MATHEMATICS NOTES FOR FORM TWO – ALL TOPICS

TOPIC 1 – EXPONENTS AND RADICALS


TOPIC 2 – ALGEBRA


TOPIC 3 – QUADRATIC EQUATIONS


TOPIC 4 – LOGARITHMS


TOPIC 5 – CONGRUENCE


TOPIC 6 – SIMILARITY


TOPIC 7 – GEOMETRIC AND TRANSFORMATIONS


TOPIC 8 – PYTHAGORAS THEOREM


TOPIC 9 – TRIGONOMETRY


TOPIC 10 – SETS


TOPIC 11 – STATISTICS

IMPORTANCE OF MATHEMATICS IN OUR DAILY LIFE – PART 2 

5. Math makes you a better cook (or baker)

With a knowledge of math, for example, you
can quickly deduce that a half-cup of flour is the same thing as eight
tablespoons of flour.  This can prove handy if you find that your half-cup
measure is missing.  Likewise, if you are cooking from a recipe that
serves 4 people, but you need to feed 8 people, your math skills tell you that
you can simply double all of the necessary ingredients.  Without math, you
may not have enough food (or have too much food) to feed your guests…

6. Great career options

Mathematics offers a great career
opportunity for students. In most careers, employers want to hire employees who
can solve complex problems. If you are good at math and have a keen ability to
solve complex problems, you are at the top of applying for many jobs. Finance
analysis and cost estimation are part of every business.

7. Math supports logical reasoning and
analytical thinking

A strong understanding of math concepts
means more than just number sense. It helps us see the pathways to a solution.
Equations and word problems need to be examined before determining the best
method for solving them. And in many cases, there’s more than one way to get to
the right answer.

It’s no surprise that logical reasoning
and analytical thinking improve alongside math skills. Logic skills are
necessary at all levels of mathematical education.

8. Math sharpens your memory

Learning mental math starts in elementary
school. Students learn addition tables, then subtraction, multiplication and
division tables. As they master those skills, they’ll begin to memorize more
tips and tricks, like adding a zero to the end when multiplying by 10. Students
will memorize algorithms and processes throughout their education.

Using your memory often keeps it sharp. As
your child grows and continues to use math skills in adulthood, their memory
will remain in tip top shape.

…………….

For larger numbers this procedure can be improved by considering multiples of one of the factors by 10, 20,…or even by higher orders of magnitude (100, 1,000,…), as necessary (in the Egyptian decimal notation, these multiples are easy to work out). Thus, one can find the product of 28 by 27 by setting out the multiples of 28 by 1, 2, 4, 8, 10, and 20. Since the entries 1, 2, 4, and 20 add up to 27, one has only to add up the corresponding multiples to find the answer.

Computations involving fractions are carried out under the restriction to unit parts (that is, fractions that in modern notation are written with 1 as the numerator). To express the result of dividing 4 by 7, for instance, which in modern notation is simply 4/7, the scribe wrote 1/2 + 1/14. The procedure for finding quotients in this form merely extends the usual method for the division of integers, where one now inspects the entries for 2/3, 1/3, 1/6, etc., and 1/2, 1/4, 1/8, etc., until the corresponding multiples of the divisor sum to the dividend. (The scribes included 2/3, one may observe, even though it is not a unit fraction.) In practice the procedure can sometimes become quite complicated (for example, the value for 2/29 is given in the Rhind papyrus as 1/24 + 1/58 + 1/174 + 1/232) and can be worked out in different ways (for example, the same 2/29 might be found as 1/15 + 1/435 or as 1/16 + 1/232 + 1/464, etc.). A considerable portion of the papyrus texts is devoted to tables to facilitate the finding of such unit-fraction values.

These elementary operations are all that one needs for solving the arithmetic problems in the papyri. For example, “to divide 6 loaves among 10 men” (Rhind papyrus, problem 3), one merely divides to get the answer 1/2 + 1/10. In one group of problems an interesting trick is used: “A quantity (aha) and its 7th together make 19—what is it?” (Rhind papyrus, problem 24). Here one first supposes the quantity to be 7: since 11/7 of it becomes 8, not 19, one takes 19/8 (that is, 2 + 1/4 + 1/8), and its multiple by 7 (16 + 1/2 + 1/8) becomes the required answer. This type of procedure (sometimes called the method of “false position” or “false assumption”) is familiar in many other arithmetic traditions (e.g., the Chinese, Hindu, Muslim, and Renaissance European), although they appear to have no direct link to the Egyptian.

Geometry

The geometric problems in the papyri seek measurements of figures, like rectangles and triangles of given base and height, by means of suitable arithmetic operations. In a more complicated problem, a rectangle is sought whose area is 12 and whose height is 1/2 + 1/4 times its base (Golenishchev papyrus, problem 6). To solve the problem, the ratio is inverted and multiplied by the area, yielding 16; the square root of the result (4) is the base of the rectangle, and 1/2 + 1/4 times 4, or 3, is the height. The entire process is analogous to the process of solving the algebraic equation for the problem (x × 3/4x = 12), though without the use of a letter for the unknown. An interesting procedure is used to find the area of the circle (Rhind papyrus, problem 50): 1/9 of the diameter is discarded, and the result is squared. For example, if the diameter is 9, the area is set equal to 64. The scribe recognized that the area of a circle is proportional to the square of the diameter and assumed for the constant of proportionality (that is, π/4) the value 64/81. This is a rather good estimate, being about 0.6 percent too large. (It is not as close, however, as the now common estimate of 31/7, first proposed by Archimedes, which is only about 0.04 percent too large.) But there is nothing in the papyri indicating that the scribes were aware that this rule was only approximate rather than exact.

A remarkable result is the rule for the volume of the truncated pyramid (Golenishchev papyrus, problem 14). The scribe assumes the height to be 6, the base to be a square of side 4, and the top a square of side 2. He multiplies one-third the height times 28, finding the volume to be 56; here 28 is computed from 2 × 2 + 2 × 4 + 4 × 4. Since this is correct, it can be assumed that the scribe also knew the general rule: A = (h/3)(a2 + ab + b2). How the scribes actually derived the rule is a matter for debate, but it is reasonable to suppose that they were aware of related rules, such as that for the volume of a pyramid: one-third the height times the area of the base.

…………….

The Secret to Success in Mathematics

There are so many people who struggle with mathematics and yet some people seem to have no trouble whatsoever. Is there something different in their DNA or some super advanced part of their brains, or do they simply know some secret trick that helps them in their understanding?

The answer is a little of the second, but mostly the last. Scientific research suggests that those who are good at mathematics do have some areas of their brain more active during mathematical activities, however, this increased brain activity could be a result of increased mathematical training. Much like how an athlete’s muscles improve with training, regular mathematical exercises help to improve the performance of those brain areas associated with mathematics. Again, like an athlete in training, there needs to be some underlying skills and a good coach. All of us possess the minimum underlying skills, but not all of us are fortunate enough to have a good coach at the critical time when we are ready to start serious mathematical training.

That’s not to say that you must have a brilliant mathematics teacher in order to succeed, but rather that when you are ready to start learning about mathematics beyond simple addition and subtraction, you need someone who can help you identify that mathematics is not a series of boring repetitive exercises from a textbook. Nor is mathematics made up of a set of discrete topics that package neatly into textbook chapters. In short, you need someone to tell you the secret to success in mathematics.

The secret to success in mathematics is understanding and accepting the following:

1. Mathematics is a language
   Mathematics is not a set of confusing hieroglyphs designed by some evil conspiracy to torture school children and university students, but rather a language built on rules. In the same way we learn any language, we must learn the alphabet and the rules used if we are to have any success in expressing our ideas or understanding others using this language. The primary rule of the language of mathematics is the “Order of Operations”.
2. Mathematics is based in logic
   Mathematical rules are firmly based in logic. For example, if we accept that   (i.e. one is less than two) and   , then it follows that   . Whilst this is a simple example, complex mathematical rules are built by joining together many more simple rules. When the logic is violated, the result can be a set of mathematics that seems to prove the ridiculous, such as, that    (see post). If you are careful about how you apply the rules and ensure you don’t violate the logic behind them, you will be unlikely to go wrong.
3. Mathematics is interconnected
   Like many things in life, mathematical ideas are often connected to more than one other idea. Techniques, such as those that apply to linear functions, often reappear in other areas of mathematics, for example, in the statistical technique of Simple Linear Regression. As such, it is dangerous to treat mathematics as a set of discrete skills, learned for one chapter of a text and forgotten shortly after the topic test. Instead, one should consider each newly acquired skill or technique as part of an arsenal or toolkit to be drawn upon for future problems.
4. Mathematics is everywhere
   Much of mathematics taught in schools and universities suffers from an inability to answer the highly intelligent question, “Where will I use this in real life?”. Unfortunately, it is often difficult to point to a real world situation and make a plausible justification for the direct use of much of the mathematics taught in high school. Pythagora’s theorem ( ) is a classic case. The ancients knew that if you took a rope, placed knots at equal distances and then used this to construct a triangle that had sides of 3 knots, 4 knots and 5 knots, then the triangle contained a right angle (very useful for constructing buildings that won’t easily fall down). Despite this, it is unlikely that many modern students will need to build a pyramid for their pharaoh, so why should they learn it? Well, Pythagora’s theorem is about more than right-angled triangles, it also tells us about the relationship between many numbers and is a theorem that is the basis of much of the mathematics used in modern computing, communications technology and cryptography.
5. Mathematicians are lazy
   Like many people, mathematicians don’t want to do more work than they have to. As such, many of the techniques employed in mathematics are about reducing a problem down to a set of previously solved sub-problems. In other words, we mathematicians want to re-use our previous work (or someone else’s work) wherever possible. What this often means is that students are often presented with almost identical problems as they learn different techniques. Unfortunately, this can lead to the perception that mathematics is nothing more than a set of boring repetitive exercises from a textbook, which is completely not true. Instead, perhaps it would be better to think that mathematics is very eco-friendly, recycling previous problems for reuse in more complex learning situations.
6. Mathematicians like simplicity
   The axiom of “The simplest solution is often the best” is truly at home in mathematics. As such, we prefer solutions that are given in the simplest form, whether that is as a surd, a fraction, a decimal or a whole number. Wherever possible, you should try to express your solutions in the simplest, but most correct, manner. For example, one should write   rather than   .
7. Mathematicians like order
   Some might suggest that mathematicians suffer from some kind of obsessive compulsive disorder, but the truth is that mathematics is based upon order, be that the counting order of numbers or the “Order of Operations”. The same is true about how we like to write equations in descending order of powers, e.g.   rather than   .
8. Statistics is a special kind of mathematics
   Statistics uses all the techniques of mathematics, from simple algebra through to calculus and beyond, but it also uses a special set of skills called “Reasoning with Uncertainty”. Statistics is a branch of mathematics devoted to the science of uncertainty (risk, odds, chance, probability, likelihood). Unfortunately, it has gained a fairly bad reputation, mostly because, like weather forecasts, it does not provide 100% guarantees about the results, only the methods used to obtain them. Instead, statistics provides insight into the possibilities, which, when combined with other facts, should lead one to a reasonable (but still possibly wrong) conclusion based upon the sample data. Mark Twain once quipped, “There are three kinds of lies: lies, damned lies and statistics”, hinting at this possibility of an incorrect conclusion. Whilst statistics is the “Science of Uncertainty”, there is no uncertainty in the science of statistics. If you perform the same statistical technique many times on the same data, it will always lead to the same conclusion. The uncertainty comes from the data and whether or not it is an accurate and representative sample of the population of interest.
9. Statisticians are “frightened” of negative numbers
   Most statistical techniques are based upon differences and as such, many of these differences are negative. Unfortunately, if you add together a bunch of positive and negative differences, many of them will cancel each other out. To avoid this, a number of statistical techniques use the trick of squaring the differences first, before the summation, and then square-rooting the result. A classic example occurs in the calculation of the standard deviation. This often leads people to comment that statisticians seem “frightened” of negative numbers.

Once you have understood and accepted the above truths about mathematics (and mathematicians), it should hopefully become clearer to you why it is that we mathematicians do certain things. Equally, these truths should help you to unlock the secret to your success in learning the techniques of mathematics and statistics.

Egyptian sekedThe Egyptians defined the seked as the ratio of the run to the rise, which is the reciprocal of the modern definition of the slope.

The Egyptians employed the equivalent of similar triangles to measure distances. For instance, the seked of a pyramid is stated as the number of palms in the horizontal corresponding to a rise of one cubit (seven palms). Thus, if the seked is 51/4 and the base is 140 cubits, the height becomes 931/3 cubits (Rhind papyrus, problem 57). The Greek sage Thales of Miletus (6th century bce) is said to have measured the height of pyramids by means of their shadows (the report derives from Hieronymus, a disciple of Aristotle in the 4th century bce). In light of the seked computations, however, this report must indicate an aspect of Egyptian surveying that extended back at least 1,000 years before the time of Thales.

Assessment of Egyptian mathematics

The papyri thus bear witness to a mathematical tradition closely tied to the practical accounting and surveying activities of the scribes. Occasionally, the scribes loosened up a bit: one problem (Rhind papyrus, problem 79), for example, seeks the total from seven houses, seven cats per house, seven mice per cat, seven ears of wheat per mouse, and seven hekat of grain per ear (result: 19,607). Certainly the scribe’s interest in progressions (for which he appears to have a rule) goes beyond practical considerations. Other than this, however, Egyptian mathematics falls firmly within the range of practice.

Even allowing for the scantiness of the documentation that survives, the Egyptian achievement in mathematics must be viewed as modest. Its most striking features are competence and continuity. The scribes managed to work out the basic arithmetic and geometry necessary for their official duties as civil managers, and their methods persisted with little evident change for at least a millennium, perhaps two. Indeed, when Egypt came under Greek domination in the Hellenistic period (from the 3rd century bce onward), the older school methods continued. Quite remarkably, the older unit-fraction methods are still prominent in Egyptian school papyri written in the demotic (Egyptian) and Greek languages as late as the 7th century ce, for example.

To the extent that Egyptian mathematics left a legacy at all, it was through its impact on the emerging Greek mathematical tradition between the 6th and 4th centuries bce. Because the documentation from this period is limited, the manner and significance of the influence can only be conjectured. But the report about Thales measuring the height of pyramids is only one of several such accounts of Greek intellectuals learning from Egyptians; Herodotus and Plato describe with approval Egyptian practices in the teaching and application of mathematics. This literary evidence has historical support, since the Greeks maintained continuous trade and military operations in Egypt from the 7th century bce onward. It is thus plausible that basic precedents for the Greeks’ earliest mathematical efforts—how they dealt with fractional parts or measured areas and volumes, or their use of ratios in connection with similar figures—came from the learning of the ancient Egyptian scribes.

Greek mathematics

The development of pure mathematics

The pre-Euclidean period

mathematicians of the Greco-Roman worldThis map spans a millennium of prominent Greco-Roman mathematicians, from Thales of Miletus (c. 600 bce) to Hypatia of Alexandria (c. 400 ce).

The Greeks divided the field of mathematics into arithmetic (the study of “multitude,” or discrete quantity) and geometry (that of “magnitude,” or continuous quantity) and considered both to have originated in practical activities. Proclus, in his Commentary on Euclid, observes that geometry—literally, “measurement of land”—first arose in surveying practices among the ancient Egyptians, for the flooding of the Nile compelled them each year to redefine the boundaries of properties. Similarly, arithmetic started with the commerce and trade of Phoenician merchants. Although Proclus wrote quite late in the ancient period (in the 5th century ce), his account drew upon views proposed much earlier—by Herodotus (mid-5th century bce), for example, and by Eudemus, a disciple of Aristotle (late 4th century bce).

However plausible, this view is difficult to check, for there is only meagre evidence of practical mathematics from the early Greek period (roughly, the 8th through the 4th century bce). Inscriptions on stone, for example, reveal use of a numeral system the same in principle as the familiar Roman numerals. Herodotus seems to have known of the abacus as an aid for computation by both Greeks and Egyptians, and about a dozen stone specimens of Greek abaci survive from the 5th and 4th centuries bce. In the surveying of new cities in the Greek colonies of the 6th and 5th centuries, there was regular use of a standard length of 70 plethra (one plethron equals 100 feet) as the diagonal of a square of side 50 plethra; in fact, the actual diagonal of the square is 50Square root of√2 plethra, so this was equivalent to using 7/5 (or 1.4) as an estimate for Square root of√2, which is now known to equal 1.414…. In the 6th century bce the engineer Eupalinus of Megara directed an aqueduct through a mountain on the island of Samos, and historians still debate how he did it. In a further indication of the practical aspects of early Greek mathematics, Plato describes in his Laws how the Egyptians drilled their children in practical problems in arithmetic and geometry; he clearly considered this a model for the Greeks to imitate.

Such hints about the nature of early Greek practical mathematics are confirmed in later sources—for example, in the arithmetic problems in papyrus texts from Ptolemaic Egypt (from the 3rd century bce onward) and the geometric manuals by Heron of Alexandria (1st century ce). In its basic manner this Greek tradition was much like the earlier traditions in Egypt and Mesopotamia. Indeed, it is likely that the Greeks borrowed from such older sources to some extent.

What was distinctive of the Greeks’ contribution to mathematics—and what in effect made them the creators of “mathematics,” as the term is usually understood—was its development as a theoretical discipline. This means two things: mathematical statements are general, and they are confirmed by proof. For example, the Mesopotamians had procedures for finding whole numbers a, b, and c for which a2 + b2 = c2 (e.g., 3, 4, 5; 5, 12, 13; or 119, 120, 169). From the Greeks came a proof of a general rule for finding all such sets of numbers (now called Pythagorean triples): if one takes two whole numbers p and q, both being even or both odd and such that pq is a square number, then a = (p − q)/2, b = Square root of√pq, and c = (p+ q)/2. As Euclid proves in Book X of the Elements, numbers of this form satisfy the relation for Pythagorean triples. Further, the Mesopotamians appear to have understood that sets of such numbers a, b, and c form the sides of right triangles, but the Greeks proved this result (Euclid, in fact, proves it twice: in Elements, Book I, proposition 47, and in a more general form in Elements, Book VI, proposition 31), and these proofs occur in the context of a systematic presentation of the properties of plane geometric figures.

The Elements, composed by Euclid of Alexandria about 300 bce, was the pivotal contribution to theoretical geometry, but the transition from practical to theoretical mathematics had occurred much earlier, sometime in the 5th century bce. Initiated by men like Pythagoras of Samos (late 6th century) and Hippocrates of Chios (late 5th century), the theoretical form of geometry was advanced by others, most prominently the Pythagorean Archytas of Tarentum, Theaetetus of Athens, and Eudoxus of Cnidus (4th century). Because the actual writings of these men do not survive, knowledge about their work depends on remarks made by later writers. While even this limited evidence reveals how heavily Euclid depended on them, it does not set out clearly the motives behind their studies.

It is thus a matter of debate how and why this theoretical transition took place. A frequently cited factor is the discovery of irrational numbers. The early Pythagoreans held that “all things are number.” This might be taken to mean that any geometric measure can be associated with some number (that is, some whole number or fraction; in modern terminology, rational number), for in Greek usage the term for number, arithmos, refers exclusively to whole numbers or, in some contexts, to ordinary fractions. This assumption is common enough in practice, as when the length of a given line is said to be so many feet plus a fractional part. However, it breaks down for the lines that form the side and diagonal of the square. (For example, if it is supposed that the ratio between the side and diagonal may be expressed as the ratio of two whole numbers, it can be shown that both of these numbers must be even. This is impossible, since every fraction may be expressed as a ratio of two whole numbers having no common factors.) Geometrically, this means that there is no length that could serve as a unit of measure of both the side and diagonal; that is, the side and diagonal cannot each equal the same length multiplied by (different) whole numbers. Accordingly, the Greeks called such pairs of lengths “incommensurable.” (In modern terminology, unlike that of the Greeks, the term “number” is applied to such quantities as Square root of√2, but they are called irrational.)

This result was already well known at the time of Plato and may well have been discovered within the school of Pythagoras in the 5th century bce, as some late authorities like Pappus of Alexandria (4th century ce) maintain. In any case, by 400 bce it was known that lines corresponding to Square root of√3, Square root of√5, and other square roots are incommensurable with a fixed unit length. The more general result, the geometric equivalent of the theorem that Square root of√p is irrational whenever p is not a rational square number, is associated with Plato’s friend Theaetetus. Both Theaetetus and Eudoxus contributed to the further study of irrationals, and their followers collected the results into a substantial theory, as represented by the 115 propositions of Book X of the Elements.

The discovery of irrationals must have affected the very nature of early mathematical research, for it made clear that arithmetic was insufficient for the purposes of geometry, despite the assumptions made in practical work. Further, once such seemingly obvious assumptions as the commensurability of all lines turned out to be in fact false, then in principle all mathematical assumptions were rendered suspect. At the least it became necessary to justify carefully all claims made about mathematics. Even more basically, it became necessary to establish what a reasoning has to be like to qualify as a proof. Apparently, Hippocrates of Chios, in the 5th century bce, and others soon after him had already begun the work of organizing geometric results into a systematic form in textbooks called “elements” (meaning “fundamental results” of geometry). These were to serve as sources for Euclid in his comprehensive textbook a century later.

The early mathematicians were not an isolated group but part of a larger, intensely competitive intellectual environment of pre-Socratic thinkers in Ionia and Italy, as well as Sophists at Athens. By insisting that only permanent things could have real existence, the philosopher Parmenides (5th century bce) called into question the most basic claims about knowledge itself. In contrast, Heracleitus (c. 500 bce) maintained that all permanence is an illusion, for the things that are perceived arise through a subtle balance of opposing tensions. What is meant by “knowledge” and “proof” thus came into debate.

Mathematical issues were often drawn into these debates. For some, like the Pythagoreans (and, later, Plato), the certainty of mathematics was held as a model for reasoning in other areas, like politics and ethics. But for others mathematics seemed prone to contradiction. Zeno of Elea (5th century bce) posed paradoxes about quantity and motion. In one such paradox it is assumed that a line can be bisected again and again without limit; if the division ultimately results in a set of points of zero length, then even infinitely many of them sum up only to zero, but, if it results in tiny line segments, then their sum will be infinite. In effect, the length of the given line must be both zero and infinite. In the 5th century bce a solution of such paradoxes was attempted by Democritus and the atomists, philosophers who held that all material bodies are ultimately made up of invisibly small “atoms” (the Greek word atomon means “indivisible”). But in geometry such a view came into conflict with the existence of incommensurable lines, since the atoms would become the measuring units of all lines, even incommensurable ones. Democritus and the Sophist Protagoras puzzled over whether the tangent to a circle meets it at a point or a line. The Sophists Antiphon and Bryson (both 5th century bce) considered how to compare the circle to polygons inscribed in it.

The pre-Socratics thus revealed difficulties in specific assumptions about the infinitely many and the infinitely small and about the relation of geometry to physical reality, as well as in more general conceptions like “existence” and “proof.” Philosophical questions such as these need not have affected the technical researches of mathematicians, but they did make them aware of difficulties that could bear on fundamental matters and so made them the more cautious in defining their subject matter.

Any such review of the possible effects of factors such as these is purely conjectural, since the sources are fragmentary and never make explicit how the mathematicians responded to the issues that were raised. But it is the particular concern over fundamental assumptions and proofs that distinguishes Greek mathematics from the earlier traditions. Plausible factors behind this concern can be identified in the special circumstances of the early Greek tradition—its technical discoveries and its cultural environment—even if it is not possible to describe in detail how these changes took place.

The Elements

The principal source for reconstructing pre-Euclidean mathematics is Euclid’s Elements, for the major part of its contents can be traced back to research from the 4th century bce and in some cases even earlier. The first four books present constructions and proofs of plane geometric figures: Book I deals with the congruence of triangles, the properties of parallel lines, and the area relations of triangles and parallelograms; Book II establishes equalities relating to squares, rectangles, and triangles; Book III covers basic properties of circles; and Book IV sets out constructions of polygons in circles. Much of the content of Books I–III was already familiar to Hippocrates, and the material of Book IV can be associated with the Pythagoreans, so that this portion of the Elements has roots in 5th-century research. It is known, however, that questions about parallels were debated in Aristotle’s school (c. 350 bce), and so it may be assumed that efforts to prove results—such as the theorem stating that for any given line and given point, there always exists a unique line through that point and parallel to the line—were tried and failed. Thus, the decision to found the theory of parallels on a postulate, as in Book I of the Elements, must have been a relatively recent development in Euclid’s time. (The postulate would later become the subject of much study, and in modern times it led to the discovery of the so-called non-Euclidean geometries.)

Book V sets out a general theory of proportion—that is, a theory that does not require any restriction to commensurable magnitudes. This general theory derives from Eudoxus. On the basis of the theory, Book VI describes the properties of similar plane rectilinear figures and so generalizes the congruence theory of Book I. It appears that the technique of similar figures was already known in the 5th century bce, even though a fully valid justification could not have been given before Eudoxus worked out his theory of proportion.

Books VII–IX deal with what the Greeks called “arithmetic,” the theory of whole numbers. It includes the properties of numerical proportions, greatest common divisors, least common multiples, and relative primes (Book VII); propositions on numerical progressions and square and cube numbers (Book VIII); and special results, like unique factorization into primes, the existence of an unlimited number of primes, and the formation of “perfect numbers”—that is, those numbers that equal the sum of their proper divisors (Book IX). In some form Book VII stems from Theaetetus and Book VIII from Archytas.

Book X presents a theory of irrational lines and derives from the work of Theaetetus and Eudoxus. The remaining books treat the geometry of solids. Book XI sets out results on solid figures analogous to those for planes in Books I and VI; Book XII proves theorems on the ratios of circles, the ratios of spheres, and the volumes of pyramids and cones; Book XIII shows how to inscribe the five regular solids—known as the Platonic solids—in a given sphere (compare the constructions of plane figures in Book IV). The measurement of curved figures in Book XII is inferred from that of rectilinear figures; for a particular curved figure, a sequence of rectilinear figures is considered in which succeeding figures in the sequence become continually closer to the curved figure; the particular method used by Euclid derives from Eudoxus. The solid constructions in Book XIII derive from Theaetetus.

In sum the Elements gathered together the whole field of elementary geometry and arithmetic that had developed in the two centuries before Euclid. Doubtless, Euclid must be credited with particular aspects of this work, certainly with its editing as a comprehensive whole. But it is not possible to identify for certain even a single one of its results as having been his discovery. Other, more advanced fields, though not touched on in the Elements, were already being vigorously studied in Euclid’s time, in some cases by Euclid himself. For these fields his textbook, true to its name, provides the appropriate “elementary” introduction.

One such field is the study of geometric constructions. Euclid, like geometers in the generation before him, divided mathematical propositions into two kinds: “theorems” and “problems.” A theorem makes the claim that all terms of a certain description have a specified property; a problem seeks the construction of a term that is to have a specified property. In the Elements all the problems are constructible on the basis of three stated postulates: that a line can be constructed by joining two given points, that a given line segment can be extended in a line indefinitely, and that a circle can be constructed with a given point as centre and a given line segment as radius. These postulates in effect restricted the constructions to the use of the so-called Euclidean tools—i.e., a compass and a straightedge or unmarked ruler.

The three classical problems

Although Euclid solves more than 100 construction problems in the Elements, many more were posed whose solutions required more than just compass and straightedge. Three such problems stimulated so much interest among later geometers that they have come to be known as the “classical problems”: doubling the cube (i.e., constructing a cube whose volume is twice that of a given cube), trisecting the angle, and squaring the circle. Even in the pre-Euclidean period the effort to construct a square equal in area to a given circle had begun. Some related results came from Hippocrates (see Sidebar: Quadrature of the Lune); others were reported from Antiphon and Bryson; and Euclid’s theorem on the circle in Elements, Book XII, proposition 2, which states that circles are in the ratio of the squares of their diameters, was important for this search. But the first actual constructions (not, it must be noted, by means of the Euclidean tools, for this is impossible) came only in the 3rd century bce. The early history of angle trisection is obscure. Presumably, it was attempted in the pre-Euclidean period, although solutions are known only from the 3rd century or later.

doubling the volume of a cubeIn the 4th century bce, Menaechmus gave a solution to the problem of doubling the volume of a cube. In particular, he showed that the intersection of any two of the three curves that he constructed (two parabolas and one hyperbola) based on a side (a) of the original cube will produce a line (x) such that the cube produced with it has twice the volume of the original cube.

There are several successful efforts at doubling the cube that date from the pre-Euclidean period, however. Hippocrates showed that the problem could be reduced to that of finding two mean proportionals: if for a given line a it is necessary to find x such that x3 = 2a3, lines x and y may be sought such that a:x = x:y = y:2a; for then a3/x3 = (a/x)3 = (a/x)(x/y)(y/2a) = a/2a = 1/2. (Note that the same argument holds for any multiplier, not just the number 2.) Thus, the cube can be doubled if it is possible to find the two mean proportionals x and y between the two given lines a and 2a. Constructions of the problem of the two means were proposed by Archytas, Eudoxus, and Menaechmus in the 4th century bce. Menaechmus, for example, constructed three curves corresponding to these same proportions: x2 = ay, y2 = 2ax, and xy = 2a2; the intersection of any two of them then produces the line x that solves the problem. Menaechmus’s curves are conic sections: the first two are parabolas, the third a hyperbola. Thus, it is often claimed that Menaechmus originated the study of the conic sections. Indeed, Proclus and his older authority, Geminus (mid-1st century ce), appear to have held this view. The evidence does not indicate how Menaechmus actually conceived of the curves, however, so it is possible that the formal study of the conic sections as such did not begin until later, near the time of Euclid. Both Euclid and an older contemporary, Aristaeus, composed treatments (now lost) of the theory of conic sections.

In seeking the solutions of problems, geometers developed a special technique, which they called “analysis.” They assumed the problem to have been solved and then, by investigating the properties of this solution, worked back to find an equivalent problem that could be solved on the basis of the givens. To obtain the formally correct solution of the original problem, then, geometers reversed the procedure: first the data were used to solve the equivalent problem derived in the analysis, and, from the solution obtained, the original problem was then solved. In contrast to analysis, this reversed procedure is called “synthesis.”

Menaechmus’s cube duplication is an example of analysis: he assumed the mean proportionals x and y and then discovered them to be equivalent to the result of intersecting the three curves whose construction he could take as known. (The synthesis consists of introducing the curves, finding their intersection, and showing that this solves the problem.) It is clear that geometers of the 4th century bce were well acquainted with this method, but Euclid provides only syntheses, never analyses, of the problems solved in the Elements. Certainly in the cases of the more complicated constructions, however, there can be little doubt that some form of analysis preceded the syntheses presented in the Elements.

Geometry in the 3rd century bce

The Elements was one of several major efforts by Euclid and others to consolidate the advances made over the 4th century bce. On the basis of these advances, Greek geometry entered its golden age in the 3rd century. This was a period rich with geometric discoveries, particularly in the solution of problems by analysis and other methods, and was dominated by the achievements of two figures: Archimedes of Syracuse (early 3rd century bce) and Apollonius of Perga (late 3rd century bce).

Archimedes

Archimedes was most noted for his use of the Eudoxean method of exhaustion in the measurement of curved surfaces and volumes and for his applications of geometry to mechanics. To him is owed the first appearance and proof of the approximation 31/7 for the ratio of the circumference to the diameter of the circle (what is now designated π). Characteristically, Archimedes went beyond familiar notions, such as that of simple approximation, to more subtle insights, like the notion of bounds. For example, he showed that the perimeters of regular polygons circumscribed about the circle eventually become less than 31/7 the diameter as the number of their sides increases (Archimedes established the result for 96-sided polygons); similarly, the perimeters of the inscribed polygons eventually become greater than 310/71. Thus, these two values are upper and lower bounds, respectively, of π

sphere with circumscribing cylinder The volume of a sphere is 4πr3/3, and the volume of the circumscribing cylinder is 2πr3. The surface area of a sphere is 4πr2, and the surface area of the circumscribing cylinder is 6πr2. Hence, any sphere has both two-thirds the volume and two-thirds the surface area of its circumscribing cylinder.

Archimedes’ result bears on the problem of circle quadrature in the light of another theorem he proved: that the area of a circle equals the area of a triangle whose height equals the radius of the circle and whose base equals its circumference. He established analogous results for the sphere showing that the volume of a sphere is equal to that of a cone whose height equals the radius of the sphere and whose base equals its surface area; the surface area of the sphere he found to be four times the area of its greatest circle. Equivalently, the volume of a sphere is shown to be two-thirds that of the cylinder which just contains it (that is, having height and diameter equal to the diameter of the sphere), while its surface is also equal to two-thirds that of the same cylinder (that is, if the circles that enclose the cylinder at top and bottom are included). The Greek historian Plutarch (early 2nd century ce) relates that Archimedes requested the figure for this theorem to be engraved on his tombstone, which is confirmed by the Roman writer Cicero (1st century bce), who actually located the tomb in 75 bce, when he was quaestor of Sicily.

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7 Strategies for Inspiring a Love of Mathematics in Your Classroom

Developing a positive relationship with mathematics early on can have a powerful and lasting impact

When young learners experience success, enjoyment, and a sense of belonging in mathematics (both at home and in the classroom), they’re far more likely to stay confident and engaged as they progress through school and into adulthood.

Creating this foundation begins with meaningful early experiences: opportunities to explore, to take risks, to feel supported, and to see themselves as capable mathematicians.

Each success, no matter how small, reinforces their confidence and strengthens the skills they need to grow.

Research consistently shows that students who enjoy mathematics or feel a sense of pride in their progress tend to achieve more over the long term.

Positive emotions fuel persistence, and each successful experience reinforces a student’s belief that they can be a capable mathematician. This is why cultivating a love of mathematics early on is so critical: it helps prevent the all-too-common cycle of students thinking, “I’m just not a math(s) person.”

Across the world, education systems are recognising the importance of strong analytical and numerical skills, especially as data and problem-solving play an increasingly central role in modern life and work. Building confidence early gives students the foundation they need to thrive in these environments.

With this in mind, here are seven strategies to help spark curiosity, confidence, and joy in mathematics – so learners feel eager and enthusiastic about engaging with the subject as they grow.

1. Use the language of mathematics in everyday situations

One of the most effective ways to build early mathematical understanding is to connect mathematics to real-world experiences.

When teachers seize these natural, everyday ‘teachable moments’, they help students recognise mathematics as meaningful, useful, and all around them.

Look for opportunities to introduce or reinforce mathematical language – such as comparisons, quantities, position words, or descriptive attributes – as students explore, play, or converse.

Simple comments like “fast,” “high,” “bigger,” “closer,” or “more” can open the door to rich mathematical thinking.

For example, if a student says, “Look, that bird is flying so fast and high!” you might extend the moment by asking:

How high do you think the bird is flying?

Is it higher than our school building?

How fast do you think it’s going?

Do you think the bird is flying faster than a car?

These small conversations help students build confidence with mathematical vocabulary while developing curiosity and reasoning.

2. Provide a variety of learning materials and approaches

Young learners benefit from experiencing mathematics in multiple ways.

Offering diverse tools, activities, and modes of exploration allows students to discover how they learn best – while keeping mathematics engaging and accessible.

Here are several ways to support rich early learning experiences:

Hands-on exploration: Encourage students to investigate patterns, shapes, or spatial relationships using real objects, building materials, or natural elements.

Collaborative activities: Learning mathematics together fosters communication, problem-solving, and shared reasoning.

Mathematics-themed books: A diverse collection of picture books – whether explicitly mathematical or not – helps students encounter mathematical ideas through stories, illustrations, and language.

Technology-enhanced learning: Digital tools can support experimentation, investigation, and immediate feedback in ways that complement hands-on learning.

Games and gamified tasks: Game elements such as goals, challenges, and rewards can motivate learners while building fluency and confidence.

Where play meets learning

Mathletics brings the joy of learning to life through engaging, curriculum-aligned activities, mathematics games, and friendly competition. Students stay motivated as they progress, earn recognition, and unlock new challenges – helping them build both confidence and capability.

Learn more about Mathletics

3. Get to know your students

Every learner brings unique strengths, experiences, interests, and ways of thinking to the classroom.

Building genuine connections – both inside and outside of lessons – helps teachers understand who students are and what motivates them.

With larger class sizes, having meaningful conversations with every student can be challenging.

Here are two practical ways to make it manageable:

Use a simple survey: At the start of the year, invite students to share their interests, preferred learning styles, and feelings about mathematics. Insights from families can also help build a fuller picture.

Create a rotation system: Choose a small group of students each week to check in with intentionally. Ask open-ended questions about what they enjoy, where they feel confident, and which areas they find challenging. Over time, you’ll build a comprehensive understanding of the whole class.

As you learn more about your students, you can amplify their strengths, adjust your explanations, and offer targeted support where needed.

4. Connect new learning to what students already know

Just as reading a book from the middle makes it difficult to understand the plot, students may struggle with mathematics when foundational ideas are missing or unclear.

To help all learners make sense of new concepts, start by uncovering what they already understand:

What prior knowledge do they have that connects to this lesson?

Are there gaps or “missing chapters” that might affect today’s learning?

Which ideas are essential for grasping the new content?

When teachers identify where students are in their learning journey, they can adapt explanations, scaffold new ideas, and support learners more effectively.

5. Show students the progress they’ve made

Celebrating growth builds motivation and confidence. Help students acknowledge their effort by making progress visible. You might:

Introduce individual or class progress charts.

Celebrate milestones during mid-year check-ins.

Award certificates, badges, or special acknowledgements for persistence and improvement.

These small moments of recognition can have a powerful impact on students’ belief in themselves as mathematicians.

6. Explore interesting careers connected to mathematics

Mathematics opens doors to a wide range of pathways – far beyond what learners might initially imagine.

Introducing students to different careers can help them see the relevance and possibilities ahead.

Share examples from STEM professions, but also from fields like design, sports, health, architecture, finance, technology, media, and more.

Tie their interests back to foundational mathematical ideas:

A student who loves design might explore proportion, symmetry, or measurement.

A sports enthusiast might investigate statistics or data analysis.

A budding chef might explore fractions, ratios, and timing.

Showing students that mathematics is woven into so many careers helps motivate them to engage deeply with the subject.

7. Foster a positive mathematical mindset

Research by Carol Dweck and others highlights how deeply a student’s mindset influences learning.

Encouraging a growth mindset helps students embrace challenges, learn from mistakes, and persist through difficulty.

Here are ways to nurture positive mathematical mindsets:

Celebrate mistakes as opportunities to learn.

Praise the process – effort, strategies, focus – rather than innate talent.

Use mixed–ability groupings to encourage peer learning and shared problem-solving.

Reflect on your own attitudes toward mathematics and how they may influence students.

Examine your assumptions about which students you think are ‘good at maths’ – all learners can grow.

Every child has the potential to flourish in mathematics. By nurturing curiosity, confidence, and perseverance, we equip them with skills that extend far beyond the classroom.

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For nearly ten years, Mathletics has been part of daily life at St Mary’s RC Primary School in Battersea.

What began as a digital tool for homework has grown into a trusted, motivating and confidence-building component of the school’s maths provision – one that the staff, students and families recognise as something bigger than just another online program.

At the heart of this long-term success is Ali Shoubber, the school’s long-standing Maths Lead.

Since joining St Mary’s nine years ago, he has overseen the school’s use of Mathletics and watched it evolve alongside students’ needs, government expectations and the growing demand for engaging, curriculum-aligned EdTech.

A trusted part of maths learning at St Mary’s

St Mary’s is a close-knit Catholic primary school with a strong commitment to nurturing confident, curious learners.

Their maths curriculum is no exception: teachers use a combination of traditional lessons, visual modelling, specialist instruction and online programs to support every child’s development.

For Ali, Mathletics is not an add-on it’s an extension of the school’s approach.

“It’s part of our wider provision. It’s not a separate thing – it’s an extension of our maths learning,” he says. “The concepts are the same whether we’re doing it on the board, in books, or on Mathletics.”

Mathletics is used flexibly across the school: for homework, in-class practice, whole-class modelling, small group intervention and even assemblies or celebrations.

This variety allows teachers to weave it naturally into their curriculum planning without the pressure of creating something new from scratch.

Motivation through progress and rewards

One of the biggest shifts Ali has seen over the past decade is how Mathletics has kept pace with what genuinely motivates children. When he first started at St Mary’s, the program served its purpose well but then something changed.

“When I first started, Mathletics was serving its purpose well. But then things kind of revolutionised – suddenly you had games like Meritopia, avatars, certificates got a rebrand and the Skills Quests challenged our more able children.”

Every week, children celebrate their progress through certificates, points, coins and levels – and St Mary’s has embraced that celebration culture wholeheartedly.

Teachers display certificates in classrooms, announce achievements in assemblies, and use the reward system as a way to recognise effort and persistence, not just correct answers.

A highlight this year was a visit from a Mathletics team member who led a whole-school assembly – complete with trophies for top performers and recognition for consistent effort.

“That was great because it took the celebration to another level,” Ali says. “It reminded them they’re part of something bigger than just St Mary’s.”

Building confidence and supporting every learner

Like many primary schools, St Mary’s teaches children with a wide range of abilities in each class. Some students arrive confident with number work: others need significant support to build foundational skills.

One of Mathletics’ biggest strengths, according to Ali, is how easily teachers can tailor the platform to meet these diverse needs.

“As a teacher, you can take a lot of ownership and tailor it to your school’s needs,” he explains. “You can assign different year groups, set groups, adjust Live Mathletics levels – there are lots of ways to personalise.”

This flexibility means teachers aren’t locked into a one-size-fits-all approach. A child who struggles with multiplication can work on targeted activities at their level, while another student tackles more complex problem-solving tasks

But this personalisation isn’t just about challenge – it’s about confidence.

Ali sees clear patterns in the data and in daily classroom interactions: children who use Mathletics regularly tend to be more self-assured in their maths learning and perform better as the year progresses.

“Children that use Mathletics regularly have positive outcomes,” he says. “Those who are getting certificates tend to do better in end-of-year assessments and are more confident.”

This isn’t just about getting questions right: it’s about creating a positive relationship with maths.

When children experience small wins regularly, their entire approach to the subject shifts. They’re more willing to try challenging problems and less likely to give up when something feels difficult.

Importantly, the program offers a balanced mix of ‘quick wins’ and deeper challenge – something Ali values.

“It’s nicely balanced. Some tasks give a confidence boost, others provide more challenge. That’s important.”

Data that empowers teachers (and supports meaningful conversations with parents)

Teachers at St Mary’s regularly use Mathletics’ reporting tools to track progress, identify patterns and prepare for parent-teacher meetings.

For Ali, the wealth of data available is one of the platform’s most practical benefits, especially when it comes to having meaningful conversations with families.

“There’s so much data,” Ali explains. “Certificates, time online, attempts – so many ways to measure progress.”

Before parents’ evening, Ali downloads the school-wide spreadsheet, filters by class and gives each teacher a snapshot of their pupils’ engagement.

This insight helps teachers celebrate progress with parents, spot patterns in how children are working, and identify students who may benefit from more structured support or encouragement at home.

“It’s always well received,” says Ali. “Parents like to see the data and it’s useful for teachers too.”

Beyond individual progress tracking, Ali also runs parent workshops on the school’s digital platforms – including Mathletics.

These sessions help families understand how the tools work, what their children are engaging with, and how they can support learning at home.

This partnership between school and home – supported by clear, accessible data – strengthens the impact of Mathletics and helps ensure that children receive consistent encouragement.

Why Mathletics has lasted a decade at St Mary’s

In an era where schools are increasingly cautious with budget decisions – and EdTech tools come and go – St Mary’s has stayed committed to Mathletics year after year.

For Ali, the reason is simple: it consistently delivers on what matters most.

“I’m very satisfied,” Ali says. “It’s well aligned and the coverage is good. We’re definitely happy and we’re keen to see what comes next.”

That forward-looking attitude reflects something important: Mathletics isn’t a static product. The program has evolved over the decade Ali has been using it, adapting to changing curricula, new teaching approaches, and what motivates today’s learners.

When asked what he’d say to other schools considering Mathletics, Ali’s advice is practical and straightforward:

“Give it a go. Get kids on it, see it for yourself – I think you’ll see a positive impact.”

As St Mary’s continues raising achievement and building a positive maths culture, Mathletics remains a trusted partner: supporting teachers, motivating learners and helping every child grow in