike terms.
• At the end, you have finished the question with a simplified form.
• Note: Always group & simplify the terms from biggest to smallest.
• 

Adding Polynomials With Brackets: 

• Adding Polynomials with brackets is similar to adding without.
• When there is an addition sign between to bracketed polynomial questions, all you have to do is take away the brackets & addition sign and simplify the question.
• 

Subtracting Polynomials With Brackets:

• When you see a subtraction sign in front of a bracketed polynomial question, you change the numbers to their opposite. So if the numbers in the brackets were originally (-4 +5), you would change it to (+4 -5).
• Then, you take the brackets away and simplify the question.
• 

Multiplying Polynomials: 

• Multiplying is just like finding the area of a rectangle.
• When you multiply polynomials, you need to remember to multiply the coefficient, keep the variable and just like in our exponent unit, add the exponents together. Example: = (because there is an invisible 1 for the exponent of the x).
• Note: That if the variables are different, you can not add them together.

Dividing Polynomials: 

• Dividing Polynomials is just like multiplying, instead you divide the coefficient and subtract the exponents. Example: =
• Note: If the variables are different, you can not subtract them.

Distribution Method: 

• Let’s say that you have a question like 3x(6+4x). In this case you would use the distribution method.
• You multiply the term on the outside of the brackets with each term on the inside of the brackets.

Example: 3x(6+4x)

(3x)(6)  (3x)(4x)

18x +