Mathematics Skills I Honor

Science & Health

Requirements

1. Know the decimal system.

   Answer: You must demonstrate to the instructor that the decimal system is a base-10 numbering system, made up of ten digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9), in which each position is worth ten times the previous one — units, tens, hundreds, thousands, and so on. It is the most widely used system in the world for everyday counting, commerce, and measurements. — The decimal system was adopted universally probably because humans have ten fingers. Positional decimal notation, which multiplies the value by the power of 10, was developed in India and brought to the Arab world and to Europe between the 7th and 13th centuries by mathematicians such as al-Khwarizmi and Fibonacci.

2. Be able to identify and classify the digits of the decimal system and represent the position of each one.

   Answer: The ten digits of the decimal system are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. As for classification, they can be even (0, 2, 4, 6, 8) or odd (1, 3, 5, 7, 9). The position of each digit defines its value (place value), from right to left: units (1), tens (10), hundreds (100), thousands (1,000), ten thousands (10,000), hundred thousands (100,000), millions (1,000,000). Each place is worth ten times the previous one. Example: in the number 4,572, the 2 occupies the units place, the 7 the tens, the 5 the hundreds, and the 4 the thousands. — The value of each digit depends on its position: the 5 is worth 5 in the units place, but is worth 500 in the hundreds place. This is the concept of positional notation. The classes are separated into groups of three places (units, thousands, millions, billions), making reading easier — hence the use of a period (thousands separator) every three digits.

3. Classify and identify the four basic operations, representing the algorithm of each one, illustrated with four examples.

   Answer: The four basic operations are: addition (+), subtraction (-), multiplication (×) and division (÷). The algorithm and terms of each, with four examples: 1) Addition — you add from the units up to the higher places, grouping into tens; terms: addends and sum/total. Ex.: 12 + 8 = 20. 2) Subtraction — you subtract from the lowest place to the highest, borrowing when necessary; terms: minuend, subtrahend and difference/remainder. Ex.: 15 - 7 = 8. 3) Multiplication — you multiply each digit and add the partial results according to their position; terms: factors (multiplicand × multiplier) and product. Ex.: 6 × 4 = 24. 4) Division — you divide the highest place of the dividend by the divisor, bringing down the next digits; terms: dividend, divisor, quotient and remainder. Ex.: 20 ÷ 5 = 4. — The terms of the operations are part of basic mathematical vocabulary. Addition and multiplication are commutative (a+b = b+a; a×b = b×a); subtraction and division are not. Division by zero is undefined — learning the standard division algorithm is the foundation for fractions, decimals, and percentages studied later.

4. Devise four practical examples in which we use the four basic operations and solve them.

   Answer: Four practical examples with solutions: 1) Addition — I bought 3 apples and got 5 more, ending up with 3 + 5 = 8 apples; 2) Subtraction — I had R$ 50 and spent R$ 18, leaving 50 - 18 = R$ 32; 3) Multiplication — 6 boxes with 12 pencils each total 6 × 12 = 72 pencils; 4) Division — 24 candies shared equally among 4 children give 24 ÷ 4 = 6 candies each. — Learning math with everyday examples sticks better than memorizing rules. The four operations appear in shopping, dividing tasks, checking change, and calculating salary. Multiplication is repeated addition (6×12 = 12+12+12+12+12+12); division is repeated subtraction — the operations are conceptually interconnected.

5. Research and present, in the form of a drawing or in writing, the possible origin of the signs for square root, division, addition, and subtraction.

   Answer: Origin of the four signs: 1) Square root (√) — the radical was introduced by the German mathematician Christoff Rudolff in 1525 (book 'Coss'); it is believed to derive from a stylized letter 'r' (from radix, root in Latin). 2) Division (÷) — the obelus sign was popularized by the Swiss Johann Rahn in 1659; its shape (two dots with a dash) resembles a generic fraction. 3) Addition (+) — it originates from the abbreviation of the Latin word 'et' (and); together with the (-), it was formalized in print by the German Johannes Widmann in 1489, from the commercial marks 'p' (plus) and 'm' (minus). 4) Subtraction (-) — also spread by Widmann in 1489, coming from the same German commercial abbreviations. — Before standardization, each mathematician invented their own symbol. The signs we use today came from centuries of European Renaissance tradition, popularized by Gutenberg's printing press (1455). Robert Recorde created the equals sign (=) in 1557 in England, claiming that 'two parallel lines are the most equal things that exist'.

6. Demonstrate in practice the solving of at least three examples of exponentiation and three examples of numerical expressions, using parentheses, brackets, and braces.

   Answer: You must present to the instructor three solved examples of exponentiation (e.g.: 2³=8, 5²=25, 3⁴=81) and three numerical expressions with parentheses, brackets, and braces (e.g.: {[(2+3)×4]-5}=15; {[(10-2)×3]+4}÷7=4; [(6²-4)+10]÷2=21), respecting the official order: first parentheses (), then brackets [], then braces {}, and within each, powers before multiplication/division before addition/subtraction. — Exponentiation is repeated multiplication: 2³ = 2×2×2 = 8. The official order in expressions: parentheses, brackets, braces; then powers and roots; then multiplications and divisions from left to right; lastly, additions and subtractions from left to right. Getting the order wrong always leads to an incorrect answer.

7. Present and solve three practical examples of situations involving fractions in the operations of addition, subtraction, multiplication, and division. Remember that for the operations of addition and subtraction we use the calculation of the LCM.

   Answer: You must present to the instructor three examples with fractions: 1) Addition/subtraction with LCM — 1/2 + 1/3 (LCM=6) → 3/6 + 2/6 = 5/6; 2) Multiplication — 2/3 × 4/5 = 8/15 (multiply numerator by numerator, denominator by denominator); 3) Division — 3/4 ÷ 1/2 = 3/4 × 2/1 = 6/4 = 3/2 (invert the second fraction and multiply). Use everyday situations (pizza, recipe, distribution). — To add or subtract fractions with different denominators, you must find the LCM (least common multiple) and convert each fraction to the new denominator. Multiplication is direct — just multiply top by top and bottom by bottom. Division becomes multiplication by inverting the second fraction: a/b ÷ c/d = a/b × d/c.

8. Present and demonstrate the solving of three problems involving percentage calculations for the purchase and sale of products, obtaining discounts.

   Answer: Three percentage problems in buying and selling: 1) An R$ 80 T-shirt with a 20% discount: 80 × 0.20 = R$ 16 discount; final price 80 - 16 = R$ 64. 2) R$ 250 sneakers with a 15% discount: 250 × 0.15 = R$ 37.50 discount; final price 250 - 37.50 = R$ 212.50. 3) A store sells for R$ 100 a product bought for R$ 60: a profit of R$ 40, which equals 40 ÷ 60 × 100 = 66.7% margin over cost. General discount calculation: value × (percentage ÷ 100) = discount amount, subtracted from the original price. — A percentage is a fraction of 100: 20% = 20/100 = 0.20. For discounts, multiply the value by the percentage and subtract. For profit margin, divide the profit by the cost (not by the sale price). Brazilian stores often present cumulative discounts — pay attention to the order in which the successive percentages are applied.

9. Present and demonstrate the solving of four examples of operations with decimal numbers in addition, subtraction, multiplication, and division, using the practical method of solving.

   Answer: You must present to the instructor four examples with decimals: 1) Addition — 2.5 + 3.75 = 6.25 (align the decimal point); 2) Subtraction — 10.8 - 4.35 = 6.45 (align the decimal point); 3) Multiplication — 2.5 × 0.4 = 1.00 = 1 (multiply as whole numbers and count the decimal places); 4) Division — 7.2 ÷ 0.3 = 24 (multiply numerator and denominator by 10 to remove the decimal point). — To add and subtract decimals, the rule is to align the decimal point. To multiply, ignore the decimal point, multiply as whole numbers, and count the total decimal places. To divide, turn the divisor into a whole number by multiplying both by 10 or 100. The decimal system was popularized in 1585 by the Dutchman Simon Stevin.

10. Present, in the form of a poster or multimedia, five examples of plane figures and five examples of solid figures, showing their characteristics and, alongside, what type of everyday object they are used as.

    Answer: You must present to the instructor a poster or multimedia presentation with five plane figures (square, circle, triangle, rectangle, and rhombus) with their sides/angles described and everyday examples (tile, plate, sliced pizza, TV screen, kite); and five solid figures (cube, sphere, cylinder, cone, pyramid) with their faces/edges described and everyday examples (die, ball, soda can, ice cream cone, Egyptian box). — Plane figures are two-dimensional (they have length and width, but no depth), whereas solid figures are three-dimensional. Recognizing geometric shapes in everyday life is the first step in geometry — and it was the starting point of the book 'The Elements' by Euclid, written around 300 BC.