Have you ever followed a trail of animal tracks? For expert animal trackers,

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1 Ever Graph Tells a Stor Describing Characteristics of Graphs Learning Goals In this lesson, ou will: Describe characteristics of graphs using mathematical terminolog. Describe a real-world situation that could be represented b a given graph. Ke Terms discrete graph continuous graph linear graph collinear points non-linear graph Have ou ever followed a trail of animal tracks? For epert animal trackers, there are man more signs to look for instead of just paw prints. Epert trackers look for rub, like when a deer scrapes velvet off its antlers. The look for chews where a twig or section of grass has been eaten. If there is a clean cut on the plant, it ma likel have been caused b an animal with incisors (like a rodent). If the plants have teeth marks all over them, those plants ma likel have been eaten b a predator. And of course, trackers look for scat, or droppings. From scat, trackers can tell an animal s shape and size and what the animal eats. Tubular scat ma come from raccoons, bears, and skunks. Teardrop-shaped scat ma come from an animal in the cat famil. How do ou follow clues in mathematics to solve problems? Incisors are the sharp teeth in humans and animals! 11 Carnegie Learning. Describing Characteristics of Graphs 55

2 Problem 1 Students describe the characteristics of graphs using the terms: discrete or continuous, linear or nonlinear, and increasing, decreasing, neither increasing or decreasing, or both increasing and decreasing. The will distinguish among the characteristics of graphs b completing a sorting activit. The activit guides students to the realization that the graphs of all sequences are discrete graphs, and linear graphs. Problem 1 Characteristics of Graphs There are man was that data can be represented through graphical displas. In this lesson, ou will eplore man characteristics of graphs. 1. Graph the first four terms of Sequence A:,,,. Let the term number represent the -coordinate, and let the term value represent the -coordinate. Then, list the coordinates of the points on our graph. Term Value Note that the term value is the number itself. The term number indicates where the term falls in the sequence (1st, nd, 3rd, and so on). Materials Scissors Term Number 7 9 Grouping Have students complete Questions 1 through with a partner. Then share the responses as a class. Have students complete Question 5 b cutting out the graphs. Share Phase, Questions 1 through What tpe of sequence is Sequence A? How can ou tell? How were ou able to graph the sequence without coordinate pairs being given? How could ou turn our discrete graph into a continuous graph? If ou connected the points, a point on the line would be (.5, 3). Wh does this point not make sense in the sequence? The points are (1, ), (, ), (3, ), and (, ).. Would it make sense to connect the points on our graph? Wh or wh not? It would not make sense to connect the points on m graph because the terms are separate points. A discrete graph is a graph of isolated points. Often, those points are counting numbers and do not consist of fractional numbers. A continuous graph is a graph with no breaks in it. The points in a continuous graph can have whole numbers and fractions to represent data points. 3. Is our graph from Question 1 discrete or continuous? Eplain our reasoning. M graph is discrete because I did not connect the points.. Are the graphs of an sequence discrete or continuous? Eplain our reasoning. The graphs of sequences are discrete because the graphs alwas have points that are separated, or isolated. 5. Carefull cut out Graphs A through L on the following pages. Time to get out our scissors. 11 Carnegie Learning 5 Chapter Linear Functions

3 11 Carnegie Learning. Describing Characteristics of Graphs 57 Note These graphs are also used in later lessons to develop the concept of functions. A B C D E F

4 5 Chapter Linear Functions 11 Carnegie Learning

5 11 Carnegie Learning. Describing Characteristics of Graphs 59 Note These graphs are also used in later lessons to develop the concept of functions. G H I J K L

6 Chapter Linear Functions 11 Carnegie Learning

7 Grouping Have students complete Questions through with a partner. Then share the responses as a class. Share Phase, Questions through How did ou determine whether the graphs were discrete or continuous? Eplain wh both lines and curves are continuous graphs. Would this graph be continuous or discrete? Eplain.. Determine if the graphs ou cut out are discrete or continuous. a. Sort the graphs into two groups: those graphs that are discrete and those graphs that are continuous. b. Record our findings in the table b writing the letter of each graph. Discrete Graphs Continuous Graphs A, F, J B, C, D, E, G, H, I, K, L 7. Determine if the graphs are increasing, decreasing, both increasing and decreasing, or neither increasing nor decreasing. a. Analze each graph from left to right. b. Sort the graphs into four groups: those that are increasing, those that are decreasing, those that are both increasing and decreasing, and those that are neither increasing nor decreasing. c. Record our findings in the table b writing the letter of each graph. Increasing Decreasing Both Increasing and Decreasing Neither Increasing nor Decreasing 11 Carnegie Learning Eplain how Graph D, the circle, is different from the other graphs that are both increasing and decreasing. Draw a discrete graph that is decreasing onl. What is the difference between a line and a series of collinear points? Eplain wh Graph L is not linear. Which of the linear graphs is also discrete and increasing onl? Which of the linear graphs is also continuous and decreasing onl? Which of the nonlinear graphs is also continuous and decreasing onl? A, B E, K C, D, H, I, L F, G, J A linear graph is a graph that is a line or a series of collinear points. Collinear points are points that lie in the same straight line. A non-linear graph is a graph that is not a line and therefore not a series of collinear points.. Determine whether Graphs A L are linear or non-linear graphs. a. Sort the graphs into two groups: those that are linear and those that are non-linear. b. Record our findings in the table b writing the letter of each graph. Linear Graph Non-linear Graph A, B, F, G, K C, D, E, H, I, J, L. Describing Characteristics of Graphs 1

8 9. Clip Graphs A L together, and keep them for Lessons and 3. You will use these graphs in another lesson. So, put them in a safe place. Problem Students are presented with two contets and two piecewise graphs that represent those contets. Because the graphs are composed of line segments onl, students will use the fact that linear graphs represent constant rates of change in order to calculate the rate of change for each piece of the graph and describe how it relates to the given contet. Grouping Have students complete Questions 1 through 3 with a partner. Then share the responses as a class. Problem Making Sense of Graphs The graph shown represents Greg s distance from home after driving for hours. Distance (mi) Time (hr) 1. Analze the graph between and hours. a. How far from home was Greg after driving for hours? Greg was 1 miles from home after hours of driving. b. How fast did Greg drive during this time? Eplain our reasoning. Greg was traveling at mph. The speed 1 miles in hours can be simplified to miles per hour. 7 9 How can ou tell b looking at the graph when Greg was traveling the fastest? Share Phase, Questions 1 and Is this graph discrete or continuous? Eplain wh our response makes sense with the given contet. Wh is this graph considered to be nonlinear? Is the graph increasing or decreasing? Eplain. What does the point (3.5, 15) represent in the contet of the problem? c. How do ou know that Greg traveled at the same rate for the first two hours? Describe in terms of the graph. The graph is increasing at the same rate. The graph is a straight line.. Analze the graph between and.5 hours. a. How far did Greg travel from home between and.5 hours? Greg did not travel an miles from home between and.5 hours. 11 Carnegie Learning Chapter Linear Functions

9 Share Phase, Question 3 How far did Greg travel before he headed back home? How can ou tell? When did Greg arrive back home? How can ou tell? How can ou tell from the graph that Greg did not travel at the same rate during his entire trip? What do the increasing segments of the graph represent in the problem contet? What do the decreasing segments of the graph represent in the problem contet? What do the horizontal segments of the graph represent in the problem contet? b. How fast did he travel during this time? Eplain our reasoning. Greg traveled at miles per hour. Because the graph is a straight horizontal line, I know that his distance did not increase or decrease from home during that time. c. Describe the shape of the graph between and.5 hours. The graph is a horizontal line. 3. Complete the table. Label each segment of the graph with letters A through G, beginning from the left. Record the time interval for each segment. Then, describe what happened in the problem situation represented b that segment of the graph. State how fast Greg traveled and in what direction (either from home or to home). Segment Time Interval (hours) A to B to.5 C.5 to.5 Description of Greg s Trip Greg traveled 1 miles from home at a rate of mph. Greg took a half-hour break when he was 1 miles from home. Greg traveled more miles from home at a rate of 3 mph. D.5 to 5.5 Greg took a one-hour break when he was 1 miles from home. E 5.5 to Greg traveled 5 miles toward home at mph. F to 9.5 Greg took a 1.5-hour break when he was 13 miles from home. G 9.5 to 1 Greg traveled 3 miles toward home at mph (or 3 miles per half hour), ending 1 miles from home. 11 Carnegie Learning. Describing Characteristics of Graphs 3

10 Grouping Have students complete Question with a partner. Then share the responses as a class. Share Phase, Question Is this graph discrete or continuous? Eplain wh our response makes sense with the given contet. Wh is this graph considered to be nonlinear? Is the graph increasing or decreasing? Eplain. What does the point (, ) represent in the contet of the problem? Where does the graph represent when the pool was empt? How can ou tell? What do the increasing segments of the graph represent in the problem contet? What do the decreasing segments of the graph represent in the problem contet? What do the horizontal segments of the graph represent in the problem contet?. The crew at the communit swimming pool prepared the pool for opening da. The graph shows the depth of water in the swimming pool after hours. Depth of water (ft) Time (hr) a. Wh do ou think the pool was emptied and then refilled? Answers will var. The bottom of the pool needed to be cleaned, repairs had to be made, or the crew wanted to fill the pool with clean water. b. Complete the table. Label each segment of the graph with letters A through E, beginning from the left. Record the time interval for each segment. Then, describe what occurred in the problem situation represented b that segment in the graph. State how fast the water level in the pool changed and whether it was being drained or filled. 11 Carnegie Learning Chapter Linear Functions

11 Segment Time Interval (hours) Description of the Water in the Pool A to The water in the pool remained at 7 feet deep. B to The pool was being drained. The depth of the water in the pool decreased at a rate of feet per hours, or.5 ft per hour. At the end of the sith hour, the depth of the water was 5 feet. C to The pool was still being drained. The depth of the water in the pool decreased at a rate of 5 feet per hours, or.5 ft per hour. At the end of the eighth hour, the depth was at feet, or the pool was empt. D to 1 The pool remained empt for hours. E 1 to The pool was being filled at a rate of 9 feet in hours, or 1.5 feet per hour. At the end of the twentieth hour, the depth was 9 feet. c. Was the pool being emptied at the same rate the entire time? Eplain using mathematics and the graph. First, the pool was being emptied at.5 feet per hour, and then it was being emptied at.5 feet per hour. The graph is not a straight line because the water level did not decrease b the same amount ever hour. d. Wh does it make sense for the graph of this situation to be continuous rather than discrete? For an given time, there is an appropriate water level. It is logical to monitor the entire draining and filling process of the pool, not just one certain time. 11 Carnegie Learning. Describing Characteristics of Graphs 5

12 Problem 3 Students are presented with a basic contet and two different numberless graphs to represent the contet. The write stories to describe the contet in more detail b interpreting the information from the graphs. Grouping Have students complete Questions 1 and with a partner. Then share the responses as a class. Share Phase, Questions 1 and How are these graphs different from the other graphs in this lesson? Are the graphs increasing or decreasing? Eplain wh our response makes sense with the given contet. How can ou tell from the graph whether the amount of popcorn is decreasing slowl or quickl? What does the horizontal portion of the graph represent in the contet of the problem? What does the vertical portion of the graph represent in the contet of the problem? Problem 3 Tell a Stor You and a friend go to the movies and decide to share a large bucket of popcorn. Write a stor to describe each graph. 1.. Amount of Popcorn Time We bu the popcorn and just take a taste when we are heading to our seats. Then, we eat it ver quickl once we sit down. Amount of Popcorn Time We did not eat an popcorn until we got to our seat. We had a few handfuls and then spilled the entire container on the floor. Be prepared to share our solutions and methods. As time increases, what happens to the amount of popcorn? 11 Carnegie Learning Chapter Linear Functions