# Algebra Chapter 4 Ms. Fisher.

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algebra Chapter 4 Ms. Fisher

Warm up: Brain Teaser If 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? 120 (100 threes in the hundreds place + 10 threes in the tens place + 10 threes in the ones place)

Agenda: Monday 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

There are many ways to identify a linear function….
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 Each domain value is paired with exactly one range value. The graph forms a line. LINEAR FUNCTION Each domain value is paired with exactly one range value . The graph is not a line. NOT A LINEAR FUNCTION The only domain value; 3 is paired with many different range values, NOT A FUNCTION

2. Identify Linear Function By Using Ordered Pairs…
-3 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. x y -2 7 -1 4 1 - 2 2 -5 -3 +1 +1 +1

Where A, B, and C are real numbers and A and B are not both 0.
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=0+3=3 3 1 Y=1+3=4 4 2 Y=2+3=5 5 Graph the following linear function: y= x + 3 (0,3) (1,4) (2,5)

Guided Practice: page234 Yes, one x for every y Yes, straight line
2. 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 6 8

Independent Practice: page 234
HW & 30-43 As always, if you complete your work early, you may: Work on a Anchor Activity Assist a classmate

Agenda: Tuesday Brain 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 & , 42, 43 Teach Chapter 4 Section 2 Whole Group Small Group: Maddie, Nikki, and Christian with Teacher Independent Work- Extension Alex & James

HW: pg & , 42, 43 15. Yes, no yes 16. Yes, yes no 17. yes, no yes 18. No yes, 2x-8y=16; A=2, B=-8, C=16 19. Yes yes, -4x+y=2; A=-4, B=1, C=2 20.Yes Yes, 2x-1/3y=-4, A=2, B=-1/3, C=-4 21.Yes yes, 3x-2y= -20, A=3, B=-2, C=-20 35.Yes, x=7, A=1, B=0, C= 36. No 37.Yes, 3x-y=1, A=3, B=-1, C= (0,7) (1,10) 38.Yes, X+Y=2; A=1, B=1, C=2 x Y=3x+7 y Y= 3(0) +7 7 1 Y= 3(1) + 7 10

4.2 Using Intercepts Objective: Find x- and y-intercepts and interpret their meanings in real world situations.

Guided Practice: Find the X and Y intercepts 5. 2x – 4y= 4
Set x= Set y=0 2(0)- 4y= x- 4(0)=4 -4y= x=4 -4y/-4=4/ x/2 =4/2 Y= x=2 (0,-1) y intercept = (2,0) X intercept = 2

Guided Practice: Use intercepts to graph the line described by each equation. 9. 4x-5y=20 4(0)- 5y = x- 5(0)=20 -5y= x=20 -5y/-5 = 20/ x/4=20/4 y= x=5 (0,-4) (5,0)

Independent Practice: HW #’s & Extension: #’s Alex & James pg 240 Complete your work early: Work on Anchor Activities

Agenda: Wednesday Brain Teaser: What is full of holes but can still hold water? A sponge! Go Over Parking Lot Questions HW pg & 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 & 24-29 13. x= -1 y=3 14. x= -5 y= -1 24. 27.

4.3 Rate of Change and Slope
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 x Slope: Rise M= y2-y Run X2-x1

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 10-30
Brain 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. R Small group: Christian & Maddie w/ Ms. Fisher Pg 257 HW 6, 7, 8-13, Extended: 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
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) 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 & Alex & James Extension: 24-35

HW: pg 263 HW 2-8 & 20-23 2. 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
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

Step 1: Find the y intercept.
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= = 2 = Given x2-x 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 & 12 Alex & 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 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
11. y= 3x y=-2x+4

Point Slope Formula: y-y1 = m (x-x1) y-1=3(x-2)
4.7 Point- Slope Form 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) 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) You can simply be given two points and asked to find the slope. From there substitute the slope and one of the points into the point-slope form. Then write the equation in point intercept form. And Graph

(1, -4) and (3,2) are on the line
Step one: find the slope M= y2-y1= 2-(-4) = 6 = 3 X2-x 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 y=3x-7 Step four: Graph Slope =3 Y intercept= -7

Use two points to Find your X and Y intercepts
(4,8) and (-1, -12) are on a line, Find the intercepts Step #1 Find the slope m= y2-y1 = = -20 = 4 x2-x 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 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 , 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
y +6= 1/5 (x-2) y=2x-1 y-5= -4 (x-1) y= -x y+7= 0 (x-3) y= -1/3x +4 Y-1 = -(x-3) point is (3, 1) xint= 3 yint=-3 slope= xint= -2 yint= 10 5. (-4,-2) slope= 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 pairs Trend Line: Best Fit line Residual: 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 1

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
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

The two lines intersect to form a 90 degree angle
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 (-1/5) (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, &34-36 Extension: Alex & James- Additional Math problems: & Leo- Independent Work- Study Island

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

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