MATHEMATICS NOTES FOR FORM ONE – ALL TOPICS

TOPIC 1 – NUMBERS


TOPIC 2 – FRACTIONS


TOPIC 3 – DECIMAL AND PERCENTAGE


TOPIC 4 – UNITS


TOPIC 5 – APPROXIMATIONS


TOPIC 6 – GEOMETRY


TOPIC 7 – ALGEBRA


TOPIC 8 – NUMBERS (II)


TOPIC 9 – RATIO, PROFIT AND LOSS


TOPIC 10 – COORDINATE GEOMETRY


TOPIC 11 – PERIMETERS AND AREAS

IMPORTANCE OF MATHEMATICS IN OUR DAILY LIFE – PART 1

 1. Learning math is good for your brain

Research conducted by Dr. Tanya
Evans of Stanford University indicates that children who know math are
able to recruit certain brain regions more reliably, and have greater gray
matter volume in those regions, than those who perform more poorly in
math.  The brain regions involved in higher math skills in high-performing
children were associated with various cognitive tasks involving visual
attention and decision-making.  While correlation may not imply causation,
this study indicates that the same brain regions that help you do math are
recruited in decision-making and attentional processes.

2. Math Helps You with Your Finances

Math is also helpful with your finance.
With the help of math, you can easily make your financial budget. You can
calculate how much money you have and how you can spend your money. Almost
every single human in the world uses math for their finance. The salaried
person uses math to calculate their expenses and salaries. On the other hand,
businessmen use math to calculate their profits and loss. They also use it to
calculate their loans and many more. It highlights the importance of business
mathematics and also plays a crucial role in business accounting.

3. Math improves problem-solving skills

At first, classic math problems like
Johnny bringing home 42 watermelons and returning 13 of them can just seem a
silly exercise. But all those math word problems our children solve really do
improve their problem solving skills. Word problems teach kids how to
pull out the important information and then manipulate it to find a solution.

Later on, complex life problems take the
place of workbooks, but problem-solving still happens the same way. When
students understand algorithms and problems more deeply, they can decode the
facts and more easily solve the issue. Real-life solutions are found with math
and logic.

4. Mathematics is applied in various
fields and disciplines, i.e., mathematical concepts and procedures are used
to solve problems in science, engineering, economics.

…………..

Being successful with math isn’t always the easiest. Math can be overwhelming to some students. Luckily, math is the kind of language that always has an answer. If a new unit begins and students are still struggling to understand the concepts from the previous one, it becomes easy to fall behind. Math isn’t an inherited skill, it’s a subject that students either quickly grasp or something that they’ll need a little extra hand with.

Math scores have been down across the board due to the COVID-19 pandemic. One education research organization predicts that the average student lost 57 to 183 days of learning in reading and from 136 to 232 days of learning in math just during the spring of 2020. There have been many learning losses due to the pandemic. Read about it here.

These tips can help students strengthen their math comprehension, develop strong math skills, and avoid falling behind in math class.

10 Tips to Help Students With Math

Doing Math Homework

Obvious, right? Even if a student understands a concept/skill clearly, doing all the assigned homework can help make the concept cement in the student’s brain.

Imagine the questions as practice test questions: complete them correctly during homework, and the student is more likely to complete them correctly during a test. Check out this article with five tips on simplifying math homework.

Know the Math Textbook

Since math is cumulative, the textbook is a chronological guide to what is coming up next. Review chapters before entering class to help prep the student’s brain for the new lesson and get a head start on seeing how new material connects to previous material.

Ask in Class

If a student feels that a new concept is harder to wrap their head around, ask for clarification in class. Not speaking up, then finding out that a student can’t complete the homework because something is confusing.

While in class, listen to other students’ questions, as they may help them understand their own or offer to complete questions on the board even if they’re unsure what they are doing. Practice makes perfect.

Understand the Method & the Process

Knowing formulas is important, but a student can’t be successful if they don’t know how or when to use them. Take the time to understand the formulas’ principles to understand math concepts truly.

Prime the Brain

Math is easier if a student’s brain is ready for it. Do a few fun brain teasers before sitting down to complete homework or study for a test to get their brain in the math mood.

Practice, Practice, Practice

If a concept is still a bit fuzzy even after the student has completed their homework, find some additional practice questions online. It is important to not only complete questions until they get the right answer, but until they understand how they got the right answer.

Don’t Stress

If a student is struggling with a question or concept, set it aside, take a break, and return to it later. If they’re still having difficulties, call up a classmate or ask a family member for help. Look online. If no one can help out, make a note of the problem and wait to ask their teacher the next day. Struggling with a problem that no one can answer will only increase frustration and cause unnecessary stress.

Slow Down

Completing work in class or finishing a test is not a race. Take time to understand, complete, and double-check their work. Taking time also lessens any chances of making silly mistakes or scribbling un legible answers.

Analyze Any Errors

When homework and tests are returned to the student, take the time to go over the wrong answers. Figure out where they went wrong and do a few practice questions to get the correct method locked in their brain. Ask the teacher if they need help figuring out their missteps.

Exercise Before Homework

Studies show that light exercise increases blood flow and improves mental clarity and the ability to concentrate, which is exactly the state a student wants their brain in before doing homework. So, go for a walk before hitting the books and can maximize your headspace and homework efforts. Check out this article on the benefits of exercising before doing homework.

Master middle school prep with our essential guide.

Contact your local centre today to learn more about how Oxford Learning can help your child to develop strong math skills. and get the most out of their education!
……..
Here are a few examples of how math is used in our daily lives:

Shopping and Budgeting:

Shopping is the most obvious scenario where we witness the application of fundamental mathematical concepts. We use math to compare prices, determine the best deal, and calculate the quantity to be purchased, the weight, the price per unit, the discount, and the overall cost of the item before making a purchase. We also use math when calculating the total purchase cost to ensure that we are sticking to the budget and have enough money to cover the cost.

Cooking and Baking:

Surprisingly, math is involved in cooking and baking. We use mathematics to measure ingredients, determine their quantity, the ratio of various components, cooking techniques, cookware to be used, and many other things. We also measure the temperature for baking. It requires a basic understanding of fractions and ratios and the ability to perform simple calculations. In addition, restaurant owners identify the correct cost of any recipe to make profits.

Navigation and Travel:

Whether it’s driving to school or planning a trip, math is essential for navigation and travel. We use mathematical concepts to measure distance, speed, and time acceleration. A driver uses math to make calculations, adjustments and change gears. Maps and GPS use geometry and trigonometry to calculate distances, while traffic patterns and travel time estimates rely on statistical analysis.

Technology:

Math is a crucial component of many technologies we use every day. For example, computer algorithms, mobile phones, internet and software use advanced mathematical concepts such as calculus, linear algebra, and algorithms to complete every task and command in a proper series of actions.

Time Management:

We all want to manage and wisely spend our time. When our day begins in the morning, we start by creating a suitable schedule for the day. To accomplish our tasks in a time-bound manner, we do straightforward calculations. Even the simple task of reading the analogue clock involves mathematics.

Foundation of Other subjects:

Though math is an exciting subject, you would be surprised to learn that it serves as the foundation for several subjects, including physics, chemistry, economics, history, accountancy, and statistics. All these subjects include math.

Sports:

Numbers are the next most important thing in sports other than enjoying the game. Math plays a significant role in the field and influences the player’s efficiency. It enhances people’s cognitive and decision-making skills and helps them call the right shots for the team. Using engineering and trigonometry, the player can identify the direction and angle that the ball will strike to score.

Music and Dancing:

Listening to music and dancing are children’s two most popular hobbies. Here, we frequently use numbers and mathematics to describe and teach music. Understanding fractions and ratios facilitate comprehension of musical note rhythm. In addition, simple mathematical procedures are used to improve coordination while dancing.

Exercising:

Math is an integral part of our workout routine. We use it to count the repetitions, calculate time, measure weights etc. We design our regimen based solely on math in accordance with our training schedule. So next time you exercise, keep track of how many times you do it.

Engineering:

Math is the basis of engineering. Every field of engineering includes mathematics. Computer science, chemical engineering, civil, mechanical, automobile and aeronautical engineering are some branches that rely heavily on math.

In conclusion, Math is a subject that we use in our every day, from simple calculations to more complex problem-solving. It is a powerful tool that helps us understand and navigate the world around us. Therefore, understanding basic math concepts is essential for making informed decisions.

……….

mathematics, the science of structure, order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects. It deals with logical reasoning and quantitative calculation, and its development has involved an increasing degree of idealization and abstraction of its subject matter. Since the 17th century, mathematics has been an indispensable adjunct to the physical sciences and technology, and in more recent times it has assumed a similar role in the quantitative aspects of the life sciences.

In many cultures—under the stimulus of the needs of practical pursuits, such as commerce and agriculture—mathematics has developed far beyond basic counting. This growth has been greatest in societies complex enough to sustain these activities and to provide leisure for contemplation and the opportunity to build on the achievements of earlier mathematicians.

All mathematical systems (for example, Euclidean geometry) are combinations of sets of axioms and of theorems that can be logically deduced from the axioms. Inquiries into the logical and philosophical basis of mathematics reduce to questions of whether the axioms of a given system ensure its completeness and its consistency. For full treatment of this aspect, see mathematics, foundations of.

This article offers a history of mathematics from ancient times to the present. As a consequence of the exponential growth of science, most mathematics has developed since the 15th century ce, and it is a historical fact that, from the 15th century to the late 20th century, new developments in mathematics were largely concentrated in Europe and North America. For these reasons, the bulk of this article is devoted to European developments since 1500.

This does not mean, however, that developments elsewhere have been unimportant. Indeed, to understand the history of mathematics in Europe, it is necessary to know its history at least in ancient Mesopotamia and Egypt, in ancient Greece, and in Islamic civilization from the 9th to the 15th century. The way in which these civilizations influenced one another and the important direct contributions Greece and Islam made to later developments are discussed in the first parts of this article.

India’s contributions to the development of contemporary mathematics were made through the considerable influence of Indian achievements on Islamic mathematics during its formative years. A separate article, South Asian mathematics, focuses on the early history of mathematics in the Indian subcontinent and the development there of the modern decimal place-value numeral system. The article East Asian mathematics covers the mostly independent development of mathematics in China, Japan, Korea, and Vietnam.

Key People:

It is important to be aware of the character of the sources for the study of the history of mathematics. The history of Mesopotamian and Egyptian mathematics is based on the extant original documents written by scribes. Although in the case of Egypt these documents are few, they are all of a type and leave little doubt that Egyptian mathematics was, on the whole, elementary and profoundly practical in its orientation. For Mesopotamian mathematics, on the other hand, there are a large number of clay tablets, which reveal mathematical achievements of a much higher order than those of the Egyptians. The tablets indicate that the Mesopotamians had a great deal of remarkable mathematical knowledge, although they offer no evidence that this knowledge was organized into a deductive system. Future research may reveal more about the early development of mathematics in Mesopotamia or about its influence on Greek mathematics, but it seems likely that this picture of Mesopotamian mathematics will stand.

From the period before Alexander the Great, no Greek mathematical documents have been preserved except for fragmentary paraphrases, and, even for the subsequent period, it is well to remember that the oldest copies of Euclid’s Elements are in Byzantine manuscripts dating from the 10th century ce. This stands in complete contrast to the situation described above for Egyptian and Babylonian documents. Although, in general outline, the present account of Greek mathematics is secure, in such important matters as the origin of the axiomatic method, the pre-Euclidean theory of ratios, and the discovery of the conic sections, historians have given competing accounts based on fragmentary texts, quotations of early writings culled from nonmathematical sources, and a considerable amount of conjecture.

Many important treatises from the early period of Islamic mathematics have not survived or have survived only in Latin translations, so that there are still many unanswered questions about the relationship between early Islamic mathematics and the mathematics of Greece and India. In addition, the amount of surviving material from later centuries is so large in comparison with that which has been studied that it is not yet possible to offer any sure judgment of what later Islamic mathematics did not contain, and therefore it is not yet possible to evaluate with any assurance what was original in European mathematics from the 11th to the 15th century.

In modern times the invention of printing has largely solved the problem of obtaining secure texts and has allowed historians of mathematics to concentrate their editorial efforts on the correspondence or the unpublished works of mathematicians. However, the exponential growth of mathematics means that, for the period from the 19th century on, historians are able to treat only the major figures in any detail. In addition, there is, as the period gets nearer the present, the problem of perspective. Mathematics, like any other human activity, has its fashions, and the nearer one is to a given period, the more likely these fashions will look like the wave of the future. For this reason, the present article makes no attempt to assess the most recent developments in the subject.

John L. Berggren

Mathematics in ancient Mesopotamia

Until the 1920s it was commonly supposed that mathematics had its birth among the ancient Greeks. What was known of earlier traditions, such as the Egyptian as represented by the Rhind papyrus (edited for the first time only in 1877), offered at best a meagre precedent. This impression gave way to a very different view as historians succeeded in deciphering and interpreting the technical materials from ancient Mesopotamia.

Owing to the durability of the Mesopotamian scribes’ clay tablets, the surviving evidence of this culture is substantial. Existing specimens of mathematics represent all the major eras—the Sumerian kingdoms of the 3rd millennium bce, the Akkadian and Babylonian regimes (2nd millennium), and the empires of the Assyrians (early 1st millennium), Persians (6th through 4th century bce), and Greeks (3rd century bce to 1st century ce). The level of competence was already high as early as the Old Babylonian dynasty, the time of the lawgiver-king Hammurabi (c. 18th century bce), but after that there were few notable advances. The application of mathematics to astronomy, however, flourished during the Persian and Seleucid (Greek) periods.

The numeral system and arithmetic operations

Unlike the Egyptians, the mathematicians of the Old Babylonian period went far beyond the immediate challenges of their official accounting duties. For example, they introduced a versatile numeral system, which, like the modern system, exploited the notion of place value, and they developed computational methods that took advantage of this means of expressing numbers; they solved linear and quadratic problems by methods much like those now used in algebra; their success with the study of what are now called Pythagorean number triples was a remarkable feat in number theory. The scribes who made such discoveries must have believed mathematics to be worthy of study in its own right, not just as a practical tool.

The older Sumerian system of numerals followed an additive decimal (base-10) principle similar to that of the Egyptians. But the Old Babylonian system converted this into a place-value system with the base of 60 (sexagesimal). The reasons for the choice of 60 are obscure, but one good mathematical reason might have been the existence of so many divisors (2, 3, 4, and 5, and some multiples) of the base, which would have greatly facilitated the operation of division. For numbers from 1 to 59, the symbolsfor 1 andfor 10 were combined in the simple additive manner (e.g.,represented 32). But to express larger values, the Babylonians applied the concept of place value. For example, 60 was written as, 70 as, 80 as, and so on. In fact,could represent any power of 60. The context determined which power was intended. By the 3rd century bce, the Babylonians appear to have developed a placeholder symbol that functioned as a zero, but its precise meaning and use is still uncertain. Furthermore, they had no mark to separate numbers into integral and fractional parts (as with the modern decimal point). Thus, the three-place numeral 3 7 30 could represent 31/8 (i.e., 3 + 7/60 + 30/602), 1871/2 (i.e., 3 × 60 + 7 + 30/60), 11,250 (i.e., 3 × 602 + 7 × 60 + 30), or a multiple of these numbers by any power of 60.

The four arithmetic operations were performed in the same way as in the modern decimal system, except that carrying occurred whenever a sum reached 60 rather than 10. Multiplication was facilitated by means of tables; one typical tablet lists the multiples of a number by 1, 2, 3,…, 19, 20, 30, 40, and 50. To multiply two numbers several places long, the scribe first broke the problem down into several multiplications, each by a one-place number, and then looked up the value of each product in the appropriate tables. He found the answer to the problem by adding up these intermediate results. These tables also assisted in division, for the values that head them were all reciprocals of regular numbers.

Regular numbers are those whose prime factors divide the base; the reciprocals of such numbers thus have only a finite number of places (by contrast, the reciprocals of nonregular numbers produce an infinitely repeating numeral). In base 10, for example, only numbers with factors of 2 and 5 (e.g., 8 or 50) are regular, and the reciprocals (1/8 = 0.125, 1/50 = 0.02) have finite expressions; but the reciprocals of other numbers (such as 3 and 7) repeat infinitelyand, respectively, where the bar indicates the digits that continually repeat). In base 60, only numbers with factors of 2, 3, and 5 are regular; for example, 6 and 54 are regular, so that their reciprocals (10 and 1 6 40) are finite. The entries in the multiplication table for 1 6 40 are thus simultaneously multiples of its reciprocal 1/54. To divide a number by any regular number, then, one can consult the table of multiples for its reciprocal.

Babylonian mathematical tablet

An interesting tablet in the collection of Yale University shows a square with its diagonals. On one side is written “30,” under one diagonal “42 25 35,” and right along the same diagonal “1 24 51 10” (i.e., 1 + 24/60 + 51/602 + 10/603). This third number is the correct value of Square root of√2 to four sexagesimal places (equivalent in the decimal system to 1.414213…, which is too low by only 1 in the seventh place), while the second number is the product of the third number and the first and so gives the length of the diagonal when the side is 30. The scribe thus appears to have known an equivalent of the familiar long method of finding square roots. An additional element of sophistication is that by choosing 30 (that is, 1/2) for the side, the scribe obtained as the diagonal the reciprocal of the value of Square root of√2 (since Square root of√2/2 = 1/Square root of√2), a result useful for purposes of division.

Geometric and algebraic problems

In a Babylonian tablet now in Berlin, the diagonal of a rectangle of sides 40 and 10 is solved as 40 + 102/(2 × 40). Here a very effective approximating rule is being used (that the square root of the sum of a2 + b2 can be estimated as a + b2/2a), the same rule found frequently in later Greek geometric writings. Both these examples for roots illustrate the Babylonians’ arithmetic approach in geometry. They also show that the Babylonians were aware of the relation between the hypotenuse and the two legs of a right triangle (now commonly known as the Pythagorean theorem) more than a thousand years before the Greeks used it.

A type of problem that occurs frequently in the Babylonian tablets seeks the base and height of a rectangle, where their product and sum have specified values. From the given information the scribe worked out the difference, since (b − h)2 = (b + h)2 − 4bh. In the same way, if the product and difference were given, the sum could be found. And, once both the sum and difference were known, each side could be determined, for 2b = (b + h) + (b − h) and 2h = (b + h) − (b − h). This procedure is equivalent to a solution of the general quadratic in one unknown. In some places, however, the Babylonian scribes solved quadratic problems in terms of a single unknown, just as would now be done by means of the quadratic formula.

Although these Babylonian quadratic procedures have often been described as the earliest appearance of algebra, there are important distinctions. The scribes lacked an algebraic symbolism; although they must certainly have understood that their solution procedures were general, they always presented them in terms of particular cases, rather than as the working through of general formulas and identities. They thus lacked the means for presenting general derivations and proofs of their solution procedures. Their use of sequential procedures rather than formulas, however, is less likely to detract from an evaluation of their effort now that algorithmic methods much like theirs have become commonplace through the development of computers.

As mentioned above, the Babylonian scribes knew that the base (b), height (h), and diagonal (d) of a rectangle satisfy the relation b2 + h2 = d2. If one selects values at random for two of the terms, the third will usually be irrational, but it is possible to find cases in which all three terms are integers: for example, 3, 4, 5 and 5, 12, 13. (Such solutions are sometimes called Pythagorean triples.) A tablet in the Columbia University Collection presents a list of 15 such triples (decimal equivalents are shown in parentheses at the right; the gaps in the expressions for h, b, and d separate the place values in the sexagesimal numerals):

(The entries in the column for h have to be computed from the values for b and d, for they do not appear on the tablet; but they must once have existed on a portion now missing.) The ordering of the lines becomes clear from another column, listing the values of d2/h2 (brackets indicate figures that are lost or illegible), which form a continually decreasing sequence: [1 59 0] 15, [1 56 56] 58 14 50 6 15,…, [1] 23 13 46 40. Accordingly, the angle formed between the diagonal and the base in this sequence increases continually from just over 45° to just under 60°. Other properties of the sequence suggest that the scribe knew the general procedure for finding all such number triples—that for any integers p and q, 2d/h = p/q + q/p and 2b/h = p/q − q/p. (In the table the implied values p and q turn out to be regular numbers falling in the standard set of reciprocals, as mentioned earlier in connection with the multiplication tables.) Scholars are still debating nuances of the construction and the intended use of this table, but no one questions the high level of expertise implied by it.

Mathematical astronomy

The sexagesimal method developed by the Babylonians has a far greater computational potential than what was actually needed for the older problem texts. With the development of mathematical astronomy in the Seleucid period, however, it became indispensable. Astronomers sought to predict future occurrences of important phenomena, such as lunar eclipses and critical points in planetary cycles (conjunctions, oppositions, stationary points, and first and last visibility). They devised a technique for computing these positions (expressed in terms of degrees of latitude and longitude, measured relative to the path of the Sun’s apparent annual motion) by successively adding appropriate terms in arithmetic progression. The results were then organized into a table listing positions as far ahead as the scribe chose. (Although the method is purely arithmetic, one can interpret it graphically: the tabulated values form a linear “zigzag” approximation to what is actually a sinusoidal variation.) While observations extending over centuries are required for finding the necessary parameters (e.g., periods, angular range between maximum and minimum values, and the like), only the computational apparatus at their disposal made the astronomers’ forecasting effort possible.

Within a relatively short time (perhaps a century or less), the elements of this system came into the hands of the Greeks. Although Hipparchus (2nd century bce) favoured the geometric approach of his Greek predecessors, he took over parameters from the Mesopotamians and adopted their sexagesimal style of computation. Through the Greeks it passed to Arab scientists during the Middle Ages and thence to Europe, where it remained prominent in mathematical astronomy during the Renaissance and the early modern period. To this day it persists in the use of minutes and seconds to measure time and angles.

Aspects of the Old Babylonian mathematics may have come to the Greeks even earlier, perhaps in the 5th century bce, the formative period of Greek geometry. There are a number of parallels that scholars have noted. For example, the Greek technique of “application of area” (see below Greek mathematics) corresponded to the Babylonian quadratic methods (although in a geometric, not arithmetic, form). Further, the Babylonian rule for estimating square roots was widely used in Greek geometric computations, and there may also have been some shared nuances of technical terminology. Although details of the timing and manner of such a transmission are obscure because of the absence of explicit documentation, it seems that Western mathematics, while stemming largely from the Greeks, is considerably indebted to the older Mesopotamians.

Mathematics in ancient Egypt

The introduction of writing in Egypt in the predynastic period (c. 3000 bce) brought with it the formation of a special class of literate professionals, the scribes. By virtue of their writing skills, the scribes took on all the duties of a civil service: record keeping, tax accounting, the management of public works (building projects and the like), even the prosecution of war through overseeing military supplies and payrolls. Young men enrolled in scribal schools to learn the essentials of the trade, which included not only reading and writing but also the basics of mathematics.

One of the texts popular as a copy exercise in the schools of the New Kingdom (13th century bce) was a satiric letter in which one scribe, Hori, taunts his rival, Amen-em-opet, for his incompetence as an adviser and manager. “You are the clever scribe at the head of the troops,” Hori chides at one point,

a ramp is to be built, 730 cubits long, 55 cubits wide, with 120 compartments—it is 60 cubits high, 30 cubits in the middle…and the generals and the scribes turn to you and say, “You are a clever scribe, your name is famous. Is there anything you don’t know? Answer us, how many bricks are needed?” Let each compartment be 30 cubits by 7 cubits.

This problem, and three others like it in the same letter, cannot be solved without further data. But the point of the humour is clear, as Hori challenges his rival with these hard, but typical, tasks.

What is known of Egyptian mathematics tallies well with the tests posed by the scribe Hori. The information comes primarily from two long papyrus documents that once served as textbooks within scribal schools. The Rhind papyrus (in the British Museum) is a copy made in the 17th century bce of a text two centuries older still. In it is found a long table of fractional parts to help with division, followed by the solutions of 84 specific problems in arithmetic and geometry. The Golenishchev papyrus (in the Moscow Museum of Fine Arts), dating from the 19th century bce, presents 25 problems of a similar type. These problems reflect well the functions the scribes would perform, for they deal with how to distribute beer and bread as wages, for example, and how to measure the areas of fields as well as the volumes of pyramids and other solids.

Ancient Egyptian numeralsAncient Egyptians customarily wrote from right to left. Because they did not have a positional system, they needed separate symbols for each power of 10.

Egyptian hieratic numeralsEgyptian hieratic numerals from a mathematical papyrus, c. 1600 bce.

The Egyptians, like the Romans after them, expressed numbers according to a decimal scheme, using separate symbols for 1, 10, 100, 1,000, and so on; each symbol appeared in the expression for a number as many times as the value it represented occurred in the number itself. For example,stood for 24. This rather cumbersome notation was used within the hieroglyphic writing found in stone inscriptions and other formal texts, but in the papyrus documents the scribes employed a more convenient abbreviated script, called hieratic writing, where, for example, 24 was written.

In such a system, addition and subtraction amount to counting how many symbols of each kind there are in the numerical expressions and then rewriting with the resulting number of symbols. The texts that survive do not reveal what, if any, special procedures the scribes used to assist in this. But for multiplication they introduced a method of successive doubling. For example, to multiply 28 by 11, one constructs a table of multiples of 28 like the following:

The several entries in the first column that together sum to 11 (i.e., 8, 2, and 1) are checked off. The product is then found by adding up the multiples corresponding to these entries; thus, 224 + 56 + 28 = 308, the desired product.

To divide 308 by 28, the Egyptians applied the same procedure in reverse. Using the same table as in the multiplication problem, one can see that 8 produces the largest multiple of 28 that is less then 308 (for the entry at 16 is already 448), and 8 is checked off. The process is then repeated, this time for the remainder (84) obtained by subtracting the entry at 8 (224) from the original number (308). This, however, is already smaller than the entry at 4, which consequently is ignored, but it is greater than the entry at 2 (56), which is then checked off. The process is repeated again for the remainder obtained by subtracting 56 from the previous remainder of 84, or 28, which also happens to exactly equal the entry at 1 and which is then checked off. The entries that have been checked off are added up, yielding the quotient: 8 + 2 + 1 = 11. (In most cases, of course, there is a remainder that is less than the divisor.)

……….

Basic Tips for Success in College

Attend class regularly and pay close attention.

Review notes the same day after class and make note of things you do not understand so you can clarify them during the next class.

Make time to study math every day. Doing homework is not enough; you need to spend time reading and thinking about the material. Highlight important material in the assigned text and notes.

Do not wait until a test or quiz to study.

Review example problems solved in the text and notes on a regular basis. Keep going back to older topics covered in the course so you do not forget the basics.

When solving problems, especially when you are not quite sure of the solution strategy, pursue every line of thinking and make note of it clearly in your notebook. When you have received the final solution, compare it against your own attempts to figure out flaws in your thinking. It is important to pay attention to your tendencies – good and bad – and make the necessary adjustments.

Pay attention to the syllabus carefully at the beginning of the semester. Review course policy as outlined in the syllabus and on tests and other assignments.

Before attending class, preview the appropriate chapter in the textbook and read over notes from the last class period.

Ask questions in class about homework, problems or material discussed in that day’s class.

Attend Office Hours regularly.

Find out about additional support available through the university and department such as the Math Center, a math tutor or the Study Skills Center.

Always review your work after you are finished, whether on a test or homework.

Show respect to your classmates and professor.

Preparing for Life After College

Develop good study skills and discipline which will serve you well in your future.

Start thinking about next steps and life after college as early as possible. By the time you are in your junior year, it is good to have some idea of possible career paths you might want to pursue.

Even as a freshman meet with the department advisor to find out about program options. The Math department offers various undergraduate, double major, and BS/MS programs which you might want to learn about. Some of these programs require you to take specific courses early on so you may complete the program on time.

Get to know the department; explore the department website and talk to faculty members, even those who are not your instructors, when you get the opportunity.

Meet with your faculty advisor and discuss courses, career options, and anything else of interest to you. Connect with advisors early and meet them at least once a semester.

Attend events organized by the department such as career workshops, graduate school preparation, seminars, etc. These provide excellent opportunities to network with others in the department and learn about opportunities you may not have known about.

Participate in the Math Community Canvas page. Review the page regularly.

Become a member of the Math Club.

By the time you are in your senior year, it is good to have a resume and portfolio ready no matter what your future path. The Math Department can help you put together a portfolio.

……..

A recipe for succeeding in mathematical research would have been a laughable oxymoron, as this is a quintessentially creative endeavor. Yet, I’d like to offer a few tips for those who are taking their first steps along this fascinating path.

1. Be stubborn and at the same time flexible. In mathematical research, unlike olympiads, solving a problem takes weeks and months rather than hours, and there is no instant gratification. Yet, you don’t want to rack your brains for too long without progress if you get stuck. Ask for help, or switch to another problem!
2. Be knowledge-seeking. If you did not make progress on your problem but learned something instead, then you did in fact make progress. Besides being intellectually rewarding by itself, at the end of the day learning always pays off in practical terms as well — it helps you obtain better results. In fact, look for directions of study that will force you to learn something! The more you learn to enjoy the process of doing mathematics (rather than the result), the better mathematician you will be.
3. Mathematical research is an intrinsically social activity. Discuss a lot, seek help/advice/feedback from others, rather than be stuck for a long time.
4. Split the problem into small, bite-size steps, or ask your mentor to do so for you. You want to have something doable on your agenda at all times.
5. Consider examples. Look for the simplest example that captures the phenomenon (Gelfand’sprinciple). Also tracing through the proof with the simplest nontrivial example is a great way of checking a proof. It often uncovers subtle errors that are harder to see in the more general context, or ways to drastically simplify the proof, by identifying parts which are a “red herring.”
6. Have several questions to think about so that you can switch from one to the other.
7. Use the Internet (Wikipediais usually good for math, even though you have to be careful with it). Also Google(or Google Scholar) keyword search is often helpful. But you have to know the right keywords, and it sometimes takes some thinking to come up with them! Also, a good source is MathOverflow, where you may ask a question and professional mathematicians will answer it online. But make sure that your question is well stated, according to the rules of MathOverflow!
8. Use analogies. There are many mathematical problems but much fewer methods for solving them. So a method used in one problem may also work in another, analogous one.
9. Do computer tests, look for patterns in data, make conjectures. The On-Line Encyclopedia of Integer Sequencesis often useful. It has an advanced tool called Superseekerwhich seeks patterns.
10. Confirm your results and proofs by computer calculations whenever possible, to avoid mistakes.
11. Keep good notes of what you are doing (preferably in LaTeX) at all times. Good bookkeeping is a big part of doing math!
12. Try to write clearly and concisely, in logical sequence. A mathematical text should be highly structured. A good way of writing, at least for beginners, is to make sure that each piece of text is a definition, proposition, proof, remark, example, question, conjecture, etc., so that the text is split into small pieces and there is little (if any) loose text that does not fall into one of these categories. About each piece of text it should be clear what its status is. Text should be proofread and edited several times after it’s written.
13. Try to understand statements and proofs of the results that you use as well as you can. Not only is it more honest and reliable, but this will also give you more power in handling the actual problem you deal with.
14. Be motivated and guided by beauty and harmony. It is the most important motivation in mathematics. If you have a proof but don’t like it, if it seems ugly, it is much more likely that it is actually wrong. And even if it’s correct, it probably will become much simpler or more powerful, and you’ll learn something if you try to understand things better so you can write a better proof. It is worth trying to understand better the things you already understand to some extent, rather than jumping forward to entirely new things. Although this seemingly slows down the process, you will surely make up for it and be rewarded further down the road.
15. Listen to your heart. As in all important things in life, what you want and what you dream about is the most essential. Try to find your own voice. The main point of mathematical research is for you to enjoy it!

…………

This chapter provides strategies to empower you to study maths in whatever form you encounter it at university. It begins by describing how maths and mathematical thinking is vital to many of the professions. It then discusses maths anxiety and presents six strategies for overcoming this common problem. Next, the chapter addresses how to approach studying maths in general, and how to approach a single module of work in maths, followed by a discussion of problem-solving and hints for success. It concludes with some tips for how to approach maths assessments. Altogether, the chapter will put you on the path to a successful encounter with maths at university.

WHO NEEDS MATHS?

Many professions are highly dependent on maths. Student scientists, engineers, and accountants may work with maths in every subject. Students in other disciplines may encounter one or more maths courses or assessments. Nursing and paramedics students will encounter medication calculations for example. Similarly, many students who wish to work in people-helping professions will need to be able to interpret statistics to ensure their approaches to problems and interventions are sound. This will include students of psychology, human services, and education. No matter your discipline, this chapter will help you approach the maths content in your courses with successful strategies and a positive attitude.

MATHS ANXIETY AND ITS IMPACT ON STUDYING MATHS

When studying maths, students often become anxious and start to overanalyse the maths content. Overanalysing can further elevate their anxiety and create difficulty in understanding the underpinning maths concepts.

The Australian Council of Educational Research (ACER) defines maths anxiety as “feelings of unease and worry experienced when thinking about mathematics or completing mathematical tasks” (Buckley, Reid, Goos, Lipp, & Thomson, 2016, p. 158). Maths anxiety causes people to have significant self-doubt in their ability to do maths, causing them great distress.

When a person has maths anxiety, their brain continually thinks about the anxiety rather than the actual maths problem. The brain allocates the working memory and other resources that it would normally use with computations of the maths problem to the anxiety itself, making it very difficult to learn or retain the relevant skills or information (Marshall, Staddon, Wilson, & Mann, 2017). Maths anxiety is highly prevalent. Sadly, between 25% to 80% of the college population in the United States of America has some form of maths anxiety and the percentage is likely to be similar in Australia, given the similarities in culture (Koch, 2018).

Symptoms and causes of maths anxiety

Maths anxiety can easily be identified. The symptoms of maths anxiety range from simple low confidence problems to more complex physical symptoms. If you experience the following, you may have maths anxiety:

low confidence and negative thoughts such as “I am no good at maths”, “I won’t be able to do this”, “I am never going to understand this maths concept” (Department of Education, 2020).

physical symptoms ranging from increased heart rate, increased breathing to a panic attack when thinking about or doing maths (Department of Education, 2020).

The symptoms of maths anxiety are triggered when doing maths or from the thought (anticipation) of doing maths. The level of anxiety will vary from person to person (Department of Education, 2020). While maths anxiety is common, it can be managed or resolved allowing you to succeed in your maths learning journey.

Strategies to reduce maths anxiety

You will be able to recognise when you are starting to feel stressed or anxious and having difficulty trying to complete maths problems. This could include avoiding maths classes, revision and assessment.  At this point, you could develop some methods to help you to relax and unwind. Some different strategies you may be able to use distraction techniques including reflecting on how you feel; leaving the room to do another activity for short periods of time; mindful breathing techniques (such as breathing in for a count of 5, and breathing out for a count of 7); or any other techniques you may already use for reducing anxiety.

It is also helpful to remind yourself of what you can do by returning to a problem that you can do before attempting the problem which caused the stress. More information about mental health resources (including sections on stress and anxiety) can be found in the chapter Successful Connections.

Maths anxiety can be managed in a positive way using six strategies. The strategies will need to be employed over time to see the results. Addressing your maths anxiety using these six strategies also allows you to study maths effectively.