Number

By the end of this unit, you will be able to:

SCO 1 Demonstrate an understanding of powers with integral bases (excluding base 0) and whole number exponents by: representing repeated multiplication using powers; using paterns to show that a power with an exponent of zero is equal to one; solve problems involving powers.
1.1  Demonstrate the differences between the exponent and the base by building models of a given power.
1.2  Explain, using repeated multiplication, the difference between two given powers in which the exponent and base are interchanged.
1.3  Express a given power as a repeated multiplication.
1.4  Express a given repeated multiplication as a power.
1.5  Explain the role of parentheses in powers by evaluating a given set of powers.
1.6  Demonstrate, using patterns, that aº is equal to 1 for a given value of a(a≠ 0).
1.7  Evaluate powers with integral bases (excluding base 0) and whole number exponents.

SCO 2 Demonstrate an understanding of operations on powers with integral bases (excluding base 0) and whole number exponents.
2.1  Explain, using examples, the exponent laws of powers with integral bases (excluding base 0) and whole number exponents.
2.2  Evaluate a given expression by applying the exponent laws.
2.3  Determine the sum of two given powers and record the process.
2.4  Determine the difference of two given powers and record the process.
2.5  Identify the error(s) in a given simplification of an expression involving powers.

SCO 3  Demonstrate an understanding of reational numbers by comparing and ordering rational numbers and solving problems that involve arithmatic operations on rational numbers.
3.1  Order a given set of rational numbers in fraction and decimal form by placing them on a number line.
3.2  Identify a rational number that is between two given rational numbers.
3.3  Solve a given problem involving operations on rational numbers in fraction or decimal form.

SCO 4 Explain and apply the order of operations, including exponents, with and without technology.
4.1 Solve a given problem by applying the order of operations without the use of technology.
4.2  Solve a given problem by applying the order of operations with the use of technology.
4.3  Identify the error in applying the order of operations in a given incorrect solution.

SCO 5 Determine the exact square root of positive rational numbers.
5.1  Determine whether or not a given rational number is a square number and explain the reasoning.
5.2  Determine the square root of a given positive rational number that is a perfect square.
5.3  Identify the error made in a given calculation of a square root.
5.4  Determine a positive rational number, given the square root of that positive rational number.

SCO 6  Determine an approximate square root of positive rational numbers
6.1  Estimate the square root of a given rational number that is not a 
perfect square, using the roots of perfect squares as benchmarks.
6.2  Determine an approximate square root of a given rational number that is not a perfect square, using technology (e.g., a calculator, a computer).
6.3  Explain why the square root of a given rational number as shown on a calculator may be an approximation.
6.4  Identify a number with a square root that is between two given numbers.



Exponents

The exponent of a number says how many times to use the number in a multiplication.

8-squared

In 82 the "2" says to use 8 twice in a multiplication, so: 82 = 8 × 8 = 64

In words: 82 could be called "8 to the power 2" or "8 to the second power", or simply "8 squared"

Exponents are also called Power


Using exponents is short hand for repeated multiplication.


Repeated multiplication can be written as an exponent or power.


Math is Fun
Written information & interactives


Parentheses and Exponents


When there is no parenthses the negative DOES NOT APPLY through the repeated multiplication - it only occurs once!


Parentheses are used when a power has a negative base to show that the negative sign is part of the base, and stays throughout the repeated multiplication.


aº always equals 1

Check out the pattern:


Laws of Exponents

Operations with exponents can be efficiently performed with the use of the laws of exponents.


Exploring Laws of Exponents: Use it 
Math Interactives - Learn Alberta

Exploring Laws of Exponents: Explore it
Math Interactives - Learn Alberta


Rational Numbers

A rational number is any number that can be written in fraction form (a/b), where b does not equal zero. Irrational numbers cannot be written as a fraction, they do not end or repeat (irrational = crazy!).

Regardless of where the (-) sign is placed each of these fractions equal -¾ or -0.75.


Comparing & Ordering Rational Numbers

Number Line Benchmarks

A Benchmark is a standard to estimate something using a few numbers.  For example below 0 and 1 are used as benchmarks to compare a variety of fractions.

Remember

  • a negative number is ALWAYS less than a positive
  • use a number line with -1, 0 and 1 marked (benchmarks)
  • find common denominators or convert fractions to decimals to compare


STEPS

1.  Convert to common standard (fraction or decimal)

2.  Draw benchmark number line

3.  Place in order on number line


Ordering Rational Numbers
Interactive - Khan Academy

Ordering Rational Numbers
Game - SoftSchools

Jeopardy - Comparing & Ordering Rational Numbers
Full class game - only works on laptops/computers


Order of Operations

BEDMAS can be used to remember the order in which to complete operations.  It is important that equations are solved following these rules.

  • Brackets group symbols so that we can treat them as a single term.
  • Exponents are performed before multiplication because exponents represent repeated multiplication.
  • Multiplication and division are performed prior to addition and subtraction because they are repeated addition and subtraction; solve as they appear.
  • Addition and subtraction occur last in the order they appear.


Exploring Order of Operations: Use it
Interactive - Alberta Education

Order of Operations
Game - 4nums


Square Roots of Rational Numbers

Squares and square roots are inverse operations - they are opposites just like addition & subtraction, and multiplication & division.


Perfect Squares are squares of the whole numbers.


Square & Square Roots
Written information: Math is Fun

IXL: Perfect Squares
Practice


To estimate the square root of numbers that do not form perfect squares, use the perfect squares as benchmarks (something to estimate from).