Page 139 - The Ontario Curriculum, Grades 11 and 12: Mathematics, 2007

A. MATHEMATICAL MODELS OVERALL EXPECTATIONS
By the end of this course, students will:
1. evaluate powers with rational exponents, simplify algebraic expressions involving exponents, and solve problems involving exponential equations graphically and using common bases;
2. describe trends based on the interpretation of graphs, compare graphs using initial conditions and rates of change, and solve problems by modelling relationships graphically and algebraically;
3. make connections between formulas and linear, quadratic, and exponential relations, solve problems using formulas arising from real-world applications, and describe applications of mathematical modelling in various occupations.
SPECIFIC EXPECTATIONS
1. Solving Exponential Equations
By the end of this course, students will:
1.1 determine, through investigation (e.g., by expanding terms and patterning), the exponent laws for multiplying and dividing algebraic expressions involving exponents [e.g., (x3)(x2), x3 ÷ x5 ] and the exponent law for simplifying algebraic expressions involving a power of a power [e.g. (x6y3)2]
1.2 simplify algebraic expressions containing inte- ger exponents using the laws of exponents
a2b5c5 Sample problem: Simplify ab–3c4 and evaluate for a = 8, b = 2, and c = – 30.
1.3 determine, through investigation using a variety of tools (e.g., calculator, paper and pencil, graphing technology) and strategies (e.g., patterning; finding values from a graph;
interpreting the exponent laws), the value of m
a power with a rational exponent (i.e., x n ,
where x > 0 and m and n are integers) Sample problem: The exponent laws suggest 1 1
that 42 x 42 = 41. What value would you
1
assign to 42 ? What value would you assign 1
to 27 3 ? Explain your reasoning. Extend your
reasoning to make a generalization about the
1
meaning of xn, where x > 0 and n is a natural number.
1.4 evaluate, with or without technology, numeri-
cal expressions involving rational exponents
1
and rational bases [e.g., 2–3, (–6)3, 42 , 1.01120]*
1.5 solve simple exponential equations numeri- cally and graphically, with technology (e.g., use systematic trial with a scientific calculator to determine the solution to the equation 1.05x = 1.276), and recognize that the solu- tions may not be exact
Sample problem: Use the graph of y = 3x to solve the equation 3x = 5.
1.6 solve problems involving exponential equa- tions arising from real-world applications by using a graph or table of values generated with technology from a given equation [e.g., h = 2(0.6)n, where h represents the height of a bouncing ball and n represents the number of bounces]
Sample problem: Dye is injected to test pan- creas function. The mass, R grams, of dye re- maining in a healthy pancreas after t minutes is given by the equation R = I(0.96)t , where
I grams is the mass of dye initially injected. If 0.50 g of dye is initially injected into a healthy pancreas, determine how much time elapses until 0.35 g remains by using a graph and/or table of values generated with technology.
MATHEMATICAL MODELS
137
*The knowledge and skills described in this expectation are to be introduced as needed, and applied and consolidated, where appropriate, throughout the course.
Foundations for College Mathematics
MAP4C