# Using Graphs to relate Two Quantities

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Using Graphs to relate Two Quantities

Goals Goal Rubric To represent mathematical relationships using graphs. Level 1 – Know the goals. Level 2 – Fully understand the goals. Level 3 – Use the goals to solve simple problems. Level 4 – Use the goals to solve more advanced problems. Level 5 – Adapts and applies the goals to different and more complex problems.

Vocabulary None

Using Graphs Graphs can be used to visually represent the relationship between two variable quantities as they each change. A graph is the most understandable way of showing how one variable changes with respect to another variable. Graphs can show changes in speed, altitude, distance, volume, time, and other variable quantities.

Graphing Relationships
State whether each word or phrase represents an amount that is increasing, decreasing, or constant. 1. stays the same 2. rises 3. drops 4. slows down constant increasing decreasing decreasing

Warm Up State whether each word or phrase represents an amount that is increasing, decreasing, or constant. 1. stays the same 2. rises 3. drops 4. slows down constant increasing decreasing decreasing

Example: Using Graphs Which graph in Example 3 could show a car sitting at a stoplight? How do you know? Graph I indicates a quantity that does not change with time. Graph II shows an increase over time. A car sitting at a stoplight would stay in the same place over time. Graph I could show a car sitting at a stoplight.

Example: Using Graphs A pelican flies above the water searching for fish. Sketch a graph of its altitude from takeoff from shore to diving to the water to catch a fish. Label each section.

Your Turn: This graph shows someone taking a walk in the neighborhood. Describe what it shows by labeling each part.

Example: To relate a graph to a given situation, use key words in the description. Each day several leaves fall from a tree. One day a gust of wind blows off many leaves. Eventually, there are no more leaves on the tree. Choose the graph that best represents the situation. Step 1 Read the graphs from left to right to show time passing.

Example: Continued Step 2 List key words in order and decide which graph shows them. Key Words Segment Description Graphs… Each day several leaves fall Wind blows off many leaves Eventually no more leaves Never horizontal Graph B Slanting downward rapidly Graphs A, B, and C Slanting downward until reaches zero Graphs A, B, and C

Example: Continued Step 3 Pick the graph that shows all the key phrases in order. Never horizontal, slanting downward rapidly, slanting downward until reaching zero. The correct graph is B.

Your Turn: The air temperature increased steadily for several hours and then remained constant. At the end of the day, the temperature increased slightly before dropping sharply. Choose the graph that best represents this situation. Step 1 Read the graphs from left to right to show time passing .

Your Turn: Continued Step 2 List key words in order and decide which graph shows them. Key Words Segment Description Graphs… Increased steadily Remained constant Increased slightly before dropping sharply Slanting upward Graph C Graphs A, B, and C Horizontal Slanting upward and then steeply downward Graph C

Your Turn: Continued Step 3 Pick the graph that shows all the key phrases in order. Slanting upward, horizontal, slanting upward and then steeply downward The correct graph is graph C.

Example: Sketching Graphs
Sketch a graph for the situation. A truck driver enters a street, drives at a constant speed, stops at a light, and then continues. As time passes during the trip (moving left to right along the x-axis), the truck's speed (y-axis) does the following: Speed Time y x • initially increases • remains constant • decreases to a stop • increases

When sketching or interpreting a graph, pay close attention to the labels on each axis.

Your Turn: • initially declines • declines more rapidly
Sketch a graph for the situation. Henry begins to drain a water tank by opening a valve. Then he opens another valve. Then he closes the first valve. He leaves the second valve open until the tank is empty. As time passes while draining the tank (moving left to right along the x-axis), the water level (y-axis) does the following: Water tank Water Level Time • initially declines • declines more rapidly • and then the decline slows down.

Example: Writing a Situation for a Graph
Write a possible situation for the given graph. Step 1 Identify labels. x-axis: time y-axis: speed Step 2 Analyze sections. over time, the speed: • initially decreases, • remains constant, • and then decreases to zero. Possible Situation: A car approaching traffic slows down, drives at a constant speed, and then slows down until coming to a complete stop.

Your Turn: Write a possible situation for the given graph.
Possible answer: The level of water in a bucket stays constant. A steady rain raises the level. The rain slows down. Someone dumps the bucket.

Example: Relating Tables and Graphs
In Algebra, we use multiple representations (verbal description, table, graph, equation) to describe data or a relationship between two variables. Marcus and Janine made the table shown below to represent the difference between their ages during different years. Which graph matches the information in the table? (Large class discussion) In Algebra, we use multiple representations to describe data or a relationship between inputs and outputs. This lesson develops the skill of transferring between these representations. The answer is D and this skill (table to graph) is fundamental to the rest of the lesson. This particular problem is taken from Kansas State Curriculum Standards (the author’s home state). It is Kansas 8th Grade Mathematics Standard 2 "Algebra", Benchmark 3 "Functions", Indicator A3 “Application: Translates between the numerical, tabular, graphical, and symbolic representations of linear relationships with integer coefficients and constants” Answer: D

Which table matches the information in the graph that Ethan made?
Your turn: Ethan heard a weather report that stated the temperature in Wichita would drop from 5 degrees Fahrenheit at a rate of 2 degrees every hour. Which table matches the information in the graph that Ethan made? Answer: A (Think, pair, share) Same type of question as the last, but backwards. The answer is A, but have students think on their own, then compare answers with a neighbor, and finally tally the results as a class. This particular problem is taken from Kansas State Curriculum Standards (the author’s home state). It is Kansas 8th Grade Mathematics Standard 2 "Algebra", Benchmark 3 "Functions", Indicator A3 “Application: Translates between the numerical, tabular, graphical, and symbolic representations of linear relationships with integer coefficients and constants” number of hours

Example: Interpreting Graphs and Tables
Reading and Interpreting Graphs and Tables Example: Interpreting Graphs and Tables Initial Cost Increasing House Value 4 8 \$0 \$100,000 \$200,000 House A Time (years) Before answering a question, students need to be able to visualize what type of graphic display is being described. One strategy is to have students insert what they comprehend the question is asking. What would the changing value of house B look like graphically? [Note: Due to the size of the graphics and the amount of text, many of the questions are developed over several slides. If instructors have difficulty in visualizing an entire question, you may wish to provide them with copies of the questions which are located in Chapter 5: Reading and Interpreting Graphs and Tables.] House A cost \$100,000 and increased in value as shown in the graph. House B cost less than house A and increased in value at a greater rate. Which graph on the next slide illustrates this situation.

Reading and Interpreting Graphs and Tables
Example: Continued (1) B A Time (years) 4 8 \$0 \$100,000 \$200,000 House A cost \$100,000 and increased in value as shown in the graph. House B cost less than house A and increased in value at a greater rate. Which One? (4) A B 4 8 \$0 \$100,000 \$200,000 Time (years) (2) \$100,000 8 4 \$0 \$200,000 B A Time (years) Seldom do texts have examples where students have to select the type of graph that depicts the scenario described. Have instructors brainstorm some ways they can better assist students in obtaining this type of skill. Why is this important? Take a look at the next question that asks students to compare data. (5) 4 8 B A Time (years) \$0 \$100,000 \$200,000 (3) Time (years) 4 8 \$0 \$100,000 \$200,000 B A

Reading and Interpreting Graphs and Tables
Your Turn: The changing values of two investments are shown in the graph below. Amount of Investment 4 8 12 \$0 \$1000 \$2000 Investment A Investment B Time (years) Review what the graph displays.

Reading and Interpreting Graphs and Tables
Your Turn: Continued How does the amount initially invested and the rate of increase for investment A compare with those of investment B? Amount of Investment 4 8 12 \$0 \$1000 \$2000 Investment A Investment B Time (years) What does the question ask?

Reading and Interpreting Graphs and Tables
Your Turn: Continued 4 8 12 \$0 \$1000 \$2000 Amount of Investment Investment A Investment B Time (years) What problems would students exhibit when answering this type of question? [Note: Instructors may respond that this looks like an organization question from the Language Arts, Writing Test Part 1 or that the verbiage is too confusing. Remember that all test items on the GED Test were normed using a norming population of high school graduating seniors. Thus, the questions are determined to be valid and reliable. Also note that the colors used within the PowerPoint are not necessarily the colors used for graphs on the GED Mathematics Test. The colors used on the test were identified to be easily viewed in print by all students. The colors used in the PowerPoint are more easily viewed from a distance.] Compared to investment B, investment A had a lesser initial investment and a lesser rate of increase. lesser initial investment and the same rate of increase. lesser initial investment and a greater rate of increase. greater initial investment and a lesser rate of increase. greater initial investment and a greater rate of increase.

Reading and Interpreting Graphs and Tables
Your Turn: The profit, in thousands of dollars, that a company expects to make from the sale of a new video game is shown in the graph. 4,000 8,000 12,000 \$0 \$200 \$400 Profit/Loss in Thousands of Dollars Video Games Sold -\$200 Interpreting values on a graph is another area that causes GED candidates difficulty. What would be the problem that students would display when interpreting this graph? What real-life scenarios may use this type of display? [Example: Graphs that show trends in earnings where companies may show a profit or loss for the quarter or year.] [Note: Have instructors identify different types of errors that students may make when interpreting this type of graph.]

Reading and Interpreting Graphs and Tables
Your Turn: Continued 4,000 8,000 12,000 \$0 \$200 \$400 Profit/Loss in Thousands of Dollars Video Games Sold -\$200 Now that instructors can view the answers, what additional problem or problems would students have when interpreting this graph? [Note: Most students selected the answer -\$150. They were able to read the graph, but did not interpret the value correctly. This is an example of a graph where students must read all of the text that describes the graph, including the headings for each of the axis.] What is the expected profit/loss before any video games are sold? (1) \$0 (2) -\$150 (3) -\$250 (4) -\$150,000 (5) -\$250,000

Your Turn: A B D C

The graph below shows the amount of gasoline in Jamie’s tank after she fills up her car. (1) What are the variables? (2) Describe how they are related at various points on the graph. Answers: (1) The variables are the amount of gas (in gallons) and time (in days). (2) The amount of gas decreases each time Jamie drives somewhere and stays constant when she is not driving.

Your Turn: Matching a Table and a Graph
The table shows the total number of customers at a car wash after 1, 2, 3, and 4 days of its grand opening. Which graph could represent the data shown in the table? Answer: A

When Malcolm jogs on the treadmill, he gradually increases his speed until he reaches a certain level. Then he jogs at this level for several minutes. Then he slows to a stop and stretches. After this he increases to a speed that is slightly lower than before and jogs at this speed for a short while before slowing to a stop again. What is a possible sketch of a graph that shows Malcolm’s jogging speed during his workout? Label each section.