Lesson Plans: Introduction Lesson Plan 1: Mathematical Modeling, Circular Movement and Transmission Ratios Lesson Plan 2: Skeeter Populations and Exponential Growth

Lesson Plan 2: Skeeter Populations and Exponential Growth

The questions below dealing with exponential functions and mathematical modeling have been selected from various state and national assessments. Although the lesson above may not fully equip students with the ability to answer all such test questions successfully, students who participate in active lessons like this one will eventually develop the conceptual understanding needed to succeed on these and other state assessment questions.

Amelia recorded the time it took for candles of different lengths to burn out. Her results are shown in the table below.

 Candle Length (in inches) 2.1 2.4 3.8 3.2 3.7 3.1 2.7 Burning Time (in minutes) 18 18 30 24 29 26 24

• On a grid, create a scatterplot of her data and draw a line of best fit for the data.

• Write an equation for your line of best fit.

• Explain what the slope represents within the context of this problem.

Solution: The graph below shows the scatterplot with a possible line of best fit.

The line of best fit in the graph above has a y-intercept at approximately (0, 3). In addition, the line appears to pass through (1, 10), so the slope of the line is . Therefore,

the equation of the line is y = 7x + 3.

Slope is the ratio of the change along the vertical axis to the change along the horizontal axis. The vertical axis represents time, and the horizontal axis represents length of the candle. Consequently, the slope represents the time required for a candle to burn a certain amount. Above, the slope was found to be 7, which means that it takes 7 minutes for a candle to burn 1 inch.

• Taken from the Texas Assessment of Knowledge and Skills, Grade 10 (2002):

Mitch wants to use 40 feet of fencing to enclose a flower garden. Which of these shapes would use all the fencing and enclose the largest area?

A. A rectangle with a length of 8 feet and a width of 12 feet
B. An isosceles right triangle with a side length of about 12 feet