Lesson Notes By Weeks and Term v5 - Grade 9

Lesson Video

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Performance objectives

• Classify numbers as rational or irrational.
• Simplify expressions using the laws of exponents.
• Express numbers in scientific notation and perform calculations.
• Solve problems in various contexts.
• Convert between standard and scientific notation.

Lesson summary

This week, we delve deeper into the fascinating world of real numbers, exponents, and scientific notation. These concepts are not just abstract mathematical ideas; they are fundamental tools used to understand and describe the world around us. From calculating interest rates on loans to understanding population growth and even comprehending the vast distances in space, these skills are essential. In South Africa, with its diverse economy and growing technological sector, a strong understanding of these concepts is crucial for future success in fields ranging from finance and engineering to science and technology.

Lesson notes

2.1 Real Numbers: A Review Real numbers encompass all rational and irrational numbers.

Rational Numbers: Numbers that can be expressed as a fraction p/q, where p and q are integers and q ≠

0. Examples include 2/3, -5, 0.75 (which is 3/4), and the square root of 9 (which is 3). Terminating and recurring decimals are also rational. For example, 0.333... (recurring) is the same as 1/

3. Irrational Numbers: Numbers that cannot be expressed as a fraction p/q. They have non-terminating, non-recurring decimal representations. Examples include √2, π (pi), and e. Important

Note: Understanding the difference between recurring and non-recurring decimals is CRUCIA

L. Recurring Decimals: Repeat in a pattern (e.g. 0.666...) and are Rational.

Non-Recurring Decimals: Do not repeat in a pattern (e.g., 3.14159265...) and are Irrational.

Example 1: Classify the following numbers as rational or irrational: a) 5 b) -3/7 c) √5 d) 0.121212... e) 0.123456789...

Solution: a) Rational (can be written as 5/1) b) Rational (already a fraction) c) Irrational (√5 is approximately 2.236..., a non-recurring, non-terminating decimal) d) Rational (recurring decimal) e) Irrational (non-recurring, non-terminating decimal) 2.2 Exponents: Powering Up Exponents (or indices) indicate how many times a base number is multiplied by itself. The laws of exponents provide rules for simplifying expressions involving exponents.

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

Quotient of Powers: a m / a n = a m-n (When dividing powers with the same base, subtract the exponents.)

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

Power of a Product: (ab) n = a n b n (When raising a product to a power, raise each factor to the power.)

Power of a Quotient: (a/b) n = a n /b n (When raising a quotient to a power, raise both the numerator and denominator to the power.)

Zero Exponent: a 0 = 1 (Any non-zero number raised to the power of 0 equals 1.)

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

Important Notes: Pay special attention to the order of operations (BODMAS/PEMDAS) when dealing with exponents in more complex expressions. Be careful with negative signs; ensure you apply the correct rule to the entire term.

Example 2: Simplify the following expressions using the laws of exponents: a) 2 3 * 2 2 b) 5 5 / 5 2 c) (3 2 ) 3 d) (2x) 3 e) 4 -2 Solution: a) 2 3 2 2 = 2 3+2 = 2 5 = 32 b) 5 5 / 5 2 = 5 5-2 = 5 3 = 125 c) (3 2 ) 3 = 3 23 = 3 6 = 729 d) (2x) 3 = 2 3 x 3 = 8x 3 e) 4 -2 = 1/4 2 = 1/16 2.3 Scientific Notation: Handling Big and Small Numbers Scientific notation is a way of expressing very large or very small numbers in a compact and convenient form. A number in scientific notation is written as: a x 10 n where: 1 ≤ |a| 6 b) 0.000047 = 4.7 x 10 -5 c) 123.45 = 1.2345 x 10 2 d) 0.5 = 5 x 10 -1 Example 4: Express the following numbers in standard form: a) 3.2 x 10 4 b) 8.1 x 10 -3 Solution: a) 3.2 x 10 4 = 32,000 b) 8.1 x 10 -3 = 0.0081 Calculations with Scientific Notation: When multiplying or dividing numbers in scientific notation, multiply or divide the 'a' values and add or subtract the exponents, respectively.

Example 5: Calculate (2 x 10 3 ) * (3 x 10 4 )

Solution: (2 3) x 10 (3+4) = 6 x 10 7 Example 6: Calculate (8 x 10 6 ) / (4 x 10 2 )

Solution: (8 / 4) x 10 (6-2) = 2 x 10 4 Guided Practice (With Solutions)

Question 1: Simplify: (3x 2 y -1 ) 2 Solution: Apply the power of a product rule: (3) 2 (x 2 ) 2 (y -1 ) 2 Simplify: 9x 4 y -2 Rewrite with positive exponents: 9x 4 / y 2

Commentary: We used the power of a product rule and the power of a power rule. We then converted the negative exponent to a positive exponent by taking the reciprocal.

Question 2: Express 0.00000015 in scientific notation.

Solution: Move the decimal point 7 places to the right to get 1.

5. Since we moved the decimal to the right, the exponent is negative.

Therefore, 0.00000015 = 1.5 x 10 -7

Commentary: We are making a small number larger, so the exponent must be negative to compensate.

Question 3: Simplify: (5 x 10 5 ) / (2.5 x 10 2 )

Solution: Divide the coefficients: 5 / 2.5 = 2 Subtract the exponents: 10 5 / 10 2 = 10 (5-2) = 10 3 Therefore, (5 x 10 5 ) / (2.5 x 10 2 ) = 2 x 10 3

Commentary: When dividing numbers in scientific notation, we divide the 'a' values and subtract the exponents of

1

0. Question 4: Which of the following is an irrational number: 22/7 or π? Explain your answer.

Solution: π is irrational. 22/7 is a rational number because it can be expressed as a fraction of two integers. π is a non-terminating, non-recurring decimal; therefore it cannot be expressed as a fraction of two integers.

Commentary: This tests understanding of the definition of rational and irrational numbers. Emphasize that 22/7 is an approximation of π, not equal.