Lesson Notes By Weeks and Term v5 - Grade 8
Integers, rational numbers and exponents (Grade 8) – Week 3 focus
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Integers, rational numbers and exponents (Grade 8) – Week 3 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: 3
Theme: General lesson support
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This week, we delve deeper into the world of numbers by exploring integers, rational numbers, and exponents. Understanding these concepts is crucial for building a strong foundation in mathematics. These concepts are not just abstract ideas; they're used every day in various aspects of life, from managing your pocket money and understanding temperatures to calculating distances and interest rates. Imagine trying to understand load shedding schedules (integers representing time differences), figuring out percentages for sales at the local shop (rational numbers representing fractions), or calculating the area of your bedroom (exponents representing repeated multiplication).
2.1 Integers Integers are whole numbers (not fractions) that can be positive, negative, or zero.
Examples: -3, -2, -1, 0, 1, 2, 3... Why are integers important? They help us represent quantities like debt (negative numbers), altitude below sea level (negative), and temperature below zero (negative).
Operations with Integers: Addition: Adding two positive integers results in a positive integer.
Example: 3 + 5 = 8 Adding two negative integers results in a negative integer.
Example: -2 + (-4) = -6 Adding a positive and a negative integer: Subtract the smaller absolute value from the larger absolute value. The result takes the sign of the integer with the larger absolute value.
Example: -7 + 3 = -4 (7-3=4, and 7 has a negative sign), 5 + (-2) = 3 (5-2=3, and 5 has a positive sign).
Subtraction: Subtracting an integer is the same as adding its opposite.
Example: 5 - 3 = 5 + (-3) = 2; 2 - (-4) = 2 + 4 = 6 Multiplication: Positive x Positive = Positive.
Example: 2 x 3 = 6 Negative x Negative = Positive.
Example: -2 x -3 = 6 Positive x Negative = Negative.
Example: 2 x -3 = -6 Negative x Positive = Negative.
Example: -2 x 3 = -6 Division: The rules for division are the same as for multiplication. Positive / Positive = Positive. Negative / Negative = Positive. Positive / Negative = Negative. Negative / Positive = Negative.
Example 1: Calculating a Bank Balance Sipho has R50 in his bank account. He spends R75 on airtime. What is his new balance?
Initial balance: R50 Spending: R75 (represented as -R75)
New balance: R50 + (-R75) = -R25 Sipho has a negative balance of R25, meaning he is overdrawn by R
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5. Example 2: Temperature Changes The temperature in Johannesburg is 25°C during the day. It drops by 30°C at night. What is the temperature at night?
Daytime temperature: 25°C Temperature drop: -30°C Night temperature: 25 + (-30) = -5°C The temperature at night is -5°C. 2.2 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 zero. This includes fractions, decimals (that terminate or repeat), and percentages. Why are rational numbers important? They allow us to represent parts of a whole, ratios, and proportions. Think about sharing a pizza (fractions), calculating discounts (percentages), or measuring ingredients in a recipe (decimals). Converting Between Fractions, Decimals, and Percentages: Fraction to Decimal: Divide the numerator (top number) by the denominator (bottom number).
Example: 1/4 = 0.25 Decimal to Fraction: Write the decimal as a fraction with a denominator of 10, 100, 1000, etc., depending on the number of decimal places. Simplify if possible.
Example: 0.75 = 75/100 = 3/4 Decimal to Percentage: Multiply the decimal by 100 and add the "%" symbol.
Example: 0.25 = 0.25 x 100% = 25% Percentage to Decimal: Divide the percentage by
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0. Example: 25% = 25/100 = 0.25 Fraction to Percentage: Convert the fraction to a decimal first, then multiply by 100%.
Example: 1/2 = 0.5 = 0.5 x 100% = 50% Example 3: Discount Calculation A pair of shoes costs R
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0. There is a 20% discount. What is the discounted price?
Discount percentage: 20% = 0.20 Discount amount: R400 x 0.20 = R80 Discounted price: R400 - R80 = R320 The discounted price of the shoes is R
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0. Example 4: Sharing a Pizza A pizza is cut into 8 slices. You eat 3 slices. What fraction of the pizza did you eat? What percentage?
Fraction of pizza eaten: 3/8 Decimal equivalent: 3/8 = 0.375 Percentage equivalent: 0.375 x 100% = 37.5% You ate 3/8 of the pizza, which is 37.5%. 2.3 Exponents An exponent (or power) indicates how many times a number (the base) is multiplied by itself. For example, in the expression 2 3 , 2 is the base, and 3 is the exponent. 2 3 means 2 x 2 x 2 =
8. Why are exponents important? They provide a concise way to express repeated multiplication, which is used in various fields like science (e.g., calculating the area of a square), finance (compound interest), and computer science. Laws of Exponents (for Grade 8, focusing on whole number exponents): Product of powers: 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 of powers: 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: (a m ) n = a mn (When raising a power to another power, multiply the exponents)
Example: (2 2 ) 3 = 2 2*3 = 2 6 = 64 Any number raised to the power of 0: a 0 = 1 (Except for a=0, which is undefined).
Example: 5 0 = 1 Example 5: Calculating Area A square has a side length of 5 cm. What is its area? Area of a square = side x side = side 2 Area = 5 2 = 5 x 5 = 25 cm 2 The area of the square is 25 cm 2 .
Example 6: Simplifying Exponential Expressions Simplify: (3 2 * 3 4 ) / 3 3 3 2 3 4 = 3 2+4 = 3 6 3 6 / 3 3 = 3 6-3 = 3 3 = 27 The simplified expression is 27.