Lesson Notes By Weeks and Term v5 - Grade 8

Lesson Video

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Performance objectives

• Perform all arithmetic operations with integers.
• Convert between fractions, decimals, and percentages.
• Simplify expressions using the laws of exponents.
• Solve multi-step problems combining these concepts.
• Use scientific notation for very large or small numbers.

Lesson summary

Integers, rational numbers, and exponents form the foundation for much of higher-level mathematics. Understanding these concepts is crucial for success in algebra, geometry, and beyond. Mastering these skills will not only help you in your mathematics studies but also equip you with valuable problem-solving skills applicable in various aspects of life, from budgeting and managing finances to understanding statistical data and interpreting scientific findings. Think about how understanding exponents helps calculate population growth, or how rational numbers are used when sharing costs and expenses with friends.

Lesson notes

Integers: Integers are whole numbers (not fractions) that can be positive, negative, or zero.

Examples: ..., -3, -2, -1, 0, 1, 2, 3, ...

Addition: When adding integers with the same sign, add their absolute values and keep the sign.

Example: -3 + (-5) = -

8. When adding integers with different signs, subtract the smaller absolute value from the larger absolute value and keep the sign of the number with the larger absolute value.

Example: -7 + 4 = -

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

Example: 5 - (-2) = 5 + 2 =

7. Multiplication and Division: When multiplying or dividing integers with the same sign, the result is positive.

Example: -4 x -3 = 12 and 10 / 2 =

5. When multiplying or dividing integers with different signs, the result is negative.

Example: -6 x 2 = -12 and 15 / -3 = -

5. 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. They include fractions, decimals (terminating or repeating), and integers.

Fractions: A fraction represents a part of a whole. The top number (numerator) indicates how many parts we have, and the bottom number (denominator) indicates the total number of parts the whole is divided into.

Decimals: Decimals are another way to represent fractions. Terminating decimals end after a finite number of digits (e.g., 0.25). Repeating decimals have a pattern of digits that repeats infinitely (e.g., 0.333...).

Percentages: Percentages are fractions out of

1

0

0. To convert a fraction to a percentage, multiply by

1

0

0. Example: 1/4 = (1/4) x 100% = 25%. To convert a decimal to a percentage, multiply by

1

0

0. Example: 0.75 = 0.75 x 100% = 75%.

Exponents: An exponent indicates how many times a base number is multiplied by itself. For example, 2 3 means 2 x 2 x 2 =

8. The '2' is the base, and the '3' is the exponent.

Laws of Exponents: Product of Powers:* a m x a n = a m+n (When multiplying powers with the same base, add the exponents).

Example: 2 2 x 2 3 = 2 2+3 = 2 5 =

3

2. 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 =

2

7. Power of a Power:* (a m ) n = a m x n (When raising a power to another power, multiply the exponents).

Example: (4 2 ) 3 = 4 2 x 3 = 4 6 =

4

0

9

6. Zero Exponent:* a 0 = 1 (Any number raised to the power of 0 equals 1, except for 0 itself).

Example: 5 0 =

1. Negative Exponent:* a -n = 1/a n (A negative exponent indicates the reciprocal of the base raised to the positive exponent).

Example: 2 -3 = 1/2 3 = 1/

8. Scientific Notation: Scientific notation is a way of expressing very large or very small numbers in a compact form. A number in scientific notation is written as a x 10 n , where 1 ≤ |a| 6 .

Example: 0.00005 can be written as 5 x 10 -5 .

Worked example

Example 1 (Integers): A weather station in Sutherland recorded a temperature of -5°C at night and 12°C during the day. What is the temperature difference between night and day?

Solution: 12 - (-5) = 12 + 5 = 17°C. The temperature difference is 17°

C. Example 2 (Rational Numbers): A shop sells a loaf of bread for R12.

5

0. If the shopkeeper wants to increase the price by 20%, what is the new price?

Solution: 20% of R12.50 = (20/100) x 12.50 = 2.

5

0. New price = R12.50 + R2.50 = R15.

0

0. Example 3 (Exponents): Simplify the expression: (2 3 x 2 -1 ) / 2 2 .

Solution: 2 3 x 2 -1 = 2 3+(-1) = 2 2 . Then, 2 2 / 2 2 = 2 2-2 = 2 0 =

1. Example 4 (Scientific Notation): The population of South Africa is approximately 60 million. Express this number in scientific notation.

Solution: 60,000,000 = 6 x 10 7

Guided Practice (With Solutions)

Question 1: Calculate: -8 + 5 - (-3)

Solution: -8 + 5 - (-3) = -8 + 5 + 3 = -3 + 3 =

0. Commentary: We first convert the subtraction of a negative number into addition. Then, we proceed from left to right.

Question 2: Convert 3/8 to a decimal and a percentage.

Solution: 3/8 = 0.375. 0.375 x 100% = 37.5%.

Therefore, 3/8 = 0.375 = 37.5%.

Commentary: To convert a fraction to a decimal, divide the numerator by the denominator. To convert a decimal to a percentage, multiply by 100%.

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

Solution: (3 2 ) -1 = 3 2 x -1 = 3 -2 . Then, 3 -2 x 3 4 = 3 -2+4 = 3 2 =

9. Commentary: We apply the power of a power rule first, then the product of powers rule.

Question 4: A water tank contains 500 liters of water. 1/5 of the water is used. How many liters are left?

Solution: 1/5 of 500 liters = (1/5) 500 = 100 liters. Water left = 500 liters - 100 liters = 400 liters

Commentary: We first find the amount of water used by calculating the fraction of the total amount. Then we subtract the used amount from the total amount to find out the remaining amount.

Independent Practice (Questions Only)

Evaluate: -12 - (-6) + 4 x (-2)

Convert 0.625 to a fraction in simplest form.

Simplify: (5 -2 x 5 5 ) / 5 2

Express 0.0000045 in scientific notation.

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

A farmer owns 3/5 of a field. He plants maize on 1/2 of his land. What fraction of the entire field is planted with maize?

Calculate: (-3) 3 + (-2) 4

Express 56,000,000,000 in scientific notation.

If a phone costs R2500 and is discounted by 15%, what is the sale price?

Simplify: 7 0 + 7 -1