This week in math 10 we learned how to write the equation of a line. We learned about 3 different ways to write an equation of a line. The 3 ways are called slope y-intercept form, point-slope form, and general form.

Slope y-intercept Form

One way to write the equation of a line is slope y-intercept form. The formula of slope y-intercept form is y = mx + b. The m stands for slope and the b stands for the y-intercept. The first thing you want to do when writing the equation of a line in slope y-intercept form is to find the slope. There are two ways to find the slope of a line. The first is to look at the line on a graph. To find the slope of the line your equation is the change in y over the change in x. A simple way we learned to do this is rise/run. The rise is the change in x and it’s how many times it goes up or down the y axis and the run is the change in x and how many times it goes across the x-axis. For example, if your two points are (2,0) and (3,2), your rise or change in y is 2 and your run or change in x is 1. Therefore, your slope would be 2/1 or 2.

The second way to find the slope of a line is if you don’t have a graph. The formula to find the slope of a line is y₂ – y₁ / x₂ – x₁. For example, if your two points are (1,5) and (5,13), you would write 13 – 5/5 – 1. You would end up with 8/4. So your slope of a line with these two points is 8/4 or 2. To find b in your slope y-intercept form equation you would either look at the graph and find the y-intercept of the line or you would have to algebraically have to solve to find b. For example, if you have a slope of 2 and a y-intercept of +3, your equation in slope y-intercept form is y = 2x + 3 or 2x + 3 =y. To check if your answer is correct you can insert the x value from a point on the graph into your equation to see if you get the y value. For example, if you have the point (5,13) you would insert the 5 into the x and see if you get 13. Your equation would be 2(5) + 3 = y and 10 + 3 = 13, so you know your equation is correct.

Point Slope Form

Another way to write the equation of a line is in point-slope form. The formula for point-slope form is m(x – x₁) = y – y₁. The first step to solving an equation in point-slope form is to find the slope of the line. Once you have found the slope of your line you would insert it in for m. The second step is to insert your x value from your coordinate into the x₁ spot and the y value of your coordinate into the y₁ spot. For example, if you have the point (1,5) and your slope is 2, you would write your equation as 2(x – 1) = y – 5, and that’s it, that is your equation in point-slope form. Point slope form is a quick and easy way to write your equation.

You can convert your point-slope form equation into a slope y-intercept form equation by using distribution and some algebra. For example, if you have 2(x – 1) = y – 5, and you use distribution you will end up with 2x – 2 = y – 5. Then you would add +5 to both sides to isolate the x. Your new look equation would look like 2x + 3 = y. Now your equation is in slope y-intercept form.

General Form

The third way we learned to write the equation of a line is in general form. The formula for writing your equation in general form is Ax + By + C = 0. Some notes when writing your equation in general form is that A has to be a positive number, if it’s negative it’s not in general form. The second thing is A, B, and C all have to be integers, which means that they can’t be fractions. Writing your equation in general form is pretty much useless because it doesn’t tell you anything about the line. For example, if you have an equation of 2x + 3y + 4, you don’t know the slope of the line or the y-intercept. Therefore, writing your line in general form is pretty much useless. To write your equation in general form when starting with a point and a y-intercept or an equation in slope y-intercept form, you have to use algebra to move the x, y, and constant all together on the same side.

In week 7 of math 10, one important thing we learned was how to find the domain and range of a relation on a graph. Another important thing we learned was how to find the x and y-intercepts of a relation on a graph.

Domain

The domain of a relation is the bit of the graph between the lowest x value and the highest x value. How to find the domain of a relation on a graph? When you look at the graph you will read it from right to left. First, you want to locate your relation. It’s often a line or curve on the graph. The domain is like the walls squishing the graph to only the lowest and highest x values. Once you have found the two points of the lowest and highest x value, you want to check if the dot on the graph is open or closed. If the dot is open you would write a less than sign and if the dot is closed or if you can’t tell you would write a less than or equal to sign. For example, if there are two open dots you would write -2 < x < 4. If there is an arrow at the end of your relation it means that the graph isn’t big enough for the relation and it will go to infinity. There are two ways to write your answer if the relation goes to infinity. The first is to write your answer just like a normal domain question except you would write the infinity symbol as your highest value. The second way to write your answer is to write x > then the lowest x value. For example x > -4. If the relation has arrows at the end of both sides then you could write your answer as -infinity < x < infinity or XER. The XER means that any number between negative infinity and positive infinity will fit in the relation. If you are given a list of coordinates instead of a relation on a graph, you just list the x value of each coordinate. You only list each value once. For example, if you have (-2,1), (1,3), (2,4). The domain would be -2, 1, 2.

Examples of the Domain on a Graph

Range

The range of a relation is the bit of the graph between the lowest y value and the highest y value. How to find the range of a relation on a graph? When you look at the graph you will read it from bottom to top. First, you want to locate your relation. It’s often a line or curve on the graph. The range is like the floor and ceiling squishing the graph to only the lowest and highest y values. Once you have found the two points of the lowest and highest y value, you want to check if the dot on the graph is open or closed. If the dot is open you would write a less than sign and if the dot is closed or if you can’t tell you would write a less than or equal to sign. For example, if there are two open dots you would write -4 < y < 3. If there is an arrow at the end of your relation it means that the graph isn’t big enough for the relation and it will go to infinity. There are two ways to write your answer if the relation goes to infinity. The first is to write your answer just like a normal range question except you would write the infinity symbol as your highest value. The second way to write your answer is to write y > then the lowest y value. For example y > -1. If the relation has arrows at the end of both sides then you could write your answer as -infinity < y < infinity or YER. The YER means that any number between negative infinity and positive infinity will fit in the relation. If you are given a list of coordinates instead of a relation on a graph, you just list the y value of each coordinate. You only list each value once. For example, if you have (-2,1), (1,3), (2,4). The range would be 1, 3, 4.

Examples of the Range on a Graph

X Intercept

The x-intercept is where the relation goes across the x-axis. When solving for the x-intercept you want to follow a couple of steps. First, you want to put a 0 in the y value. For example, if you have 3x + 2y = 3, you would place a 0 in the y value making it 3x + 2(0) = 3. Once you have gotten rid of the y you want to isolate the x. To do this you will divide both sides by the coefficient of the x. For example, if your question is 3x = 3, you would divide both sides by 3, 3x/3, and 3/3. Then you are left with x = 1. Lastly, you would write a coordinate with your x value in the 1st spot and 0 in the y value. For example, (1,0).

Y Intercept

The y-intercept is where the relation goes across the y axis. When solving for the y-intercept you want to follow a couple of steps. First, you want to put a 0 in the x value. For example, if you have 3x + 2y = 4, you would place a 0 in the x value making it 3(0) + 2y = 4. Once you have gotten rid of the x you want to isolate the y. Just like when you solve for the x-axis, you want to divide both sides by the coefficient in front of the y. Lastly, you would write your coordinate with 0 in the x value and your value for y in the second spot. For example, (0,2).

In week 6 of math 10, the most important thing we learned about was factoring 1 2 3. Factoring 1 2 3 is a saying to help you remember what to look for when factoring a polynomial.

Step 1

The first step in factoring 1 2 3 is 1. The first step is to find out if any of the terms have anything in common. For example, if you have a polynomial of 2x + 4. Both terms have 2 in common. So you would divide both terms by 2. Once you have removed everything the terms have in common, you place what they have in common outside a set of brackets and what multiples by the thing in common to get the starting binomial on the inside. For example, if you have 2x + 4, and you divide 2x/2 and 4/2, you would end up with 2(x+2). Once you have done this you check to see if you can factor anymore. How do you know if you can factor again? If the terms inside the brackets have something in common then you can factor again, but if they have nothing in common they can’t be broken down anymore. In this example, 2(x+2), x and 2 don’t have anything in common, so the number is not factorable or prime.

Step 2

The second step in factoring 1 2 3 is 2. Are there two terms? If your answer is no then you move onto step 3. If your answer is yes you have to check if there is a difference of squares. A difference of squares is when both terms are perfect squares. For example, 2, 9, 16, 25, are all examples of perfect squares. The other thing to check for to see if it is a difference of squares is is it a subtraction question. For example, 4 – 9 would be a difference of squares because 4, , and 9 are all perfect squares and there is a subtraction sign. However, 4 + 9 would not be a difference of squares because all though 4, , and 9 are perfect squares they are adding together which would not be a difference of squares. When there is a difference of squares are factors they are factored into pairs of binomials called conjugates. Conjugates are binomials that share identical terms but have a different operation in the middle. For example, (x+2)(x-2) and (3x+1)(3x-1) are both conjugates

Step 3

The last step or step 3 in factoring 1 2 3 is 3. Are there three terms? If the answer is yes you want to look at the format of the trinomial. If it is a simple trinomial which means it starts with an , any number with x, and a number, you may be able to do a trick to factor it quicker. The trick is that are there 2 numbers that add together to make the number with x, that also multiply together to make the constant number. For example, if you have + 7x + 12. Are there two numbers that add together to make 7 and multiply together to make 12? If you don’t know off the top of your head you can make a list of all the possible combinations of numbers that multiply to 12. There is 1 x 12, 2 x 6, and 3 x 4. Now we can see that 3 and 4 add together to make 7 and multiply together to make 12. So you would right your answer as (x+3)(x+4). To check your answer you can expand and use FOIL or distribution to check your answer.

If there are three terms but it’s not a simple trinomial you can use the box method. We know from expanding using the box method that there are patterns to where the numbers are placed in the box. For example, the is placed in the top left of the box, the x’s are placed on diagonals, and the constant is placed in the bottom right. When you have your trinomial you want to place the in the top left and the constant in the bottom right. Now you are left over with the x. In the box, the two terms diagonal of each other can multiply together and equal the same number as the two terms on the other diagonal. For example, if you have + 13x + 15. The can multiply together with the 15 to make 30. Now you have to split the 13x into the other two boxes in a way where they multiply together to make 30. To make 30 you can split 13 into 10 x 3. Then you want to put 3x and 10x into your two empty boxes, it doesn’t matter where you place them. Lastly, you want to find the dimensions of the box by finding two numbers that multiply together to make the term on the inside of the box.

Questions that involve more than 1 step

In week 5 of Math 10, the most important thing we learned was how to multiply polynomials. The 3 ways we learned to multiply were with algebra tiles, area models, and FOIL/Distribution.

How to solve with algebra tiles?

When solving a question with algebra tiles you first need to draw a t-chart and place one polynomial going across the top and one polynomial going down the side. Next, you need to fill in the chart combining together the algebra tile on top and the algebra tile on the side. Note when you combine the two together, the sides will match together perfectly. When you combine the sides two x’s make an , an x and a 1 make an x, and two 1’s make a 1. Once you have filled in your chart you rewrite your equation using the algebra tiles inside the chart. Algebra tiles are a great visual way to solve questions, however, they have some disadvantages. One disadvantage is, you can only solve a question using algebra tiles when the degree is equal to one. For example, (x+3)(2x-6) would work, but (3-3x+2)(2-2x+6) would not work because it has a degree higher than 1.

Another reason people don’t like using algebra tiles is that it’s a lot of drawing. However, we learned a way to draw algebra tiles that involved less drawing. When drawing your algebra tiles instead of drawing a t-chart, you would draw a rectangle with one polynomial across the top and one down the side. Next, you would draw a line across the rectangle every time you got to the end of an x or 1 piece on the side of the rectangle. Then you would draw a line down the rectangle every time you got to the end of an x or 1 piece at the top of the rectangle.

Lastly, if the two outer pieces are the same, both positive or both negative you would shade the tile in. If the two outer pieces are different colors you wouldn’t shade in the inside pieces and they would stay white. Then combine your like terms and rewrite your answer.

How to solve with an area model?

Another visual way of solving questions that involve multiplying 2 binomials is an area model. To complete an area model you need to follow a couple of steps. First, you need to draw a box and cut it into pieces depending on how many terms your polynomials have. For example, if you are multiplying a binomial by a binomial you would draw a vertical line and a horizontal line splitting your box into 4 smaller squares. Next, you have to split your 2 binomials into their individual terms. For example, if you had (2x+4)(-5x-9), you would split them up into 2x, +4, -5x, and -9. Once you have split up your binomials you will place one of the individual terms on top of the first column and the other term on top of the second column. Then you would place one of the individual terms from the other binomial next to the top row and the other next to the bottom row.

Next, you have to fill the grid. To fill the grid you have to multiply 2 individual terms together. For example, if you are filling in the first box you would multiply the term above the first column and the term outside the first row. Lastly, you combine the like terms and rewrite your simplified answer.

How to solve with FOIL/Distribution?

Solving using FOIL and distribution are the nonvisual ways to solve questions that involve multiplying polynomials. FOIL stands for First, Outside, Inside, Last, and only works when you are multiplying binomials. To solve using FOIL you will multiply the terms together in the order of FOIL. First, the first means you are multiplying the first terms in each binomial. Outside means, you are multiplying the two outside terms which would be the first term in the first binomial and the second term in the second binomial. Inside means, you are multiplying the two inside terms which would be the second term of the first binomial and the first term of the second binomial. Last means, that you are multiplying the two last terms in both binomials. FOIL would not work when multiplying polynomials with more terms because if you were to follow FOIL some terms would be missed. Once you have multiplied using FOIL you combine like terms and rewrite your simplified answer.

Distribution works when solving all questions that involve multiplying polynomials. Distribution is just multiplying each term of one polynomial by the terms of the other. For example, if you have (2a+5)(a+4), you would distribute (a+4) to both 2a and +5, and would end up with 2a(a+4)+5(a+4). Then you use distribution and multiply. Lastly, you combine like terms and rewrite your simplified answer.

In week 4 of math 10, the most important thing we learned was multiplying a polynomial by a monomial.

What are monomials and polynomials?

Monomial – A monomial is an algebraic expression that consists of only one term. For example, 7, x, 5x, 0.75, and 2 are all monomials.

Polynomial – A polynomial is an algebraic expression that is made up of variables, constants, and exponents. A polynomial contains many terms. For example, 2y + 8z, +9x-3, and (5a-9b-2c) + (c – 7b – 3a), are all polynomials.

How to solve using algebra tiles?

Before you can solve equations with algebra tiles you need to know what they are and how they work. There are 3 types of algebra tiles that we have learned about and use to solve equations. The first is a big square. A big square represents $latex x^2″, a rectangle represents x, and a little square represents 1. The shaded algebra tiles are positive and the white algebra tiles are negative. For example, if you have a white rectangle and shaded little square, you would have -x and 1. To solve a multiplication question between a polynomial and a monomial you need to follow a couple of steps.

First, you need to draw a t-chart and place your monomial on the top and your monomial going down the side

Next, you need to fill in the chart combining together the algebra tile on top and the algebra tile on the side. Note when combining the two together the sides will match together perfectly. For example, if you are multiplying a positive x and a positive 1, your answer will be positive x. The sides will match because the length of the rectangle will match with the length of the x and the width of the rectangle will match with the width of the 1. When you combine the sides two x’s make an , an x and a 1 make an x, and two 1’s make a 1.

Lastly, once you have filled in your chart you rewrite your equation using the algebra tiles on the inside of the chart.

Bonus – If you want to check if your answer is right pick a number and insert it in all the variable spots. If your answer is correct the total on the left side of the equation will equal the total on the right side of the equation.

Solving without algebra tiles.

When solving a question that has a polynomial multiplied by a monomial, and you want to solve it without algebra tiles, all you need to do is use the distributive law. You will be given an algebraic expression with the monomial on the outside of brackets and the polynomial on the inside. For example, 2x(3x+5). The distributive law tells you to distribute/multiply everything inside the bracket by what’s outside. In this question, the 2x would be doing the distributing, so you would end up with 2x x 3x, and 2x x 5. You would simplify this to 6+10x.

In week 3 of Math 10, the most important things we learned about were naming the sides of a right-angle triangle and “SOH-CAH-TOA”, an acronym to help us find the side lengths and angles in a right-angle triangle.

How do you name a right-angle triangle?

In a right angle-triangle, there is one of three names a side can be called. The three names of the side lengths are hypotenuse, opposite, and adjacent. The name of the side depends on where the reference angle is located. The reference angle can never the right angle of 90 degrees. The hypotenuse is always the longest side and the side across from the 90-degree angle. The opposite or adjacent side depends on where the reference angle is. The opposite side is the side across from the reference angle and the adjacent side is the side next to the right-angle and the reference angle.

What is SOH-CAH-TOA?

When trying to find the value of an unknown side length of a right-angle triangle there are three types of trig functions. The names of the three trig functions are sine, cosine, and tangent. “SOH-CAH-TOA” helps you figure out what trig function to use. Let’s start with SOH. SOH tells us which function to use, the first letter is S which means we would use sine. The OH tells us which side lengths are being divided. In SOH it tells us that the opposite is being divided by the hypotenuse. In CAH, you would use the function cosine, and the adjacent is being divided by the hypotenuse. In TOA, you would use the function tangent, and the opposite side is being divided by the adjacent side.

Example of how SOH-CAH-TOA can be used to find an unknown side length

First, you want to draw your triangle and label your sides. Hypotenuse, Opposite, and Adjacent. Next, you want to find out what side you are looking for. In this question, we are looking for x which is on the adjacent side. Then find what sides are needed in order to solve for side length x. In this question, the two sides needed to solve for x are the hypotenuse and adjacent.

Knowing “SOH-CAH-TOA,” we need to find which function is needed to find a side length with the two sides hypotenuse and adjacent.

We can see that “CAH” has both the hypotenuse and adjacent, so this means to find the side length x we are going to use the cosine function.

Next, you are going to write out your ratio and equation. Depending on where the variable is will depend on how you solve it. If the variable is on top then you will just multiply both sides by the denominator number to isolate x. If the variable is on the bottom you will have your function and x switch places, again this will isolate x.

Once you have rewritten the formula, you can solve it. In this question, I have rounded my answer to the nearest tenth of a decimal, and because I have rounded my answer I have put a dot over the equation sign showing it’s not an exact answer.

Example of how SOH-CAH-TOA can be used to find an unknown angle

First, you want to draw your triangle and label your sides. Hypotenuse, Opposite, and Adjacent. Next, find which two side lengths are needed to solve for the angle. In this question, the hypotenuse and opposite sides are needed.

Knowing “SOH-CAH-TOA,” we need to find which function is needed to find angle x with the two sides hypotenuse and opposite.

We can see that “SOH” has both the hypotenuse and opposite, so this means to find angle x we are going to use the sine function.

Next, you’re going to write your ratio and equation. When trying to find the degrees of an unknown angle you are going to move the function to the opposite side of the equation and turn it to its opposite. In this question, we would change the sign to inverse sine.

Now, you’re going to rewrite your formula and solve it. In this question, I have rounded the angle to the nearest degree, so I have put a dot over the equal sign showing that it’s not an exact answer.

In week 2 of Math 10, the most important things we learned about were negative exponents and how to solve questions that involve negative exponents, and scientific notation.

What are Negative Exponents?

Negative exponents are also known as integral exponents. The integral exponent law says that when it’s a negative exponent the copies of the base are in the other part of the fraction. Negative exponents are just reciprocals of their positive counterparts. For example, is 81 or can be written as a fraction of 81/1. Whereas, 3^-4 is 1/81. 81/1 and 1/81 are reciprocals of each other.

The negative exponent just tells you that the exponent is on the wrong side of the fraction. A negative exponent can move from the numerator to the denominator and the denominator to the numerator.

Once you have moved the negative exponent to the other side of the fraction it becomes positive. For example, if you have 3x^-2, you would rewrite your question as 3x^-2/1, then you would move the x^-2 to the denominator and your final answer would be 3/x^2. When solving with negative numbers you always want to have your final answer in positive exponents.

Examples:

What is Scientific Notation?

Scientific notation is also called standard form. To write in scientific notation numbers are written as a number times 10 to the power of something. For example, normally you would write a number like 700, wherein scientific notation you would write that as 7 x . Why is 700 written like this? It’s written like this because 7oo = 7 x 100, 100 = , and finally, you get 7 x . Scientific notation is written in 2 parts: the digits, with the decimal point placed after the first digit. Followed by x 10 to a power. The power tells you how many times to move the decimal point. If the power is a positive number the decimal point will move to the right and if the power is negative the decimal point will move to the left. For example, if you have 7 x your decimal will move to the right and you will get 700. If you have 7 x 10^-2 your decimal will move to the left and you will get 0.07. If your number is shown in standard form. For example, 950 instead of 9.5 x . To find your answer all you need to do is move the decimal either right or left depending on your number to in front of the first digit. The number of times you moved the decimal is going to be the exponent on the 10.

Examples:

In week 1 of Math 10, the two most important things that I learned were how to find the lowest common multiple (LCM) and the greatest common factor (GCF) of two or more numbers using prime factorization.

How to find the LCM?

To find the LCM of two numbers you start by writing out your numbers and below your numbers making a factor tree to find the factors of your numbers. To complete a factor tree you would write two numbers branching out from your number that multiple together to make that number. For example, if your number is 15 you would write 5 and 3 branching out underneath 15.

Once you have found all your factors for both numbers you want to circle all the prime numbers. You want to circle your prime numbers so they are easier to find for the next step. Once you have completed your factor tree and circled all your prime numbers, you want to rewrite your number and the factors side by side next to it under your factor tree. For example, if your number was 150 you would write 150 = 2 x 3 x 5 x 5.

You do this for both numbers. Once you have completed this step you want to circle the common pairs. When circling common pairs you want to circle them vertically or diagonally, you don’t want to circle common pairs in the same row.

Then you place each common pair, written once inside brackets, and the other factors are placed outside the brackets. For example, if you have a common pair of 2, 3, 5, and your other factors are 2, 5, 7, you would write in brackets (2 x 3 x 5), and on the outside, you would write 2, 5, 7, and you would finish with (2 x 3 x 5) 2 x 5 x 7.  Lastly to find your LCM you multiple all the factors together.

Examples

How to find the GCF?

To find the GCF of two numbers you start by writing out your numbers and below your numbers making a factor tree to find the factors of your numbers. Once you have created your factor tree you want to circle all the prime numbers so they are easier to find for the next step. Once you have circled all of the prime numbers you want to write any prime numbers they both share or common factors under your factor trees. For example, if both numbers share a 5 and 9 you would write those two numbers under your factor trees. Lastly multiple your common factors together to find the GCF of your two numbers.

Examples