# IGCSE Higher: Standard Form

## Scheme of work: IGCSE Higher: Year 10: Term 4: Standard Form

#### Prerequisite Knowledge

1. Understanding Powers of Ten
• Concept: Knowing how to express large and small numbers using powers of ten.
• Example: $10^3 = 1000, \quad 10^{-2} = 0.01$
2. Writing Numbers in Standard Form
• Concept: Converting numbers to and from standard form, where a number is written as $$a \times 10^n$$ with $$1 \leq a < 10$$ and $$n$$ an integer.
• Example: $4500 = 4.5 \times 10^3, \quad 0.0032 = 3.2 \times 10^{-3}$

#### Success Criteria

1. Add and Subtract Numbers in Standard Form
• Objective: Students should be able to add and subtract numbers in standard form with the same or different powers of ten.
• Example: Add $$2.5 \times 10^3$$ and $$3.7 \times 10^2$$: $2.5 \times 10^3 + 0.37 \times 10^3 = (2.5 + 0.37) \times 10^3 = 2.87 \times 10^3$
2. Multiply and Divide Numbers in Standard Form
• Objective: Students should be able to multiply and divide numbers in standard form correctly.
• Example: Multiply $$4 \times 10^5$$ by $$2 \times 10^3$$: $(4 \times 10^5) \times (2 \times 10^3) = 8 \times 10^{5+3} = 8 \times 10^8$ Divide $$6 \times 10^7$$ by $$3 \times 10^2$$: $\frac{6 \times 10^7}{3 \times 10^2} = 2 \times 10^{7-2} = 2 \times 10^5$
3. Solve Real-Life Problems Using Standard Form
• Objective: Students should be able to apply standard form to solve real-life problems involving very large or very small numbers.
• Example: Calculate the distance traveled by light in one year (light year) given that the speed of light is $$3 \times 10^8$$ m/s and there are $$3.15 \times 10^7$$ seconds in a year: $\text{Distance} = 3 \times 10^8 \times 3.15 \times 10^7 = 9.45 \times 10^{15} \text{ meters}$

#### Key Concepts

1. Adding and Subtracting in Standard Form
• Concept: To add or subtract numbers in standard form, the powers of ten must be the same. If not, convert them so they are the same.
• Example: Add $$2.5 \times 10^3$$ and $$3.7 \times 10^2$$: $2.5 \times 10^3 + 0.37 \times 10^3 = (2.5 + 0.37) \times 10^3 = 2.87 \times 10^3$
2. Multiplying and Dividing in Standard Form
• Concept: To multiply or divide numbers in standard form, multiply or divide the coefficients and add or subtract the exponents of the powers of ten.
• Example: Multiply $$4 \times 10^5$$ by $$2 \times 10^3$$: $(4 \times 10^5) \times (2 \times 10^3) = 8 \times 10^{5+3} = 8 \times 10^8$ Divide $$6 \times 10^7$$ by $$3 \times 10^2$$: $\frac{6 \times 10^7}{3 \times 10^2} = 2 \times 10^{7-2} = 2 \times 10^5$
3. Solving Real-Life Problems Using Standard Form
• Concept: Applying standard form to solve problems involving very large or very small numbers in practical contexts.
• Example: Calculate the distance traveled by light in one year (light year) given that the speed of light is $$3 \times 10^8$$ m/s and there are $$3.15 \times 10^7$$ seconds in a year: $\text{Distance} = 3 \times 10^8 \times 3.15 \times 10^7 = 9.45 \times 10^{15} \text{ meters}$

#### Common Misconceptions

1. Errors in Adding and Subtracting
• Common Mistake: Students might add or subtract the coefficients without adjusting the exponents to be the same.
• Example: Adding $$2.5 \times 10^3$$ and $$3.7 \times 10^2$$ directly without converting: $\text{Incorrect: } 2.5 \times 10^3 + 3.7 \times 10^2 = 6.2 \times 10^3 \\ \text{Correct: } 2.5 \times 10^3 + 0.37 \times 10^3 = 2.87 \times 10^3$
2. Errors in Multiplying and Dividing
• Common Mistake: Students might incorrectly multiply or divide the coefficients or incorrectly add or subtract the exponents.
• Example: Multiplying $$4 \times 10^5$$ by $$2 \times 10^3$$ incorrectly: $\text{Incorrect: } (4 \times 10^5) \times (2 \times 10^3) = 8 \times 10^{5+3} = 8 \times 10^{15} \\ \text{Correct: } (4 \times 10^5) \times (2 \times 10^3) = 8 \times 10^{5+3} = 8 \times 10^8$
3. Incorrect Conversion to Standard Form
• Common Mistake: Students might incorrectly convert numbers to standard form by not ensuring the coefficient is between 1 and 10.
• Example: Incorrectly writing 4500 as $$45 \times 10^2$$ instead of $$4.5 \times 10^3$$.

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