Mathematics 8

In this lesson students will learn to represent, compare, and order rational numbers.

In this lesson students will review the concept of place value to thousandths.

In this lesson students will review operations with integers, using a variety of tools (e.g., two color counters, number lines).

Subtracting a negative integer is like adding a positive integer.

Example: 2 − (−5) = 2 + 5 = 7

Students will review rules for multiplying and dividing integers. Students will represent the multiplication of integers using a variety of tools (e.g., two color counters, number lines).

Important to remember:

Students will learn to:

- express repeated multiplication using exponential notation

- represent whole numbers in expanded form using powers of ten

Exponential Notation: a⁵                      Repeated Multiplication: a × a × a × a × a

Students will learn rules for multiplying and dividing powers.

Product Rule: Multiplying Powers

When multiplying powers: keep the base and add the exponents.

a² × a³ = a²⁺³ = a⁵

A = l × w  = 4 m × 2 m = (4 × 2)(m¹ × m¹) = 8(m^{1 + 1}) = 8 m²

Students will learn to express standard form as a scientific notation and convert the scientific notation to its standard form

The scientific notation becomes very useful when dealing with very small or very large numbers. For example one of the most significant numbers in science is a mole. To have a dozen eggs is to have 12 eggs, to have a mole of eggs is to have 602 200 000 000 000 000 000 000 eggs. That is a lot of eggs and a lot of zeros. To make this huge number easier to represent we can write it in scientific notation as 6.02 × 10²³ eggs. The mole is commonly used for atoms and molecules, and not eggs, in chemistry.

On the other hand, bacteria are tiny, around 0.000 006 m.

Students will learn to find the perfect square root.

Students will evaluate expressions that involve integers, including expressions that contain brackets and exponents, using order of operations.

The order of operation is: Brackets (or Parenthesis), Exponents and Roots, Divisio, Multiplication, Addition and Subtraction.

If the correct order is not followed, the answer will be wrong.

Solve:   9 − 2 × 3 = 7 × 3 = 21 − WRONG ORDER

              9 − 2 × 3 = 9 − 6 = 3 − CORRECT ORDER

Students will represent, compare, and order fractional amounts with unlike denominators, including proper and improper fractions and mixed numbers.

Proper and Improper Fractions and Mixed Numbers:

Students will learn to demonstrate and explain the concept of equivalent fractions.

Equivalent Fractions are different fractions that represent the same number.

Students will learn to determine the Greatest Common Factor (GCF) and common multiples using the prime factorization of numbers.

Find all the prime numbers (factors) for number 48:

The prime factorization of 48 is $2 \times 2 \times 2 \times 2 \times 3$

Students will learn to reduce fractions to lowest terms by dividing the numerator and denominator by the Greatest Common Factor (GCF).

Find the Greatest Common Factor for numbers 12 and 24:

8   = 2 × 4 = 2 × 2 × 2

16 = 4 × 4 = 2 × 2 × 2 × 2

GCF = 2 × 2 × 2 = 8

Students will learn to add and subtract fractions with same denominators.

When adding fractions with same denominators add the numerators and keep the denominator.

When subtracting fractions with same denominators subtract the numerators and keep the denominator.

Students will learn to add and subtract fractions with different denominators.

When adding fractions with different denominators, first change the denominators to the same number, and then add the numerators and keep the same denominator.

Students will learn to multiply fractions.

When multiplying fractions multiply their numerators and multiply their denominators.

Students will learn to divide fractions and to represent the division of fractions using tiles.

When dividing fractions change division to multiplication and invert (flip) the second fraction: $$\frac{1}{1}\div\frac{1}{4} = \frac{1}{1} \times\frac{4}{1}$$ Then, multiply fractions by multiplying numerators and denominators: $$\frac{1}{1} \times\frac{4}{1} = \frac{4}{1} = 4$$

Students will learn about the relation between the amount of one thing compared to the amount of another thing

A ratio is a relation between the amount of one thing compared to the amount of another thing.

Ratios do not have any units. The colon (:) is the symbol used to show a ratio.

There are 7 bananas for every 3 monkeys in the picture below.

The ratio of bananas to monkeys is 7 to 3 or 7 : 3.

Students will learn about a special ratio called rate. Students will learn to calculate rate, such as speed of driving or speed of reading.

A ratio is a relation between the amount of one thing compared to the amount of another thing.

A rate is a special kind of ratio that always has units, because the amounts that are divided are measured in different units.

Students will learn to convert between percents, decimal numbers and fractions.

Percent is a number expressed as a fraction of 100. The percent sign (%) is the symbol used for a percentage.

Students will solve problems involving percent in real life situations.

Calculating Final Prices

Sunglasses cost \$45.00. The Harmonized Sales Tax (HST) is 13% in Ontario. What is the total cost of sunglasses including tax?

sunglasses: \$45.00                                                                                                                                                        tax rate: 13%

To calculate a final price:

The total cost of sunglasses is \$50.85.

Students will learn to represent, compare, and order rational numbers (i.e., positive and negative fractions and decimals to thousandths and percentages).

Without using a number line, determine whether 12.34 is greater than 12.3.

In this lesson students will become familiar with converting metric units, such as meters, grams and liters.

The metric system is the decimal measuring system based on the meter, liter, and gram as units of length, capacity, and weight or mass. The system is very simple and logical because it is based on multiples of 10.

At this time there are three countries in the world that do not use the metric system: the United States of America, Liberia, and Myanmar (Burma). Even though Canada accepted the metric system in 1970s, many Canadians are unfamiliar with units such as kilograms (for weight) or meters (for height).

The metric system is very important in science and trade and all students need to become familiar with the conversions of metric units.

Metric Units

In this lesson students will learn to evaluate algebraic expressions and equations.

An algebraic expression is made of numbers, variables (like $x$ or $y$) and operations (addition, subtraction, multiplication and division).

A variable is a letter that represents one or more numbers.

Evaluate the expression:                $3x + 2$                   when $x = 5$

Substitute 5 for $x$:                           3(5) + 2                    note: 3(5) is the same as 3 times 5

Solve:                                                15 + 2 = 17

For the equation to be true, the left side (LS) and the right side (RS) must be equal.

Evaluate the equation  when $x$ = 6:        $y=\frac{x}{2} - 2$           

Substitute 6 for $x$:                                     $y=\frac{6}{2} - 2$   

Solve:                                                          $y = 3-2 = 1$

Students will learn to recognize and collect like terms in the algebraic expression.

To simplify an expression we combine like terms.

A term is made of a coefficient and a variable. Like terms are terms that have the same variable with the same exponent. \[ \text{a coefficient} \longrightarrow \enspace \underbrace{7x}_{\text{a term}} \enspace \longleftarrow \text{a variable} \] These are examples of like terms: \[ \underbrace{5, 9, 19, 0, 1, 89}_{\text{all are numbers}} \qquad \underbrace{7x, 93x, 17x, x, 11x}_{\text{all have "} \alpha \text{" in term}} \qquad \underbrace{23x^2, 981x^2, 10x^2, x^2}_{\text{all have "} \alpha^2 \text{" in item}} \]

In this lesson students will learn to solve equations.

Solve using the opposite operation.

Subtract 5 from each side: \[ x + 5 - 5 = 9 - 5 \] \[ x = 4 \]

1. In the first reading focus on the meaning of the question. Try to understand what is going on and what you have to do.
2. The second reading should be careful. Isolate specific information from the question, and highlight or copy important data. Translate the verbal expression into algebraic form.
3. In the third reading verify the data and algebraic expression.

In this lesson students will learn the angle relationships for parallel lines and transversals and how to use these relationships to find the missing angles.

Data is a set of qualitative or quantitative values.

Categorical data is qualitative and can be grouped into categories or groups, such as age, education, gender, eye or hair color.

The quantitative data can be measured and includes discrete and continuous sets of values.

Discrete data includes the counted object and has specific numerical values.

Continuous data is measured and has a continuous range.

Probability describes how likely it is that something is going to happen.