# IGCSE Mathematics Foundation: Standard Form

## Scheme of work: IGCSE Foundation: Year 11: Term 1: Standard Form

#### Prerequisite Knowledge

1. Knowing the Power Rules of Indices
• Concept: Understanding how to apply the power rules for indices, including multiplication, division, and raising powers to powers.
• Example: $a^m \times a^n = a^{m+n}$ $\frac{a^m}{a^n} = a^{m-n}$ $(a^m)^n = a^{mn}$ $\text{For example: } 2^3 \times 2^2 = 2^{3+2} = 2^5 = 32$ $\frac{2^5}{2^2} = 2^{5-2} = 2^3 = 8$ $(2^2)^3 = 2^{2 \times 3} = 2^6 = 64$
2. Multiplying and Dividing by Powers of Ten
• Concept: Understanding how to multiply and divide numbers by powers of ten.
• Example: $5 \times 10^3 = 5000$ $7.2 \div 10^2 = 0.072$

#### Success Criteria

1. Calculate with and Interpret Numbers in the Form $$a \times 10^n$$ where $$n$$ is an Integer and $$1 \leq a < 10$$
• Objective: Accurately perform calculations and interpret results involving numbers in standard form.
• Example: $(2 \times 10^3) \times (3 \times 10^2) = 6 \times 10^5$ $\frac{4 \times 10^6}{2 \times 10^3} = 2 \times 10^3$
2. Convert Between Numbers Written in Ordinary and Standard Form
• Objective: Convert numbers from ordinary form to standard form and vice versa.
• Example: $\text{Convert } 4500 \text{ to standard form: } 4.5 \times 10^3$ $\text{Convert } 3.2 \times 10^4 \text{ to ordinary form: } 32000$
3. Use Written Methods to Calculate with Numbers in Standard Form
• Objective: Apply written methods to perform calculations with numbers in standard form.
• Example: $\text{Multiply } (4 \times 10^2) \times (5 \times 10^3).$ $\text{Multiply the coefficients: } 4 \times 5 = 20.$ $\text{Add the exponents: } 10^2 \times 10^3 = 10^{2+3} = 10^5.$ $\text{Combine: } 20 \times 10^5 = 2 \times 10^6.$
4. Use Calculator Methods to Calculate with Numbers in Standard Form
• Objective: Use a scientific calculator to perform calculations with numbers in standard form.
• Example: $\text{Calculate } (6 \times 10^4) \div (2 \times 10^2) \text{ using a calculator}.$ $\text{Input } 6 \times 10^4 \div 2 \times 10^2 \text{ and obtain the result } 3 \times 10^2.$

#### Key Concepts

1. Calculate with and Interpret Numbers in the Form $$a \times 10^n$$ where $$n$$ is an Integer and $$1 \leq a < 10$$
• Concept: Standard form is a way of writing very large or very small numbers using powers of ten, which helps in simplifying calculations and understanding scales.
• Example: $\text{For } (2 \times 10^3) \times (3 \times 10^2), \text{ multiply the coefficients: } 2 \times 3 = 6.$ $\text{Add the exponents: } 10^3 \times 10^2 = 10^{3+2} = 10^5.$ $\text{Result: } 6 \times 10^5.$
2. Convert Between Numbers Written in Ordinary and Standard Form
• Concept: Converting between ordinary and standard form involves understanding the place value of digits and shifting the decimal point appropriately.
• Example: $\text{To convert } 4500 \text{ to standard form: } 4.5 \times 10^3.$ $\text{To convert } 3.2 \times 10^4 \text{ to ordinary form: } 32000.$
3. Use Written Methods to Calculate with Numbers in Standard Form
• Concept: Written methods for standard form calculations involve multiplying/dividing the coefficients and adding/subtracting the exponents.
• Example: $\text{Multiply } (4 \times 10^2) \times (5 \times 10^3).$ $\text{Multiply the coefficients: } 4 \times 5 = 20.$ $\text{Add the exponents: } 10^2 \times 10^3 = 10^{2+3} = 10^5.$ $\text{Combine: } 20 \times 10^5 = 2 \times 10^6.$
4. Use Calculator Methods to Calculate with Numbers in Standard Form
• Concept: Scientific calculators have specific functions for handling numbers in standard form, which help in performing accurate calculations.
• Example: $\text{To calculate } (6 \times 10^4) \div (2 \times 10^2) \text{ using a calculator}.$ $\text{Input: } 6 \times 10^4 \div 2 \times 10^2.$ $\text{Result: } 3 \times 10^2.$

#### Common Misconceptions

1. Calculate with and Interpret Numbers in the Form $$a \times 10^n$$ where $$n$$ is an Integer and $$1 \leq a < 10$$
• Common Mistake: Incorrectly adding or subtracting the exponents instead of multiplying or dividing them.
• Example: $\text{Incorrect: } (2 \times 10^3) \times (3 \times 10^2) = 6 \times 10^{3 \times 2} = 6 \times 10^6.$ $\text{Correct: } (2 \times 10^3) \times (3 \times 10^2) = 6 \times 10^{3+2} = 6 \times 10^5.$
2. Convert Between Numbers Written in Ordinary and Standard Form
• Common Mistake: Incorrectly shifting the decimal point when converting between forms.
• Example: $\text{Incorrect: Converting } 4500 \text{ to standard form as } 45 \times 10^2.$ $\text{Correct: Converting } 4500 \text{ to standard form as } 4.5 \times 10^3.$ $\text{Incorrect: Converting } 3.2 \times 10^4 \text{ to ordinary form as } 3200.$ $\text{Correct: Converting } 3.2 \times 10^4 \text{ to ordinary form as } 32000.$
3. Use Written Methods to Calculate with Numbers in Standard Form
• Common Mistake: Incorrectly multiplying or dividing the coefficients or misapplying the exponent rules.
• Example: $\text{Incorrect: } (4 \times 10^2) \times (5 \times 10^3) = 20 \times 10^6.$ $\text{Correct: Multiply the coefficients: } 4 \times 5 = 20.$ $\text{Add the exponents: } 10^2 \times 10^3 = 10^{2+3} = 10^5.$ $\text{Combine: } 20 \times 10^5 = 2 \times 10^6.$
4. Use Calculator Methods to Calculate with Numbers in Standard Form
• Common Mistake: Entering the numbers incorrectly into the calculator or not using the scientific notation function correctly.
• Example: $\text{Incorrect: Entering } 6 \times 10^4 \div 2 \times 10^2 \text{ as } 6 \times 10^4 \div 2 \times 10^2 \text{ without parentheses.}$ $\text{Correct: Using parentheses or scientific notation function: } \text{(} 6 \times 10^4 \text{)} \div \text{(} 2 \times 10^2 \text{)} \text{ or using the } EE \text{ or } EXP \text{ button on the calculator}.$ $\text{Result: } 3 \times 10^2.$

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