ALGEBRA CHAPTER 4

Ms. Fisher

120 (100 THREES IN THE HUNDREDS PLACE + 10 THREES IN THE TENS PLACE + 10 THREES IN THE ONES PLACE)

Warm up: Brain TeaserIf you wrote all of the numbers from 300 to 400 on a piece of paper, how many times would you have written the number 3?

Agenda: Monday 10-27-2014

Objective: Identify linear functions & linear equations Go over Chapter 3 Test- Great Job!!! Make-ups Go with Marge to library: Nikki & Christian

Begin Ch4- Identifying Linear Functions Whole Group Instruction Independent Practice

Function: A graph is a function if each domain (x-value) is paired with exactly one range (y-value).

Linear function: a function whose graph forms a straight line.

There are many ways to identify a linear function….

1. Identify a linear function by it’s graph

A. Each domain value is paired with exactly one range value. The graph forms a line. LINEAR FUNCTION

B. Each domain value is paired with exactly one range value . The graph is not a line. NOT A LINEAR FUNCTION

C. The only domain value; 3 is paired with many different range values, NOT A FUNCTION

YOU CAN IDENTIFY A LINEAR FUNCTION BY LOOKING AT A TABLE OR A LIST OF ORDERED PAIRS. IN A LINEAR FUNCTION, A CONSTANT CHANGE IN X CORRESPONDS TO A CONSTANT CHANGE IN Y.

IN THIS TABLE, A CONSTANT CHANGE OF +1 IN X CORRESPONDS TO A CONSTANT CHANGE OF -3 IN Y. THESE POINTS SATISFY A LINEAR FUNCTION.

2. Identify Linear Function By Using Ordered Pairs…

-3

-3

x y

-2 7

-1 4

0 1

1 - 2

2 -5

+1

-3

+1

+1

3. Identify if a function is linear by… Can it be written in standard form?

Ax + By = C

Where A, B, and C are real numbers and A and B are not both 0.

Notice that when a linear equation is written in standard form

X and y both have exponents of 1

X and y are not multiplied together

X and y do not appear in denominators, exponents or radical signs.

Linear Not Linear

3x + 2y = 10 X^3 +y = -1

Y-2 = 3x 3xy +x = 1

-y= 5x X + 6/y= 12

**Remember** A function may or may not be linear

y=x + 2 is a linear function y= x^2 is NOT a linear function because it

does not form a straight line

(You only need 2 ordered pairs to graph a line)

(0,3) (1,4)(2,5)

Graph the following linear function: y= x + 3

x y= x + 3 y

0 Y=0+3=3 3

1 Y=1+3=4 4

2 Y=2+3=5 5

Guided Practice: page2342. Does the following graph represent a function?

Yes, one x for every y

Is the function linear?

Yes, straight line

5. Tell whether the given ordered pairs satisfy a linear function.

Yes, X difference -1, Y difference +2

x 5 4 3 2 1

y 0 2 4 6 8

Independent Practice: page 234

HW 15-24 & 30-43

As always, if you complete your work early, you may: Work on a Anchor Activity Assist a classmate

Agenda: 10-28 TuesdayBrain Teaser: I come in different shapes and sizes. Parts of me are curved, other parts are straight. You can put me anywhere you like, but there is only one right place for me. What am I?

A jigsaw puzzle piece

Go Over Parking Lot Questions HW pg 234 15-24 & 30-38, 42, 43

Teach Chapter 4 Section 2 Whole Group

Small Group: Maddie, Nikki, and Christian with Teacher

Independent Work- Extension Alex & James

HW: pg 234 15-24 & 30-38, 42, 4315. Yes, no 22.yes

16. Yes, yes 23. no

17. yes, no 24. yes

18. No 30.yes, 2x-8y=16; A=2, B=-8, C=16

19. Yes 31. yes, -4x+y=2; A=-4, B=1, C=2

20.Yes 32.Yes, 2x-1/3y=-4, A=2, B=-1/3, C=-4

21.Yes 34. yes, 3x-2y= -20, A=3, B=-2, C=-20

35.Yes, x=7, A=1, B=0, C=7 42.

36. No

37.Yes, 3x-y=1, A=3, B=-1, C=1 (0,7) (1,10)

38.Yes, X+Y=2; A=1, B=1, C=2

x Y=3x+7 y

0 Y= 3(0) +7 7

1 Y= 3(1) + 7 10

OBJECTIVE: FIND X- AND Y-INTERCEPTS AND INTERPRET THEIR MEANINGS IN REAL WORLD SITUATIONS.

4.2 Using Intercepts

Guided Practice:Find the X and Y intercepts

5. 2x – 4y= 4

Set x=0 Set y=0

2(0)- 4y=4 2x- 4(0)=4

-4y=4 2x=4

-4y/-4=4/-4 2x/2 =4/2

Y=-1 x=2

(0,-1) y intercept = -1 (2,0) X intercept = 2

Guided Practice:

Use intercepts to graph the line described by each equation.

9. 4x-5y=20

4(0)- 5y = 20 4x- 5(0)=20

-5y=20 4x=20

-5y/-5 = 20/-5 4x/4=20/4

y= -4 x=5

(0,-4) (5,0)

Independent Practice: HW #’s 13-21 & 24-29

Extension: #’s 34-37 Alex & James pg 240

Complete your work early: Work on Anchor Activities

Agenda: 10-29 WednesdayBrain Teaser: What is full of holes but can still hold water?

A sponge!

Go Over Parking Lot Questions HW pg 240 13-21 & 24-29 Teach Chapter 4 Section 3 Whole Group Small Group: Maddie, Nikki, and Christian with Teacher Independent Work- Extension Alex & James

Homework Page 240 13-21 & 24-2913. x= -1 y=3

14. x= -5 y= -1 24. 27.

15. x=-4 y=2

16. x=2 y=4

17. x= -4 y=2

18. x= -8 y=2

19. x=2 y=8

20. x=5 y= -15

21. x=1/8 y= -1

OBJECTIVE: FIND RATES OF CHANGE AND SLOPES

RATE OF CHANGE: IS S RATIO THAT COMPARES THE AMOUNT OF CHANGE IN A DEPENDENT VARIABLE TO THE AMOUNT OF CHANGE IN AN INDEPENDENT VARIABLE.

RATE OF CHANGE: CHANGE IN DEPENDENT VARIABLES Y CHANGE IN INDEPENDENT VARIABLES XSLOPE: RISE M= Y2-Y1 RUN X2-X1

4.3 Rate of Change and Slope

Independent Practice: HW: #’s 4-11 14-17 & 26

Extended: Alex & James #’s 12 & 13

Complete work early… Anchor Activities!

4.4 THE SLOPE FORMULA THURS 10-30Brain Teaser:

I come in different shapes and sizes. Parts of me are curved, other parts are straight. You can put me anywhere you like, but there is only one right place for me. What am I?

A jigsaw puzzle piece

PAGE 257 EVERYONE TRY #4 TOGETHER & TRY #8 TOGETHER HINT Y=MX+ B

NIKKI: WITH MRS. RSMALL GROUP: CHRISTIAN & MADDIE W/ MS. FISHERPG 257 HW 6, 7, 8-13, 15-17EXTENDED: ALEX & JAMES HW PLUS 18-22

4.5 Direct Variation 11-3 Monday

Direct variation: is a special type of linear relationship that can be written in the form y=kx where k is a nonzero constant

Example: 1 cup of rice makes 5 servings

The number of servings varies directly

with the # of cups

Therefore, it is a direct variation!

Look at the equation and tell me if it represents a direct variation?

If an equation can be written in the form: y=kx

then it represents a direct variation

Example: y= 10x Yes

y= -2x + 10 No

If a relationship is a direct variation y/x is equal to the constant variation.

Each y value is 6 times the corresponding x value. Therefore, the constant variation is 6.

If this relationship is a direct relationship, y/x will equal the constant variation which is 6.

y/x= 6/1 =6

y/x= 18/3=6

x 1 3 5

y 6 18 30

Graphing Direct Relations

x Y=6x (x,y)

0 Y= 6 (0) (0,0)

1 Y= 6 (1) (1,6)

2 Y= 6 (2) (2,12)

Independent Practice:

Small group: Maddie, Christian, & Nikki Page 263 HW 2-8 & 20-23

Alex & James Extension: 24-35

HW: pg 263 HW 2-8 & 20-232. No

3. yes, -4

4. yes; -1

5. No

6. Yes

7. 18

8. 4

20. y=5x

k=5 graph shows slope= 5

21. y= -3x k= -3 graph shows slope -3

Tuesday Nov 4th

Agenda:

Open Book to page 267

Ready to move on?

Take Quiz on lessons 4-1

AGENDA:MONDAY NOVEMBER 10TH

Objective: Write a linear equation in slope intercept form

Teach section 4.6 Slope- Intercept Form pg 268 in textbook

Give Leo laptop- Independent Math Work – No more downloads please…

Whole group instruction

Small group instruction

Extension for Alex & James

Slope intercept form: y=mx + b

m= slope

b= y intercept

Write the equation that describes each line in slope intercept form:

1. Slope= -12, y intercept = -1/2

y= -12x-1/2

Writing Linear Equation in Slope-Intercept Form (When given Graph and 2 points)

Step 1: Find the y intercept.

The graph crosses the y-axis at (0,1) so b=1

2. Find the slope. The line contains the points (0,1) and (1,3)

m= y2-y1 m= 3-1 = 2 = 2 Given

x2-x1 1-0 1

Answer:

y= 2x + 1

Use slope Intercept Form to Create a Graph

Given: y=4x-3

Step 1: Identify Slope= 4/1

Step 2: Identify y- intercept= -3 which means one point is (0, -3)

Now, how do you create a graph?

Use the slope to find the second point. You need two points to create a straight line!

y x

Slope is 4/1 which means you go up 4 and to the right one

x, y

(0,-3)

x x

0+1= 1

y y

-3+4=1

Second point is (1,1)

INDEPENDENT WORK TIME

SMALL GROUP: MADDIE, CHRISTIAN, NIKKI PG HW 272 3,4,6-9, 11 & 12ALEX & JAMES EXTENSION: 14-17, 23-25, 32-34

AGENDA: TUES NOV 11TH

• Go over HW Parking lot Questions

• Teach lesson 4.7- Point Slope Form

• Objective Today: Graph a line and Write a linear equation using point-slope form

• Whole Group Instruction

• Small Group Instruction

• Independent Work Time

Page 272 3,4,6-9, 11 & 12 HW Answers 4.6

3. slope=5

Slope is positive

Up 5

Over/right 1

y intercept= -1

(0,-1)

Page 272 3,4,6-9, 11 & 12 HW Answers 4.6

4. Slope= -2 -2/1

Meaning go down 2

go to the right 1

Y intercept = 2

Meaning (0,2)

Page 272 3,4,6-9, 11 & 12 HW Answers 4.6

6. y= 8x+2

7. y=-3

8. y=5x-3

9. y=-2x-1

11. y= 3x-1 12.y=-2x+4

4.7 Point- Slope Form 11-11 Tues

If you know the slope and any point on the line, you can write an equation of the line by using the slope formula.

Example:

Suppose a line has a slope of 3 and contains point (2,1) Let(x,y) be any other point on the line

Point Slope Formula: y-y1 = m (x-x1)

y-1=3(x-2)

Now, how would you convert this to slope intercept form? y=mx + b

y-1 = 3x -6 (Distribute)

+1 +1

y= 3x-5

Now that you know: y intercept formula: y=mx + b The slope formula: m = y2-y1

x2-x1 Point slope formula: y-y1= m(x-x1)

A. You can simply be given two points and asked to find the slope.

B. From there substitute the slope and one of the points into the point-slope form. Then write the equation in point intercept form.

C. And Graph

1. (1, -4) and (3,2) are on the line

Step one: find the slope

M= y2-y1= 2-(-4) = 6 = 3

X2-x1 3-1 2

Step two: Substitute the slope and one of the points into the point slope form.

Slope=3 point(3,2)

y-y1=m(x-x1)

y-2=3(x-3)

Step three: Write the equation in slope intercept form y=mx + b

Distribute y-2= 3(x-3)

y-2= 3x-9

+2 +2

y=3x-7

Step four: Graph

Slope =3

Y intercept= -7

Use two points to Find your X and Y intercepts

1. (4,8) and (-1, -12) are on a line, Find the intercepts

Step #1 Find the slope

m= y2-y1 = -12-8 = -20 = 4

x2-x1 -1-4 -5

Step Two: Write in Point Slope Form and convert it into slope intercept form

y-y1 = m (x-x1)

Y-8=4(x-4) Point slope form

Step 3: convert it to slope intercept form…

1st distribute

Y-8 = 4x-16

+8 +8

y= 4x-8

Step 4:

Now, find your Intercepts:Y=4x-8

Y intercept x intercept(0,-8) plug in zero y=4x-8 0= 4x-8 8=4x 2=x (2,0)

Independent Practice

Small group: Maddie, Christian, Nickki

HW: pg 279 1-3, 4-6,8-10, &13-15

Extension Alex & James: 17-19, 44-46

AGENDA: WED NOV 12TH

• Go over HW Parking lot Questions

• Teach lesson 4.8- Line of Best Fit

• Objective Today: Determine a line of best fit for a set of linear data.

• Whole Group Instruction

• Small Group Instruction

• Independent Work Time

HW: pg 279 1-3, 4-6,8-10, &13-15 ANSWERS CHECK YOUR WORK

1. y +6= 1/5 (x-2) 8.y=2x-1

2. y-5= -4 (x-1) 9. y= -x

3. y+7= 0 (x-3) 10. y= -1/3x +4

4. Y-1 = -(x-3) point is (3, 1) 13. xint= 3 yint=-3

slope= -1 14. xint= -2 yint= 10

5. (-4,-2) slope= -2 15. xint= -1 yint= 3

6. (0,-3) slope= -1/2

SCATTER PLOTS 4.8

SCATTER PLOT: SHOWS TWO DATA SETS AS ONE SET OF ORDERED PAIRSTREND LINE: BEST FIT LINERESIDUAL: ONE WAY TO EVALUATE HOW WELL A LINE FITS A DATA SET IS TO USE RESIDUALS THE VERTICAL DISTANCE BETWEEN DATA POINTS AND A LINE OF FIT. THE CLOSER THE SUM OF THE SQUARED RESIDUALS IS TO 0, THE BETTER THE LINE FITS THE DATA.

Question:

Two lines of fit for this data are:

Y= -1/2x + 6 and y =-x + 8

For each line find the sum of the squares of the residuals.

Which line is a better fit?

x 2 4 6 8

y 3 6 1 4

Friday Nov 14th

Section 4.9 Slopes of Parallel & Perpendicular Lines

Objective: Identify and Graph parallel & Perpendicular lines

Take just a minute and think… What does parallel mean? What does perpendicular mean?

Take a look around our classroom.

The ceiling is parallel to the floor, while the side while is perpendicular to the floor and ceiling.

Can you name any other examples looking around the room?

What does the word parallel mean? Two lines have NO points in common Two lines have the same slope Two lines lie in the same plane

How do we Identify which lines are parallel?

Make sure they have the same slope

Example: y= 1/2x -5

y= 1/2x +1

These two lines are parallel

What does the word perpendicular mean? Two non-vertical lines are perpendicular if… The two lines intersect to form a 90 degree angle

The product of their slopes is -1

Ex:

Y= -1/5x +1

Y= 5x -1 (-1/5) (5)= -1

X=4 undefined line

Y=2 slope of zero

Intersects perpendicularly

BUT the product of their

Slope is not -1.

Vertical lines are

Perpendicular to horizontal lines

Write an equation in slope intercept form for the line that passes through point(4,5) and is parallel to the line described by y=5x+10

Parallel means has same slope

Slope = 5

Givens: Slope and a point (4,5)

Use point slope formula

y-y1=m (x-x1)

y-5= 5 (x-4)

y-5= 5x -20

y=5x-15

Write an equation in slope intercept form for the line that passes through point(3,2) and is perpendicular to the line described by y=3x-1

Perpendicular lines have a slope product of -1

Slope of known line is 3. Other line must have a slope of -1/3 because 3(-1/3)= -1

Givens: Slope and a point (3,2)

Use point slope formula

y-y1=m (x-x1)

y-2= -1/3 (x-3)

y-2= -1/3x +1

y= -1/3x +3

Independent Work Time:

Small group work: Maddie, Nikki, Christian with Ms. Fisher

Page 297 HW #’s 2, 3, 5,6,18, 22-24 &34-36

Extension: Alex & James- Additional Math problems: 28-30 & 40-42

Leo- Independent Work- Study Island

Review for Ch 4 Test!!!!(Section 4.10 OMIT)