Lesson Notes By Weeks and Term v5 - Grade 8
Integers, rational numbers and exponents (Grade 8) – Week 5 focus
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Integers, rational numbers and exponents (Grade 8) – Week 5 focus
Download the Lessonotes Mobile South Africa app for faster lesson access on Android and iPhone.
Subject: Mathematics
Class: Grade 8
Term: 1st Term
Week: 5
Theme: General lesson support
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This week, we delve deeper into the fascinating world of numbers, specifically integers, rational numbers, and exponents. Understanding these concepts is crucial not just for mathematics, but also for making informed decisions in everyday life. From managing your pocket money and calculating savings to understanding scientific data and even coding, a solid grasp of these concepts is essential. Think about using integers to understand profit and loss in a small business run by a family member, or using exponents to calculate the growth of a loan. This knowledge empowers you!
2.1 Integers: Integers are whole numbers (no fractions or decimals) and their opposites. They can be positive, negative, or zero.
Examples: ... -3, -2, -1, 0, 1, 2, 3...
Number Line Representation: The number line is a visual way to represent integers. Zero is in the middle, positive numbers are to the right, and negative numbers are to the left. The further to the right a number is, the greater its value. The further to the left a number is, the smaller its value.
Example: Compare -3 and 1 on a number line. 1 is to the right of -3, therefore 1 > -
3. Operations with Integers: Addition: Adding two positive integers: The result is positive. (e.g., 2 + 3 = 5)
Adding two negative integers: The result is negative. (e.g., -2 + (-3) = -5)
Adding a positive and a negative integer: Subtract the smaller absolute value from the larger absolute value. The result takes the sign of the number with the larger absolute value. (e.g., -5 + 2 = -3; 7 + (-4) = 3)
Subtraction: Subtracting an integer is the same as adding its opposite. (e.g., 5 - 3 = 5 + (-3) = 2; 2 - (-4) = 2 + 4 = 6)
Multiplication: Positive x Positive = Positive (e.g., 2 x 3 = 6) Negative x Negative = Positive (e.g., -2 x -3 = 6) Positive x Negative = Negative (e.g., 2 x -3 = -6) Negative x Positive = Negative (e.g., -2 x 3 = -6)
Division: Follow the same sign rules as multiplication. (e.g., 6 / 2 = 3; -6 / -2 = 3; 6 / -2 = -3; -6 / 2 = -3) 2.2 Rational Numbers: A rational number is any number 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, 0.25 (which is 1/4), -1.5 (which is -3/2)
Operations with Rational Numbers: The rules for addition, subtraction, multiplication, and division of fractions apply here. Remember to find a common denominator when adding or subtracting fractions.
Example: Add 1/2 and 1/
3. Solution: Find a common denominator: The least common multiple of 2 and 3 is
6. Convert the fractions: 1/2 = 3/6 and 1/3 = 2/
6. Add the fractions: 3/6 + 2/6 = 5/
6. Example: Multiply 2/5 and -3/
4. Solution: Multiply the numerators: 2 x -3 = -
6. Multiply the denominators: 5 x 4 =
2
0. Simplify the fraction: -6/20 = -3/10. 2.3 Exponents: An exponent indicates how many times a base number is multiplied by itself. For example, in 2 3 , 2 is the base and 3 is the exponent. This means 2 x 2 x 2 =
8. Laws of Exponents (Integer Exponents): Product Rule: a m a n = a m+n (When multiplying powers with the same base, add the exponents)
Example: 2 2 2 3 = 2 2+3 = 2 5 = 32 Quotient Rule: a m / a n = a m-n (When dividing powers with the same base, subtract the exponents)
Example:* 3 5 / 3 2 = 3 5-2 = 3 3 = 27 Power of a Power Rule: (a m ) n = a mn (When raising a power to another power, multiply the exponents)
Example: (2 3 ) 2 = 2 32 = 2 6 = 64 Zero Exponent Rule: a 0 = 1 (Any number raised to the power of 0 equals 1, except 0 itself which is undefined)
Example:* 5 0 = 1 Negative Exponent Rule: a -n = 1/a n (A number raised to a negative exponent is the reciprocal of the number raised to the positive exponent)
Example:* 2 -3 = 1/2 3 = 1/8 2.4 Order of Operations (BODMAS/PEMDAS): Remember to follow the order of operations when solving problems involving multiple operations: Brackets (or Parentheses) Orders (or Exponents) Division and Multiplication (from left to right) Addition and Subtraction (from left to right)
Example: Solve: 2 + 3 x (4 - 1) 2 Solution: Brackets: 4 - 1 = 3 Exponents: 3 2 = 9 Multiplication: 3 x 9 = 27 Addition: 2 + 27 = 29 Guided Practice (With Solutions)
Question 1: Represent -2, 0, and 3 on a number line.
Solution: Draw a number line. Mark 0 in the center. Mark -2 two units to the left of
0. Mark 3 three units to the right of
0. Commentary: This reinforces the understanding of integer placement on a number line.
Question 2: Evaluate: -5 + 8 - (-2)
Solution: -5 + 8 - (-2) = -5 + 8 + 2 = 3 + 2 =
5. Commentary: Remember that subtracting a negative number is the same as adding its positive counterpart.
Question 3: Simplify: (3 2 x 3 4 ) / 3 3 Solution: (3 2 x 3 4 ) / 3 3 = 3 2+4 / 3 3 = 3 6 / 3 3 = 3 6-3 = 3 3 =
2
7. Commentary: This uses both the product and quotient rules of exponents.
Question 4: Evaluate: 1/4 + 2/3 Solution: Find a common denominator, which is 12. 1/4 = 3/12 and 2/3 = 8/
1
2. Therefore, 3/12 + 8/12 = 11/
1
2. Commentary: A good example of adding rational numbers.
Question 5: Solve: 10 - 2 x (3 + 1)
Solution: Following BODMAS: 10 - 2 x (3 + 1) = 10 - 2 x 4 = 10 - 8 =
2. Commentary: Emphasis on the order of operations. Independent Practice (Questions Only) Order the following integers from least to greatest: -7, 4, -1, 0, -3, 2 Calculate: -12 ÷ 3 + 5 x (-2)
Simplify: (5 7 / 5 2 ) x 5 -1 Calculate: 3/5 - 1/2 A thermometer reads -5°C in the morning. By noon, the temperature has risen by 8°
C. What is the temperature at noon?
Simplify: (4 2 ) 3 / 4 4 Evaluate: -3/4 x 8/9 Simplify: 7 0 + 7 1 Solve: 20 ÷ (2 + 3) x 2 - 4 Represent -1.5 and 2.75 on a number line.