gr9_Exponents

= Exponents =

//It is expected that students will://
 * **Prescribed Learning Outcomes**

- representing repeated multiplication using powers - using patterns to show that a power with an exponent of zero is equal to one - solving problems involving powers [C, CN, PS, R]
 * A1** demonstrate an understanding of powers with integral bases (excluding base 0) and whole number exponents by

[C, CN, PS, R, T]
 * A2** demonstrate an understanding of operations on powers with integral bases (excluding base 0) and whole number exponents


 * A3** Explain and apply the order of operations, including exponents, with and without technology [PS, T]


 * C2** Determine the surface area of composite 3-D objects to solve problems [C, CN, PS, R, V]

|| For example: (-2)^5 = -2 x -2 x -2 x -2 x -2 -(2)^5 = -1 x 2 x 2 x 2 x 2 x 2 -demonstrate the difference between the exponent and the base -express a given power (with integral and rational bases) as a repeated multiplication -express a given repeated multiplication as a power -explain why b^0=1 using patterns -evaluate powers with integral bases (excluding base 0) and whole number exponents. ||
 * **Planning for Assessment** || **Assessment Strategies** ||
 * * Have students use repeated multiplication to evaluate powers with negative bases and determine a rule for when the exponent is odd or even.
 * Have students explain the difference between the exponent and the base of powers such as 2^3 and 3^2 using repeated multiplication then have students give a pictorial representation using volume and area.
 * Have students express repeated multiplication involving negative bases and non-negative bases as powers to determine the role of parentheses when evaluating powers.
 * Have students use repeated multiplication to explain the difference between 3y^4 and (3y)^4 and evaluate for y=2.
 * Have students evaluate b^4, b^3, b^2, b^1 for b=2, 3, and 4 and then work in groups to discuss the pattern and predict the answer to b^0 for b=2,3, and 4. Then generalize a rule for b^0 for all b not equal to zero. || Look for evidence that students are able to:
 * * Have students work in groups to discover the exponent laws by using repeated multiplication to evaluate products of powers, quotients of powers, and powers of powers without technology and expressing the pattern as a law..
 * Provide the students with the 5 exponent laws of powers with integral bases as mathematical sentences. Then ask the students to explain the laws using English sentences providing examples and non-examples.

For example:

Given (a^m)(a^n)=a^(m+n)

Possible Student Response: The Product Law states what when MULTIPLYING powers of the SAME base, keep the base and ADD the exponents.

Two examples are : (3^4)(3^2)=3^6 and (x^3)(x^4)=x^7.

Note: It is important to introduce here the idea of powers with variable bases (despite the fact that PLO A1 clearly states "demonstrate an understanding of powers with integral bases") as it is a prerequisite for PLO B7 "model record and explain the operations of multiplication and division of polynomial expressions"**

Two non-examples are:

(3^2)(2^3) In this case the bases are not the same so the exponent laws do not apply, instead apply the order of operations to evaluate.

(3^2) + (3^4) In this case it is a sum of powers so the exponent laws do not apply, instead apply the order of operations to evaluate.

-simplify an expression involving powers with integral bases and whole number exponents by applying the exponent laws. -evaluate a given expression by applying the exponent laws and the order of operations without technology. -solve a given problem by applying the order operations with the use of technology. || a) 0.3^2 or 0.2^3 b) (-2.1)^2 or -2.1^2 c) (-3.5)3 or (-3.5)^2 d) 4.6^7 or (-5.1)^6 Then have them verify their answers by using a calculator. || Assess students based on their abilities to: -solve a given problem by applying the order operations with the use of technology. || __Word Form__ -with entries such as Four cubed, Fifth power of 2, ten to the exponent four, __Repeated Multiplication__ -with entries such as 2 x 2, __Exponential Form__ -with entries like 5^3 and __Decimal Form__ -with entries such as 8 and 36. (Note that 36 could represent the 6 x 6 or -6 x 6) and have students fill in the blanks || Have students write an essay about all the things that they have learned about exponents. They should be able to: provide a definition of a power, exponent and base; explain the exponent laws, providing examples that include integral bases (excluding base 0) and stating common mistakes; explain the role of parenthesis in powers, providing examples and stating common mistakes; and explain how to evaluate sums and differences of powers using order of operations. || -evaluate a given expression by applying the exponent laws and the order of operations without technology. -determine the area of overlap in a given concrete composite 3-D object (cubes), and explain its effect on determining the surface area. -determine the surface area of a given concrete composite 3-D object (cubes), ||
 * Give students a selection of problems with solutions, involving the exponent laws, sums and difference of powers, and order of operations involving exponents, some of which have an error in them. Ask students to find which problems have a mistake, identify the mistakes, explain the errors in thinking that were made and provide correct solutions for each problem || Assess students based on their abilities to:
 * * Show students how to check their work by using calculators to evaluate expressions involving operations on powers. Include questions that require the appropriate use of brackets such as -3^2 and (-3)^2. Then have students work in pairs to make up questions to challenge their partner's skills with a calculator. Look for use of brackets, exponents, and negative coefficients. Confirm that the authors of the questions can solve them and explain the solutions to their partners.
 * Present the students with the following powers and have them predict which is larger.
 * * Have students write daily entries into a journal about new concepts that they have learned about powers.
 * Create a chart with column titles of :
 * * Have students represent the area of squares, and volume and surface area of cubes using powers and then evaluate.
 * Give students a diagram of a tower of cubes with decreasing side lengths, such as 5, 4, 3, 2, and 1. Then have students work with a partner to express the volume and exposed surface area of the tower using powers and then evaluating. || Assess students based on their abilities to: