Mathematics Skills II Honor

Science & Health

Requirements

1. Know the four basic operations.

   Answer: The 4 basic operations: addition (+), subtraction (-), multiplication (×) and division (÷). Addition sums values; subtraction takes away; multiplication is repeated addition. — Addition/subtraction were invented 5,000+ years ago. Multiplication is a shortcut for addition. Division is the inverse of multiplication. Calculators make it easier, but knowing how to do it mentally is essential for practical life.

2. Explain and present the history of the square root and solve two practical examples of root extraction.

   Answer: History: the square root emerged in Babylon (~1800 BC), with tablets that calculated the sides of squares from the area; the Egyptians and Greeks used it in geometry, and Pythagoras applied it to triangles. The symbol √ was created by Christoff Rudolff in 1525 (a distortion of the letter 'r' from radix, root). The square root is finding the number that, multiplied by itself, gives the given value. Practical examples: 1) √81 = 9, because 9 x 9 = 81. 2) √144 = 12, because 12 x 12 = 144. (To check: raise the result to the second power and see if it returns to the original number.) — The square root is the basis of geometry (calculating the side of a square from the area). Pythagoras used it for theorems. The Babylonian algorithm approximates non-exact roots. Modern calculators solve any root in seconds.

3. Present and solve two simple examples of exponentiation with integer numbers using positive and negative exponents.

   Answer: Exponentiation is multiplying the base by itself N times. Positive: 2³ = 2×2×2 = 8. Negative: 2⁻² = 1/2² = 1/4 = 0.25. — Scientific notation uses powers of 10 (10⁶ = 1,000,000). In computing, bytes are powers of 2 (2¹⁰ = 1024 = 1KB). A fractional exponent relates to roots (x^½ = √x). A fundamental concept in science.

4. Present, in the form of a drawing or collages, three practical examples in which we use negative and positive integers in our daily life.

   Answer: Negative numbers arose to represent debt and deficit. They were only accepted in the 18th century by European mathematicians. Today they are fundamental in science, finance, navigation. GPS uses negative coordinates for south/west.

5. Demonstrate the ability to solve a numerical expression involving negative and positive integers. Show two examples.

   Answer: Solve expressions with the rules of signs. Ex 1: (-3) + 5 = 2 (positive dominated). Ex 2: (-7) - (-4) = -7 + 4 = -3. Rules: same signs add; different signs subtract. — Mnemonic PEMDAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction. A minus sign in front of parentheses switches all the internal signs. Calculators follow the order correctly; mental calculation requires practice.

6. Research and present in writing the main fractions of our daily life and in which situations we use each of them.

   Answer: Common fractions: 1/2 (half — half an hour, half a kilo), 1/4 (quarter — quarters of a liter, quarter-final), 1/3 (third — dividing a pizza, a third part), 3/4 (three quarters — recipes), 2/3 (two thirds — qualified majority). — Fractions came from Ancient Egypt (the Rhind papyrus, 1,650 BC). Decimal fractions (1.5 = 3/2) make calculations easier. In recipes, 1/2 cup is a common unit. In statistics, 2/3 and 3/5 are milestones. In politics, 2/3 is a special quorum.

7. Demonstrate the ability to solve the four basic operations involving fractions, including the calculation of the LCM in the case of addition and subtraction and finally the simplification when possible.

   Answer: Addition/subtraction: find the LCM of the denominators. Ex: 1/2 + 1/3 = 3/6 + 2/6 = 5/6. Multiplication: numerator×numerator / denominator×denominator. — The LCM (least common multiple) equalizes denominators. The GCD (greatest common divisor) simplifies fractions. Fraction calculators solve them, but knowing the technique cements the learning. Equivalent fractions: 2/4 = 1/2.

8. Present, in the form of a poster, the main plane figures with their characteristics and demonstrate how to calculate their area and perimeter.

   Answer: Square: 4 equal sides; A=s², P=4s. Rectangle: 2 pairs of sides; A=b×h, P=2(b+h). Triangle: 3 sides; A=(b×h)/2. Circle: A=πr², P=2πr (circumference). Make a poster with the formulas and drawings of the figures. — π (pi) is the constant 3.14159... Triangles have specific formulas (Heron's for 3 sides). Trapezoid: A=((B+b)×h)/2. Archimedes gave the method for the area of the circle 2,300 years ago. Geometry is the basis of architecture.

9. Demonstrate the ability to convert the main units of measurement: meters, square meters (m2), kg, grams, and cubic meters (m3). Present three conversion examples.

   Answer: 1m = 100cm = 1,000mm. 1m² = 10,000cm² (not 100). 1m³ = 1,000,000cm³. 1kg = 1,000g. Examples: 2.5m = 250cm; 3kg = 3,000g; 5m² = 50,000cm². — Square units raise the factor to the second power (cm² × 10,000). Cubic units raise it to the third power. The International System (SI) uses powers of 10. Imperial (inches, pounds) is still used in the USA. Brazil adopted the SI in 1862 — Law No. 1,157.

10. Present three examples of equations involving the letter x and solve each one, giving the correct solution.

    Answer: An equation has a letter (variable) that represents an unknown value. Ex 1: x + 5 = 12 → x = 7. Ex 2: 2x = 16 → x = 8. Ex 3: x/3 = 5 → x = 15. — Equations were developed by al-Khwarizmi in the 9th century (origin of the word 'algebra'). Principle: what you do to one side, you do to the other. Quadratic equations (x²) use the Bhaskara formula. The basis of all advanced mathematics.