Aliquippa School District Manual42.doc · Web viewUnit 4 – Algebra Concepts Length of section 4-1 Coordinate System 3 days 4-2 Relations 3 days 4-3 Linear Equations 5 days 4.1

Unit 4 – Algebra Concepts Length of section

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

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Review QuestionWhat way do we move on the number line for a negative number? Left

DiscussionWhat is GPS? Global Positioning SystemHow 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.

SWBAT plot points on a coordinate planeSWBAT state the location of a point on a graph

DefinitionsCoordinate System – “the graph thing”X-axis – horizontal lineY-axis – vertical lineOrigin – point where the two lines meetQuadrants – four sections of the graph

Draw and label picture based on the definitions above.

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Section 4-1: Coordinate System (Day 1) (CCSS: Prepares for 8.F.3)

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)

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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 ______

163

Section 4-1 In-Class Assignment (Day 1)

SHAPE \* MERGEFORMAT

Review QuestionHow would you plot the point (-3, -4)? Left 3 units, Down 4 unitsWhat quadrant would that be located? III

DiscussionHas 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.

SWBAT plot points on a coordinate planeSWBAT state the location of a point on a graph

ActivityEach 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.


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Review QuestionHow would you plot the point (-3, 6)? Left 3 units, Up 6 unitsWhat quadrant would that be located? II

DiscussionWhat 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.

SWBAT plot points on a graphing calculator or graph paperSWBAT create 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 L23. Graph

Example 2: Plot (13, -35) (18, -3) (-50, 20) using the graphing calculator1. Stat – edit – enter data in L1 and L22. GraphHow many points should you see? 3Why 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. Window2. Change: xmin, xmax, xscl ymin, ymax, yscl

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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.


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.

166

Section 4-1 Homework (Day 3)


Review QuestionHow would you plot the point (-5, -8)? Left 5 units, Down 8 unitsWhat quadrant would that be located? III

DiscussionI 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”

SWBAT state the domain, range, and inverse of a relationSWBAT express a relation as a table, map, graph, or ordered pair

DefinitionsRelation – set of ordered pairsDomain – x valuesRange – y valuesInverse – 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

510

-83

-2 12

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Section 4-2: Relations (Day 1) (CCSS: 8.F.1, 8.F.2)

-8

5

-2 5

12

10

3

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

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DiscussionCan 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 = 5Range: All Reals

3. Domain: All Reals Range: y > 0


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)

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Section 4-2 Homework (Day

Express each table, map, or graph as a relation. 5.

x y-3 65 1-8 2-3 5

6.

7.

State the domain and range of each graph.

8. 9.

10. 11.

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2

-4 1

-7

5


Review QuestionWhat is a relation? Set of pointsWhat is domain? ‘x’ values

DiscussionWhy 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.

SWBAT solve an equation given a domain

Example 1: y = 4xHow many answers are there to this equation? Infinite; (1, 4) (2, 8) (3, 12) …y = 4(1) y = 4(2) y = 4(3)y = 4 y = 8 y = 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 = -12 y = -4 y = 0 y = 8(-3, -12) (-1, -4) (0, 0) (2, 8)

What does it look like? Set of points

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Notice the difference. This answer is just a set of points.

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Example 3: y = 2x + 3; D = {-2, 0, 4}How many answers are there? 3; (-2, -1) (0, 3) (4, 11) y = 2(-2) + 3 y = 2(0) + 3 y = 2(4) + 3y = -1 y = 3 y = 11(-2, -1) (0, 3) (2, 11)

What do they look like? Set of pointsNotice 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) + 2 y = -3(0) + 2 y = -3(1/2) + 2y = 14 y = 2 y = 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)


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

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Section 4-2 Homework

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.

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DiscussionHow many solutions are there for the equation y = -3x + 2? InfiniteHow many solutions are there for the equation y = -3x + 2; D = {-3, -1, 2} ? Three

SWBAT tell 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) + 3 y = 2(-1) + 3 y = 2(0) + 3 y = 2(3) + 3y = -1 y = 1 y = 3 y = 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) + 5 y = -3(2) + 5 y = -3(0) + 5 y = -3(3) + 5y = 8 y = -1 y = 5 y = -4(-1, 8) (2, -1) (0, 5) (3, -4)


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)

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Section 4-2 In-Class Assignment

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.

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DiscussionLast 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 01 42 83 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? LineAlgebra 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.

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.

DefinitionLinear – having a constant rate of change

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x y0 1 1 3 2 5 3 7

Section 4-3: Linear Equations (Day 1) (CCSS: 8.EE.5, 8.EE.6, 8.F.1,

+ 1 + 2

SWBAT identify a linear relationship

Example 1: Proportional? Linear?

x y0 31 62 93 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’.


x y0 144 118 812 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’.


x y0 03 66 129 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’.


x y0 33 66 189 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’.

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You Try!Proportional? Linear?1.

x y0 02 104 206 30

Proportional, Linear

2. x y0 32 64 96 12

Not proportional, Linear


Determine if each relation is proportional, linear, neither, or both.

1.

2. (0, 3) (4, 5) (8, 7) (12, 9)

3.

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x y0 01 22 43 6

x y0 24 68 14

12 18


4. (4, 5) (12, 6) (20, 7) (28, 8)

5. (

6. (0, 13) (4, 9) (8, 5) (12, 1)

7.

8. (2, 5) (4, 9) (6, 13) (8, 17)

180

x y2 64 126 188 24

x y0 31 42 63 9


The introduction to this lesson uses graphing calculators.

Review QuestionWhat does linear mean? LineHow do you determine if a relation is linear? The rate at which the x’s and y’s are changing is constant.

DiscussionYesterday, 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?

Yes Yes

No Yes

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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 Yes 2. y = x2 + 2x + 3 No

3. y = No 4. y = -2x – 3 Yes

5. y = 2 Yes 6. 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.

SWBAT determine whether an equation is linear or not

DefinitionTo 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 Yes b. y = -3x + 5 Yesc. y = x2 + x + 2 No d. No

e. x = 7 Yes f. y = 5 Yes


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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

7. y = x3 + x2 – 2x + 2 8. y = 5x + 7

9. 10.

11. Determine if each relation is proportional or linear.

a.

183

x y0 02 44 86 12



b. (0, 5) (3, 10) (6, 15) (9, 20)

12. For each of the following equations, find the solutions given the restricted domain without a calculator. Show all of your substitutions and work.

a. y = 4x + 1; D = {-3, 1, 4} b. y = -4x + 5; D = {-2, 0, 4}

13. Find the solution set for each equation, given the replacement set.

a. y = 3x + 4; (3, 10) (1, 7) (0, 2) (1, 7) b. y = -3x – 1; (4, -13) (0, 0) (-2, 5 ) (3, 3)

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Review QuestionHow do you determine if something is a line? The exponents on the variables are 1.

DiscussionThe last couple of days we determined if something was a line. Today we are going to try to write an equation of a line. This is going to be tricky. Let’s try an easy one first. What would be the appropriate equation for this relation? (Use guess and check.)

x y0 11 32 53 7

y = 2x + 1Notice this equation “works” for each one of the previous points. Let’s try a more difficult one:

x y0 53 76 99 11

The correct equation is 2/3x + 5. We wouldn’t get this equation by using guess and check. So let’s try to figure out where each of the numbers come from. In this equation, the ‘2/3’ represents the constant ratio between the x’s and y’s and the ‘5’ represents the starting point. We are going to apply these same concepts to writing a linear equation.

SWBAT write a linear equation given a relation

Example 1: Write a linear equation. Notice this is a linear equation. The x’s and y’s are changing at a constant rate. To write a linear equation, we need to figure out two things: the constant rate and the starting point.

x y0 41 62 83 10

y = __ x + __

185

Section 4-3: Linear Equations (Day 3) (CCSS: 8.EE.5, 8.EE.6, 8.F.1, 8.F.2,

+ 3 + 2

There are two blanks that we are looking for in this equation. The second blank is easy. It is the starting point. In other words, it is the value of ‘y’ when ‘x’ is zero. Therefore, the second blank is ‘4’. To get the first blank it is the change in ‘y’ over the change in ‘x’. Notice the ‘y’ values are changing by ‘2’ and the ‘x’ values are changing by ‘1’. In other words, when ‘y’ goes up ‘2’, ‘x’ goes up ‘1’. Therefore, the constant rate is . So, the relation is starting at ‘1’ and increasing by 2/1. The final equation is y = 2x + 4.

Example 2: Write a linear equation.Notice this is a linear equation. The x’s and y’s are changing at a constant rate. To write a linear equation, we need to figure out two things: the constant rate and the starting point.

x y

4 15

8 20

12 25

y = __ x + __

There are two blanks that we are looking for in this equation. The second blank is the starting point. Notice that they don’t give it to us. We need to figure it out. It is the value of ‘y’ when ‘x’ is zero. Following the pattern, ‘y’ will be ‘10’ when ‘x’ is ‘0’. Therefore, the second blank is ‘10’. To get the first blank it is the change in ‘y’ over the change in ‘x’. Notice the ‘y’ values are changing by ‘5’ and the ‘x’ values are changing by ‘4’. In other words when ‘y’ goes up ‘5’, ‘x’ goes up ‘4’. So, the relation is

starting at ‘10’ and increasing by 5/4. The final equation is .

Example 3: Write a linear equation.Notice this is a linear equation. The x’s and y’s are changing at a constant rate. To write a linear equation, we need to figure out two things: the constant rate and the starting point.

x y

5 30

10 28

15 26

y = __ x + __

There are two blanks that we are looking for in this equation. The second blank is the starting point. Notice that they don’t give it to us. We need to figure it out. It is the value of ‘y’ when ‘x’ is zero. Following the pattern, ‘y’ will be ‘32’ when ‘x’ is ‘0’. Therefore, the second blank is ‘32’. To get the first blank it is the change in ‘y’ over the change in ‘x’. Notice the ‘y’ values are changing by ‘-2’ and the

186

‘x’ values are changing by ‘5’. In other words ‘y’ goes down ‘2’, ‘x’ goes up ‘5’. So, the relation is

starting at ‘32’ and decreasing by -2/5. The final equation is .

187

Example 4: Write a linear equation.Notice this is a linear equation. The x’s and y’s are changing at a constant rate. To write a linear equation, we need to figure out two things: the constant rate and the starting point.What is different about this relation? Trying to find the ‘0’ will be tricky

x y

2 1

6 7

10 13

14 19y = __ x + __

There are two blanks that we are looking for in this equation. The second blank is the starting point. Notice that they don’t give it to us. We need to figure it out. It is the value of ‘y’ when ‘x’ is zero. Following the pattern, ‘y’ will be ‘-5’ when ‘x’ is ‘-2’. We want the value of ‘y’ when ‘x’ is zero. So, we have to split the difference to find the value of ‘y’ when ‘x’ is ‘0’. Therefore, the second blank is ‘-2’. To get the first blank it is the change in ‘y’ over the change in ‘x’. Notice the ‘y’ values are changing by ‘6’ and the ‘x’ values are changing by ‘4’. When ‘y’ goes up ‘6’, ‘x’ goes up ‘6’. In other words, the first blank is 6/4. So, the relation is starting at ‘-2’ and increasing by 4/2. The final equation is

.


Write a linear equation to represent the relation.

1.

2. (2, -5) (4, -1) (6, 3) (8, 7)

3.

188

x y0 61 102 143 18

x y5 22

10 1815 14


4. (4, 5) (12, 6) (20, 7) (28, 8)

5.

6. (4, 3) (6, 8) (8, 13) (10, 18)


1.

2. (2, 5) (4, 8) (6, 11) (8, 14)

3.

4. (3, 8) (9, 10) (15, 12) (21, 14)

189

x y2 55 88 11

11 14

x y0 181 132 83 3

x y3 126 79 2


5.

6. (6, 12) (5, 6) (4, 9)

190

x y2 66 9

10 1214 15


Review QuestionHow do we write a linear equation from a relation?Figure out the relationship between the ‘x’s’ and ‘y’s’. Then find ‘y’s’ value when ‘x’ is ‘0’.

DiscussionWhat is different about this relation? It doesn’t appear that the change in ‘x’ and ‘y’ is consistentLet’s take a closer look. The relationship between the x’s and y’s are 3/1, 6/2, and 9/3. These all represent a rate 3/1. Think of different people walking at rates of 3 miles/1 hr, 6 miles/2 hr, and 9 miles/ 3 hr. Notice that they are all walking at the same rate. Therefore, the pattern is linear. Remember back in the beginning of this section we said: The thing that makes a line, a line is that it increases or decreases at the same rate.

x y1 22 54 117 20

SWBAT write a linear equation given a relation

Example 1: Write a linear equation.

x y1 22 54 117 20

y = __x + __

There are two blanks that we are looking for in this equation. Let’s figure out the first blank.Notice that we are trying to find the relationship between the change in ‘y’ and ‘x’. This relationship is 3/1, 6/2, and 9/3. These all represent a 3/1. In other words, the y’s increase by ‘3’ every time the x’s change by ‘1’. Therefore, the first blank is 3/1. The second blank is the starting point. This is the value of ‘y’ when ‘x’ is zero. Following the pattern, ‘y’ will be ‘-1’ when ‘x’ is ‘0’. Therefore, the second blank is ‘-1’. So, the final equation is y = 3x – 1.

Example 2: Write a linear equation.

x y

1 5

3 7

4 8

191


10 14

y = __ x + __

192

There are two blanks that we are looking for in this equation. Let’s figure out the first blank.Notice that we are trying to find the relationship between ‘y’ and ‘x’. This relationship is 2/2, 1/1, and 6/6. These all represent a 1/1. In other words, the y’s increase by ‘1’ every time the x’s change by ‘1’. Therefore, the first blank is 1/1. The second blank is the starting point. This is the value of ‘y’ when ‘x’ is zero. Following the pattern, ‘y’ will be ‘3’ when ‘x’ is ‘-1’. We want the value of ‘y’ when ‘x’ is zero. So, we have to split the difference to find the value of ‘y’ when ‘x’ is ‘0’. Therefore, the second blank is ‘4’. So, the final equation is y = x + 4.

You Try!Write the appropriate linear equation.1. (1, 12) (2, 9) (3, 6) y = -3x + 152. (2, 5) (6, 11) (10, 17) y = 6/4x + 23. (1, 5) (2, 8) (4, 14) y = 3x + 24. (2, 5) (5, 14) (6, 17) (12, 35) y = 3x – 2



1. 2. (3, 5) (9,

1) (15, -3)

(21, -7)

3. 4. (5, 12) (10, 18)

(15, 24) (20, 30)

5. 6. (1, 3) (2, 8)

(3, 13) (4, 18)

193

x y0 41 102 163 22

x y4 208 14

12 8

x y1 02 34 97 18


194

7. 8. (2, 10) (4, 7)

(6, 4) (8, 1)

9. 10. (2, 8) (6, 13)

(10, 18) (14, 23)

195

x y0 41 62 83 10

x y2 04 28 6

12 10


Review QuestionWhat does linear mean? LineHow do you determine if an equation is linear? The exponents on the variables are ‘1’.

DiscussionFor the past couple of days, we were writing linear equations. Remember the equations were written: y = __x + __.

What did the first blank represent? The rate at which the x’s and y’s were changing.What did the second blank represent? The starting point.

For now we will refer to these values as constant rate and starting point. In the weeks to come, you will see that these are the two most important values in Algebra. They will have fancy names.

We are going to use these two ideas to help us graph linear equations. Hopefully, seeing a picture of these values will help you understand what they mean.

SWBAT graph a line

Example 1: Graph: y = 2x + 3We know that ‘3’ is the starting point. So we will put a point at (0, 3)Next we know that ‘2’ is really 2/1.We know that the ‘2’ represents how much the ‘y’ is changing.We know that the ‘1’ represents how much the ‘x’ is changing.So we will go up ‘2’ and over ‘1’ to get the second point.

Example 2: Graph: y = -3x – 1 We know that ‘-1’ is the starting point. So we will put a point at (0, -1)Next we know that ‘-3’ is really -3/1.We know that the ‘-3’ represents how much the ‘y’ is changing.We know that the ‘1’ represents how much the ‘x’ is changing.So we will go down ‘3’ and over ‘1’ to get the second point.

Example 3: Graph:

We know that ‘2’ is the starting point. So we will put a point at (0, 2)We know that the ‘1’ represents how much the ‘y’ is changing.We know that the ‘3’ represents how much the ‘x’ is changing.So we will go up ‘1’ and over ‘3’ to get the second point.

196


Example 4: Graph: y = 4This equation is really y = 0x + 4We know that ‘4’ is the starting point. So we will put a point at (0, 4)Next we know that ‘0’ is really 0/1.We know that the ‘0’ represents how much the ‘y’ is changing.We know that the ‘1’ represents how much the ‘x’ is changing.So we will go up ‘0’ and over ‘1’ to get the second point.

You Try!Graph each of the following lines by finding two points.1. y = 4x + 1 Start at 1, go up 4 over 12. y = -2x – 4 Start at -4, go down 2 over 1

3. Start at 2, go up 1 over 2

4. y = 2 Start at 2, go up 0 over 1

5. Start at 5, go down 2 over 3

6. y = -4x2 + 2 Not linear (Suckers!)7. y = x – 5 Start at -5, go up 1 over 1


If the equation is linear, graph it.

1. y = 2x + 1 2. y = -x – 4

3. y = x + 2 4. y = x4 + 5

5. y = -3 6. y = 1

7. 8.

9. 10. y = 5x + 1

11. y = -3x – 6 12.

13. y = x3 14. y = 3x – 3

15. y = 5 16. y = -4x + 2

197


Review QuestionHow do find the value that goes in the first blank for y = __x + __?Find the change in the y’s and then find the change in the x’s. Discussion This value is one of the most important values in all of mathematics. It is called slope. We are going to focus on this value for a few days so you can really understand what it means. Here we go…

How would you compare these two sled riding hills? The first hill is steeper.

In math we don’t use “steep”. Instead we use the word slope.

What is causing the first hill to be steeper than the second? The amount that the y value changes is bigger than the amount that the x value changes. Show this in the

drawings above. So to calculate how steep a line is (slope), we must compare the changes in y and x. How do you find the change in height (y)? SubtractHow do you find the change in horizontal distance (x)? Subtract This is exactly what we did to find the first blank in our equation: y = __x + __. We also did this when we were graphing. This is the value that told us how far up and over to go.

How would you compare the slopes (steepness) of line 1 and 2?

The steepness is the same but the direction is different.Therefore, lines 1 and 2 have the same numeric slope but line 2 is negative.Notice the negative doesn’t have anything to do with the steepness of the line, but purely the direction of the line.

A positive slope creates a line that goes up/right. A negative slope creates a line that goes down/right.

198

Section 4-4: Slope (Day 1) (CCSS: 8.EE.5, 8.EE.6,

y

x

y x

2 1

Slope – steepness and direction of a line

SWBAT calculate the slope of a line base on a graph

Example 1: Find the slope of the line.

m = 2/6 Example 2: Find the slope of the line.

You can choose any two points on the line. We will get the same answer no matter what. We know this because of the definition of linear. The rate of change has to be constant everywhere.

The two furthest points give us a slope of -6/4. This is because we go down 6 then over 4.The two closer points give us a slope of -3/2. This is because we go down 3 then over 2.Notice that -6/4 is equal to -3/2.

Example 3: Draw a line that has a slope of .

Example 4: Draw a line that has a slope of .

199

2

6

2

7

2

5

You Try!1. Draw a line that has a slope of .

2. Draw a line that has a slope of .

3. Draw a line that has a slope of 4.

4. Draw a line that has a slope of -3.


Find the slope of the line.

1. 2.

3. 4.

5. 6.

200


3 ft

24 in

5 1

2

6

2

4

6

5

7. 8.

9. Change in y: -8 in 10. Change in x: 3 feet Change in x: 5 in Change in y: 4 feet

11. Change in y: 8 feet 12. Change in y: -300 cm Change in x: -2 feet Change in x: -4 m

13. Draw a line that has a slope of . 14. Draw a line that has a slope of .

15. Draw a line that has a slope of 5. 16. Draw a line that has a slope of -7.

201

3 in

3 in

Review QuestionWhat does slope mean? Steepness and direction of a lineHow do you find slope? Compare the changes in y’s and x’sHow do you define direction? Positive – up/right, Negative – down/right

Discussion So to calculate how steep a line is (slope), we must compare the changes in y and x.

How do you find the change in height (y)? SubtractHow do you find the change in horizontal distance (x)? SubtractRemember we just did this… y = __x + __. This was the value that we put into the first blank.

SWBAT calculate the slope given two points

Definition

Slope: * You can use y2 –y1. We are just trying to simplify the process.

A positive slope creates a line that goes up/right. A negative slope creates a line that goes down/right.

Example 1: Find the slope between (4, 12) (2, 1).

What direction does the line go? Up/RightGraph the two points to confirm answer.

Example 2: Find the slope between (-4, 1) (2, 2).


Example 3: Find the slope between (2, 1) (-1, 8).

What direction does the line go? Down/RightGraph the two points to confirm answer.

You Try! Calculate the slope and direction of the line. Then graph.1. (4, 8) (3, 2) m = 6/1 2. (1, 5) (7, 4) m = 1/-63. (-1, 2) (1, -4) m = 6/-2 4. (-2, 1) (3, 8) m = 7/5

202



For each problem:a. Find the slope.b. Describe the line as up/right or down/right.c. Graph the two points.

1. (6, 8) (2, 7) 2. (8, 8) (6, 1)

3. (2, 6) (3, 1) 4. (-4, -8) (1, 4)

5. (10, 5) (-4, 1) 6. (-2, 1) (3, -5)

7. (1, -5) (-4, 6) 8. (-4, -2) (-3, -5)

9. (10, 8) (-2, -7) 10. (8, 9) (3, 2)


11. 12.

13. Draw a line that has a slope of . 14. Draw a line that has a slope of -6.

203


4 ft

36 in

Review QuestionWhat is 0 ÷ 2? 0; You divide $0 into 2 people. Each person gets $0.What is 2 ÷ 0? Undefined; You divide $2 into 0 people. You can’t do that.

DiscussionWe know that if the change in y is bigger than the change in x the line is steep.We also know that if the change in x is bigger than the change in y the line is flat.

What if the change in y’s and x’s are the same? It would give us a slope of 1What kind of line would that give us? “Average” We will say any line with a slope greater than one is steep and less than one is flat.

We understand different scales. For example, we understand the grading scale: 0% to 100%. We are going to try to understand the scale for slopes. It is going to be a bit confusing but let’s try.

Estimate the slope of each of the lines starting with the middle line and work your way left and right.

m = 4/0 m = 4/2 = 2 m = 2/2 = 1 m = 1/4 = .25 m = 0/4 = 0

Notice that a horizontal line’s slope is zero. If you go “half way up” from horizontal, the slope is just one. You would think that a vertical line would be two. But it is not.

How steep is a vertical line? It is so steep that we can’t put a number on it. It is undefined.How steep is a horizontal line? It is so flat that it is zero.

SWBAT calculate the slope of horizontal and vertical lines

DefinitionsHorizontal Line – slope of zeroVertical Line – undefined slope

204


2

2

4

2 4 1

x

y

y

x x

y

Example 1: Find the slope between (4, 5) (4, 8). Then graph.

The slope is undefined. It is a vertical line.

Example 2: Find the slope between (5, 3) (2, 3). Then graph.

The slope is zero. It is a horizontal line.

You Try!Calculate the slope and direction. Then describe the steepness of the line (flat, steep, average, horizontal, or vertical). Then graph.1. (6, 8) (4, 8) m = 0, Horizontal 2. (6, 2) (3, 4) m = -2/3, Flat3. (7, 5) (7, 4) m = undefined, Vertical 4. (3, 3) (4, 8) m = 5/1, Steep


Calculate the slope and direction. Then describe the steepness of the line (flat, steep, average, horizontal, or vertical). Then graph.

1. (6, 2) (2, 1) 2. (4, 2) (3, 1)

3. (1, 6) (3, 8) 4. (2, 4) (2, 1)

5. (4, 6) (-3, 6) 6. (2, 0) (0, 8)

7. (8, 3) (2, 4) 8. (-4, 1) (0, 2)

9. (-1, 6) (4, 6) 10. (3, 6) (3, 8)

205



11. 12.

Circle the line that has the bigger slope.

13. 14.

Which line has a bigger slope?

15. (4, 8) (5, 10) or (5, 15) (3, 9)

16. (-2, 4) (5, -4) or (8, -3) (-1, 6)

206

10

12

Review QuestionWhat is the slope of a horizontal line? ZeroWhat is the slope of a vertical line? Undefined

DiscussionWe have been calculating the slope between two points.Find the slope between (-4, 3) (3, 5).


Today, we are going to do the opposite. I will give you a slope and you will generate two points. For

example, find two points that will give you a slope of .

SWBAT generate two points given a particular slope

Example 1: Find two points that will give a slope of .

We must choose ‘y’ values that differ by 2.We must choose ‘x’ values that differ by 3.(4, 8) (1, 6)

Example 2: Find two points that will give a slope of .

We must choose ‘y’ values that differ by -4.We must choose ‘x’ values that differ by 5.(6, 2) (1, 6)

Example 3: Find two points that will make a vertical line.We must choose ‘y’ values that differ by any amount.We must choose ‘x’ values that differ by 0.(8, 2) (8, 3)

Example 4: Find two points that will make a horizontal line.We must choose ‘y’ values that differ by 0.We must choose ‘x’ values that differ by any amount.(2, 2) (5, 2)

207


You Try!Find two points that will give the following slope. Then graph your two points to confirm.

1. m = (5, 6) (2, 5) 2. m = 0 (2, 3) (4, 3)

3. m = undefined (4, 5) (4, 2) 4. m = (6, 5) (2, 8)


Find two points that will give the following slope. Then graph your two points to confirm.

1. m = 2. m = 0

3. m = undefined 4. m =

Find two points that will give the following type of slope. Then graph your two points to confirm.

5. Steep 6. Vertical line

7. Horizontal line 8. Not steep

Calculate the slope and direction. Then describe the steepness of the line (flat, steep, average, horizontal, or vertical). Then graph.

9. (5, 3) (2, 1) 10. (6, 2) (3, 2)

11. (1, 5) (1, 8) 12. (-2, -4) (2, 1)


13. 14.

208


8 ft

7 ft


Review QuestionWhat does slope mean? Steepness and direction of a line

Discussion We are going to learn the slope intercept equation today.What two things do you think we need to know to use this equation? Slope, intercept

Remember: y = __x + __We used this equation to graph. The second blank told us where to start. The first blank told us how far up and over to go. For example when we graph y = 3x + 2. We would start at 2 then go up 3 and over 1.

We now know that the first __ is slope. The second blank has a fancy name too. It is called the y-intercept.

What does intercept mean? The place where a line touches an axis. What does the y-intercept mean? The place where line touches the y-axis.

What is the y intercept?

1. 2. 3. (5, 2) (0, -1)

y-int = -1

y-int = -3 y-int = 1

SWBAT write an equation of a line using the slope intercept equation

Definitionsy-intercept – place where line touches the y-axis

y = mx + b (Slope intercept equation)m = slopeb = y-intercept

Example 1: Write an equation of a line in slope intercept form with a slope of -2 and a y-intercept of 8. y = mx + by = -2x + 8

209

Section 4-5: Slope Intercept Equation (Day 1) (CCSS: 8.EE.5, 8.EE.6, 8.F.2, 8.F.3, 8.F.4, 8.SP.3)

Example 2: Write an equation of a line in slope intercept form that goes through the point (0, -4) and has

a slope of .

y = mx + b

Example 3: Write an equation of a line in slope intercept form that goes through the points (0, 3) and (5, 1). What two things do we need to know in order to write an equation of a line? Slope, y-intercept

y = mx + b

Example 4: Write an equation of a horizontal line that goes through the point (0, 5). The slope of a horizontal line is zero.y = 0x + 5y = 5

You Try!Write an equation of a line in slope intercept form with the following conditions. 1. m = 4, y-int = -2 y = 4x – 2 2. Horizontal line that touches the y-axis at -3. y = 0x – 3; y = -3 3. (5, 5) (0, 2) y = 3/3x + 2 4. Horizontal line that goes through the point (0, -6). y = -6

5. Write an equation of a line that goes through the point (0, 2) and has a slope of . y = 1/5x + 2


Write an equation of the line in slope intercept form with the given conditions.

1. Slope: 2, y-intercept: -6 2. Slope: -3, y-intercept: 5

3. Horizontal line that touches the y axis at 2. 4. Slope: , y-intercept: 3

5. (0, 4) (3, -1) 6. Slope: , y-intercept: 0

7. Slope: -1, y-intercept: -6

210


8. A line that goes through the point (0, -3) and has a slope of .

9. Slope: 0.5, y-intercept: 7.5

10. (5, 8) (0, -3)

11. Which line has a bigger slope?

a. (4, 2) (6, 12) or y = 3x + 7

b. y = -4x + 5 or y = -2x + 1

211


Review QuestionWhat two things do you need to know to use the slope intercept equation? Slope, y-interceptWhat is slope? Constant rate of change, change is ‘y’ over change in ‘x’What is the y-intercept? The place where a line touches the y-axis.

Discussion The slope intercept equation is going to be one of the most important equations in math over the next few years. The concepts of slope and intercepts are also going to be important concepts over the next few years. So let’s take some time to practice using these concepts today.

SWBAT write an equation of a line using the slope intercept equation

Definitionsy = mx + b (Slope intercept equation)m = slopeb = y-intercept

Example 1: Write an equation of a line in slope intercept form that goes through the points (0, -2) and (3, 6). What two things do we need to know in order to write an equation of a line? Slope, y-intercept

y = mx + b

You Try!Write an equation of a line in slope intercept form with the following conditions. 1. m = 2, y-int = 3 y = 2x + 3 2. Horizontal line that touches the y-axis at 5. y = 0x + 5; y = 5 3. (5, -8) (0, 3) y = -11/5x + 3 4. Horizontal line that goes through the point (0, -2). y = -2

5. Write an equation of a line that goes through the point (0, -5) and has a slope of . y = 1/5x – 5


212


Write an equation of the line in slope intercept form with the given conditions.

1. Slope: 3, y-intercept: 4 2. Slope: -3, y-intercept: -1

3. Horizontal line that touches the y-axis at 4. 4. Slope: , y-intercept: -2

5. (0, 8) (3, -1) 6. Slope: , y-intercept: 0

7. Slope: -6, y-intercept: -6

8. A line that goes through the point (0, 6) and has a slope of .

9. Slope: 0.25, y-intercept: -1.5

10. (1, 8) (0, 0)

11. Which line has a bigger slope?

a. (1, 5) (4, 17) or y = 3x + 7

b. y = 5x + 5 or y = 8x + 7

213



Review QuestionWhat is the slope intercept equation? y = mx + bWhat does each letter represent?y is ym is slopex is xb is the y-intercept

Discussion Why is this equation so easy? You just plug in the slope and y-intercept.

SWBAT graph an equation of a line using the slope intercept equation

Example 1: Graph: y = 3x + 1How would you graph this line in previous sections? Start at 1, then go up 3 and over 1We are going to do the exact same thing is this section. We are repeating this skill for a few reasons:1. We know why we are doing it. 2. We know the fancy names (slope, y-intercept).3. This equation and concepts are super important in the future.How could you graph this line using the slope intercept equation?Start at (0, 1) because that is the y-intercept then go up three and over one using the slope.

Example 2: Graph: y = -2x – 4 To graph start at (0, -4) because that is the y-intercept then go down two and over one using the slope.

DiscussionWhat is different about the graphs in examples 1 and 2? One is going up; one is going down.What is causing this to happen? One has a positive slope; one has a negative slope.

214


Example 3: Graph:

To graph start at (0, -2) because that is the y-intercept then go down three and over four using the slope.

You Try!Graph each line.1. y = 2x – 5 Start at (0, -5) then go up 2 over 1

2. Start at (0, 5) then go down 2 over 5

3. y = 4x + 1 Start at (0, 1) then go up 4 over 1

4. Start at (0, 5) then go up 2 over 3

5. y = -4x – 5 Start at (0, -5) then go down 4 over 16. y = 0x + 5 Start at (0, 5) then go up 0 over 1


Graph each equation.

1. y = 3x + 1 2. y = x – 2

3. y = -4x + 1 4. y = -x + 2

5. 6.

7. y = -3x – 2 8. y = 2x – 3

9. y = 2x + 3 10. y = -5x + 1

11. y = -x + 3 12.

215


(-5, 2)

(0, -2)

Write the equation of the line in slope intercept form. Then graph.13. Horizontal line that touches the y axis at 2.

14. Slope: , y-intercept: 2

15. (0, -1) (3, 4)


17. Write an equation of a line that passes through the origin with slope 3.

18. (3, 9) (0, 6)

Write an equation of the line shown in each graph.

19. 20.

216


Review QuestionWhat is the slope intercept equation? y = mx + bWhat does each letter represent?y is ym is slopex is xb is the y-intercept

Discussion How do you use the slope intercept equation to graph y = 4x – 2?Start at (0, -2). Then go up 4 and over 1.Why is this equation so easy? You just start at the y-intercept then use the slope.

SWBAT graph an equation of a line using the slope intercept equation

Example 1: y + 2x = 8What is different about this equation? It’s not in the slope intercept form. After some manipulation: y = -2x + 8 To graph start at (0, 8) because that is the y-intercept then go down two and over one using the slope.

Example 2: y – 3x = -2What is different about this equation? It’s not in the slope intercept form. After some manipulation: y = 3x – 2 To graph start at (0, -2) because that is the y-intercept then go up three and over one using the slope.

217


You Try!Graph each line.1. y = 3x – 2 Start at (0, -2) then go up 3 over 1

2. Start at (0, 5) then go down 1 over 4

3. y + 3x = 1 Start at (0, 1) then go down 3 over 1

4. Start at (0, 5) then go up 2 over 5

5. y – 4x = -2 Start at (0, -2) then go up 4 over 16. y + x = -3 Start at (0, -3) then go up 1 over 1



1. y = 2x + 2 2. y = x – 4

3. y – 4x = 1 4. y + x = 3

5. 6.

7. y = -4x – 3 8. y = 5x – 2

9. y – 2x = -3 10. y + 5x = -1

11. y = -x + 1 12.

218


(-6, 1)

(0, -1)

Write the equation of the line in slope intercept form. Then graph.13. Horizontal line that touches the y axis at -3.


15. (0, -3) (3, 8)


17. Write an equation of a line that passes through the origin with slope -4.

18. (3, 9) (0, 3)

Write an equation of the line shown in each graph.

19. 20.

219


Review QuestionHow do you use the slope intercept equation to graph y = 3x + 2?Start at (0, 2). Then go up 3 and over 1.

Discussion Back in Unit 3, we were writing equations based on real life situations. For example: a phone costs $250 plus $80 per month. Write an equation to calculate how much it will cost after 5 months.

C = $250 + $80m

Let’s write this equation a little bit differently: y = 80m + 250.Does this format look familiar? It should. It is the slope intercept equation.Notice the $250 represents the starting point (y-intercept) and the $80 represents the amount the bill increases each month (slope).

Today we are going to write equations based on real life examples. We are going to identify the different parts of the examples and how they relate to slope and the y-intercept.

SWBAT write an equation in slope intercept form based on a real life example

Example 1: Timmy borrowed $24,000 to buy a new car. He pays $400/month.

a. Write an equation in slope intercept form to represent this situation. y = -$400x + $24,000

b. Graph.

c. What does the slope represent? The decrease in the amount you owe each month.

d. What does the y-intercept represent? The starting amount that you owe.

e. How much will he owe after 8 months?y = –$400m + $24,000y = –$400(8) + $24,000 = -$3,200 + $24,000 = $20,800

f. How long will it take to pay off the car?y = –$400m + $24,0000 = –$400m + $24,000-$24,000 = -$400m -$400 -$400 60 = m

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1. y = 2x + 6 2. y = -3x – 4

3. 4. y + 4x = -2

5. y = 2x 6.

Write the equation of the line in slope intercept form. Then graph.7. Horizontal line that touches the y axis at 4.


9. (0, 4) (5, 2)

10. A line that goes through the point (0, -3) and has a slope of .

11. Write an equation of a line that passes through the origin with slope -1.

12. (4, 9) (0, 3)

13. Shirley’s phone bill is $79.99/month plus $.99/app she downloads.

a. Write an equation in slope intercept form to represent this situation.

b. Graph.

c. What does the slope represent?

d. What does the y-intercept represent?

e. How much will she owe if she downloads 12 apps?

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14. It is currently 36° F outside. The temperature is supposed to drop 4° F per hour. a. Write an equation in slope intercept form to represent this situation.

b. Graph.



e. What will the temperature be after 3 hours?

f. How many hours will it take until it is 0° F.

15. Tommy saved $120. He saves another $40/month for his chores.

a. Write an equation in slope intercept form to represent this situation.

b. Graph.



e. How much money will he have after 4 weeks?

f. How long will it take until he has $520?

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Review QuestionHow do you use the slope intercept equation to graph y = -4x – 2?Start at (0, -2). Then go down four and over 1.

Discussion Let’s say that we collected some data on the number of hours the students in our class studied and the grade they received. When plotted on a graph, would the data points be in a straight line? NoWhat are some things that would cause the data to be erratic? Difficulty of material, student

SWBAT read and create scatter plots

DefinitionScatter plot – graph that shows the relationship of two sets of data (each piece of data is a point)

Clusters – data points that are grouped togetherOutlier – data point away from the others

Identify any clusters or outliers in the graph above by circling them.

Example 1: What was the highest score in the class? 95%How many students studied for one hour? 4How many students scored above 75%? 12

Example 2: Draw a scatter plot of the following data set. Identify any clusters or outliers.

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Time(minutes)

Money Left

(mall)10 $20030 $15540 $14570 $8580 $8090 $5

Section 4-6: Scatter Plots (Day 1) (CCSS: 8.SP.1, 8.SP.2, 8.SP.3)

Time Studying vs. Test Score

6065707580859095

100

0 1 2 3 4

Time studying (Hours)

Test

Sco

re (P

erce

nt)

Test score

Things to remember when creating a scatter plot:1. Choose good starting and ending points for each axis.2. Choose sensible scales.3. Time always goes on the x-axis.

You Try!1. Create a scatter plot based on the following sets of data. Identify any clusters or outliers.

2. Create a scatter plot based on the following sets of data. Identify any clusters or outliers.

DiscussionHow does this relate to our study of lines? The data is almost in a line pattern.Notice the data sets are sort of in a line pattern. They are not perfectly linear as that requires the ‘x’ and ‘y’ values to increase or decrease at a constant rate. Tomorrow we will try to summarize these data sets by drawing a line that best fits data sets.


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Time(hours)

Money Earned

1 $82 $154 $355 $456 $509 $85

Age of Car

Value of Car

1 $15,0002 $12,5003 $10,0006 $7,50010 $5,00015 $375

1. The following data sets represent the amount of time students spend playing Playstation and their average grades:

Time (hours) 1 2 3 4 5 6 7 8

Average Grade 96 98 85 83 78 81 68 65

a. Draw a scatter plot based on the data sets. Circle any outliers or clusters.

b. Predict the average grade for a student that plays for 15 hours.

c. Predict what the time would be when a student started to fail.

2. The following data sets represent the amount of time driving in a car and how far you traveled.

Time (hours) 1 2 3 4 5 6 7 8

Distance Traveled 45 105 140 210 270 325 385 420

a. Draw a scatter plot based on the data sets. Circle any outliers or clusters.

b. Predict the total distance traveled after 12 hours.

c. Predict the amount of time necessary to drive 800 miles.

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Review QuestionWhy couldn’t you use a pie graph or bar graph for the two homework problems? Two sets of dataWhen should we use a scatter plot? When we are graphing two sets of data

Discussion What is the difference between the two scatter plots in each of your two homework problems? DirectionWhat is causing this to happen? Data; slope

SWBAT identify positive/negative relationshipsSWBAT draw a best fit line

DefinitionsPositive Relationship/Slope - up/right, x increases/y increases

What would happen if x and y were decreasing? It would still be positive. Think of a slope of a negative over a negative. It is still a positive slope.

Example: Driving Time and Distance Although this data isn’t perfectly linear, it is considered to have a linear association because that is the basic pattern of the data.What would cause this data not to be in a “perfect” line? Traffic, pace of driving, red lights

Can someone give me another example of a positive relationship?

Negative Relationship/Slope - down/right, x increases/y decreases or vice versa

Example: Time and Battery Life on IphoneAlthough this data isn’t perfectly linear, it is considered to have a linear association because that is the basic pattern of the data.What would cause this data not to be in a “perfect” line? Sleep mode, using the web, taking pictures

Can someone give me another example of a negative relationship?226


x

y

x

y

What type of relationship exists in homework examples 1 and 2? #1 negative, #2 positive

No Relationship - scattered

Example: Example: hair color, grades

Can someone give me another example of a scattered relationship?

You Try!Determine whether a scatter plot of the data for the following might show a positive, negative, or no

relationship. 1. Time spent in the gym and your strength. Positive2. The amount of songs on your iPod and the amount of space left. Negative3. Total text messages and your bill. No relationship (if you have unlimited plan)

DefinitionBest fit line – line that summarizes the data set

Best fit lines are used to summarize data sets that have a linear association (like above). Not all data sets have a linear association. Why wouldn’t the following graph have a linear association?

Its overall pattern is a curve. For now, we would say that this has a non-linear association. We are going to focus on linear associations because that has been our focus all unit.

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x

y

x

y

Things to remember:1. Follow the basic direction of the data.2. Same amount of points above and below the line.3. Draw line through as many points as possible.

Example 1: Draw a best fit line

Example 2: Draw a best fit line.

You Try!Draw a best fit line for each of your two homework problems.


1. Determine whether a scatter plot of the data for the following might show a positive, negative, or no relationship.

a. Age of a car and value of the car. b. The size of a family and the weekly grocery bill.c. The size of a car and the cost.d. A person’s weight and percent body fat.e. Time spent playing video games and time spent on outdoor activity.

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Time Studying vs. Test Score

6065707580859095

100

0 1 2 3 4

Time studying (Hours)

Test

Sco

re (P

erce

nt)

Test score


2. Draw a best fit line for each graph.

a.

b.

3. The following data set represents the average salary (in thousands) for people who have a four year college degree.

Year 1999 2000 2001 2002 2003 2004 2005 2006

Salary(Thousands) 52 55 58 62 62 66 67 69

a. Draw a scatter plot based on the data set.

b. Draw a best fit line.

c. What type of relationship exists between the two sets of data?

d. Predict the average salary in 2010.

e. Predict the year in which salaries will be $80,000.

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4. The following data set represents the miles traveled and how much gas is left.



c. What type of relationship exists?

d. How many gallons should be left after you travel 300 miles?

e. How far did you travel if you have 12 gallons left?

f. What factors cause the points not to be in a straight line?

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Miles Traveled 25 100 110 140 220

Gallons of gas 14 10 8 7 4


Review QuestionExplain how to draw a best fit line. 1. Follow the basic direction of the data.2. Same amount of points above and below the line.3. Draw line through as many points as possible.

Discussion What two things do you need to write an equation of a line? Slope, y-interceptWhat would the equation of this best fit line be?

y = __x + __The y-intercept is 250 (starting amount). Then calculate the slope by using two points on the best fit line. (40, 150) and (70, 90)

(Spend $2/minute)

y = -2x + 250

SWBAT write the equation of the best fit line

Example 1: Days and money saved.Draw a scatter plot and best fit line. Then write the equation of the line.

DAYS Money Saved

2 2012 3517 3825 6535 7540 80

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y = __x + __The y-intercept is 15 (starting amount). Then calculate the slope by using two points on the best fit line.(0, 15) and (35, 75)

(Save $1.7/day)

You Try!Write the equation of the best fit line for problems 3 and 4 from last night’s homework. What do the slope and y-intercept represent in these equations?


1. Determine whether a scatter plot of the data for the following might show a positive, negative, or no relationship.

a. The height of a person and their shoe size. b. The amount a student talks in class and their grade.c. The amount of shots a basketball player takes and the amount of shots they make.d. The color of someone’s shoes and their grade.

2. Write the equation of the best fit line.a.

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b.

3. The following data set represents the grade a person is in and their IQ.

Grade 5 6 7 8 9 10 11 12

IQ 75 78 85 100 102 120 125 140

a. Draw a scatter plot and a best fit line based on the data set.

b. Write the equation of the best fit line.

c. What do the slope and y-intercept represent in this equation?

d. What type of relationship exists between the two sets of data?

e. Predict the IQ after two years of college.

4. The following data set represents the amount of songs on your IPod and the amount of space left.

a. Draw a scatter plot and best fit line based on the data set.


c. What do the slope and y-intercept represent in this equation?


e. Predict the amount of space that would be left with 1200 songs.

f. What factors would cause the points not to be in a straight line?

g. How would the best fit line and slope change if we added the point 1700 songs, 4.0 gigs?

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Songs 100 250 300 400 650 750 1000

Space Left (Gigs) 7.5 7.1 6.7 6.3 5.4 5.1 4.3


Review QuestionHow do you write the equation of the best fit line?Use y = mx + b. Find the y-intercept (b). Then locate two points on the line. Then find the slope (m) between the two points.

Discussion Scatter plots and best fit lines are used in engineering. When engineers are designing roadways they must calculate how many lanes of traffic and traffic lights are needed. In order to do this, they collect data. They collect data on how many cars are added to the roads for different size housing plans. This data is then graphed on a scatter plot.

Once the data is graphed, a best fit line and equation are developed. The engineers use this equation the next time someone wants to put in a housing plan. They enter the amount of new homes into the best fit equation and get a value for how many new cars the development will add to the current roadways.The engineers use this information to figure out how many new lanes and lights will be needed.

SWBAT will make up a data set that represents a positive/negative relationship

ActivityIf you truly understand something, then you can talk freely about it. Specifically, you should be able to come up with your own explanations about the topic. This is what we will be doing today.

1. Make up a data set (at least 10 points) that has a positive relationship. Then do the following:

a. Write a sentence describing your data set.b. List your data set in a table. c. Make a scatter plot.d. Draw a best fit line.e. Find the equation of the best fit line.f. What do the slope and y-intercept represent in this equation?

2. Make up a data set (at least 10 points) that has a negative relationship. Then do the following:

a. Write a sentence describing your data set.b. List your data set in a table. c. Make a scatter plot.d. Draw a best fit line.e. Find the equation of the best fit line.f. What do the slope and y-intercept represent in this equation?


234


Section 4-6: Scatter Plots (Day 5) (CCSS: 8.SP.1,

Review QuestionWhat does a graph of a positive relationship look like? Up/rightWhat does a graph of a negative relationship look like? Down/right

DiscussionWhen engineers construct a building or bridge they need to figure out how strong each beam needs to be based on the amount weight and stress that will be placed on it. How do they know that the beams will be strong enough? They do tests.

The results from these tests help engineers select the best materials and sizes for beams. The activity that we will be doing today will simulate these tests. SWBAT collect data using a lab set upSWBAT construct a scatter plot based on the data collectedSWBAT identify positive and negative relationshipsSWBAT draw and write equations for best fit linesSWBAT make predictions about the data

ActivityFirst, make two stacks of books of equal height. Place a piece of spaghetti so that it forms a bridge that connects the two stacks of books. Then punch holes on the opposite sides of a Dixie cup. Then tie the string through the holes. Hang your cup at the center of your spaghetti beam. Support the beam between the stacks of books so that it overlaps each stack by about 1 inch. Put another book on each stack to hold the beam in place. Put pennies in the cup one at a time until the beam breaks.

What did we learn day?

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Section 4-6 Activity 1. Record the data for 1-6 pieces of spaghetti. 2. Graph the data.

3. What type of relationship exists between the two data sets?

4. Does the data have a linear association?

5. Draw a best fit line. Then write the equation of that line.

6. What do the slope and y-intercept represent in this equation?

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Pieces ofSpaghetti

Number of Pennies

7. Some of the data points are close to the best fit line, while others are not. What is causing these points to be away from the line?

8. Predict how many pennies 12 pieces of spaghetti would hold?

9. Predict how many pieces of spaghetti would be needed in order to hold $5 worth of pennies?

10. How would the results change if you used half pieces of spaghetti instead of full pieces?

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Review QuestionWhat is different about a positive and negative relationship? Direction, data, slope

DiscussionIn this section, we are going to start to talk about functions. A function is a graph with a special quality. We will get into that tomorrow. Today, let’s make sure we understand the basics of a graph. Let’s start with understanding how a situation in real life is modeled by a graph.

Sketch a graph to model each situation.1. The height of your grass over the days of summer. (It is not just increasing all of the time.)

2. Your age over the course of the years. (It could be the greatest integer function as well.)

Another basic characteristic of a graph is where it is increasing and decreasing. This is usually pretty easy to identify. Let’s make sure we can identify where a graph increases and decreases.

1.

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Section 4-7: Functions (Day 1) (CCSS: 8.F.1, 8.F.3, 8.F.5)

2.

SWBAT graph a real life exampleSWBAT identify where the graph is increasing or decreasing

Examples: Linear or not? Where is it increasing and decreasing? Provide a real life example.

Examples: Linear or not?, Domain/Range?, Where is it increasing and decreasing?, maximums/minimums?, Provide a real life example.

1. 2.

Linear, D: Reals, R: Reals Linear, D: x = 3, R: RealsAlways decreasing Not increasing or decrasing

Business losing $ over time Profits of different businesses after 3 yrs

3. 4.

Not Linear, D: Reals, R: y > 0 Linear, D: Reals, R: y = 2Decreasing < 0, Increasing > 0, Min at 0 Not increasing or decreasingTemperature during 24 hr period Age of 2nd graders Temperature during 24 hr period Age of 2nd graders


239

For each example, draw a graph that would represent the situation.1. The month of the year and the temperature.2. Number of pieces of pizza ordered and cost.3. Your height from the age of zero to thirteen.4. Speed of your car on a drive from your house to grandmother’s house.5. Your distance from the ground as you ride a Ferris wheel.

For each of the following examples:a. Decide if the graph is linear or not.b. State the domain and range.c. Determine where the graph is increasing or decreasing. State a maximum or minimum.d. Give a real life example of what the graph could represent.

6. 7.

8. 9.

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Review QuestionHow do you know if an equation is linear?To be linear – ‘x’ and ‘y’ have exponents of 1 when the equation is in “y =” form* The word linear simply means that something is a line.

Today, we are going to talk about whether or not something is a function. This is totally different. Let’s not get them confused.

DiscussionWhen we substitute a value in y = 2x +1 for ‘x’, how many answers do we get? 1Can someone think of an equation that would give two answers?

SWBAT state whether something is a function given a map, set of points, table, or equation

Definitionfunction – relation where every ‘x’ has exactly one ‘y’

Three women go to the hospital to have a baby. One has a single baby, one has twins, and one has triplets. Which one(s) would be considered to be a “function?” The woman with one baby.

This is my analogy for a function. One woman; one baby is a function.

Example 1: Function or not? Yes, every ‘x’ has one exactly one ‘y’.

Example 2: Function or not? Yes, every ‘x’ has one exactly one ‘y’.

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1

2

3

4

5

6

1

2

3

4

5


Example 3: Function or not? No, every ‘x’ has two ‘y’s’.

Example 4: Function or not? (2,3) (3,0) (5,2) (-1,-2) Yes, every ‘x’ has one exactly one ‘y’.

Example 5: Function or not? No, one ‘x’ has two ‘y’s’.

x y-3 -15 02 65 -3

Example 6: Function or not? y = 3x + 8 Yes, every ‘x’ has one exactly one ‘y’.Does every ‘x’ have exactly one ‘y’ value?

Example 7: Function or not? x = 3 No, every the ‘x’ value of ‘3’ has infinite ‘y’ values.Does every ‘x’ have exactly one ‘y’ value?


242

1

4

16

4

5

6

-1

1

-2

2

-4

4

Determine whether each relation is a function.

1.

2.

3.

4.

x y-3 -15 02 65 -1

5.

x y-4 -15 2-6 67 -1

243


2

4

6

4

5

6

-1

4

6

2

3

-1

7

2

4

5

6. (1, 2) (3, 4) (5, 6) 7. (2, -3) (5, -4) (2, -1) (7, 2)

8. y = 2x + 8 9. y = 5

10. x = -2 11. y = x2


1.

2.

3.

4.

x y2 -1-3 02 65 -1

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1

-4

6

2

3

6

-1

5

6

2

5

-1

1

3

1

9

5.

x y-4 1-5 2-6 37 4

6. (-1, 2) (3, 4) (-1, 6) 7. (1, 3) (5, 4) (2, -1) (7, 3)

8. y = -x + 1 9. y = 2

10. y = x3 11. x = -2

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Review QuestionHow do you know if something is a function? Every ‘x’ has exactly one ‘y’ value.

DiscussionYesterday, we figured out if a map, t-chart, points, or equation was a function.What is another way to represent a set of points? Graph

Today we will decide if something is a function based on a graph.

1. Function or not? Yes, every ‘x’ has exactly one ‘y’.

2. Function or not? No, some ‘x’s’ have two ‘y’s’. For example when ‘x’ is 2, there are two ‘y’ values. The ‘y’ values are 2 and -2.

SWBAT state whether something is a function given a graph

DefinitionVertical Line Test – if a vertical line touches the graph only once, it is a function

Example 1: Function or not? Yes, pass the VLT and every ‘x’ has one exactly one ‘y’.Is every line a function? No, not a vertical line.

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Example 2: Function or not? No, it doesn’t pass the VLT and every ‘x’ doesn’t have exactly one ‘y’.

Hmmm?!?Function or not? Yes, every ‘x’ has exactly one ‘y’.

Function or not? No, one ‘x’ has two ‘y’s’.



1.

247


1

4

8

6

7

8

2.

3.

4.

X y1 -12 04 65 -1

5.

X y-4 -10 22 64 -1

6. (1, 4) (3, 6) (5, 8) 7. (4, -1) (5, -5) (4, -1) (7, 2)

8. 9. y = -7

10. x = 1 11. y = x5 + 3x + 5

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-1

4

6

2

3

2

4

5

For problems 12 – 19, state whether the graph is a function or not. Then state the domain and range.12. 13.

14. 15.

16. 17.

18. 19.

20. For each example, draw a graph that would represent the situation.a. Your age and your salary. b. The temperature over the course of a day. c. The month and your math grade.

21. For each of the following examples:a. Decide if the graph is linear or not.b. Determine where the graph is increasing or decreasing.c. Give a real life example of what the graph could represent.

a. b.

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Review QuestionWhat are the two ways that you can tell if something is a function?The definition: every ‘x’ has one ‘y’. The vertical line test.

SWBAT study for the Unit 4 Test

DiscussionHow do you study for a test? The students either flip through their notebooks at home or do not study at all. So today we are going to study in class.

How should you study for a test? The students should start by listing the topics.

What topics are on the test? List them on the board- Coordinate System - Relations- Linear Equations- Slope- Slope Intercept Form- Scatter Plots - Functions

How could you study these topics? Do practice problems; study the topics that you are weak on

Practice Problems Have the students do the following problems. They can do them on the dry erase boards or as an assignment. Have students place dry erase boards on the chalk trough. Have one of the groups explain their solution.

1. Find the solutions to y = 2x – 3 given a domain of {-2, 1, 3}.

2. Linear or not? If yes, then graph by plotting two points.

a. y = 3x + 1 b.

c. y + 2x = 3 d. y = x2 – 1

e. y = 5

3. Write the equation of the line in slope intercept form (y = mx + b). Notice we are always trying to find the slope and y-intercept in each problem.

a. (3, 2) (6, 7) (9, 12) (12, 17)

b. (2, 10) (6, 18) (10, 26)

250

Unit 4 Review

c. Slope: , y-intercept: 2

d. Horizontal line that touches the y axis at 2.

e. (0, 4) (2, 5)

f. A line that goes through the point (0, -5) and has a slope of .

4. The following data set represents the average salary (in thousands) for teachers.

Years of Teaching 1 2 5 8 12 15 20 30

Salary(Thousands) 34 36 45 50 52 85 88 90



c. Write the equation of the best fit line.


5. Function or not? a.

b.

c. x = -2

6. After you do the review problems, pick out one or two topics that you are weak on and find three problems from your notes or homework and do them.


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x Y1 12 23 34 4

1. Which ordered pair is not represented in the graph?a. (-3, 2) b. (-2, 0) c. (3, 2) d. (-5, 2)

2. What are the slope and y-intercept represented in the graph? a. m = -1, b = -2 b. m = 1, b = -2 c. m = 1, b = 2 d. m = -1, b = 2

3. What is an equation for the line that passes through the coordinates (2, 0) and (0, 3)?a. y = -3/2x + 3 b. y = -2/3x + 2 c. y = -3/2x – 3 d. y = -2/3x – 2

4. Which of the following doesn’t show ‘y’ as a function of ‘x’?a. b.

c. d.

252

Unit 4 Standardized Test

5. Which equation represents the following relation: (0, -3) (1, -1) (2, 1) (3, 3) (4, 5)?a. y = x + 2 b. y = 2x c. y = 2x – 3 d. y = x – 3

The following problem requires a detailed explanation of the solution. This should include all calculations and explanations.

6. Give the following relation (2, 4) (-3, 4) (2, 7) (4, -2)

a. Using the definition of a function, explain why the relation above isn’t a function.

b. Using the vertical line test, show why the relation above isn’t a function.

c. Make up a relation with four points that is a function.

d. Make up an equation that represents a function. Show a graph of your equation to prove that it is a function.

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UNIT 4 CUMULATIVE

SWBAT do a cumulative review

DiscussionWhat does cumulative mean?All of the material up to this point.

Our goal is to remember as much mathematics that we can by the end of the year. The best way to do this is to take time and review after each unit. So today we will take time and look back on the first three units.

Does anyone remember what the first four units were about? Let’s figure it out together.1. Problem Solving2. Numbers/Operations3. Pre-Algebra4. Algebra

Things to Remember:1. Reinforce test taking strategies: guess/check, eliminate possibilities, work backwards, and estimating.2. Reinforce the importance of retaining information from previous units.3. Reinforce connections being made among units.

In-Class

1. What two consecutive odd numbers add up to 28? a. 11, 17 b. 10, 18 c. 12, 16 d. 13, 15

2. Who is the youngest basketball player? a. Justin Bieber b. Drake c. Lebron James d. Michael Jordan

3. What is the next term: 1, 4, 7, 10, ___? a. 13 b. 14 c. 15 d. 17

4. What is the next term: 2, 6, 18, 54, ___? a. 132 b. 142 c. 152 d. 162

5. Tommy had $95. He spent $5.75 at Wendy’s and $14.99 on a t-shirt. About how much money does he have left? a. $45 b. $55 c. $65 d. $75

6. What set(s) of numbers does ‘7.2’ belong? a. C, W, I, R b. W, I, R c. I, R d. R

7. What set(s) of numbers does ‘3’ belong? a. C, W, I, R b. W, I, R c. I, R d. R

8. What number is the smallest 15%, .12, 18%, ?

a. 15% b. .12 c. 18% d.1/10

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9. -8 + 10 = a. 2 b. -2 c. 18 d. -18

10.

a. -1/120 b. -6/5 c. 10/40 d. -10/40

11. Which of the following is equal to (4)3? a. -64 b. 64 c. 12 d. 10

12. Which of the following is equal to ? a. 1/9 b. -1/9 c. -9 d. 9

13. Which of the following is equal to ? a. 30 b. 31 c. 36 d. 38



16. Which of the following is equal to 3.2 x 103? a. .0032 b. 32 c. 320 d. 3200

17. Which of the following is equal to 8.1 x 10-4? a. .00081 b. 81000 c. .081 d. 85

18. Which of the following is 234 in scientific notation? a. 2.34 b. 2.34 x 102 c. 234 x 103 d. 2.34 x 103

19. Which of the following is .0136 in scientific notation? a. 1.36 x 102 b. 1.36 x 10-5 c. 1.36 x 10-2 d.1.3 x 103

20. 28 – (10 + 3) + 22 a. 10 b. 28 c. 37 d. -17

21. 4x + 12y + 8x – 6y a. 12x + 6y b. 12x + 18y c. 11x + 10y d. -5x + 10y

22. 2(4x + 5) a. 6x + 7 b. 8x + 10 c. 8x + 5 d. 8x

23.

a. All Reals b. 18 c. 52 d. 24

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24. 6x + 2 = 6x + 4 a. Empty Set b. All Reals c. 4 d. 5

25. 3(2x + 4) = x + 8 + 3x a. Empty Set b. All Reals c. -2 d. 2

26. Johnny has $150. He makes $7.50/hour. How many hours will it take for him to save $255? a. 10 b. 14 c. 16 d. 20

27. Solve y = 3x – 4; given a domain of {-2, 0, 4}. a. (-2, -10) (0, -4) (4, 8) b. (-2, 10) (0, 4) (4, 8) c. (-2, -10) (0, -4) (4, 0) d. (-10, -2) (4, 1) (1, 3)

28. Which point is a solution to the following equation: y = 4x + 1? a. (2, -1) b. (0, 5) c. (9, 2) d. (1, 5)

29. Which equation is not a linear equation?

a. 2x2 + 3y = 5 b. x + y = 5 c. x = 2 d.

30. Write an equation for the following relation: (2, 10) (4, 7) (6, 4)

a. b. y = 4x + 10 c. d.

31. y = -2x – 4 a. b. c. d.

32. x = 3 a. b. c. d.

33. Which of the following is a function? a. (1,4) (2,5) (3,6) (1,-5) b. (1,4) (2,5) (3,6) (2,3) c. (1,4) (2,5) (3,6) d. (1,2) (2,3) (1,4)

34. Which of the following is a function?

a. b. c. d.

35. Write an equation of a line that contains the points (6, 5) and (0, 3).

a. b. y = 4x + 3 c. d.

36. A scatter plot of the days of school you miss and your grade would be what type of relationship? a. Positive b. Negative c. Scattered d. Weathered

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Unit 4 Hand-In This problem set is intended to challenge the students and encourage students to apply a deep

understanding of problem–solving skills.

Include a picture for each problem.

1. Make up a relation that would not be linear. Explain why it would not be linear using the definition of linear and by using a graph.

2. Make up a relation that would be linear. Explain why it would linear using the definition of linear and by using a graph.

3. Explain how to calculate the slope between two points.

4. Find the slope between .

5. Explain how to find the equation of a line given two points if one of the points represents the y-intercept.

6. Find the equation of a line that goes through the points (0, .6) and (-3, -5). Then graph.

7. Given an example of a positive relationship between two sets of data. Prove it is positive by show that as one decreases so does the other.

8. Make up a relation that would not be a function. Explain why it would not be a function using the definition of function and by using a graph.

9. Make up a relation that would be a function. Explain why it would be a function using the definition of function and by using a graph.

10. Make up an equation that would not be a function. Explain why it would not be a function using the definition of function and by using a graph.

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Major League Baseball Project

1. Collect data. (Enter the data in table on the next page)

a. Find the payroll of the 30 major league baseball teams last year.

b. Find the number of wins for each team.

c. Find which teams made the playoffs.

2. Draw a scatter plot using the teams’ payrolls (x-axis) and the teams’ wins (y-axis). Circle the teams that made the playoffs.

a. Draw a best fit line.


c. What type of relationship exists between the two sets of data? (Write a sentence summarizing the relationship between payroll and number of wins)

d. Using your best fit line equation, predict the amount of wins that you should have if you spend $100 million.

e. Using your best fit line equations, predict the amount of money that you would have to spend in order to win 110 games.

f. What factors would cause the points not to be in a straight line?

3. Write a five-paragraph essay discussing the following topics: the relationship between the teams’ payrolls and their chances of making the playoffs why a high payroll does not guarantee a team will make the playoffs the factors (other than money) that affect a team’s number of wins

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Unit 4 Project

Team Payroll Wins Playoffs (Y or N)

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Documents

Aliquippa School District Manual42.doc · Web viewUnit 4 – Algebra Concepts Length of section 4-1 Coordinate System 3 days 4-2 Relations 3 days 4-3 Linear Equations 5 days 4.1