Lesson Notes By Weeks and Term v5 - Grade 8

Lesson Video

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Performance objectives

• Define and identify integers and rational numbers.
• Perform all four basic operations with integers and rational numbers.
• Express numbers in exponential form.
• Compare and order numbers on a number line.
• Solve simple word problems.

Lesson summary

This week, we're diving into the world of numbers, specifically integers, rational numbers, and exponents. Understanding these concepts is like learning the alphabet of mathematics. Just as you need letters to form words and sentences, you need these fundamental number types and operations to tackle more complex mathematical problems later on. These skills are crucial for everything from managing your pocket money to understanding financial concepts like interest rates and budgeting when you are older.

Lesson notes

Integers: Integers are whole numbers, both positive and negative, including zero. They do not include fractions or decimals.

Examples: ..., -3, -2, -1, 0, 1, 2, 3, ... A positive integer is any whole number greater than zero (1, 2, 3, ...). A negative integer is any whole number less than zero (-1, -2, -3, ...). Zero (0) is an integer, but it's neither positive nor negative.

Why integers matter: Think about temperature: it can be below zero degrees Celsius, especially in places like Sutherland.

Or bank balances: you can have money (positive integer) or be in debt (negative integer). Even measuring height relative to sea level involves integers – above sea level is positive, below is negative.

Rational Numbers: Rational numbers are numbers that can be expressed as a fraction p/q, where p and q are integers and q is not equal to zero.

Examples:* 1/2, -3/4, 5, 0.75 (which is 3/4), -2 (which is -2/1), 0 (which is 0/1).

Why rational numbers matter:* Most measurements we take in daily life are not whole numbers. The weight of a bag of mealie meal, the length of a piece of fabric, the amount of petrol in your car's tank – these are all rational numbers. Important

Note: Decimals that terminate (like 0.25) or repeat (like 0.333...) are rational numbers because they can be written as fractions. For example, 0.25 = 1/4 and 0.333... = 1/

3. Operations with Integers and Rational Numbers: Addition: Integers:* Adding two positive integers results in a positive integer. Adding two negative integers results in a negative integer. When adding a positive and a negative integer, you find the difference between their absolute values and use the sign of the integer with the larger absolute value.

Example 1:* -5 + (-3) = -8 (Both negative, so add the absolute values and keep the negative sign)

Example 2:* 7 + (-2) = 5 (Find the difference between 7 and 2, which is

5. Since 7 is larger than 2, the result is positive.)

Example 3:* -8 + 3 = -5 (Find the difference between 8 and 3, which is

5. Since 8 is larger than 3, the result is negative.)

Rational Numbers:* To add rational numbers, they need to have a common denominator.

Example 1:* 1/4 + 2/4 = 3/4 (Common denominator is 4, so simply add the numerators)

Example 2:* 1/2 + 1/3 = 3/6 + 2/6 = 5/6 (Find the least common denominator, which is 6, then convert the fractions and add)

Subtraction: Integers:* Subtracting an integer is the same as adding its opposite.

Example 1:* 5 - 3 = 5 + (-3) = 2 Example 2:* 2 - 5 = 2 + (-5) = -3 Example 3:* -4 - (-2) = -4 + 2 = -2 Rational Numbers:* Similar to addition, find a common denominator before subtracting.

Example 1:* 3/4 - 1/4 = 2/4 = 1/2 Example 2:* 2/3 - 1/2 = 4/6 - 3/6 = 1/6 Multiplication: Integers:* A positive times a positive is positive. A negative times a negative is positive. A positive times a negative is negative. A negative times a positive is negative.

Example 1:* 3 x 4 = 12 Example 2:* -2 x -5 = 10 Example 3:* 6 x -2 = -12 Example 4:* -4 x 3 = -12 Rational Numbers:* Multiply the numerators and multiply the denominators.

Example 1:* 1/2 x 2/3 = (1 x 2) / (2 x 3) = 2/6 = 1/3 Example 2:* -3/4 x 1/5 = (-3 x 1) / (4 x 5) = -3/20 Division: Integers:* The rules for the sign of the result are the same as for multiplication.

Example 1:* 8 ÷ 2 = 4 Example 2:* -10 ÷ -5 = 2 Example 3:* 12 ÷ -3 = -4 Example 4:* -15 ÷ 5 = -3 Rational Numbers:* To divide by a fraction, you multiply by its reciprocal (flip the fraction).

Example 1:* 1/2 ÷ 2/3 = 1/2 x 3/2 = (1 x 3) / (2 x 2) = 3/4 Example 2:* -1/4 ÷ 1/2 = -1/4 x 2/1 = (-1 x 2) / (4 x 1) = -2/4 = -1/2 Exponents (Positive Integer Exponents): An exponent tells you how many times to multiply a number (called the base) by itself.

Example:* 2 3 means 2 x 2 x 2 =

8. Here, 2 is the base and 3 is the exponent.

Vocabulary:* 2 3 is read as "2 to the power of 3" or "2 cubed". 5 2 is read as "5 to the power of 2" or "5 squared".

Why exponents matter:* Exponents allow us to write repeated multiplication in a more compact form. They are used in calculating areas, volumes, and are essential in science.

Example with units:* If a square room has sides of 3 metres each, the area of the room is 3 2 square metres = 9 square metres. Guided Practice (With Solutions)

Question 1: Arrange the following integers in ascending order: -5, 2, 0, -1, 4 Solution: -5, -1, 0, 2,

4. Ascending order means from smallest to largest. On the number line, numbers to the left are smaller than numbers to the right.

Question 2: Calculate: -3 + 7 - 2 Solution: Step 1: -3 + 7 = 4 Step 2: 4 - 2 = 2 Therefore, -3 + 7 - 2 =

2. Remember to work from left to right.

Question 3: Simplify: (1/2) x (-4/5)

Solution: (1/2) x (-4/5) = (1 x -4) / (2 x 5) = -4/10 = -2/

5. Remember to multiply the numerators together and the denominators together. Also, a positive times a negative is a negative.

Question 4: Evaluate: 3 4 Solution: 3 4 = 3 x 3 x 3 x 3 =

8

1. The exponent 4 means we multiply the base 3 by itself four times.