L INEAR E QUATIONS AND T HEIR G RAPHS 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
C URRENT G RAD S TUDENTS AT OSU L IMA IN TACCC – P ROFESSOR H EA -J IN L EE Carrie Becker Kristi Spencer BS in Education from BA from MVNU in BGSU in 2000 1985 Master of Education Master of Education from BGSU in 2004 from BGSU in 2004 HS Math Teacher at HS Math Teacher at Wapakoneta Wapakoneta 11 years of teaching 25 years of teaching experience experience
C ONTINUOUS VS . D ISCRETE G RAPHING A CTIVITY Objective: Students will understand the difference between continuous and discrete graphs and when to use them.
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 ACTIVITY OPENER like when it is increasing or decreasing and that there is a difference in the steepness when the increase or decrease is sharp verses gradual. We usually have a small discussion about Graph C and what it means.
A CTIVITY I NSTRUCTIONS FOR P ART 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.
A CTIVITY I NSTRUCTIONS FOR P ART 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.
W HAT HAVE YOU LEARNED ? Continuous Graphs Discrete Graphs When to use? When to use? Why? Why?
E XTENDING A T EXTBOOK P ROBLEM Objective: To expose students to as many situations as possible that can be expressed as linear equations or be represented pictorially with linear graphs.
I assign this after teaching the first lesson in Algebra 1. W RITING AND E VALUATING E XPRESSIONS 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.
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 This is the graph and the cost of the plan as your y-coordinate. that I created to go with the book’s 280 question about 270 260 phone plans. 250 240 Things we discuss: 230 E XTENDING THE Q UESTION 220 • Point of 210 Intersection 200 • Intercepts 190 180 • Ordered pairs and Cost of Plan 170 their meanings 160 • Whether or not to 150 connect the points 140 130 and/or to extend the 120 lines 110 100 90 80 70 60 50 40 30 20 10 0 0 100 200 300 400 500 600 700 Number of Minutes
H OW CAN YOU EXTEND THESE PROBLEMS ?
C ONNECTING L INEAR E QUATIONS AND T HEIR S OLUTIONS TO T HEIR G RAPHS Objective: To reinforce what you have taught: points on graphs represent solutions to a • problem ordered pairs are values which make the • equation a true statement
Graph y = 3x + 5 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?
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?
W HAT CAN YOU TALK ABOUT WITH YOUR STUDENTS USING THESE ITEMS ?
I NVESTIGATION TO U NDERSTAND P ARALLEL AND P ERPENDICULAR L INES AND T HEIR E QUATIONS Objective: To investigate the relationship of slopes involving parallel and perpendicular lines.
A CTIVITY I NSTRUCTIONS : 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.
Directions: Graph the points and use a ruler to draw the line that passes through them. Use the designated color to draw each line. P ARALLEL L INES I NVESTIGATION 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) This first GREEN: (1, 1) (2, – 2) C: ( – 2, 4) (0, – 2) page is used to reinforce C graphing. A The students already have knowledge of what parallel lines look like. B
2 The equation of Line A is y x 1 . 5 On this page, I 2 P ARALLEL L INES I NVESTIGATION – P AGE 2 The equation of Line B is y x 2 . 3 am reinforcing The equation of line C is y 3 x 2 . how to write Directions: Use the points given to write the equation of each line in slope- the equation intercept form. RED LINE BROWN LINE GREEN LINE of a line when given two points. I have them write the pairs of equations that are parallel to help them Directions: Use your graph to help answer the following questions. answer the 1. Which colored line is parallel to line A? _____________ What are the equations of these 2 lines? questions on the next page. 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?
Directions: Use the equations of each pair of parallel lines to answer the following On this page, I questions. P ARALLEL L INES I NVESTIGATION – P AGE 3 am hoping 1. What do you notice about the slopes in each pair of equations? that the students can make some 2. What do you notice about the y-intercepts of in each pair of equations? conclusions about the equations of 3. What general statement can you make about the equations of parallel lines in relation to y = mx +b ? parallel lines. After making conclusions, I Directions: Answer the following the questions using the knowledge you gained from your investigation. test for 1. Are y 3 x 7 and y 3 x 8 parallel to each other? YES or NO understanding 2 3 2. Are y x 2 and y x 1 parallel to each other? YES or NO by asking a 3 2 3. Name 3 lines that are parallel to y 2 x 3 . few questions. 4. Name 3 lines that are not parallel to y 5 x 2 .
Directions: Graph the points and use a ruler to draw the line that passes through them. Use the designated color to draw each line. P ERPENDICULAR L INES I NVESTIGATION BLUE: (0, 2) (2, – 1) Given Lines and Their Points This first PURPLE: ( – 3, 6) ( – 6, 5) A: (0, 1) ( – 5, 3) page is used B: (3, 0) ( – 6, – 6) ORANGE: (4, 0) (6, 5) C: ( – 2, 4) (0, – 2) to reinforce graphing. C The students already have A knowledge of what perpendicular lines look like. B
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