Mathematics Past Paper PDF

LESSON 1: THE NATURE OF MATHEMATICS Introduction Mathematics is the science of numbers, shapes, patterns and quantities and the relationship between them. Mathematics as a science means that it is a systematic body of knowledge which focuses on a specific study. It is the study of the relationships...

LESSON 1: THE NATURE OF MATHEMATICS Introduction Mathematics is the science of numbers, shapes, patterns and quantities and the relationship between them. Mathematics as a science means that it is a systematic body of knowledge which focuses on a specific study. It is the study of the relationships among numbers, quantities and shapes. It includes arithmetic, algebra, trigonometry, geometry, statistics and calculus Mathematics nurtures human characteristics like power of creativity, reasoning, critical thinking, spatial thinking and others It provides the opportunity to solve both simple and complex problems in many real-world contexts using a variety of strategies. Mathematics is a universal way to make sense of the world and to communicate understanding of concepts and rules using the mathematical symbols, signs, proofs, language and conventions. 1.1 MATHEMATICS AS PATTERNS IN NATURE AND THE WORLD Patterns in nature are visible regularities found in the natural world. These patterns persist in different context and can be modelled mathematically. The pictures below show some of the basic patterns you can observe in our nature. Zebra Stripes Sunflower Honeycomb Snowflakes W. Gary Smith adopts eight patterns in his landscape work, namely: scattered, fractured, mosaic, naturalistic drift, serpentine, spiral, radial and dendritic. These patterns occur in plants, animals, rocks formations, river flow, stars or in human creations (Goral,2017). PATTERNS FOUND IN NATURE Common patterns appear in nature, just like what we see when we look closely at plants, flowers, animals, and even at our bodies. These common patterns are all incorporated in many natural things. WAVES AND DUNES A wave is any form of disturbance that carries energy as it moves. Waves are of different kinds: mechanical waves which propagate through a medium ---- air or water, making it oscillate as waves pass by. Wind waves, on the other hand, are surface waves that create the chaotic patterns of the sea. Similarly, water waves are created by energy passing through water causing it to move in a circular motion. Likewise, ripple patterns and dunes are formed by sand wind as they pass over the sand. SPOTS AND STRIPES We can see patterns like spots on the skin of a giraffe. On the other hand, stripes are visible on the skin of a zebra. Patterns like spots and stripes that are commonly present in different organisms are results of a reaction-diffusion system (Turing, 1952). The size and the shape of the pattern depend on how fast the chemicals diffuse and how strongly they interact. MILKY WAY The spiral patterns exist on the scale of the cosmos to the minuscule forms of microscopic animals on earth. The Milky Way that contains our Solar System is a barred spiral galaxy with a band of bright stars emerging from the center running across the middle of it. Spiral patterns are also common and noticeable among plants and some animals. Spirals appear in many plants such as pinecones, pineapples, and sunflowers. On the other hand, animals like ram and kudu also have spiral patterns on their horns. SYMMETRIES In mathematics, if a figure can be folded or divided into two with two halves which are the same, such figure is called a symmetric figure. Symmetry has a vital role in pattern formation. It is used to classify and organize information about patterns by classifying the motion or deformation of both pattern structures and processes. There are many kinds of symmetry, and the most important ones are reflections, rotations, and translations. These kinds of symmetries are less formally called flips, turns, and slides. REFLECTION SYMMETRY Sometimes called line symmetry or mirror symmetry, captures symmetries when the left half of a pattern is the same as the right half. ROTATIONS, also known as rotational symmetry, captures symmetries when it still looks the same after some rotation (of less than one full turn). The degree of rotational symmetry of an object is recognized by the number of distinct orientations in which it looks the same for each rotation. TRANSLATIONS. This is another type of symmetry. Translational symmetry exists in patterns that we see in nature and in man-made objects. Translations acquire symmetries when units are repeated and turn out having identical figures, like the bees’ honeycomb with hexagonal tiles. SYMMETRIES IN NATURE From the structure of subatomic particles to that of the entire universe, symmetry is present. The presence of symmetries in nature does not only attract our visual sense, but also plays an integral and prominent role in the way our life works. HUMAN BODY The human body is one of the pieces of evidence that there is symmetry in nature. Our body exhibits bilateral symmetry. It can be divided into two identical halves. ANIMAL MOVEMENT The symmetry of motion is present in animal movements. When animals move, we can see that their movements also exhibit symmetry. SUNFLOWER One of the most interesting things about a sunflower is  that it contains both radial and bilateral symmetry. What appears to be "petals" in the outer ring are actually small flowers also known as ray florets. These small flowers are bilaterally symmetrical. On the other hand, the dark inner ring of the sunflower is a cluster of radially symmetrical disk florets. SNOWFLAKES Snowflakes have six-fold radial symmetry. The ice crystals that make up the snowflakes are symmetrical or patterned. The intricate shape of a single arm of a snowflake is very much similar to the other arms. This only proves that symmetry is present in a snowflake. FIBONACCI IN NATURE By learning about nature, it becomes gradually evident that the nature is essentially mathematical, and this is one of the reasons why explaining nature is dependent on mathematics. Mathematics has the power to unveil the inherent beauty of the natural world. In describing the amazing variety of phenomena in nature we stumble to discover the existence of Fibonacci numbers. It turns out that the Fibonacci numbers appear from the smallest up to the biggest objects in the natural world. This presence of Fibonacci numbers in nature, which was once existed realm mathematician’s curiously, is considered as one of the biggest mysteries why some patterns in nature is Fibonacci. But one thing is definitely made certain, and that what seemed solely mathematical is also natural. For instance, many flowers display figures adorned with numbers of petals that are in the Fibonacci sequence. The classic five-petal flowers are said to be the most common among them. These include the buttercup, columbine, and hibiscus. Aside from those flowers with five petals, eight-petal flowers like clematis and delphinium also have the Fibonacci numbers, while ragwort and marigold have thirteen. These numbers are all Fibonacci numbers. As we have discussed in the preceding lesson, human mind is capable of identifying and organizing patterns. We were also to realized that there are structures and patterns in nature that we don’t usually draw attention to. Likewise, we arrived at a position that in nature, some things follow mathematical sequences and one of them follow the Fibonacci sequence. We noticed that these sequences is observable in some flower petals, on the spirals of some shells and even on sunflower seeds. It is amazing to think that the Fibonacci sequence is dramatically present in nature and it opens the door to understand seriously the nature of sequence. SEQUENCE Sequence refers to an ordered list of numbers called terms, that may have repeated values. The arrangement of these terms is set by a definite rule. (Mathematics in the Modern World, 14th Edition, Aufmann, RN. et al.). Cosider the given below example: 1, 3, 5, 7, … (1stterm) (2nd term) (3rd term) (4th term) As shown above, the elements in the sequence are called terms. It is called sequence because the list is ordered and it follows a certain kind of pattern that must be recognized in order to see the expanse. The three dots at the end of the visible patterns means that the sequence is infinite. There are different types of sequence and the most common are the arithmetic sequence, geometric sequence, harmonic sequence, and Fibonacci sequence. ARITHMETIC SEQUENCE. It is a sequence of numbers that follows a definite pattern. To determine if the series of numbers follow an arithmetic sequence, check the difference between two consecutive terms. If common difference is observed, then definitely arithmetic sequence governed the pattern. To clearly illustrate the arrangement, consider the example below: Notice in the given example above, the common difference between two consecutive terms in the sequence is two. The common difference is the clue that must be figure out in a pattern in order to recognize it as an arithmetic sequence. GEOMETRIC SEQUENCE. If in the arithmetic sequence we need to check for the common difference, in geometric sequence we need to look for the common ratio. The illustrated in the example below, geometric sequence is not as obvious as the arithmetic sequence. All possibilities must be explored until some patterns of uniformity can intelligently be struck. At first it may seem like pattern less. But only by digging a little bit deeper that we can finally delve the constancy. That is ,…generate 4, 4, 4,… HARMONIC SEQUENCE. In the sequence, the reciprocal of the terms behaved in a manner like arithmetic sequence. Consider the example below and notice an interesting pattern in the series. With this pattern, the reciprocal appears like arithmetic sequence. Only in recognizing the appearance that we can finally decode the sequencing the govern the series. FIBONACCI SEQUENCE. This specific sequence was named after an Italian mathematician Leonardo Pisano Bigollo (1170 - 1250). He discovered the sequence while he was studying rabbits. The Fibonacci sequence is a series of numbers governed by some unusual arithmetic rule. The sequence is organized in a way a number can be obtained by adding the two previous numbers. Notice that the number 2 is actually the sum of 1 and 1. Also the 5th term which is number 5 is based on addition of the two previous terms 2, and 3. That is the kind of pattern being generated by the Fibonacci sequence. It is infinite in expanse and it was once purely maintained claim as a mathematical and mental exercise but later on the it was observed that the ownership of this pattern was also being claimed by some species of flowers, petals, pineapple, pine cone, cabbages and some shells. 1, 1, 2, 3, 5, 8, 13, 21, … The amazing grandeur of Fibonacci sequence was also discovered in the structure of Golden rectangle. The golden rectangle is made up of squares whose sizes, surprisingly is also behaving similar to the Fibonacci sequence. Take a serious look at the figure: THE GOLDEN RATIO  As we can see in the figure, there is no complexity in forming a spiral with the use of the golden rectangle starting from one of the sides of the first Fibonacci square going to the edges of each of the next squares. This golden rectangle shows that the Fibonacci sequence is not only about sequence of numbers of some sort but it is also a geometric sequence observing a rectangle ratio. The spiral line generated by the ratio is generously scattered around from infinite to infinitesimal. By dividing two consecutive terms of the sequence, we can approximate Φ = 1.618034. It is also noted that as you continue dividing bigger numbers of the sequence, the estimation will be more accurate and closer to the actual value. The golden ratio can be observed in the golden spiral. The golden spiral (as shown below) grows at a rate that is equal to the value of the golden ratio (Φ). One great example of a golden spiral that can be seen in the nature is the Nautilus Shell. Mathematics helps organize patterns and regularities in the world. The geometry of most patterns in nature can be associated, either directly or indirectly to mathematical numbers. The limit and extent to which natural patterns adhere to mathematical series and numbers are amazing. Mathematics helps predict the behavior of nature and phenomena in the world. It helps control nature and occurrences in the world for the good of mankind. Because of its numerous applications, mathematics becomes indispensable Mathematics, being a science of patterns, helps students to utilize, recognize and generalize patterns that exist in numbers, in shapes and in the world around them. Students with such skills are better problem solvers, and have a better sense and appreciation of nature and the world. Hence, they should have opportunities to analyze, synthesize and create a variety of patterns and to use pattern-based thinking to understand and represent mathematical and other real-world phenomena. These explorations present unlimited opportunities for problem solving. verifying generalizations and building mathematics and scientific competence.  LESON 2: MATHEMATICAL LANGUAGE AND SYMBOLS Language is the reason why people can communicate and express themselves. Language is like internet that connects people in the world. MATHEMATICS HAS ITS OWN LANGUAGE Since it is a language also, mathematics is very essential in communicating important ideas. But most mathematical language is in a form of symbols. Example: “Five added by three is eight” We could translate this in symbol as “5 + 3 = 8” Mathematical Language  the system used to communicate mathematical ideas.  It consists of some natural language using technical terms (mathematical terms) and grammatical conventions that are uncommon to mathematical discourse, supplemented by a highly specialized symbolic notation for mathematical formulas.  Mathematical notation used formulas has its own grammar and is shared by mathematicians anywhere in the globe. CHARACTERISTICS OF MATHEMATICAL LANGUAGE The language of mathematics makes it easy to express the kinds of thoughts that mathematicians like to express. It is: 1. Precise (it is precise because it can be stated clearly) + means add, - means subtract, x means multiplication and means divide. - There is only a certain answer to a certain question or problem. 2. Concise (able to say things briefly) Two times eight plus two is equal to eighteen 2 x 8 + 2 = 18 3. Powerful (it is powerful because it is capable of expressing complex ideas into simpler forms.) The sum of a number x and two is equal to twelve. x + 2 = 12 VOCABULARY VS. SENTENCES Every language has its vocabulary (the words), and its rules for combining these words into complete thoughts (the sentences). Mathematics is no exception. As a first step in discussing the mathematical language, we will make a very broad classification between the ‘nouns’ of mathematics (used to name mathematical objects of interest) and the ‘sentences’ of mathematics (which state complete mathematical thoughts)’ You must study the Mathematics Vocabulary! Student must learn on how to use correctly the language of Mathematics, when and where to use and figuring out the incorrect uses. Students must show the relationship or connections the mathematics language with the natural language.  Students must look backward or study the history of Mathematics in order to understand more deeply why Mathematics is important in their daily lives. MATHEMATICAL EXPRESSION AND SENTENCE  A mathematical expression is a finite combination of symbols that is well-defined according to rules that depend on the context. Does not contain complete thought cannot be determined to be true or false. - It is a correct arrangement of mathematical symbols used to represent a mathematical object of interest. Example:  A mathematical sentence must state a complete thought and can be determined as true, false, sometimes true, or sometimes false. - it is correct arrangement of mathematical symbols that states a complete thought it makes sense to ask about the TRUETH of a sentence.  =, Example: Connectives A question commonly encountered, when presenting the sentence example 1 + 2 = 3 is that; If = is the verb, then what is + ? The answer is the symbol + is what we called a connective which is used to connect objects of a given type to get a ‘compound’ object of the same type. Here, the numbers 1 and 2 are connected to give the new number 1 + 2. In English, this is the connector “and”. Cat is a noun, dog is a noun, cat and dog is a ‘compound’ noun. Activity time! Write E for expression and S for sentence. 1. 2. X = 2y 3. X+2 4. The capital of the Philippines is Manila 5. Rizal Park is in Cebu 6. 5+3=8 7. 1+ 1+ 1+ 1+ 1 8. Mathematics is the science of numbers, shapes, patterns and quantities and the relationship between them. 9. 7+2 10. MATHEMATICAL CONVENTION   Accepted rules and practice of spelling writing and punctuations.  Symbol used in writing mathematical expressions and sentences, including their meaning and rules in writing. Conventions in mathematics, some commonly used symbols, its meaning and an example a) Sets and Logic b) Basic Operations and Relational Symbols c) Set of Numbers Activity time! Let x be a number. Translate each phrase or sentence into a mathematical expression or equation. 1.Twelve more than a number. Ans.: 2.Eight minus a number. Ans.: 3.An unknown quantity less fourteen. Ans.: 4.Six times a number is fifty-four. Ans.: 5. Two ninths of a number is eleven. Ans.: 6. Three more than seven times a number is nine more than five times the number. Ans.: 7. Twice a number less eight is equal to one more than three times the number. Ans.: Explore! 1. Make an advanced reading on the concept of Mathematical Language. 2. Review the concepts of:  Sets  Relation  Functionss