4-1 Coordinate System / 3 days
4-2 Relations / 3 days
4-3 Linear Equations / 5 days
4.1 – 4.3 Quiz / 1 day
4-4 Slope / 4 days
4-5 Slope Intercept Equation / 5 days
4-6 Scatter Plots / 5 days
4-7 Functions / 3 days
Test Review / 1 day
Test / 1 day
Cumulative Review / 1 day
Unit Project / 4 days
Total days in Unit 4 - Algebra = 36 days
Review Question
What way do we move on the number line for a negative number?Left
Discussion
What is GPS? Global Positioning System
How does it work? A satellite sends a signal down to your phone or device. Then your device sends a signal back. Based on the amount of time that it takes the signal to get back to the satellite, the satellite can calculate where you are. Specifically, it knows how far up/down and left/right you are on the earth. It bases this on the longitude and latitude lines on the earth. The satellite can give you the exact coordinates of your location and the location of where you are heading.
This is basically what the coordinate system is. It is a set of horizontal and vertical lines that allows us to find the location of any point, line, or graph.
A majority of Algebra is the study of lines. This unit is going to be an introduction to lines. We are going to talk about points, slopes, and graphing lines. These are three major topics in Algebra I. We need to start the discussion with points because they are what make up lines.
SWBATplot points on a coordinate plane
SWBATstate the location of a point on a graph
Definitions
Coordinate System– “the graph thing”
X-axis– horizontal line
Y-axis– vertical line
Origin– point where the two lines meet
Quadrants – four sections of the graph
Draw and label picture based on the definitions above.
Every point has two parts: the x-coordinate and the y-coordinate. The x tells you how far to go left and right. The y tells you how far to go up and down
(x, y)
A little hint to help remember: Run then Jump.
Example 1: Graph the following points.
(2, 3) (-3,1) (5,0) (0,-3) (-2,-3) (4,-2) (3.1, -4.8)
Example 2: State the location of each point.
A(-5,2) B(-2,0) C(-2,-3) D(0,3) E(3,2) F(3,-4)
What did we learn today?
Graph and label each point.
Then state the quadrant.
1. A (4, -3)2. B (5, 4)
3. C (-1, 7)4. D (2, 8)
5. E (-6, -6)6. F (-5, 3)
7. G (2.2, -7.4)8. H ()
9. I (0, -4)10. J (3, 0)
Write the ordered pair for each point graphed on the coordinate plane. Then state the quadrant.
11. J______
12. K______
13. L______
14. M______
15. N______
16. O______
17. P______
Review Question
How would you plot the point (-3, -4)? Left 3 units, Down 4 units
What quadrant would that be located? III
Discussion
Has anyone ever played or heard of the game Battleship?
How does the game work?
Before play begins, each player secretly arranges their ships (5 ships) on their grid. Each ship occupies a number of consecutive squares on the grid, arranged either horizontally or vertically. The number of squares for each ship is determined by the type of the ship. The ships cannot overlap (i.e., only one ship can occupy any given square in the grid). The types and numbers of ships allowed are the same for each player. These may vary depending on the rules.
After the ships have been positioned, the game proceeds in a series of rounds. In each round, each player takes a turn to announce a target square in the opponent's grid which is to be shot at. The opponent announces whether or not the square is occupied by a ship, and if it is a "hit" they mark this on their own primary grid. The attacking player notes the hit or miss on their own "tracking" grid, in order to build up a picture of the opponent's fleet.
When all of the squares of a ship have been hit, the ship is sunk, and the ship's owner announces this (e.g. "You sank my battleship!"). If all of a player's ships have been sunk, the game is over and their opponent wins.
We are going to play a version of this game today.
SWBATplot points on a coordinate plane
SWBATstate the location of a point on a graph
Activity
Each student receives a piece of graph paper. They will mark the x-axis and y-axis from -20 to 20. Each student will plot four different line segments on their graph paper. This will represent their 4 ships. Each segment will be either horizontal or vertical. The segments can’t overlap each other. Each segment will be a different length: two units long, three units long, four units long, and five units long.
Then each student will partner with another student. Each student will receive another piece of graph paper to record their guesses. The students will follow the rules from the Battleship game above.
What did we learn today?
Review Question
How would you plot the point (-3, 6)? Left 3 units, Up 6 units
What quadrant would that be located? II
Discussion
What is difficult about taking a group shot of 10 of your friends at a dance with your smart phone?
Fitting them all into the picture
When you take the picture, you must make sure that you zoom out enough to fit all of them in the picture. Also, you try to zoom out enough to make sure that there is some space all the way around the edge of the picture so it looks nice. You wouldn’t cut off someone’s arm because you didn’t zoom out enough. Can someone show me a good picture of a group of friends where you properly zoomed out to fit everyone nicely in the picture?
We are going to be doing the same thing today with graphing points on the coordinate plane. If you were trying to plot the points (1, 2) (-65, 125) and (55, -83), you would have to make sure that the picture was “zoomed out” enough to see all of the points. You can accomplish this on a graph by adjusting the scale that you use. On the graphing calculator, you will “zoom out” by adjusting the window. This is the portion of the graph that you see on the calculator.
SWBATplot points on a graphing calculator or graph paper
SWBATcreate an appropriate window or scale based on a data set
Example 1: Plot (-2, 5) using the graphing calculator
1. Turn stat plot on
2. Stat – edit – enter data in L1 and L2
3. Graph
Example 2: Plot (13, -35) (18, -3) (-50, 20) using the graphing calculator
1. Stat – edit – enter data in L1 and L2
2. Graph
How many points should you see? 3
Why can’t we see all of the points? Screen isn’t big enough
Now think of each point as a couple that we are trying to fit into the picture. The x’s are the boys and the y’s are the girls. Let’s make the screen (window) big enough that everyone fits into the picture.
Changing the window and scale:
1. Window
2. Change:
xmin, xmax, xscl
ymin, ymax, yscl
You Try!
1. Plot the following points. Create an appropriate window or scale.
(-11, 1) (-11, 3) (-11, 5) (-11, 7) (-11, 9) (-8, 5) (-5, 5) (-2, 1) (-2, 3)
(-2, 5) (-2, 7) (-2, 9) (2, 1) (2, 3)(2, 5) (2, 7) (2, 9)
2. What does it say? HI
3. Try to get the initial (at least 8 points) of your first name in quadrants one and four with an appropriate window.
What did we learn today?
1. Given the following points, fill in reasonable values for an appropriate window.
(10, 25) (-12, 36) (1, -10) (5, 4)
XMin = XMax = XScale =
YMin = YMax = YScale =
2. Given the following points, fill in reasonable values for an appropriate window.
(-24, 5) (-10, 6) (0, -22) (15, 4)
XMin = XMax = XScale =
YMin = YMax = YScale =
3. Given the following points, fill in reasonable values for an appropriate window.
(100, 250) (-125, 50) (10, -100) (50, 75)
XMin = XMax = XScale =
YMin = YMax = YScale =
4. Given the following points, fill in reasonable values for an appropriate window.
(-15, -45) (-12, -36) (-28, -24) (-5, -4)
XMin = XMax = XScale =
YMin = YMax = YScale =
5. Given the following points, fill in reasonable values for an appropriate window.
(55, 35) (25, 45) (75, 40) (5, 5)
XMin = XMax = XScale =
YMin = YMax = YScale =
6. Draw your own picture on a piece of graph paper. It must contain at least 20 points. Your picture must be in all four quadrants. Label each one of the points and list the corresponding coordinates down the right hand side of the paper.
Review Question
How would you plot the point (-5, -8)? Left 5 units, Down 8 units
What quadrant would that be located? III
Discussion
I can’t think of a good one to start the lesson. But I have a good one for the end of the lesson. Be patient.
Do you know why you wouldn’t be good doctors? No “patients”
SWBATstate thedomain,range, and inverse of a relation
SWBATexpress a relation as a table, map, graph, or ordered pair
Definitions
Relation – set of ordered pairs
Domain – x values
Range – y values
Inverse – switching x’s and y’s
Example 1: State the domain, range, and inverse of the following relation:
(-2, 5) (5, 10) (-8, 3) (-2, 12)
D: {-2, 5, -8}
R: {5, 10, 3, 12}
I: (5, -2) (10, 5) (3, -8) (12, -2)
Example 1 (Continued): Write the previous relation as a table, map, and graph.
Table:Map:Graph:
x / y-2 / 5
5 / 10
-8 / 3
-2 / 12
You Try!
1. State the domain, range, and inverse for the following relation: (2, 1) (1, -5) (-4, 3) (4, 1).
D = {2, 1, -4, 4}
R = {1, -5, 3, 1}
I = (1, 2) (5, 1) (3, -4) (1, 4)
2. Express the relation as a table, map, and graph.
See Above
Discussion
Can you figure out the domain and range for the following graphs?
1. Domain: All Reals
Range: All Reals
Is there any way to have a line whose domain and range are not all reals? How? See Below.
2.Domain: x = 5
Range: All Reals
3.Domain: All Reals
Range: y 0
What did we learn today?
State the domain, range, and inverse for the following relations. Then express the relation as a table, map, and graph.
1. (5, 2) (-5, 0) (6, 4) (2, 7)
2. (3, 8) (3, 7) (2, -9) (1, -9)
3. (0, 2) (-5, 1) (0, 6) (-1, 9)
4. (7, 6) (3, 4) (4, 5) (-2, 6) (-3,2)
Express each table, map, or graph as a relation.
5.
x / y-3 / 6
5 / 1
-8 / 2
-3 / 5
6.
7.
State the domain and range of each graph.
8. 9.
10. 11.
Review Question
What is a relation?Set of points
What is domain?‘x’ values
Discussion
Why don’t you take your parents’ car and drive around?
Because your parents put restrictions on what you can do and you listen to them.
Notice that you do things because your parents say so. Also, notice that your parents put restrictions on what you do.I am going to do the same thing today. First, I am going to put certain restrictions on what the domain is allowed to be. Also, the domain is going to be certain things because I said so. Down the road these restrictions will be lifted and you can choose your own domain.
SWBATsolve an equation given a domain
Example 1: y = 4x
How many answers are there to this equation? Infinite; (1, 4) (2, 8) (3, 12) …
y = 4(1) y = 4(2)y = 4(3)
y = 4y = 8y = 12
(1, 4) (2, 8)(3, 12)
What do these solutions look like on a graph? Line
Example 2: y = 4x; D = {-3, -1, 0, 2}
How many answers are there? 4; (-3, -12) (-1, -4) (0, 0) (2, 8)
y = 4(-3) y = 4(-1)y = 4(0)y = 4(2)
y = -12y = -4y = 0y = 8
(-3, -12)(-1, -4)(0, 0)(2, 8)
What does it look like? Set of points
Notice the difference. This answer is just a set of points.
Example 3: y = 2x + 3; D = {-2,0, 4}
How many answers are there? 3; (-2, -1)(0, 3) (4, 11)
y = 2(-2) + 3y = 2(0) + 3y = 2(4) + 3
y = -1y = 3y = 11
(-2, -1)(0, 3)(2, 11)
What do they look like? Set of points
Notice the difference. This answer is just a set of points.
Example 4: y = -3x + 2; D = {-4, 0, }
How many answers are there? 3; (-4, 14) (0, 2) ( , )
y = -3(-4) + 2y = -3(0) + 2y = -3(1/2) + 2
y = 14y = 2y = 1/2
(-4, 14)(0, 2)(1/2, 1/2)
You Try!
1. y = 2x; D = {-2, 1, 0, } (-2, -4) (1, 2) (0, 0) (1/2, 1)
2. y = 4x – 2; D = {-4, 1, 2} (-4, -18) (1, 2) (2, 6)
3. y = 2x + 5; D = {-1, 0, 5} (-1, 3) (0, 5) (5, 15)
4. y = -3x + 2; D = {-2, 0, 1, 5} (-2, 8) (0, 2) (1, -1) (5, -13)
What did we learn today?
1. What does it mean to have a restricted domain?
Solve each equation if the domain is {-2, -1, 1, 3, 4}
2. y = 2x + 3 3. y = -3x + 1
4. y = 4x – 5 5. y = x + 4
6. y = 2x – 5 7. y = 4x + 8
Solve each equation for the given domain. Graph the solution set.
8. y = 3x + 1; D = {-3, 0, 1, 4}
9. y = ; D = {-4, 0, 1, 4}
What is the domain and range for each of the following graphs?
10. 11.
Review Question
What is a relation?Set of points
What is domain?‘x’ values
Discussion
How many solutions are there for the equation y = -3x + 2? Infinite
How many solutions are there for the equation y = -3x + 2; D = {-3, -1, 2}? Three
SWBATtell which answers are appropriate for a given equation
Example 1: Which of the ordered pairs are a solution to y = 2x + 3? (-2, -1) (-1,-3) (0, 4) (3, 9)
y = 2(-2) + 3y = 2(-1) + 3y = 2(0) + 3y = 2(3) + 3
y = -1y = 1y = 3y = 9
(-2, -1)(-1, 1)(0, 3)(3, 9)
Example 2: Which of the ordered pairs are a solution to y = -3x + 5? (-1, 8) (2, 11) (0, 5) (3, 14)
y = -3(-1) + 5y = -3(2) + 5y = -3(0) + 5y = -3(3) + 5
y = 8y = -1y = 5y = -4
(-1, 8)(2, -1)(0, 5)(3, -4)
What did we learn today?
Find the solution set for each equation, given the replacement set.
1. y = 4x + 1; (2, -1) (1, 5) (9, 2) (0, 1)
2. y = 8 – 3x; (4, -4) (8, 0) (2, 2) (3, 3)
3. y = x; (-1, -1) (2, -1) (2, 4) (2, 2)
4. y = x + 6; (3, 9) (2, 8) (-2, -4) (4, 10)
5. y = -3x + 4; (0, 4) (4, 10) (2, 10) (2, 2)
6. y = 5x; (0, 5) (-3, 5) (5, 25) (1, 5)
7. y = -4x + 2; (2, -6) (0, 4) (1, 2) (3, 14)
8. y = -2x – 4; (0, -4) (2, -8) (4, -12) (-6, 8)
Solve each equation for the given domain. Graph the solution set.
9. y = 5x + 2; D = {-3, 0, 1, 4}
10. y = -2x – 5; D = {-4, 0, 4}
11. Given the following points, fill in reasonable values for an appropriate window.
(80, 75) (-45, 111) (1, -10) (15, 0)
XMin = XMax = XScale =
YMin = YMax = YScale =
12. Given the following points, fill in reasonable values for an appropriate window.
(-124, 5) (-10, 86) (200, -22) (15, 84)
XMin = XMax = XScale =
YMin = YMax = YScale =
13. State the domain, range, and inverse for the following relation. Then express the relation as a table, map, and graph. (-5, 1) (-5, 4) (6, 4) (2, 0)
14. What is the domain and range for each of the following graphs?
a. b.
Review Question
What is a relation? Set of points
What is domain? ‘x’ values
Discussion
Last year, we discussed proportional relationships. Does anyone remember what proportional means?
Having a constant ratio
Let’s take a look at an example from that unit:
x 4
x / y0 / 0
1 / 4
2 / 8
3 / 12
This relationship is proportional because ratios between the ‘x’ and ‘y’ values are all times 4.
This unit we are going to discuss linear relationships. We are going to write their equations as well. They are very similar to proportional relationship but are a little more difficult.
What does the word linear mean? Line
Algebra I is the study of lines. We need to be able to recognize if something is linear. So let’s take a look at another relation.
x / y0 / 1
1 / 3
2 / 5
3 / 7
Is this relation proportional? No; the ratio between the x’s and y’s isn’t constant. (x3, x2.5, x2.3)
Let’s take a closer look at what is going on. Notice as the x’s increase by ‘1’,the y’s increase by ‘2’. So a relationship does exist. The relationship involves a constant rate of change (not ratio). In other words, as the x’s increase at a constant rate the y’s increase at a constant rate. So instead of looking for a pattern between the x’s and y’s (left to right in the table), we are going to look for a pattern between the way the x’s and y’s are increasing together (up and down in the table). If the rate at which the x’s and y’s are changing is constant (like above), then the relationship is linear.
I know that this is a bit difficult but just stay with me. It will get easier. Over the next few sections, we will bring all of these ideas together and it will make sense. For now let’s just try to recognize linear relationships.
Definition
Linear – having a constant rate of change
SWBATidentify a linear relationship
Example 1: Proportional? Linear?
x / y0 / 3
1 / 6
2 / 9
3 / 12
The relation isn’t proportional because the ratio between ‘x’ and ‘y’ isn’t constant. (x6, x4.5, x4)
The relation is linear because the rate at which the x’s and y’s are changing is constant. The y’s increase by ‘3’ every time the x’s increase by ‘1’.
Example 2: Proportional? Linear?
x / y0 / 14
4 / 11
8 / 8
12 / 5
The relation isn’t proportional because the ratio between ‘x’ and ‘y’ isn’t constant. (x2.75, x1, x1.6)
The relation is linear because the rate at which the x’s and y’s are changing is constant. The y’s decrease by ‘3’ every time the x’s increase by ‘4’.
Example 3: Proportional? Linear?
x / y0 / 0
3 / 6
6 / 12
9 / 18
The relation is proportional because the ratio between ‘x’ and ‘y’ is constant. (x2, x2, x2)
The relation is linear because the rate at which the x’s and y’s are changing is constant. The y’s increase by ‘4’ every time the x’s increase by ‘3’.
Example 4: Proportional? Linear?
x / y0 / 3
3 / 6
6 / 18
9 / 36
The relation isn’t proportional because the ratio between ‘x’ and ‘y’ is constant. (x2, x3, x4)
The relation isn’t linear because the rate at which the x’s and y’s are changing is constant. The y’s increase by a different amount every time the x’s increase by ‘3’.
You Try!
Proportional? Linear?
1.
x / y0 / 0
2 / 10
4 / 20
6 / 30
Proportional, Linear
2.
x / y0 / 3
2 / 6
4 / 9
6 / 12
Not proportional, Linear
What did we learn today?
Determine if each relation is proportional, linear, neither, or both.
1.
x / y0 / 0
1 / 2
2 / 4
3 / 6
2. (0, 3)
(4, 5)
(8, 7)
(12, 9)
3.
x / y0 / 2
4 / 6
8 / 14
12 / 18
4. (4, 5)
(12, 6)
(20, 7)
(28, 8)
5. (
x / y2 / 6
4 / 12
6 / 18
8 / 24
6. (0, 13)
(4, 9)
(8, 5)
(12, 1)
7.
x / y0 / 3
1 / 4
2 / 6
3 / 9
8. (2, 5)
(4, 9)
(6, 13)
(8, 17)
The introduction to this lesson uses graphing calculators.
Review Question
What does linear mean? Line
How do you determine if a relation is linear?
The rate at which the x’s and y’s are changing is constant.
Discussion
Yesterday, we determined if a relation (set of points) is linear. Today we are going to determine if a graph and equation is linear. This should be easy because we all know what a line looks like but let’s try to see how it fits into yesterday’s lesson. We said that to be linear the rate at which the x’s and y’s are changing is constant. Notice that amount that we are going over and up is constant. This is what creates a line.
If I gave you different graphs, could you tell which ones are linear?
Linear or Not?
YesYes
NoYes
No No
If I were to give you different equations, could you tell which ones are linear? Let’s use our graphing calculator to look at their graphs to figure out how we can tell if something is linear based on its equation.
Linear or Not?
1. y = 3x + 2 Yes2. y = x2 + 2x + 3 No
3. y = No 4. y = -2x – 3Yes
5. y = 2Yes6. y = x3 – 2 No
So how do we know if an equation is a line?
Can anyone come up with a rule? The exponents have to be ‘1’ when the equation is in “y =” form.
Why do you think that the exponents have to be ‘1’?
In order to be a line, something must increase/decrease at the same rate. If the exponent is something other than one, then that something will increase or decrease by a different amount. Therefore, that something would not be a line.
SWBATdetermine whether an equation is linear or not
Definition
To be linear –‘x’ and ‘y’ have exponents of 1; no x’s or y’s on the bottom
* The word linear simply means that something is a line.
Example 1: Linear or not? (Check your results on the calculator.)
a. y = 4x – 2 Yesb. y =-3x + 5 Yes
c. y = x2 + x + 2 Nod. No
e. x = 7 Yesf. y = 5 Yes
What did we learn today?
Determine if each equation is a line. Then confirm your answer on the calculator.
1. y = 5x – 3 2. y = 4x + 3
3. y = x2 + x + 2 4. x = 7
5. y = 5 6.
7. y = -2x + 4 8. y = 4x – 2
9. 10.
Determine if each equation is a line. Then confirm your answer on the calculator.
1. y = 6x – 3 2. x =
3. y = 3 4. y= -3x + 6
5. 6. y = x + 2