Content Area: Math

 

Index: 4.3B Grade 8 CPI 1

 

Standard: 4.3 - Patterns and Algebra

 

Strand: B - Functions and Relationships

 

Cumulative Progress Indicator: 1 -  The student will graph functions, and understand and describe their general behavior

·        Equations involving two variables

·        Rates of change (informal notion of slope)

 

Grade: 8

 

Sample Activities:

 

·        Students investigate graphs without numbers. For example, they may study a graph that shows how far Olivia has walked on a trip from home to the store and back, where time is shown on the horizontal axis and the distance covered is on the vertical axis. Students tell a story about her trip, noting that where the graph is horizontal, she has stopped for some reason. In addition, their stories account for those parts of the graph that are steeper, by explaining why Olivia is walking faster (e.g., she is running from a dog), and those parts of the graph that are not as steep, by explaining why Olivia is walking slower (e.g., she is going up a hill).

 

·        Students use probes and graphing calculators or computers to collect data involving two variables for several different science experiments (such as measuring the time and distance that a toy car rolls down an inclined plane, or the temperature of a beaker of water when ice cubes are added). They look at the data that has been collected in tabular form and as a graph on a coordinate grid. They classify the graphs as straight or curved lines and as increasing (direct variation), decreasing (inverse variation), or mixed. For those graphs that are straight lines, the students try to match the graph by entering and graphing a suitable equation.

 

·        Given several nonlinear functions, such as y = x^2, y = 3x^2, y = x^2 + 1, y = x^3, or y = 16/x, students create a table of values for each and use graphing calculators to graph them.

 

·        Students describe what happens when a ball is tossed into the air, experimenting with a ball as needed. They make a graph that shows the height of the ball at different times and discuss what makes the ball come back down. They also consider the speed of the ball: when is it going fastest? slowest? With some help from the teacher, they make a graph showing the speed of the ball over time.

 

·        Students use probes and graphing calculators or computers to collect data involving two variables for several different science experiments (such as measuring the time and distance that a toy car rolls down an inclined plane or measuring the brightness of a light bulb as the distance from the light bulb increases or measuring the temperature of a beaker of water when ice cubes are added). They look at the data that has been collected intabular form and as a graph on a coordinate grid. They classify the graphs as straight or curved lines and as increasing (direct variation), decreasing (inverse variation), or mixed. For those graphs that are straight lines, the students try to match the graph by entering and graphing a suitable equation.

 

·        Students measure the temperature of boiling water as it cools in a cup. They make a table showing the temperature at five-minute intervals for an hour. Then they graph the results and make observations about the shape of the graph, such as "the temperature went down the most in the first few minutes," "it cooled more slowly after more time had passed," or "it's not a linear relationship." The students also predict what the graph would look like if they continued to collect data for another twelve hours.

 

·        Students make Ferris wheel models from paper plates (with notches cut to represent the cars). They use the models to make a table showing the height above the ground (desk) of a person on a Ferris wheel at specified time intervals (time needed for next chair to move to loading position). After collecting data through two or three complete turns of the wheel, they make a graph of time versus height. In their math notebooks, they respond to questions about their graphs: Why doesn't the graph start at zero? What is the maximum height? Why does the shape of the graph repeat? The students learn that this graph represents a periodic function.

 

·        Students compare two ways of cooling a glass of soda, adding lots of ice at the beginning or adding one cube at a time at one minute intervals. Each student first makes a prediction about which cools the soda faster, and the class summarizes the predictions. Then the teacher collects the data, using probes and graphing calculators or computers and displays the results in table and graph form on the overhead. The students compare the graphs and write their conclusions in their math notebooks. They discuss the reasons for any difference between these two methods with their science teacher.

 

·        Students compute the average speed of a toy car as it travels down a ramp by dividing the length of the ramp by the time the car takes to travel the ramp. They try different angles for the ramp, recording their results. They make a graph of average speed vs. angle and discuss whether this graph is linear.

 

·        Students make a graph that shows the minimum wage from the time it was first instituted until the present day. Some of the students begin by simply plotting points and connecting them but soon realize that the minimum wage was constant for a time and then abruptly jumped up. They decide that parts of this graph are like horizontal lines. They look for other examples of "step functions."