Grade Level: Math A-Grade 9 (Algebra) Time Span: 4 …math.buffalostate.edu/~it/projects/Hudson.pdfAmy Hudson-Burnham Graphing Linear Equations December 1, 2005 I2T2 Final Project

Amy Hudson-Burnham Graphing Linear Equations December 1, 2005 I2T2 Final Project

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Grade Level: Math A-Grade 9 (Algebra) Time Span: 4 Blocks (83 Minutes Each) Not Including an chapter assessment Technology or Tools:

• Graphing calculator • CBR • Geometer’s Sketchpad

Overall Objectives: The student will be able to: • Plot Points in a coordinate plane • Draw a scatter plot • Graph a linear equations using a table of values • Difference between horizontal line & vertical lines and how to graph them • Find slope given two points on a line • Discover how to graph a line given a slope and/or y-intercept • Tell what a parallel line is and the relation ship of the slope

NCTM Standards:


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• Algebra • Connections • Data Analysis • Communication • Problem Solving

NYS Standards:

• Standard 3- Operations • Standard 4- Modeling & Multiple Representation • Standard 5- Measurement • Standard 6- Uncertainty • Standard 7- Patterns & Functions

Resources Used: McDougal Littel, Algebra 1, by Larson, Boswell, Kanold, & Stiff, Chapter 4 pages 202-249, 2001 Incentive Publications, Book of Math, by Frank, pages 162-163, 1998 Key Curriculum Press, Exploring Algebra, by Chanan, Bergofsky, & Bennett, Chapter 2 pages 42-44, 2002


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Outline of Chapter

Day Topics to be covered

Day 1 Review of Graphing Basics • what we need to label • graphing coordinate points

Scatter Plots What graphing calculator buttons mean

Day 2 Slope • Investigate with GSP slope of: 1, 0, -1,

Undefined, +, & - slopes • Discover with GSP 2 different slope

formulas • Find the slope of a line that goes through 2

points • Rate of change real life problems

Day 3 Graphing Linear Equations

• Rewrite equations in function form • Graph a line using a table of values

Day 4 Graphing Linear Equations • Distance Match with CBR • Graph a line using slope intercept form


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Amy Hudson-Burnham Lesson plan Course IR December 1, 2005 Concept: Scatter Plots (Day 1):

1) Goals and objectives of this lesson? What do I want the students to learn? • Review of basics

i. Labeling axes ii. Plotting Points

• Scatter Plots i. Collect data

ii. Use graphing calculator to create a scatter plot iii. Transfer data from graphing calculator onto paper

• Check to see if a coordinate point is a solution • Graphing calculator basics

i. What does each button do?

2) Goals support the state frameworks:

Standard 1: Students use mathematical analysis to pose questions, seek answers, and develop solutions Standard 2: Students will understand mathematics and become mathematically confident be communicating and reasoning mathematically, by applying mathematics in real-life settings, and solving problems through number systems and algebra. Standard 6: Students will understand common themes that connect mathematics, science, and technology

3) Events of instruction: ( 83 minute lesson, 1 day lesson)

• Anticipatory set • Go over homework • Go over graphing calculator basics • Have students complete pulse patrol to gather data • Walk students through creating a scatter plot with their data • Verify if a point is a solution notes • Go through as a class what the graphing calculator buttons mean

4) Assessment:

1) Homework (from McDougall Littell Workbook) 2) Quizzes 3) Unit tests 4) Student to teacher questions 5) Daily Quiz

5) Use of Assessment: • Work with students after school that are having difficulties


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• Re-teach any area that may need more time. • Assign students extra help time to math clinic • Students have opportunity to earn back extra points through make up quizzes and corrections

Concept: Slope (Day 2):

1) Goals and objectives of this lesson? What do I want the students to learn? • Slope

i. Positive ii. Negative

iii. Zero iv. Undefined

• Slope formula • Find the slope that goes through 2 points • Rate of change

2) Goals support the state frameworks:

Standard 1: Students use mathematical analysis to pose questions, seek answers, and develop solutions Standard 2: Students will understand mathematics and become mathematically confident be communicating and reasoning mathematically, by applying mathematics in real-life settings, and solving problems through number systems and algebra. Standard 6: Students will understand common themes that connect mathematics, science, and technology

3) Events of instruction: ( 83 minute lesson, 1 day lesson)

• Anticipatory set • Go over homework • Students will investigate with GSP (using note sheet)

i. Positive slopes, negative slopes, undefined slope, & undefined ii. Compare steepness

iii. Discover slope formulas • They will use slope formula that they just discovered to find slope of line that goes between 2

points • Use slope formula to answer rate of change questions

4) Assessment: 6) Homework (from McDougall Littell Workbook) 7) Quizzes 8) Unit tests 9) Student to teacher questions 10) Daily Quiz

5) Use of Assessment:


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• Work with students after school that are having difficulties • Re-teach any area that may need more time. • Assign students extra help time to math clinic • Students have opportunity to earn back extra points through make up quizzes and corrections

Concept: Graphing lines (Day 3):

1) Goals and objectives of this lesson? What do I want the students to learn? • To put equations into function form • Graph a line using a table of values

2) Goals support the state frameworks: Standard 1: Students use mathematical analysis to pose questions, seek answers, and develop solutions Standard 2: Students will understand mathematics and become mathematically confident be communicating and reasoning mathematically, by applying mathematics in real-life settings, and solving problems through number systems and algebra. Standard 6: Students will understand common themes that connect mathematics, science, and technology

3) Events of instruction: ( 83 minute lesson, 2 day lesson, first day)

• Anticipatory set • Go over homework • Talk about y=mx+b format • Practice putting several equations into function form (in note packet) • Graph line using table of values

4) Events of instruction: ( 83 minute lesson, 2 day lesson, second day) • Anticipatory set • Go over homework • Practice writing equations in function form • Distance match on with CBR • Have students create a list of steps for graphing a line using slope intercept • Graph lines using slope intercept format (in note packet)

5) Assessment: • Homework ( from McDougall Littell Workbook) • Quizzes • Unit tests • Student to teacher questions • Daily Quiz

6) Use of Assessment: • Work with students after school that are having difficulties • Re-teach any area that may need more time.


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• Assign students extra help time to math clinic • Students have opportunity to earn back extra points through make up quizzes and corrections

Chapter 4

Graphing Linear Equations & Functions


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Warm-up:

1. Simplify:

2323622

!+!++ xxxx

2. Is 3=x a solution for the following equation?

3753 !=+ xx

3. Solve for x:

( ) 28435 =++ xx

Chapter 4.1


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Coordinate and Scatter Plots

A _____________ _______________ is formed by two real number lines that Intersect

at a right angle.

Each point in a coordinate plane corresponds to an ______________ _______________

of real numbers

The first number in an ordered pair is the ______________ and the second number is the

________________.

Example: ( )2,3 ! 3 is the _____________ -2 is the ____________

Coordinate Plane:

Identify the:


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• y-axis

• x-axis

• origin

Example:

Label the

• y-axis

• x-axis

Plot the following points:

• ( )4,3A

• ( )2,1!B

• ( )3,4!C

• ( )6,0D

• ( )5,3 !E

A _______________ __________________ is a graph containing several points that

represent real-life data.

Our graphing calculator will help us to create a scatter plot after we collect the data.

Pulse Patrol

Directions


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• To locate your pulse lay 2 fingers on the side of your neck besides the adam’s

apple and press lightly to feel a heart beat.

• Fill in the chart below by doing your activity then taking your pulse after the

activity (example running in place, walking, or jumping jacks)

• Take your pulse for 10 seconds then multiply your pulse by 6 to get the beats

per minute.

• Record your results in the chart below; this will be used to create our scatter

plots

Time Beats Per Minute

Resting Pulse

After 1 Minutes

After 3 Minute

After 5 Minutes

After 7 Minutes

After 9 minutes

Steps to making your scatter plot on your graphing calculator:

Step 1: We need to enter our data into the calculator. Press Stat


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Step 2: Go to Edit and hit enter Step 3: Enter time into one list and your results into the other list Step 4: 2nd Press y= Step 5: Go to plot 1 and hit enter


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Step 6: Turn the plot on by highlighting the on button and pressing enter Type: we want scatter plot which is in the first column first row X-list: enter the list where you put your times in Y-list: enter the list where you put your data Mark: I like to use the first option but it’s up to you Step 7: Go to Zoom, down to 0, then hit enter Step 7: You should see a graph similar to mine but that with your data


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The graphing calculator can also help to find the best fit line which we will look at in the future. Goal 2: Graphing Linear Equations

A ________________ of an equation makes an equation true no matter how many variables

there are.

The ______________ of an equation shows all possible solutions.

What does Linear mean?

*** If a point lies on the line it is a solution for the equation.

Examples: Verify if the coordinate point is a solution for the given linear equation:


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1. 63 =+ yx is ( )2,1 a solution?

2. 63 =+ yx is ( )3,3! a solution?

What is a line?

Lets look at our graphing calculator to see what all those buttons do:

Answer the following questions:

What does the Y= button do?

What does the Window button do?


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What does the Zoom button do? What does the 2nd graph button do? Quiz of the day Name:____________________________________


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Chapter 4.1 Date:____________________________________

Block:____________________________________

1. What would you be doing if you were looking at the following screen: ( 2 points)

2. Enter the following information into your lists to create a scatter plot. Be sure you can see all points on your calculator. What would your window be so you can see all points? (2 points)

Quiz of the day Name: ___________Key____________________


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Chapter 4.1 Date:____________________________________

Block:____________________________________

3. What would you be doing if you were looking at the following screen: ( 2 points)

Answers will vary

4. Enter the following information into your lists to create a scatter plot. Be sure you can see all points on your calculator. What would your window be so you can see all points? (2 points)


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Chapter 4


Warm-up:

4. Evaluate:

( ) 8146 +!+!

5. Solve for t ?

103

2!=t

6. Is ( )4,3 ! a solution for the following equation?

5.115.2 != xy

What is slope?


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Where in real life do we use slope? We will use Geometer’s Sketchpad to investigate slope: Steps for Today’s Sketch:

1. Open a new sketch

2. Using the line tool draw a line

3. With the line still selected measure the slope from the Measure Menu

4. Choose Snap Points from the Graph Menu (this will allow points to move to

locations with integer coordinates)


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5. Drag one of your points around the coordinate plane to answer the following

questions:

a. Describe the lines that have the following slopes & draw a picture:

I. A slope of 1

II. A slope of -1

III. A slope of 0

IV. An undefined slope


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V. Any positive slope (look at lots of different lines)

VI. Any negative slope (look at lots of different lines)

b. How do the lines with slope greater than 1 compare to lines with a slope of 1? How do lines with a slope between 0 and 1 compare to lines with a slope of 1?


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c. Fill in the following chart by moving point A to the specific location and recording the coordinates of point B

Slope Coordinate of A

Coordinates of B

Run Rise

2 ( )0,0

-3 ( )3,2

0 ( )4,1!

1.5 ( )5,2 !

Undefined ( )1,3

0.25 ( )2,4

What do you notice about the slope compared to the rise and the run? What do you notice about the slope compared to the coordinate points of A and B?


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If we had to slope into a formula format it would be: We use the letter ___________ to represent slope. Examples:

1. Find the slope of the line that passes through the points (-2,2) and (3,4) 2. Find the slope of the line that passes through the point (2,4) and the origin

3. Find the slope of the line that passes through the points (2,4) and (2,1)

4. Find the slope of the line that passes through the points (-8,9) and (-7,9)


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*** If you have different signs in the numerator and the denominator then your line will have a ____________ slope. *** If you have same signs in the numerator and the denominator then your line will have a ____________ slope. Fill in the following graphical organizer to summarize today’s learning.

Slope


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At one point we talked about how we can talk about slope in real life. When we are talking about slope in real life we call it the _______________ of ______________. Examples Answer the following real life slope problems:

1. You are parachuting from a plane. After 5 seconds from jumping from the plane you open your parachute at a height of 2500 feet above the ground. Thirty seconds later you calculate that you are height of 2155 above the ground.

a. What is your rate of change?

b. When will you reach the ground?


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2. In 1975 a 4 function calculator costs $125. In 1995 that same calculator costs $5. Find the average rate of change in the cost in dollars per year.

3. In 1995 the library had 16,000 books on its shelves. In 1999 the library had 19,000 books on its shelves. Find the average rate of change in the number of books per year.

4. At UMMC there were 600 births in 1998. In 2000, there were 550 births. Find the average rate of change in the number of births per year.


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Quiz of the day Name:____________________________________

Chapter 4.4 Date:____________________________________

Block:____________________________________

1. What is the slope of a line that passes through the following points: ( 2 points) (3,0) and (-2,9)

2. In 1997 gas was $1.19 a gallon and in 2005 gas is $2.42 a gallon. What is the average rate of change in gallons per year? (3 points)


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Quiz of the day Name: _____________Key___________________

Chapter 4.4 Date:____________________________________

Block:____________________________________

3. What is the slope of a line that passes through the following points: ( 2 points) (3,0) and (-2,9)

5

9!=m

4. In 1997 gas was $1.19 a gallon and in 2005 gas is $2.42 a gallon. What is the average rate of change in gallons per year? (3 points)

15375.8

23.1

19972005

19.142.2==

!

!=m

it went up approximately 15 cents per year


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Chapter 4


Warm-up:

1. Simplify:

( ) 53723 +!+! xx

2. Find the slope of the line that goes through the following points:

(1,2) and (5,-6)

3. Solve for x

8542 !=+ xx


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Goal 1: Graphing linear lines Lets look at the y= screen on our graphing calculator

What do you notice? In order to graph a function we must have the equation as a function of x (meaning we solve for __________) HELPFUL HINT: Always have the x come first because we are putting the function in bmxy +=


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Examples: Let’s practice solving the following equations as a function of x: 1. xy 32 =+ 2. 123 =+! yx 3. 5=!! yx 4. 484 =+ yx 5. 1955 =+ yx 6. 1

2

5

2

1=+ yx


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7. 1236 =! yx 8. 1585 =!! yx There are 2 different ways we are going to be able to graph a line: 1. 2. Lets look at the first way: Steps: 1. rewrite equation in function form 2. enter the function into y= 3. Press 2nd Graph to get the table function 4. Plot Points


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Examples Graph the following lines using method 1 1. 23 != xy 2. 824 =! yx We just finished the distance match game:


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If I were to give you an equation in bmxy += form the ‘m’ would be and the ‘b’ would be:

=

=

+=

b

m

bmxy

Examples: Identify the slope and y-intercept from the following equations: 1.

=

=

+=

b

m

xy 32

2.

=

=

!=

b

m

xy 24

3.

=

=

+=

b

m

xy2

3

2

1

Graphing lines using Method 2:


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Steps:

1. Rewrite the equation so that it is in function form

2. Identify the slope and the y-intercept

3.

4.

5. Example: Graph the following equations using Method 2 1. 12 != xy


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2. 23 =+ yx 3. Graph the following lines on the same coordinate plane

1236

1642

=+!

=!

yx

xy


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What do we notice about the lines you just graphed? What do we know about parallel lines? Goal 3: Graphing Horizontal & Vertical Lines When a line crosses the y-axis ONLY then it has an equation like 7=y What type of line will this be? Check it out on your graphing calculator! When a line crosses the x-axis ONLY then it has an equation like 3=x What type of line will this be?


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Examples: Graph the following lines: 1. 3=x 2. 7=y


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Chapter 4.2 & 4.6 Day 1 Date:____________________________________

Block:____________________________________

1. Rewrite in function form: ( 2 points) 1239 =! yx

2. Graph the following line using a table of values (4 points) 52 != xy


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Quiz of the day Name: __________Key______________________

Chapter 4.2 & 4.6 Day 1 Date:____________________________________

Block:____________________________________

1. Rewrite in function form: ( 2 points) 1239 =! yx

43: != xyanswer

2. Graph the following line using a table of values (4 points) 52 != xy


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Chapter 4.2 & 4.6 Day 2 Date:____________________________________

Block:____________________________________

1. Graph the following lines on same coordinate plane: ( 6 points)

612

4

=+

=

xy

y

What is the coordinate point where they cross?


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Quiz of the day Name: ______________Key__________________

Chapter 4.2 & 4.6 Day 2 Date:____________________________________

Block:____________________________________

1. Graph the following lines on same coordinate plane: ( 6 points)

612

4

=+

=

xy

y

What is the coordinate point where they cross? (-2,4)

Documents

Grade Level: Math A-Grade 9 (Algebra) Time Span: 4 …math.buffalostate.edu/~it/projects/Hudson.pdfAmy Hudson-Burnham Graphing Linear Equations December 1, 2005 I2T2 Final Project