# Grades 9/10/11 Number Sense and Algebraic Expressions

Exponent laws. Manipulating expressions including polynomials, radical and rational expressions. Investigating prime factorization.

This is one of seven strands of the CEMC Grade 9/10/11 courseware. The other strands and more information about this courseware is available on the Grade 9/10/11 homepage.

The alternative formats for some lessons are currently under construction. These will be completed in the coming months.

### Units

## Unit 1: Exponents

This lesson will examine the relationship between the exponential representation of length, area, and volume. Algebraic expressions will be represented in both exponential form and expanded form.

This lesson introduces the exponent rules for multiplying and dividing monomials.

In this lesson, we will continue working with operations involving exponents and explore the power of a power rule.

In this lesson, we will examine powers with positive and negative integer bases. We will also explore the exponent rule for an exponent of zero and will examine powers with negative integer exponents.

In this lesson, we will define the \(n\)^{th} root of a number and explore positive rational exponents of the form \(\dfrac{1}{n}\).

In this lesson, we will simplify and evaluate positive rational exponents of the form \(\dfrac{a}{n}\) and negative rational exponents of the form \(\dfrac{-a}{n}\).

This lesson puts together all of the Exponent Rules learned to simplify algebraic expressions. We will also use exponents and exponent rules to evaluate expressions.

All of the pencil and paper practice exercises, answers, and solutions for this unit are reproduced here.

This is a collection of additional, and sometimes challenging, problems that extend the material covered in this unit, connect material from different lessons, and further explore real-world applications.

## Unit 2: Manipulating Algebraic Expressions

This lesson will review the concept of a variable. Students will classify polynomials based on the number of terms and will state the degree of the polynomial.

In this lesson, students will learn to identify like terms. They will also add and subtract polynomials by collecting like terms.

In this lesson, students will learn to multiply a polynomial by a monomial using the distributive property.

In this lesson, students will apply the Distributive Property to multiply a polynomial by a polynomial.

In this lesson, students will simplify polynomials using addition, subtraction, and multiplication. They will also determine if two algebraic expressions are equivalent.

All of the pencil and paper practice exercises, answers, and solutions for this unit are reproduced here.

This is a collection of additional, and sometimes challenging, problems that extend the material covered in this unit, connect material from different lessons, and further explore real-world applications.

## Unit 3: Radicals and Rational Functions

In this lesson, students will simplify and order radicals involving integers and rational numbers. Technology will be used to estimate the value of a radical.

In this lesson, students will simplify radical expressions involving addition, subtraction, and multiplication. They will also learn how to rationalize the denominator.

In this lesson, students will define extraneous roots and will solve radical equations algebraically and graphically. They will identify restrictions on the domain and any extraneous roots and will solve real world problems.

In this lesson, students will be introduced to rational expressions. They will identify and simplify rational expressions and state the restrictions on the variable values.

In this lesson, students will simplify rational expressions by multiplying and dividing and will state the restrictions on the variable values.

In this lesson, students will determine a common denominator for rational expressions. They will use this common denominator to add and subtract rational expressions. Students will combine operations to simplify rational expressions.

All of the pencil and paper practice exercises, answers, and solutions for this unit are reproduced here.

This is a collection of additional, and sometimes challenging, problems that extend the material covered in this unit, connect material from different lessons, and further explore real-world applications.

## Unit 4: Prime Factorization

In this lesson, students will define prime and composite numbers. They will learn to write composite numbers as a product of its prime factors.

In this lesson, students will use prime factorization to determine the Greatest Common Factor (GCF) and the Least Common Multiple (LCM). They will solve word problems involving GCF and LCM.