MYP 2 Unit 1: Numerical and Abstract Reasoning - Rational Numbers, Ratios, and Percentages

Unit 1 Overview: Numerical and Abstract Reasoning

• Grade Level: MYP 2.

• Key Concept: Relationship.

• Related Concepts:

  • Patterns.

  • Approximation.

  • Simplification.

• Global Context: Globalization and sustainability.

• Exploration Areas: Markets, commodities, and commercialization; commonality, diversity, and interconnection.

• Statement of Inquiry: Patterns found in markets, commodities, and commercialization help to understand the relationship between globalization and sustainability.

Real Numbers: Rational and Irrational Numbers

• Definition of Rational Numbers: A rational number is any number that can be represented in the form $\frac{p}{q}$ where $q \neq 0$.

• Conditions for Identification:

  • It is represented as $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$.

  • The ratio $\frac{p}{q}$ can be simplified and represented in decimal form.

• Key Properties of Rational Numbers:

  • Includes positive numbers, negative numbers, and zero.

  • Every integer is a rational number because any integer $a$ can be written as $\frac{a}{1}$. Examples include $1 = \frac{1}{1}$, $2 = \frac{2}{1}$, $-3 = \frac{-3}{1}$, and $0 = \frac{0}{1}$.

  • Not all rational numbers are integers. Examples: $\frac{3}{2}$ and $\frac{-5}{3}$.

  • Every fraction is a rational number. If $\frac{a}{b}$ is a fraction, then $a$ and $b$ are natural numbers (which are also integers), making the quotient a rational number.

  • Not all rational numbers are fractions. For example, $\frac{2}{-3}$ is a rational number but technically not a fraction because its denominator is not a natural number ($-3$ is an integer but not a natural number).

  • Every mixed fraction is a rational number because it can be expressed as an improper fraction (a quotient of two integers).

Rational Numbers as Decimals

• Classification: Any rational number can be written as either a terminating decimal or a recurring (repeating) decimal.

• Terminating Decimals: These occurs when the division of the numerator by the denominator results in a remainder of $0$.

  • Example 1: Convert $\frac{5}{8}$ to a decimal. Calculation: $\frac{5}{8} = 0.625$. This is a terminating decimal.

• Recurring (Repeating) Decimals: These occurs when the remainders begin to repeat after some point.

  • Example 2: Is $0.15151515...$ a rational number? Yes, because it is non-terminating but recurring.

  • Example 3: Convert $\frac{1}{3}$ to a decimal. Calculation: $\frac{1}{3} = 0.333333...$. This is a recurring decimal.

Irrational Numbers

• Definition: Some numbers cannot be written as a ratio of two integers. These are called Irrational Numbers. Their decimal representation goes on forever without repeating a pattern.

• Famous Irrational Numbers:

  • Pi (\pi): Calculated to over a quadrillion decimal places with no pattern. First digits: $3.1415926535897932384626433832795...$

  • Euler's Number (e): First digits: $2.7182818284590452353602874713527...$

  • Golden Ratio: First digits: $1.61803398874989484820...$

• Other Examples:

  • Many square roots and cube roots: $\sqrt{3} = 1.7320508075688772935274463415059...$

  • $\sqrt{99} = 9.9498743710661995473447982100121...$

  • Patterned but non-repeating decimals: $3.01001000100001...$ and $-2.34344344434444...$

Arithmetic Operations on Rational Numbers

• Addition (Same Denominators): To add rational numbers like $\frac{2}{9}$ and $\frac{3}{9}$, add the numerators and keep the common denominator: $\frac{2+3}{9} = \frac{5}{9}$.

• Addition (Different Denominators):

  • Find the Least Common Multiple (LCM) of the denominators.

  • Convert rational numbers to equivalent fractions with the LCM as the denominator.

  • Add the numerators. Example: To add $\frac{5}{6}$ and $\frac{3}{9}$, identify the LCM of $6$ and $9$ as $18$.

• Subtraction: Uses the same method as addition.

  • Example: Subtract $\frac{3}{4}$ from $\frac{5}{6}$.

  • $LCM(6, 4) = 12$.

  • Convert: $\frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}$.

  • Convert: $\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}$.

  • Calculate: $\frac{10-9}{12} = \frac{1}{12}$.

• Multiplication:

  • The product equals simple multiplication: $\frac{\text{product of numerators}}{\text{product of denominators}}$.

  • To multiply a rational number by an integer, multiply the integer with the numerator and keep the denominator unchanged.

  • Example: Multiply $\frac{3}{8}$ and $\frac{-9}{11}$. Result: $\frac{3 \times (-9)}{8 \times 11} = \frac{-27}{88}$.

• Division:

  • Involves the concept of the reciprocal (swapping the numerator and denominator).

  • The product of a number and its reciprocal is always $1$. Example: $\frac{7}{6}$ is the reciprocal of $\frac{6}{7}$.

  • Procedure: Change the division sign $\div$ to multiplication $\times$, then take the reciprocal of the number following the sign.

  • Example: Divide $\frac{11}{12}$ by $\frac{7}{4}$.

  • Reciprocal of $\frac{7}{4}$ is $\frac{4}{7}$.

  • Calculate: $\frac{11}{12} \times \frac{4}{7} = \frac{44}{84}$, which simplifies to $\frac{11}{21}$.

Ratio and Proportion

• Ratio: Comparison of two quantities of the same kind, expressed as a fraction or quotient. A ratio of $x$ to $y$ is denoted as $x : y$.

• Equivalent Ratios: Formed by multiplying or dividing both terms by the same non-zero number.

  • Example: $1 : 3 = 2 : 6 = 3 : 9$.

  • Example: $4 : 5 = 12 : 15 = 16 : 20$.

• Proportion: An equation stating that two ratios are equivalent ($a : b = c : d$ or $a : b :: c : d$).

  • Cross-Product Property: When four terms are in proportion, the product of the extremes ($1^{\text{st}}$ and $4^{\text{th}}$ terms) equals the product of the means ($2^{\text{nd}}$ and $3^{\text{rd}}$ terms).

  • $ad = bc$.

  • Example: Prove $16 : 12$ and $4 : 3$ are in proportion.

  • Means: $12 \times 4 = 48$.

  • Extremes: $16 \times 3 = 48$. Since $48 = 48$, they are proportional.

  • Example: Find the missing number in $3 : 4 = 12 : a$.

  • $3 \times a = 4 \times 12 \rightarrow 3a = 48 \rightarrow a = \frac{48}{3} = 16$.

Percentage

• Definition: Derived from Latin 'per centum', meaning "per hundred" or "for every hundred." Symbol: $\%$.

• Finding Percent of a Number ($x$): $\frac{P}{100} \times x$.

  • Example 1: Find $40\%$ of $240$. Calculation: $\frac{40 \times 240}{100} = 96$.

  • Example 2: Find $10\%$ of $1\,\text{hour}$. Calculation: $1\,\text{hour} = 60\,\text{minutes}$. $10\% \text{ of } 60 = \frac{10}{100} \times 60 = 6\,\text{minutes}$.

• Conversions:

  • Percentage to Fraction: Remove $\%$ sign and divide the number by $100$, then reduce to lowest terms. Example: $5\% = \frac{5}{100} = \frac{1}{20}$. Example: $25\% = \frac{25}{100} = \frac{1}{4}$.

  • Fraction to Percentage: Multiply the fraction by $100$ and add the $\%$ symbol. Example: $\frac{x}{y} = (\frac{x}{y} \times 100)\%$.

  • Ratio to Percentage: Convert ratio to fraction, then multiply by $100$. Example: Ratio $8 : 25$ becomes $\frac{8}{25} \times 100 = 32\%$.

• Application Examples:

  • Example 3 (Seating): A hall has $600$ seats, $75\%$ are filled.

    • Filled: $\frac{75}{100} \times 600 = 450$ seats.

    • Vacant: $600 - 450 = 150$ or $25\% \text{ of } 600 = 150$.

  • Example 4 (Missing Sum): $15\%$ of a sum is $\$225$.

    • $\frac{15}{100} \times m = 225 \rightarrow m = \frac{225 \times 100}{15} = 1500$. Sum = $\$1500$.

  • Example 5 (Exam Marks): Ashley secured $332$ marks, representing $83\%$.

    • $\frac{83}{100} \times m = 332 \rightarrow m = \frac{332 \times 100}{83} = 400$. Maximum marks = $400$.

Real-World Scenarios and Comparison

• Terry's Stock Market:

  • Day 1 decrease: $0.5\%$.

  • Day 2 decrease: $0.45$.

  • Comparison: Convert $0.45$ to percentage: $0.45 \times 100 = 45\%$.

  • Comparison: $0.5\% < 45\%$. The second day (represented by $0.45$) had a significantly worse decrease.

• Inequality Comparison: Compare $0.56$ and $\frac{4}{5}$.

  • Convert $\frac{4}{5}$ to a fraction with common denominator or decimal. $\frac{4 \times 5}{5 \times 5} = \frac{20}{25}$ (Note: Transcript intended comparison through decimal/fraction conversion).

  • $0.56 < 0.80$ (since $\frac{4}{5} = 0.8$). Result: $0.56 < \frac{4}{5}$.

• Subway Car Statistics:

  • $45$ males and $60$ females in a car traveling $2.5\,\text{miles}$ in $5\,\text{minutes}$.

  • Ratio males to females: $45 : 60 = 3 : 4$.

  • Speed: $\frac{2.5\,\text{miles}}{5\,\text{minutes}} = 0.5\,\text{mile per minute}$.

• Turf Cost: Unit rate is calculated in dollars per square foot using a ratio table.

Resources and References

• Admin. "Rational Numbers - Definition, Types, Properties & Examples." BYJUS.

• Math Only Math. "Express Rational Numbers in Terminating and Non-Terminating Decimals: Examples."

• Foundation, CK-12. "12 Foundation." CK, Middle School Math Concepts Grade 8.

• Toppr. "Rational Numbers Examples: Operations, Concepts, Properties and Videos."

• Admin. "Arithmetic Operations on Rational Numbers with Examples." BYJUS.

• Bullseye. "Ratio and Proportion."

• Big Idea Math. "Ratio and proportion big idea math."

• Toppr. "Ratio And Percentage: Definitions, Formulas, Videos and Solved Examples."

• Math Only Math. "Ratio into Percentage: Ratios as Fraction Percentage: Ratios into Decimal Percent."

• Eduplace. "Grade 5: Fractions, Ratios, Rates, and Percents."

• BBC Bitesize. "Finding the percentage of a quantity - Percentages."