Pacing Guide Secondary II: Block 2, Part A Extending the Number System
Recommended Time Frame: 3 weeks
Start Date: October 29, 2012
Estimated End Date: Nov. 13, 2012 Actual End Date:
Background/Helpful Information: The first instructional block expanded the kinds of functions and contexts that students understand and
expanded their skills around manipulating functions. The second instruction block builds on the first by turning the focus of attention to solving
equations, inequalities, and systems of equations involving exponential and quadratic expressions. This new focus on numerical solutions demands
that the number system be extended in useful ways. First, integers can be thought of as a special subset of polynomials (e.g. ax3 + bx2 + cx + d
reduces to a four-digit integer, abcd, when a, b, c, d are all non-negative and x = 10.) Second, the real numbers can be thought of as a subset of
complex numbers. Earlier in their mathematical learning, student discovered that, to have a solution to x + 1 = 0, we need negative numbers. Now,
in order to solve x2 + 1 = 0 (and similar equations), we need to expand our number system to include the complex numbers. The real number line is
expanded to the complex plane, where each point (a, b) in the complex plane represents a unique complex number, a + bi. Finally, real-world
contexts of exponential functions demand that we be able to expand the domain from integers to rational numbers (for example, in a context
whereby a population is doubling every 3 years, we would like to be able to figure what the population might be after 1 year—by multiplying the
starting population, A, by 21/3.) This first part of block 2 develops these three extensions of the number system and gives the students practice doing
operations with these new extensions.
CURRICULUM
INSTRUCTION
ASSESSMENT
Rational Exponents
N.RN.1: Explain how the definition of the meaning of rational
exponents follows from extending the properties of integer exponents to
those values, allowing for a notation for radicals in terms of rational
exponents. For example, we define 51/3 to be the cube root of 5 because
we want (51/3)3 = (51/3) 3 to hold, so (51/3) 3 must equal 5.
N.RN.2: Rewrite expressions involving radicals and rational exponents
using the properties of exponents.
Properties of Rational and Irrational Numbers
N.RN.3: Explain why sums and products of rational numbers are
rational, that the sum of a rational and an irrational number is
irrational, and that the product of a nonzero rational number and an
irrational number is irrational. [Connect to physical situations, e.g.,
finding the perimeter of a square of area 2]
Operations with Complex Numbers
N.CN.1: Know there is a complex number i such that i2 = -1, and
every complex number has the form a + bi with a and b real.
N.CN.2: Use the relation i2 = -1 and the commutative,
associative, and distributive properties to add, subtract, and
multiply complex numbers. [Limit to multiplications that involve
i2 as the highest power of i.]
Operations with Polynomials
A.APR.1: Understand that polynomials form a system analogous
to the integers, namely, they are closed under the operations of
addition, subtraction, and multiplication; add, subtract, and
multiply polynomials. [Focus on polynomial expressions that
simplify to forms that are linear or quadratic in a positive
integer power of x.]