Category: Math 10

Systems:

This week we started on our last chapter in math 10, systems of linear equations. We learned about the 3 different ways to solve these equations. The 3 ways are, graphing, substitution, and elimination, my favourite being elimination.

Graphing example:

• Always make both given equations into slope y-intercept form (y=mx+b)
• m= slope & b= y intercept
• only use this when you have small and simple numbers (no fractions or decimals as the y intercept because they can be hard to graph)
• graph both lines with the 2 bits of information (slope and y-intercept)
• find the spot where the 2 lines intersect and find its coordinates (this is your answer)

Substitution example:

• It doesn’t matter what variable you iscolate first
• To get the second variable (x or y) substitute the number you got on the previous step into the x or y place and solve again
• ALWAYS check your answers by putting the coordinates back into both equations and solving to see if the equation works
• Need to know BEDMAS and algebra to solve

Elimination example:

• your goal is to make a zero pair with one of the 2 variables (doesn’t matter what one)
• zero pair= when 2 numbers add to zero so they just cancel out and disappear
• to make a zero pair you need one positive and one negative of the same number and variable
• like in substitution when you have solved for one variable put in back into the equation to solve for the other
• example of zero pair= -2x & 2x
• if there arnt already any zero pairs you can make one by multiplying an entire equation by the same number
• after cancelling out a zero pair add the 2 equations together
• after adding and finding the first variable solve for the other variable by inputing the x or y into the other equation
• ALWAYS CHECK your answer by inputing both variables into their places and solving

This week we learned about equations, and how all of the numbers and variables mean something. The number beside the x in an equation is equal to the slope and the last number is the y intercept. In a example below I am going to show and explain how a t-chart, graph and equation can all represent the same thing. If you have one of these things (a graph or t-chart or equation), you can make the other 2 with just that information. I found it cool how all of these are intertwined and how they are related. Even though they all look very different they contain and display the same information, it doesnt matter how you present it, in either of the 3 ways the viewer will take out the same data.

Things to know:

• the number beside the x is the slope
• the last number that you add or subtract is the y-intercept
• the slope is equal to rise/run
• if the slope is a whole number you put that number over 1 (run=1)
• in a t-chart when displaying the data the rise=y and run=x (add the run of the slopes fraction to the first column which is x, and the top number of the slopes fraction to the second y column of the t-chart)

      

Comparing different methods to find slope:

At the beginning of this week we learned how to find the slope of 2 points without a graph. Before, we were taught to count the rise and run from the boxes on a grid but without the grid lines it makes it pretty challenging to determine the slope of a line. This is where we would use a formula, this does the same thing it is just a different way to find slope. The only requirement is that we are given 2 coordinates with 1 x and y value each.

In an example below I will use the same two coordinates on a graph and in a formula to prove that you will still end up with the same slope no matter what method you use.

*the formula is y(#1)  –  y(#2) / x(#1   –   x(#2) /// the first y in coordinate 1 subtract the y in coordinate 2 divided by the first x in coordinate 1 subtract the x in the second coordinate*

What you need to know:

• the first number in a coordinate is x
• the second number in a coordinate is y
• the second coordinate is 2
• the first coordinate is 1
• two negatives equals a positive (1-(-2) = 1+2)
• always y/x (remember this by rise=y & run=x // (rise/run)

Example:

Comparing slope:

We also learned about the term “collinear” this is a word used a lot in this unit of slope so it is important to understand its meaning. This term is referring to the relationship between 2 or more points/coordinates on a graph. In other words you are looking a weather the points all have the same slope. This also requires the use of the slope formula when you aren’t given a graph. The example below is demonstrating how to solve a question that is asking if points are collinear, with and without a graph (use of formula).

Collinear: is the slope of 3 or more points the same (do they line up to make a straight line)

Slope of a line:

This week in Math 10 I learned all about the slope of a line. We would use this when trying to find, locate, or describe, a line or point on a graph. The idea and method of slope is fairly simple as long as you understand the basics and know how to identify the different types of slopes.

*The slope is basically directions to find any line or point on a graph*

Things you need to know:

• slope= rise over run (rise/run)
• rise= up and down (vertical/y-axis)
• run= side to side (horizontal/x-axis)
• always start with the point on the left of the graph, because you read left to right!
• a nice point on a graph is a point located directly on the grid lines and not in between (no decimals)
• positive slope = diagonal line pointing towards the right
• negative slope = diagonal line pointing towards the left
• undefined slope = straight horizontal line (parallel to the x-axis)
• zero slope = straight vertical line (paralell to the y-axis)

How to find the slope of a line:

1. find 2 nice points on the line
2. connect the lines with a right angled triangle
3. determine if the slope is negative or positive (is the line pointing left or right? right= positive// left= negative)
4. figure out the vertical change (up and down) (rise)
5. figure out the horizontal change (side to side) (run)
6. put the numbers in the fraction (rise/run) and divide or reduce to lowest terms if and when possible

Examples:

This week we started a new unit of graphing. Along with this unit comes more vocabulary than others. This is something I used to help me with the relationships between all of the words and how they are related. This photo is showing the way the x and y are connected and intertwined throughout this unit. In the blue is everything related to x and in pink is everything with y.

the x is the -> input, independent variable, horizontal axis, domain, and the first number in a ordered pair

the y is the -> output, dependent variable, vertical axis, range, and the second and last number in an ordered pair

at first I struggled with remembering what words mean the same thing. Vocabulary is a huge part of this unit so I needed toremember the relationship between the words and the x and y. This was my way of remembering, to create a photo.

This week we learned how to factor “ugly” trinomials. These are trinomials that you cannot use that pattern to factor. The pattern is that one of the factors off the last term would add to the middle term. For ugly triennials this is not the case. At first I struggled a bit with this new concept, from knowing when to use it, toactually factoring it and knowing if its right. I was going to go early to math to get extra hep but i was determined to teach and figure stout myself. Here is the method I use to factor these trinomials that you can’t use the pattern for. And also how to know when you can’t use the pattern and how to check if its correct.

   

This week in polynomials we covered many different methods for ways to solve different polynomial questions. I had one question I was meaning to ask, it was “what method do we use in certain situations, and know do we know when to use each different one”. Before asking I was determined to try my best to figure it out myself. I took a bit of extra time to look at these different methods and investigate my question. After a bit of extra work I answered my own question, making a few notes of what to look for when trying to figure out what method to use. I made a video explaining only a few of the methods I looked into. I tried to explain my way of knowing how you know when to use each different method.

 

Different solving methods:

• the pattern
• GCF (greatest common factor)
• conjugates (zero pairs or difference of squares)

 

The pattern:

1. has to factor to a simple binomial
2. no leading coefficient (GCF first to reduce and simplify)
3. has to have x² in front (no exponent higher than 2)  if so reduce with GCF first
4. the last term has to have a factor that adds up to the middle term

GCF (greatest common factor)

1. all numbers have to have something in common
2. all numbers have to be divisible by the same # or variable
3. used to simplify the polynomial so you can do the pattern after

Conjugates:

1. no middle term
2. a factor of the negative last term has to equal 0
3. can use the pattern if you recognize that the middle term has t0 add to 0

 

This week in math 10 we learned how to find missing angles, which was differs from last week where we had to find the opposite, missing side lengths. Here is a short video I made explaining how to find a missing angle when only given 2 side lengths.