Math 10 Common

Unit 2 Algebra and Number

General Outcome: Develop algebraic reasoning and number sense.

In this unit, you will solve problems involving ...

Chapter 4 Exponents and Radicals

Big Ideas

When you have completed this chapter, you will be able to...

Specific Outcome: It is expected that students will:

1. Demonstrate an understanding of factors of whole numbers by determining the:

Open Specific Outcome 2.1 General Notes

General Notes

Prior knowledge from previous grade levels includes:

  • factors of numbers (Grade 7)
  • perfect squares and square roots (Grade 8)
  • equivalent fractions (Grade 7).

Students are expected to develop a conceptual understanding of factors, multiples, square roots and cube roots, without the use of technology.

Students may not be familiar with the terms least common multiple and greatest common factor as these terms are not formally introduced in previous grade levels.

Students will be familiar with the use of a common denominator in adding or subtracting fractions but may not necessarily be familiar with the lowest common denominator.

A prime number is defined as a positive integer with two distinct divisors —1 and the number itself; thus, 1 is not a prime number.

In addition to listing the prime factors of a whole number, students are expected to determine the prime factorization of a composite number.

There is a subtle distinction between the following statements.

  • Determine a number whose square is 16; i.e., solve the equation x2 = 16
  • Determine the square root of 16; i.e. determine √ 16  

The correct response to the first question is ± 4, whereas the correct response to the second question is 4 (or + 4). The convention for the use of the square root symbol is that the symbol by itself ,e.g., the square root of 16 ( ), refers specifically to the principle (positive) square root of the number.

The symbol for the negative square root includes a minus sign before the symbol; e.g.,the negative square root of 16 is -√ 16   .

Open Achievement Indicators

Achievement Indicators

Acceptable Standard

Standard of Excellence
2.1.1 Determine the prime factors of a whole number **
2.1.2 Explain why the number 0 to 1 have no prime numbers **
2.1.3 Determine using a variety of strategies, the greatest common factor or least common multiple of a set of whole numbers, and explain the process. **
2.1.4 Determine, concretely, whether a given whole number is a perfect square, a perfect cube or neither. **
2.1.5 Determine, using a variety of strategies, the square root of a perfect square, and explain the process. **
2.1.6 Determine, using a variety of strategies, the cube root of a perfect cube, and explain the process. **
2.1.7 Solve problems that involve prime factors, greatest common factors, least common multiples, square roots or cube roots **

Specific Outcome: It is expected that students will:

2. Demonstrate an understanding of irrational numbers by:

[CN, ME, R, V] [ICT: C6-2.3]

Open Specific Outcome 2.2 General Notes

General Notes

Prior knowledge from previous grade levels includes:

  • rational numbers (Grade 9)
  • operations on rational numbers, including order of operations (Grade 9)
  • square roots of positive rational numbers that are perfect squares (Grade 9)
  • approximations of square roots of positive rational numbers that are not perfect squares (Grade 9).

Operations on radicals, other than those required for simplifying radicals, are not part of this outcome and will be introduced in Mathematics 20-1 and Mathematics 20-2.

Students are expected to estimate, within reason, the value of a radical, using perfect square numbers, e.g.,1, 4, 9, 16, 25 ..., as benchmarks.

A benchmark is a standard against which something can be measured or assessed.

In Achievement Indicator 2.8, students should be encouraged to develop their own graphic organizer to show the relationship among the subsets of the real numbers.

Open Achievement Indicators

Achievement Indicators

Acceptable Standard

Standard of Excellence
2.2.1 Sort a set of numbers into rational and irrational numbers. **
2.2.2 Determine an approximate value of a given irrational number. Express to the nearest whole number. Express to the nearest tenth.
2.2.3 Approximate the locations of irrational numbers on a number line, using a variety of strategies and explain the reasoning. **
2.2.4 Order a set of irrational numbers on a number line. **
2.2.5 Express a radical as a mixed radical in simplest form (limited to numerical radicands). **
2.2.6 Express a mixed radical as an entire radical (limited to numerical radicands) **
2.2.7 Explain, using examples, the meaning of the index of a radical. **
2.2.8 Represent, using a graphic organizer, the relationship among the subsets of the real numbers (natural, whole, integer, rational and irrational). **

Specific Outcome: It is expected that students will:

3. Demonstrate an understanding of powers with integral and rational exponents.

[C, CN, PS, R]

Open Specific Outcome 2.3 General Notes

General Notes

Prior knowledge from previous grade levels includes:

  • powers with integral bases, excluding base 0, and whole number exponents (Grade 9)
  • exponent laws for whole number exponents (Grade 9).

Technology [T] has not been identified as one of the mathematical processes to be emphasized in completing this outcome. Students are expected to apply the exponent laws without relying on the use of technology.

It is important that students are able to provide explanations for the restrictions on the variables in the definitions and exponent laws in Achievement Indicators 3.1, 3.2 and 3.3. In particular, in Achievement Indicator 3.2, students are expected to explain when a ≥ 0:

  • If n is even, then a≥ 0.
  • If n is odd, then there is no restriction on the variable a.

Students are expected to perform simple operations with rational numbers. This outcome is not intended to assess operations with rational numbers but may be used to reinforce students’ understanding of rational numbers.

Exponents should be restricted to simple rational numbers.

Open Achievement Indicators

Achievement Indicators

Acceptable Standard

Standard of Excellence
2.3.1 Explain, using patterns, why a -n = 1/a n , a ≠ 0 **
2.3.2 Explain, using patterns, why a 1/n = n √a , n > 0. **
2.3.3 Apply the exponent laws:
  • (a m )(a n ) = a m+n
  • a m ÷ a n = a m - n
  • (ab) m = a m b m
  • (a m ) n = a m + n
  • ( a / b ) m = a m / b m, b ≠ 0
To expressions with rational and variable bases and integral and rational exponents, and explain the reasoning 		**
2.3.4 Express powers with rational exponents as radicals and vice versa. Use exponents such as 1/n. Use exponents such as m/n where m ≠ 1.
2.3.5 Solve a problem that involves exponent laws or radicals Solve simple problems. Solve complex problems that involve applying more than one exponent law.
2.3.6 Identify and correct errors in a simplification of an expression that involves powers. **

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4.1 Square Roots and Roots

Focus on...

Perfect square

A number that can be expressed as the products of two equal factors.


What is the difference between 12 and -12 and (-1)2 ?


Perfect square

A number that can be expressed as the products of two equal factors.

Square root

One of two equal factors of a number

Squares and square roots are opposites (they cancel each other).

Why can't your calculator solve: √ -1   ?

Perfect cube

A number that is the product of three equal factors

What is the difference between 13 and -13 and (-1)3 ?

Cube root

One of three equal factors of a number

Can you take the cube root of a negative number?

What is the rule about the roots of negative numbers?

Prime factorization

The process of writing a number written as a product of its prime factors

Prime factors are used to find square roots and cube roots.


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4.2 Integral Exponents

Focus on...

Exponent Laws

Product of powers (am )(an ) = am+n
Quotient of powers am ÷ an = am-n
Power of a power (am )n = amn
Power of a product (ab)m = ambm
Power of a quotient (a/b)n = an/bn , b ≠0
Zero exponent a0 = 1, a ≠ 0

A power with a negative exponent can be written as a power with a positive exponent by inverting it.
x-1 = 1/x
1/x-1 = x

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4.3 Rational Exponents

Focus on...

Square root is the same as a power of a half.
√ x   = x1/2
Cube root is the same as a power of a third.
3√ x   = x1/3

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4.4 Irrational Numbers

Focus on...

Number Sets - groupings of numbers that were developed together.

Natural numbers - counting numbers, used by cave people to count rabbits etc...
N ∈ 1, 2, 3, 4 ... ∞

Whole numbers - natural numbers plus zero, used to keep track of money, first calculations, etc...
W ∈ 0, 1, 2, 3, 4 ... ∞
W ∈ 0, N

Integers - positive and negatives, used to keep track of debt
I ∈ - ∞, ... -3, -2, -1, 0, 1, 2, 3, ..., ∞
I ∈ ± ∞

Rational numbers - all fractions or all repeating or terminating decimals
Q ∈ a/b, b ≠ 0,

Irrational numbers - all non-repeating, non-terminating decimals

Real numbers - all rational and irrational numbers that can be found on a number line

Non-real numbers - imaginary numbers including the square root of negatives.

Diagram of number sets

Radical

Consists of a root symbol, an index, and a radicand.

n√x   where n is the index and x is the radicand
n√x m   = xm/n

Mixed radical
The product of a rational number and a radical.

Entire radical
The product of 1 and a radical.
Mixed radical → 2√3 = √4 × √3 = √12 ← Entire radical