Summary: Systems of Equations- Graphing

Systems of Equations- Graphing                                November 28, 2011

OBJECTIVE: SWBAT SOLVE SYSTEMS BY GRAPHING, ELIMINATION AND SUBSTITUTION 

Today we learned about the three different types of systems of equations. The three methods are graphing, elimination and substitution. When graphing, always convert equations to y=mx+b. When you graph both lines, look for the point of intersection. This point is your solution set. A solution set is a solution to the equations that you graphed; it must satisfy both. Parallel lines have no solution; they have a solution, but they don’t share the same solution. You graphing your lines you should start labeling them. An example is shown below.

EXAMPLE:

Summary: Systems of Equations- Substitution

Systems of Equations- Substitution                        November 28, 2011

OBJECTIVE: SWBAT SOLVE SYSTEMS BY GRAPHING, ELIMINATION AND SUBSTITUTION

Today we learned about the three different types of systems of equations. The three methods are graphing, elimination and substitution. Today we focused on substitution and graphing. For substitution, it might be easier to use different colors and draw lines to separate the parts you have already done. You don’t have to do this; this is optional, but this usually helps others. There are four steps to remember when doing substitution. 1) Pick a variable-either x or y; 2) Solve the variable; 3) Substitute into an equation; 4) Combine Like Terms- CLT. An example is shown below on how to do this.  

EXAMPLE:

Summary: Scatter Plots and Trend Lines

Scatter Plots and Trend Lines                                         November 1, 2011

OBJECTIVE: SWBAT DISCUSS TRENDS AND TREND LINES; IDENTIFY SCATTER PLOTS

Today we learned about scatter plots, trends and trend lines. Scatter plots and trend lines are like correlations. Correlations are how things are related to each other. Scatter plot is a graph of scattered points and trend line is drawing a line in the middle of all the scattered points. Basically, you want to include as most of the dots as possible. To know if the trend line is positive, the change in y and the change in x is either both positive or both negative. The trend line is negative if the change in y is positive and the change in x is negative. An example is shown below.

EXAMPLE:

Summary: Continuous and Discrete

Continuous and Discrete                                                                  October 27, 2011

OBJECTIVE: SWBAT DIFFERENCIATE BETWEEN CONTINUOUS/ DISCRETE

Today we learned about continuous and discrete. Discrete means that the item has to be a whole number. Discrete usually occurs when we are counting using whole numbers. Continuous can be fractional numbers and whole numbers. Basically, something that is measurable: height, weight, and time. If the numbers can be fractions and decimals then it is probably continuous. Some examples are shown below.

EXAMPLE:

Classify each set as discrete or continuous-

1. The number of suitcases lost by an airline.

2. The height of corn plants.

3. The number of ears of corn produced.

4. The number of green M&M’s in a bag.

5. The time it takes for a car battery to die.

6. The Production of tomatoes by weight.

1) Discrete. The number of suitcases lost must be a whole number.

2) Continuous. The height of corn plants can take on infinitely many values (any decimal is possible).

3) Discrete. The number of ears of corn must be a whole number. 

4) Discrete. The number of green M&M's must be a whole number. 

5) Continuous. The amount of time can take on infinitely many values (any decimal is possible).

6) Continuous. The weight of the tomatoes can take on infinitely many values (any decimal is possible). 

Summary: Shading Inequalities

Shading Inequalities                                                                 October 25, 2011

OBJECTIVE: SWBAT RECALL INEQUALITIES AND HOW TO SOLVE THEM

Today we learned more about inequalities and about how to shade the graph. For shading a graph, you either shade with or without the point you are using to substitute it in an equation. For example, let’s use the inequality y<4x-3. For x and y we will substitute in the points (0, 0). Now the equation reads 0<4(0)-3. Solve that and it reads 0<-3. If this inequality is true then you will shade the part that INCLUDES the points (0, 0). If the inequality is false then you would shade AWAY from the point.

EXAMPLE:

y<4x-3

Summary: Graphing Inequalities

Graphing Inequalities                                                                 October 25, 2011

OBJECTIVE: SWBAT RECALL INEQUALITIES AND HOW TO SOLVE THEM

Today we learned more about inequalities and about how to graph them. We learned that for inequalities with < and > that they had a dashed line. Then for ≤ and ≥ they have solid, as well as =. Graphing inequalities is actually easy. You graph it like a regular slope intercept equation. You just have to make the “y” an isolated variable as it is in slope intercept form. An example is shown below.

EXAMPLE:

y+3<4x à y<4x-3

Summary: Inequalities

Inequalities                                                                 October 24, 2011

OBJECTIVE: SWBAT RECALL INEQUALITIES AND HOW TO SOLVE THEM

Today we relearned inequalities. I actually remembered this from when we were in sixth grade and that all we had to do was just remember what the sign were. Those signs were: less than of equal to (≤), less than (<), greater than or equal to (≥), and greater than (>). We also learned that when you graph inequalities, you use a dashed line, not a regular line. Also we learned that when you’re writing inequalities on the number line there are open and closed circles. Open circles mean less than or greater than and closed circles mean it has the equal to part with it. An example is shown below.

EXAMPLE:

  3 ≤ x-2 ----------------------------> Your Problem

+2   +2-----------------------------> Add 2 to 3 and -2

5 ≤ x -----------------------> Answer used to graph on number line

Summary: Linear Equations

Linear Equations                                                                 October 13, 2011

OBJECTIVE: SWBAT WRITE, GRAPH, AND CREATE LINEAR EQUATIONS IN THE THREE FORMS

Basically all we did was review slope intercept (SI), point slope (PS), and standard form (STD). Along with that, we practiced changing from one form to another. For example, we did a problem for PS then changed it to SI then changed it to STD. Now, changing from PS to SI is quite easy actually but, changing from SI to STD. Now, that is a whole another story because I get it but, then I don’t get it. It looks hardier than it actually is but, I think I can manage to it.

EXAMPLE:

(3, 4) b=-5

1. Point Slope (PS)-

*To find your slope do y1-y2 over x1-x2, when you have two points*

-5-4/0-3 equals -9/-3 which equals positive 3

Now just substitute the numbers into the formula-

y-4=3(x-3)

2. Slope Intercept (SI)-

*Substitute in the numbers for slope and y-intercept*

y=3x-5

3. Standard Form (STD)-

*If your slope is positive the y will be negative*

**If your y is negative then C might be positive**

3x 1y=

Now, just figure out the sign and if the C is negative or positive

3x-1y=5 or 3x-y=5

   -y is negative because it will then make -3x positive and make 5 negative

Summary: Perpendicular and Parallel Lines

Perpendicular and Parallel Line                          October 13, 2011

OBJECTIVE: SWBAT DESCRIBE THE SIMILARITIES OF ┴ (PERPENDICULAR LINES) AND ║ (PARALLEL LINES)

Today in class we learned about perpendicular and parallel lines. This was actually kind of easy. All you have to know for parallel lines is that lines are parallel if they have the same slope. For perpendicular lines, if they have the same y-intercept they will eventually intersect. Also, for perpendicular lines, you use multiplicative inverse for the slope. For example, 1/2, the inverse is -2/1 because you want the opposite of ½ so it has to be negative. Some examples are shown below.

EXAMPLES:

Parallel Lines-

~Remember to make lines parallel, they have to have the same slope~

y=2/3x+4

y=2/3x+6

*To make this parallel with the equation above, change the y intercept*

Perpendicular Lines-

~Remember to make lines perpendicular, you have to use multiplicative inverse~

y=3x+2

y=-1/3x+2

*If they have the same y-intercept, they will eventually intersect*

Summary: Standard Form

Standard Form                                                                  October 5, 2011

OBJECTIVE: SWBAT USE WHITE BOARDS TO GROUP LINES USING POINT SLOPE FORM

Today in class, we went over standard form again and learned how to actually do it this time. Another name for standard form is STD, not the disease though. The formula for standard form is Ax+By=C. Learning this was very confusing. It still is but, I can understand it better than before. An example is shown below about how to do this.

EXAMPLE:

Ax+By=C

***m= -Ax/B; b= C/B***

3x+2y=10

2y=-3x+10

    -Subtract 3x from both side

y=-3x/2+ 10/2

    -Divide everything by two

y=-3x/2+5

    -Reduce and you are done!

    -You know you are done if the answer looks like y=mx+b

Summary: Point Slope Countined

Point Slope Continued                                                                  October 3, 2011

OBJECTIVE: SWBAT GRAPH A LINE USING POINT SLOPE FORM

Today in class, we reviewed what point slope is. We practiced how to write point slope into slope intercept. It was actually pretty easy to understand. But it just took a long time to get it. We then took a quiz on this, which I failed. I know that I could have passed if Mr. Sudovsky gave us more time. But, I am just glad that he didn’t take it up as a grade.

EXAMPLE:

Slope- 1/10; Point- (10, 6)

y-6=1/10(x-10)----> Point Slope 

y=1/10(x-10)+6

y=1/10x-1{10/10}+6

y=1/10x+5----> Slope Intercept

Summary: Point Slope

Point Slope                                                                  September 30, 2011

OBJECTIVE: SWBAT REVIEW CFA TO GAIN BETTER UNDERSTANDING OF ALGEBRA USAGE IN REAL WORLD APPLICATIONS

Today in class we learned about point slope. Point slope is where they give you the point and the slope. The formula for point slope is y-y1=m(x-x1). It is hard to explain so; I will show you in the example instead. We also learned about quadratics, standard and some linear equations with how their graphs look like. The formula for standard is Ax+By=C. Some of the linear equations were y=x²+4, y=x³, y=x+2, and y=1x1.

EXAMPLE:

Summary: Slopes

Slopes                                                                   September 14, 2011

OBJECTIVE: SWBAT REVIEW UNIT 1 EXAM AND RECOGNIZE KEY CONCEPTS MOVING FORWARD

Today in class we learned more about slopes. We learned how to do the input and output table for them also. We had a question where it asked what the “y” intercept was. The answer was (0,?) because the definition of intercept is where the line meets the graph. Any time the given question is what is the x intercept or the y intercept the answer is (0,?) or (?,0).  An example is shown below.

EXAMPLE:

Summary: Plotting a Graph

Plotting a Graph                                                                  September 13, 2011

OBJECTIVE: SWBAT USE DOMAIN/RANGE OF A FUNCTION AND PLOT THEM ON A GRAPH  

Today in class we learned how to plot a graph. We learned the words intercept, origin, and slope. The intercept is where the line or graph “intercepts” the axis. The origin is the starting point, not the center. To make a line you need three points; when reading the line you read it from left to right. The slope is always a fraction and you need to draw arrows at the end of the lines you draw. The reason is because the line is continuous and never ends. There are two steps to plotting a graph. Step one: locate the intercept and step two: use slope. An example is show below.

EXAMPLE:

Summary: Multiplying Matrices

Multiplying Matrices-                              August 31, 2011

OBJECTIVE: SWBAT EVALUATE MATRICES BY APPLYING THE SCALER MULTIPLICATION METHOD 

Today in class we learned how to multiply matrices. At first, I did not get it at all; to me it was just a bunch of numbers thrown into a bracket to make matrices. But later on when the video started to explain how to do everything; I got it. It looked hard at first but then when we did it, I saw just how easy it was. The only things that still throw me off are the negatives but, I think I got a hang of it now. An example is shown below.

EXAMPLE

Summary: Matrices Continued

Matrices Continued-                                                            August 27, 2011

OBJECTIVE: SWBAT DEFINE AND NAME MATRICES GIVEN A NUMERAL SET OF DATA

Today we learned what equal matrices are. Equal matrices are two matrices that have the same dimension and each element of one matrix is equal to the corresponding element of the other matrix. Basically, it’s like saying 'copy and paste'. We also learned what determinants are; a determinant represents a single number. We obtain this value by multiplying and adding its elements in a special way, which is multiplying diagonally. Examples are shown below.

EXAMPLES

Summary: Matrices

Matrices-                                                                                     August 26, 2011
OBJECTIVE: SWBAT DEFINE AND NAME MATRICES GIVEN A NUMERAL SET OF DATA

Today we learned what matrices are and how to add and subtract them. A matrix is a rectangular arrangement of numbers into rows and columns. To add and subtract matrices they have to have the same number of rows and columns. It is hard to explain how to do them so I will give you an example below. We also learned what an element is and what a dimension is. An element is each value in a matrix; either a number or a constant. A dimension is the number of rows by the number of columns in a matrix.

EXAMPLES:  

 

Summary: Function Notation

Function Notation-                                                                             August 18, 2011

OBJECTIVE: SWBAT DEFINE, EXPAND, AND CREATE RELATIONS THAT ARE FUNCTIONS AND NOT FUNCTIONS

Today we learned about function notation and how to use it. Function notation is f(x)=y and is read as "f of x". For example, y=2x+1 can also be written as f(x)=2x+1. Examples of how to solve problems for function notation is shown below. The domain cannot contain values for which the range is undefined (0).

EXAMPLE:

f(x)=4x²-2x+5              f(4)                       
     =4(4)²-2(4)+5
     =4(16)-2(4)+5
     =64-8+5
     =56+5
     =61

Summary: Vertical Line Test

Vertical Line Test-                                                                       August 17, 2011
OBJECTIVE: SWBAT UNDERSTAND AND USE COMPATIBLE NUMBERS TO ASSIST IN PROBLEM SOLVING

Today we reviewed what a function is and what a relation is. A function is when every x has its own y and a relation is a set of ordered pairs. We also learned about the vertical line test is. A vertical line test is when you draw a line on the graph of the relation and the vertical lines cross two different points then the relation is not a function. We also learned how to tell which relation is a function.

EXAMPLE:Which relations below are functions?

A: Names and social security numbers
B: Addresses and names
C: (2,4) (-2,5) (3,7)
D: (4,1) (4,3) (5,6)
E: (2,5) (3,5) (4,5)

Summary: Relations & Functions

Relation and Functions-                                                              August 16, 2011
OBJECTIVE: SWBAT UNDERSTAND AND USE COMPATIBLE NUMBERS TO ASSIST IN PROBLEM SOLVING

Today we learned what a relation is, what a function is, and the names for "x" and "y". The names for "x" is domain, abscisses, input, and independent; for "y" is range, ordinates, output, and dependent. A relation is a set of ordered pairs. A function is a relation in which each element of the domain is paired with exactly one element of the range. Basically, for every "x" there is only one "y".

EXAMPLE:

If x is a positive integer less than 6 and y=5+x, list the ordered pairs that satisfy the relation. State the domain and range of the relation.

y=5+x    x<6~54321                              

D	1	2	3	4	5
R	6	7	8	9	10

   

Summary: Cornell Notes & Love

Cornell Notes and Love-                                                  August 15, 2011

OBJECTIVE: SWBAT DEFINE LOVE

Today we learned to define the word "love". There are many synonyms for the word like: caring, affection, family, word, feeling, something you give/ receive, paper, etc. We also learned how to set up for Cornell Notes in class. The objective goes on the first line; date in the upper right hand corner; summary on the bottom of page; and vocabulary, examples, definitions and questions in the space between the spine and the red margin.