Number Skills in Context

Computation and Financial Maths: Number Skills in Context

• Number skills are essential for success in algebra and in many university and college courses
• Number skills are important in professions, trades, and for home renovations
• Jewelers use ratios to mix gold with other metals to to strengthen it.
  • For example, mixing gold, silver, and copper at a ratio of 15:17:8.
• Medical scientists use ratio skills in hospital pathology labs to prepare reagents and solutions for analyzing blood samples.
• Nurses use percentages, fractions, and rates to calculate medication dosages.
• Chefs calculate with decimals, fractions, ratios, and percentages in recipes and temperature conversions.
• Farmers use rate calculations to manage water usage, such as pump flow rates in L/s, drip irrigation rates in L/h, and irrigator rates in acres/h.
• Daily life applications of number skills include calculating weekly pay increases after tax and calculating a car’s price increase including interest.

Chapter 1 Contents and NSW Syllabus

• Chapter 1 covers computations with integers, decimal places, significant figures, rational numbers, fractions, ratios, rates, best buys, percentages, money, income, PAYG tax, simple interest, compound interest, and depreciation.
• Students develop understanding and fluency in mathematics by exploring and connecting concepts, applying techniques, and communicating clearly.
• Students solve financial problems involving simple interest, earning, and spending (MA5-FIN-C-01).
• Students solve financial problems involving compound interest and depreciation (MA5-FIN-C-02).

Section 1A: Computations with Integers

• The learning intentions of this section are learning the mathematical operations applied to negative numbers and notation for powers and roots and applying operations of order of operations
• Intends to consolidate Stage 4 concepts used in Stages 5 and 6.
• Applies learning to algebraic expressions that expertise with integers is assumed knowledge for Stage 6 Advanced and may also be required in non-calculator examinations such as NAPLAN and industry aptitude tests
• Mathematicians developed number systems, Egyptians using hieroglyphics for whole numbers and fractions, Babylonians using a base-60 place-value system, and ancient Chinese and Indians developing systems using negative numbers.
• Positive and negative numbers, fractions (rational numbers), and non-fractional numbers (irrational numbers, like π and √2) are included in the base 10 decimal system (Hindu-Arabic system).
• Real numbers are all the numbers in the number system, while imaginary numbers are excluded.

Key Ideas Involving Integers

• Integers consist of all whole numbers and their negatives, including zero.
• Rules for negative numbers in addition: a + (-b) = a - b
• Rules for negative numbers in subtraction: a - (-b) = a + b
• Rules for negative numbers in multiplication: a × (-b) = −ab and -a × (-b) = ab
• Rules for negative numbers in division: a ÷ (-b) = -a/b and -a ÷ (-b) = a/b
• Squares and Cubes: a² = a × a, √a² = a (if a ≥ 0) and a³ = a × a × a, ³√a³ = a
• The lowest common multiple (LCM) is the smallest multiple shared by two numbers.
• The greatest common factor (HCF) of two numbers is the largest factor shared by them.
• Prime numbershave only two factors: 1 and the number itself.
• Composite number have more than two factors
• Order of operations: Deal with brackets first, then indices, then multiplication and division (left to right), then addition and subtraction (left to right)

Section 1B: Decimal Places and Significant Figures

• The learning intentions are consolidating decimal rounding, introducing significant figures, and rounding numbers to a specified number of significant figures.
• Section consolidates Stage 4 decimals and introduces Stage 5 significant figures; Section 6G extends significant figures to scientific notation that is assumed knowledge for Stage 6 Advanced and may be required in non-calculator examinations such as NAPLAN and industry aptitude tests
• Numbers are rounded to a level of accuracy, with decimals often rounded to a certain number of decimal places.
• Digits in science and engineering numbers hold significance, and numbers are rounded to within a specific number of significant figures, also known as "sig. fig." or "s.f.".
• When rounding to a required number of decimal places, the digit in the required place is located.
• When rounding to a required number of decimal places, the digit is left as is if the next digit is 4 or less, while it is increased by 1 if the next digit is 5 or more.
• To round a number to a required number of significant figures, the first non-zero digit is located, and then digits are counted from there. The number is rounded at the required digit, and non-significant digits to the left of a decimal point are replaced with zeros

Section 1C: Rational Numbers

• Section is mostly a revision of Stage 4 ideas, but may introduce the concept of irrational numbers with intention to understand the difference between both, convert fractions and decimals and use notations for recurring decimals and to compare factions
• Irrational numbers appear later in trigonometry. Expertise with fractions is assumed in Stage 6 Advanced courses non-calculator exams such as in NAPLAN and industry aptitude tests
• Pythagoras found some numbers couldn't be expressed as fractions or rational numbers in 500 BCE, these are irrational numbers which when written decimals continue forever and don't show a pattern
• To represent these kinds non-terminating numbers, special symbols √ and π are used with rational numbers can be expressed as a fraction
• A infinite decimal is one in which the decimal places continue indefinitely.
• Equivalent fractions have the same value.
• b is a proper fraction then a < b.
• A mixed numeral is written as a whole number plus a proper fraction.
• Fractions can be compared using a common denominator, found using the lowest common multiple (LCM). A dot or bar shows a pattern in recurring decimal number,

Section 1D: Computation with Fractions.

• Learning intentions*
• Understand that to add or subtract fractions a common denominator is required.
• Be able to add and subtract fractions, including those in mixed numeral form, by using a LCD.
• That to multiply fractions it is simpler to first cancel any common factors between numerators and denominators.
• Know that to divide a number by a fraction we multiply by the reciprocal of the fraction.
• Know to express mixed numerals as improper fractions before multiplying or dividing and to be able to multiply and divide fractions
• Past, present and future learning*
• This section consolidates Stage 4 concepts which are used in Stages 5 and 6 • Some of these questions are more challenging than those in our Stage 4 books • In Chapter 2 of this book, we apply this learning to algebraic fractions • Expertise with fractions is assumed knowledge for Stage 6 Advanced and may also be required in non-calculator examinations such as NAPLAN and industry aptitude tests
• When adding or subtracting fraction, you need to first convert fraction to an equivalent with have the same denominator: lowest common denominator (LCD).
• In multiplying fractions, multiply the numerators and denominators after converting mix numerals to improper and cancelling highest common factor (HCF)
• Reciprocal of a number multiplied by the number itself is equal to 1. and dividing is multiplying by its reciprocal
• Whole numbers can also be written using a denominator of 1.

Section 1E : Ratios, Rates and Best Buys

• Learning intentions* • To revise and understand use and applications of rates and ratios • Simplify ratios, recognize equivalent ratios and divide a given quantity in ratios • Simplify rates and recognize equivalent rates •Solve problems involving ratios and rates with the unitary method •Solve problems including speed, best buys and more
• Past, present and future learning* • This section consolidates Stage 4 concepts which are used in Stages 5 and 6 • Some of these questions are more challenging than those in our Stage 4 books • The Path Topic Variation and Rates of Change can be found in Chapter 7 of our Year 10 book • Expertise with ratios and rates is assumed knowledge for Stage 6 Advanced and may also be required in non-calculator examinations such as NAPLAN and industry aptitude tests
• Fractions, ratios and rates are used to compare quantities. Example could be to make a petrol of 2 parts oil to 25 parts petrol, which is shown in an oil to petrol ratio of 225
• Ratios are used to compare quantities with the same units.
• Ratio of a and b is is "a:b" and should stay in simplest whole numbers possible without common factors
• The unitary method finds the value for one part of the whole
• Rates can have more than just similar units
• A rate has one quantity usually compared to a single unit. Can be used for purchasing.

Section 1F: Percentages and Money.

• Intention to convert between fractions, decimals and percentages while being able to describe one quantity as another's percentage, find percentage, or solve to find initial amount of a percentage.
• Previously learnt Stage 4 concepts are consolidated and assumed knowledge in the Stage 6 Advanced tests
• Percentages are a number relative to 100. (Per centum in latin, or out of 100 in English)
• Number can be percentage by multiplying x 100 or made a decimal by dividing to 100
• A known percentage of a number uses Multiplication
• (Unitary Method) Use the unitary method or inverse operations when trying to find an origin number
• Make sure have the same units when expressing one amount into a different percentage

Section 1G: Percentage Increase and Decrease

• Learning intentions* To be able to calculate increase or decrease of given amount in percentage and vice versa
• Past, present and future learning:*

• This section consolidates Stage 4 concepts which are used in Stages 5 and 6 Some of these questions are more challenging than those in our Stage 4 books • These concepts are not revised in our Year 10 book • Expertise with percentages is assumed knowledge for Stage 6 Advanced and may also be required in non-calculator examinations such as NAPLAN and industry aptitude tests

• Percentages can describe an amount's change in amount
• Price of a car can describe increase of 5%, shirt might be market down for 30%- decreases come from the original amount calculated
• To increase percentage in percentage, you need to multiply its given percentage
• to decrease need to minus percentage from the given percentage
• To find a percentage change, you need to divide what u want too change/ orginal amount = X 100

Section 1H: Profits and Discounts

• The learning intentions are to understand profits, discounts and losses and apply this while solving problems.
• Stage 4 concepts are consolidated to be used in Stages 5 and 6
• Expertise in percentages are needed in Stage 6 Advanced NAPLAN tests
• Profits and discounts are used in finance
• Profit is the amount made from a sale and has "Profit = selling price - cost price" Mark up- is price added to sale- has "Selling Price= cost+ markup"
• Percentage loss or profit can be worked out from dividing the profit or loss from cost then multiplying to 100.
• Discount is an specific amount an item was market down
• "New price= original price- discount" and also "Discount= % discount*original price"

Section 1I: Income

• Learning intentions*

• To understand the different ways that worker can be paid

• Use rates and percentages to cal salaries, comissions

• To be able to make conversions between per hr,day , week,month and year

• Past, present and future lerning*

• This section introduces stage core topic called Financial Mathematics A. (no revision on 10yr)

• Most income come from paid work

• Many prof workers receive a fix annual salary

• Casual workers receive wage where paid out per hour

• Sales gets payed via weekly fee or % of commission from sales they make

• Have to pay living costs such as electricity, rent , groceries, tax etc.

Section 1J: Understanding the PAYG (Pay As You Go) Income Tax System

• To know and understand the meanings of gross income, net income and taxable income and the formula of it The Australian Taxation Office (ATO) collects taxes for the government to pay for resources to help society
• In Australia, the financial year starts in July and ends in June, in the following year
• The 2023’2024 Medicare levy costs 2% of our taxable income, but this can possibly lead to receive tax refund or be in tax liability.

Section 1K: Simple Interest

Key concepts of simple interest, its formula and its calculations:

• Simple interest is interest calculated only on the principal amount.
• Apply this formula: I = Prn where: • 𝐼 is the amount of simple interest (in $) • 𝑃 is the principal amount (in $). • 𝑟 is the interest rate per time period (expressed as a fraction or decimal). • 𝑛 is the time periods.
• Formula to calculate total invested after simple rate: A = P + I

Section 1LM: Compound Interes and Depreciation

• Compound Interest section consolidates with Mathematics B and helps to form new formulas in this section
• Also look at appreciation which looks at the total amount the money lost at the end of the year
• A = P(1+r)^n and must minus interest rate (p) with the total money (amount) and its (rate) percentage all to its (num of periods or compounding terms
• And know that "amount= final amount-principal" is the formula for this total rate