Number Skills in Context

Number Skills in Context

• Number skills are essential for algebra, university, college, professions, trades, and home renovation success.
• Jewellers mix gold with metals in ratios such as 15:17:8 (gold:silver:copper) for strength.
• Medical scientists in pathology labs use ratio skills when preparing reagents.
• Nurses use percentages, fractions, and rates to calculate medication dosages.
• Chefs use decimals, percentages, and ratios for recipe scaling and temperature conversions.
• Farmers utilize rate calculations such as pump flow, drip irrigation, and travelling irrigator rates to efficiently manage water.
• Number skills enable calculation of weekly pay increases after tax and car price increases including interest.

Chapter Contents Summary

• 1A Computations with integers (CONSOLIDATING): Working with integers.
• 1B Decimal places and significant figures: Rounding decimals and figures.
• 1C Rational numbers (CONSOLIDATING): Simplifying rationals
• 1D Computation with fractions (CONSOLIDATING): Working with fractions.
• 1E Ratios, rates and best buys (CONSOLIDATING): Comparing rates
• 1F Percentages and money (CONSOLIDATING): Calculations wit percentages
• 1G Percentage increase and decrease (CONSOLIDATING): Performing calculations with percentages
• 1H Profits and discounts (CONSOLIDATING): Making profits and discounts using calculations
• 1I Income: Understanding income.
• 1J The PAYG income tax system: Calculating income tax.
• 1K Simple interest: Working with simple interest
• 1L Compound interest and depreciation: Calculating compound interest and depreciation
• 1M Using a formula for compound interest and depreciation: Utilizing a formula for simple interest

NSW Syllabus Covered

• Develop mathematical skills through exploring concepts and applying techniques to solve problems, while communicating reasoning.
• Solve financial problems related to simple interest, earning, and spending (MA5-FIN-C-01).
• Tackle financial problems involving compound interest and depreciation (MA5-FIN-C-02).

Online Resources

• Interactive textbook includes HOTmaths content, video demonstrations and auto-marked quizzes.

1A: Computations with Integers

• Learning intentions include knowing rules for operations with negative numbers, powers, roots and order of operations.
• Math knowledge is consolidated to be used in later stages
• Egyptian hieroglyphics recorded whole numbers and fractions.
• Babylonians used a base-60 place-value system.
• Ancient Chinese and Indians used negative numbers.
• The current base-10 decimal system includes positive/negative numbers, fractions, and irrational numbers.
• Real numbers include all numbers except imaginary numbers.

Key Ideas - Integers

• Integers: ..., -3, -2, -1, 0, 1, 2, 3, ...
• a + (-b) = a - b (e.g., 5 + (-2) = 3)
• a - (-b) = a + b (e.g., 5 - (-2) = 7)
• a × (-b) = -ab (e.g., 3 × (-2) = -6)
• -a × (-b) = ab (e.g., -4 × (-3) = 12)
• a ÷ (-b) = -a/b (e.g., 8 ÷ (-4) = -2)
• -a ÷ (-b) = a/b (e.g., -8 ÷ (-4) = 2)
• a² = a × a, √a² = a (if a ≥ 0) (e.g., 6² = 36, √36 = 6)
• a³ = a × a × a, √a³ = a (e.g., 4³ = 64, √64 = 4)
• Lowest Common Multiple (LCM): The smallest multiple shared by two numbers. LCM of 6 and 9 is 18.
• Highest Common Factor (HCF): The largest factor shared by two numbers. HCF of 24 and 30 is 6.
• Prime numbers have only two factors: 1 and itself. The number 1 is not considered prime.
• Composite numbers have more than two factors.
• Order of Operations: Brackets, Indices, Multiplication/Division (left to right), Addition/Subtraction (left to right).

1B: Decimal Places and Significant Figures

• Numbers are rounded based on required accuracy levels.
• Decimal places used to round off numbers, like timing a 100 m sprint in 9.94 seconds.
• Civil engineers are able to design a road cutting accurately with calculations to two or three significant figures.

Key Ideas - Rounding

• To round to a required decimal place, locate the digit and consider the next (critical) digit.
• Round down if the critical digit is 4 or less; round up if 5 or more.
• Ex: 1.543 rounds to 1.54, and 32.9283 rounds to 32.93 (two decimal places).

Key Ideas - Signifigant Figures

• To round to a required number of significant figures:
  • Locate the first non-zero digit.
  • Count the required number of digits including zeros.
  • Stop and round the last digit; replace non-significant digits to the left of the decimal with zeros.
• Ex: 2.5391 ≈ 2.54, 0.002713 ≈ 0.00271, and 568810 ≈ 569000 (all rounded to three significant figures

1C: Rational Numbers

• Around 500 BCE, it was found that irrational numbers lacked fraction expression, needing symbols such as the square root and π.
• Rational numbers have terminating or repeating decimal patterns for representation as fractions.

Key Ideas - Rational Numbers

• Infinite vs. Terminating vs. Recurring Decimals:
  • The first continues indefinitely with no pattern, the second comes to an end in a division, the third repeats a pattern.
• Equivalent fractions: Have equal values
• Simplifying: the numerator/denominator are divided by their highest common factor.
• Proper vs. Improper vs. Mixed Fractions
  • Proper a<b,
  • Improper a≥b.
  • Mixed comprised of a whole number plus a proper fraction.
• Comparing Fractions: Compare using common denominators.
• Recurring Decimals: Are indicated with dot or bar usage.

1D: Computation with Fractions

• To add or subtract fractions, establish equivalent fractions through like denominators
• Multiplication of fractions involves improper conversions and canceling for calculation ease, as well as multiplying by a reciprocal

Key Ideas - Fractions

• Equivalent Fractions: First convert each fraction to an equivalent fraction that has the same denominator.
• Then choose the lowest common denominator (LCD) and add or subtract the numerators, and retain the denominator
• Convert mixed numerals to improper fraction before multiplying and cancel the highest common factor between numerator and denominator before multiplying
• Divide a number by a fraction by muliplting the number by reciprocals
• To divide a number by a fraction, multiply by its reciprocal. For example: 3 ÷ 5/6 becomes 3 × 6/5.
• Whole Numbers: Can be written by a denominator of 1 so whole numbers can be turned into fractions. For example: 3 = 3/1

1E: Ratios, Rates, and Best Buys

• Fractions, ratios, and rates compare quantities.
• Leaf blowers might require 1/27 litres of oil
• Math is used to calculate ratios such a Fan speed in 1000 revolutions/min.
• Fuel efficiency is a rate, usually given in liters per 100km

Key Ideas - Rates, Rations and Best Buys

• Comparisons: Ratios with the same units.
• Ratios in simplest form: Whole number without common factor
• Unitary Method: Determines a part's value of a total
• Rate: compares related units with different quantities
• Rate: expressed with 1 quantity per different unit as 50 km/h
• Ratios and Best Buy: Rates are used to determine the best buy when purchasing products.

1F: Percentages and Money

• Percentages are a number against 100.
• "Per cent" originates Latin "per centum", meaning "out of 100."

Key Ideas - Percentages

• A number by a percentage by multiplicatio.
• For percentages, divide by 100 by.
• To express a number as a fraction, multiply by dividing by a decimal.
• Use the unitary method for percentages for unit of division.

1G: Percentage Increase and Decrease

• Percentages: Used for quantitative descriptions
• Increase or decrease is calculated on the original amount.

Key Ideas - Percentages Pt2

• Increasing by A Percentage : Amount × 100%
• Decreasing by Given Percentage: Amount × 100% - Given Percentage

1H: Profits and Discounts

• Profits and Losses: calculated using percentages

Key Ideas : Profit Discount and Losses

• Profit: Total amount of money made on the sale
• Mark-ip: Amount to the cost price selling by mark-up
• Percentage Profit: By dividing profit and losses by the cost price from 100 or loss times
• Discount: Divided by how much is market down amount times discount/ original price

1I: Tax and Salaries

• Salaries is used depending on the rate of work and is payed using the number to which many are paid
• Tax is payed to the government dependig on person taxable/ income

Key Ideas- Tax and Salaies

• Workers earned a wage and a salarie
• Wage Salary is dependengt on fix rated
• Salaries are at a fixed yearly amount paid each month
• Some are paid through a bonus like a leave loading
• Income is people earnings and can have net amount and gross amount which can take out all tax

1K: Simple Interest

• Banks offer investments at a rate per annum, as well as loans .
• Loans all increase in price due to interest
• In simple rates a percentage of borrowers are calculated

Key Ideas- Simple Interest

• The principal account is constant even through interest
• The formula to find the value as A= P plus 1, as p is the principal investment i is for an amount loaned per, as P initialy number one times period .