L INEAR E QUATIONS AND T HEIR G RAPHS Continuous vs. Discrete Graphs - - PowerPoint PPT Presentation

LINEAR EQUATIONS AND THEIR GRAPHS

Continuous vs. Discrete Graphs Activity Extending a Textbook Problem to Incorporate Linear Equations Connecting Linear Equations and Their Solutions to Their Graphs (Textbook Faults) Investigation to Understand Parallel and Perpendicular Lines and Their Equations Presenters: Carrie Becker and Kristi Spencer

CURRENT GRAD STUDENTS AT OSU LIMA

IN TACCC – PROFESSOR HEA-JIN LEE

 BS in Education from

BGSU in 2000

 Master of Education

from BGSU in 2004

 HS Math Teacher at

Wapakoneta

 11 years of teaching

experience

 BA from MVNU in

1985

 Master of Education

from BGSU in 2004

 HS Math Teacher at

Wapakoneta

 25 years of teaching

experience

Carrie Becker Kristi Spencer

CONTINUOUS VS. DISCRETE GRAPHING ACTIVITY

Objective: Students will understand the difference between continuous and discrete graphs and when to use them.

ACTIVITY OPENER

I use this as my class opener on the day of this activity. The goal is to get the students thinking about what a graph looks like when it is increasing or decreasing and that there is a difference in the steepness when the increase

• r decrease is sharp

verses gradual. We usually have a small discussion about Graph C and what it means.

ACTIVITY INSTRUCTIONS FOR PART 1:

 Find a partner and clear your work area.  Each pair will receive an orange set of activity

cards.

 Each person will receive an Activity Handout

(Speed vs. Time and Quantity vs. Time)

 With your partner read each of the stories and

examine the graphs. Put one graph with each

• story. (Even though it may seem like a story

could go with a couple of graphs, in the end there is only one way to match up the graphs.)

 Once every pair has completed this, we will come

back together as a group to share what we have found.

 After we are done sharing we will complete the

Speed vs. Time Activity Handout.

ACTIVITY INSTRUCTIONS FOR PART 2:

 Place all of the orange cards back in the baggie.  Each pair will receive a green set of activity

cards.

 With your partner read each of the stories and

examine the graphs. Put one graph with each

• story. (Even though it may seem like a story

could go with a couple of graphs, in the end there is only one way to match up the graphs.)

 Once every pair has completed this, we will come

back together as a group to share what we have found.

 After we are done sharing we will complete the

Quantity vs. Time Activity Handout.

WHAT HAVE YOU LEARNED?

 When to use?  Why?  When to use?  Why?

Continuous Graphs Discrete Graphs

EXTENDING A TEXTBOOK PROBLEM

Objective: To expose students to as many situations as possible that can be expressed as linear equations or be represented pictorially with linear graphs.

WRITING AND EVALUATING EXPRESSIONS

I assign this after teaching the first lesson in Algebra 1. The next day in class we work on graphing the 2 phone plans on the same coordinate plane; discussing what the points on the line represent and how they can tell just by looking at the graph which plan is cheaper at any amount of minutes. The students then re-evaluate their response to Question 4 and try to give me their best “4-point” answer.

EXTENDING THE QUESTION

This is the graph that I created to go with the book’s question about phone plans. Things we discuss:

• Point of

Intersection

• Intercepts
• Ordered pairs and

their meanings

• Whether or not to

connect the points and/or to extend the lines

Directions: Plot the points for Plan A and draw a RED line through them. Then plot the points for Plan B and draw a BLUE line through them. Use the number of minutes as your x-coordinate and the cost of the plan as your y-coordinate.

280 270 260 250 240 230 220 210 200 190 180 170 160 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10

Cost of Plan Number of Minutes

100 200 300 400 500 600 700

HOW CAN YOU EXTEND THESE PROBLEMS?

CONNECTING LINEAR EQUATIONS

AND THEIR SOLUTIONS TO THEIR

GRAPHS

Objective: To reinforce what you have taught:

• points on graphs represent solutions to a

problem

• rdered pairs are values which make the

equation a true statement

 Clarify textbook directions. What do they mean?  Why do we graph linear equations?  What does the line represent?  What are ordered pairs in relation to the equation?  Reinforce the relationship between an equation,

its graph and its solutions by continuing to ask probing quesitons

 What does this point on the graph represent?  What happens if the values of an ordered pair are

substituted into the equation?

 Why isn’t an ordered pair on the graph?  What would happen if the values of an ordered pair

that was not on the graph were substituted into the equation?

Graph y = 3x + 5

 Talk about domain and range.  Why are domain and range important?  How can stating the domain and range of a graph

make it easier to graph?

A mail-order company charges $5 per order plus $2 per item in the order.

 Write a rule?  What do your variables mean?  What is your domain?  What is your range?  How can you graph this?

WHAT CAN YOU TALK ABOUT WITH YOUR

STUDENTS USING THESE ITEMS?

INVESTIGATION TO UNDERSTAND PARALLEL AND PERPENDICULAR LINES AND THEIR EQUATIONS

Objective: To investigate the relationship of slopes involving parallel and perpendicular lines.

ACTIVITY INSTRUCTIONS:

 Find a partner and clear your work area.  Each pair will need a straightedge.  Each person will receive a graph and a set of

worksheets relating to parallel lines.

 With your partner complete these worksheets.  Once every pair has completed this, we will come

back together as a group to share what we have found.

 After we are done sharing we will complete a

second set of worksheets relating to perpendicular lines.

PARALLEL LINES INVESTIGATION This first page is used to reinforce graphing. The students already have knowledge of what parallel lines look like.

Directions: Graph the points and use a ruler to draw the line that passes through

• them. Use the designated color to draw each line.

RED: (–3, 2) (0, 4) Given Lines and Their Points BROWN: (–5, –1) (5, –5) A: (0, 1) (–5, 3) B: (3, 0) (–6, –6) GREEN: (1, 1) (2, –2) C: (–2, 4) (0, –2)

A B C

PARALLEL LINES INVESTIGATION – PAGE 2 On this page, I am reinforcing how to write the equation

• f a line when

given two points. I have them write the pairs

• f equations

that are parallel to help them answer the questions on the next page.

The equation of Line A is    2 1 5 y x . The equation of Line B is   2 2 3 y x . The equation of line C is    3 2 y x . Directions: Use the points given to write the equation of each line in slope- intercept form. RED LINE BROWN LINE GREEN LINE Directions: Use your graph to help answer the following questions. 1. Which colored line is parallel to line A? _____________ What are the equations of these 2 lines? 2. Which colored line is parallel to line B? _____________ What are the equations of these 2 lines? 3. Which colored line is parallel to line C? _____________ What are the equations of these 2 lines?

PARALLEL LINES INVESTIGATION – PAGE 3 On this page, I am hoping that the students can make some conclusions about the equations of parallel lines. After making conclusions, I test for understanding by asking a few questions.

Directions: Use the equations of each pair of parallel lines to answer the following questions.

• 1. What do you notice about the slopes in each pair of equations?
• 2. What do you notice about the y-intercepts of in each pair of equations?
• 3. What general statement can you make about the equations of parallel lines in

relation to y = mx +b ? Directions: Answer the following the questions using the knowledge you gained from your investigation.

• 1. Are

  3 7 y x and   3 8 y x parallel to each other? YES or NO

• 2. Are

  2 2 3 y x and   3 1 2 y x parallel to each other? YES or NO

• 3. Name 3 lines that are parallel to

  2 3 y x

.

• 4. Name 3 lines that are not parallel to

  5 2 y x .

PERPENDICULAR LINES INVESTIGATION This first page is used to reinforce graphing. The students already have knowledge of what perpendicular lines look like.

Directions: Graph the points and use a ruler to draw the line that passes through

• them. Use the designated color to draw each line.

BLUE: (0, 2) (2, –1) Given Lines and Their Points PURPLE: (–3, 6) (–6, 5) A: (0, 1) (–5, 3) B: (3, 0) (–6, –6) ORANGE: (4, 0) (6, 5) C: (–2, 4) (0, –2)

A B C

PERPENDICULAR LINES INVESTIGATION – PAGE 2 On this page, I am reinforcing how to write the equation

• f a line when

given two points. I have them write the pairs

• f equations

that are perpendicular to help them answer the questions on the next page.

The equation of Line A is    2 1 5 y x . The equation of Line B is   2 2 3 y x . The equation of line C is    3 2 y x . Directions: Use the points given to write the equation of each line in slope- intercept form. BLUE LINE PURPLE LINE ORANGE LINE Directions: Use your graph to help answer the following questions. 1. Which colored line is perpendicular to line A? _____________ What are the equations of these 2 lines? 2. Which colored line is perpendicular to line B? _____________ What are the equations of these 2 lines? 3. Which colored line is perpendicular to line C? _____________ What are the equations of these 2 lines?

PERPENDICULAR LINES INVESTIGATION – PAGE 3 On this page, I am hoping that the students can make some conclusions about the equations of perpendicular lines. After making conclusions, I test for understanding by asking a few questions.

Directions: Use the equations of each pair of perpendicular lines to answer the following questions.

• 1. What do you notice about the slopes in each pair of equations?
• 2. What do you notice about the y-intercepts of in each pair of equations?
• 3. What general statement can you make about the equations of perpendicular

lines in relation to y = mx +b ? Directions: Answer the following the questions using the knowledge you gained from your investigation.

• 1. Are

  3 7 y x and   3 8 y x perpendicular to each other? YES or NO

• 2. Are

  2 2 3 y x and    3 1 2 y x perpendicular to each other? YES or NO

• 3. Name 3 lines that are perpendicular to

  2 3 y x

.

• 4. Name 3 lines that are not perpendicular to

  5 2 y x .

Visit the website below to download any worksheet or handout as a PDF.

http://web.wapak.org/hs/StaffInfo/Math/Mrs.%20Becker.htm

Questions or Comments can be sent to Carrie Becker. beckca@wapak.org Thanks for coming to our session. We hope it was beneficial.