Understand that repeated multiplication defines an exponential function Understand the algebra of exponents and that the inverse of an exponent is a logarithm Understand the conversion of exponential form into logarithmic form and vice versa Understand the application of exponential and logarithmic functions to various contexts
Conceptual Framework
relationships change model representation in the context of Scientific & Technical Innovation
Statements of Inquiry
Growth or decay dependent on the size of the entity changing can be modelled mathematically and applied in a diverse range of real-world contexts
Factual
How do we generalize repeated multiplication to model increasing or decreasing change?
Conceptual
How can we manipulate the exponential form to analyze increasing or decreasing change?
Debatable
To what extent do real-world phenomena continue to change with fixed ratios?
Description
The focus of this unit will be on exponential and logarithmic functions, their representation and application to model change by a constant factor. Students will review the fundamental concept of a mathematical inverse and appreciate the need for an inverse to find an exponent. Once familiar with the representation of repeated multiplication, students will investigate graphical transformations of exponential functions. Students will study skills in the manipulation of exponential form and will solve exponential equations graphically and algebraically with the use of logarithms. Students will also apply skills and concepts to model real life exponential relationships including compound interest, depreciation, tuning musical instruments, carbon-dating and logarithmic relationships including the Richter scale, the pH scale, decibels, and Newton’s rate of cooling.
Learning Outcomes
Understand that repeated multiplication is exponentiation
Understand that repeated multiplication by a number greater than one is growth and that repeated multiplication by a number between zero and one is decay
Generalise patterns formed by repeated multiplication as exponential functions
Compare exponential change with linear and quadratic change
Define the domain and range of an exponential function
Appreciate the difference between exponential growth and decay
Interpret change factors from exponential models
Estimate unknown exponents in exponential expressions by intuition and thus appreciate the need for the inverse of an exponent (i.e. logarithms)
Evaluate logarithms
Apply properties of logarithms for simplification and evaluation
Understand e as the natural rate of change
Solve for unknown exponents in exponential functions graphically
Solve for unknown exponents in exponential functions algebraically by manipulating and equating bases
Solve for unknown exponents in exponential functions algebraically by manipulating and converting to logarithmic form
Graph and transform exponential and logarithmic functions
Be able to determine theoretical exponential functions from change factors and initial amounts
Be able to fit exponential models to approximately exponential data