Assigment

Multiples The products of a number with the natural numbers 1, 2, 3, 4, 5, ... are called the multiples of the number. For example:

So, the multiples of 7 are 7, 14, 21, 28, and so on. Note:

The multiples of a number are obtained by multiplying the number by each of the natural numbers. For example: • • •

multiples of 2 are 2, 4, 6, 8, … multiples of 3 are 3, 6, 9, 12, … multiples of 4 are 4, 8, 12, 16, …

Example 1 Write down the first ten multiples of 5. Solution:

The first ten multiples of 5 are 5, 10 15, 20, 25, 30, 35, 40, 45, 50.

Common Multiples Multiples that are common to two or more numbers are said to be common multiples. E.g. Multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18, … Multiples of 3 are 3, 6, 9, 12, 15, 18, … So, common multiples of 2 and 3 are 6, 12, 18, … Example 2 Find the common multiples of 4 and 6.

Solution:

Multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, … Multiples of 6 are 6, 12, 18, 24, 30, 36, … So, the common multiples of 4 and 6 are 12, 24, 36, …

Lowest Common Multiple The smallest common multiple of two or more numbers is called the lowest common multiple (LCM). E.g. Multiples of 8 are 8, 16, 24, 32, … Multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, …

In general:

To find the lowest common multiple (LCM) of two or more numbers, list the multiples of the larger number and stop when you find a multiple of the other number. This is the LCM. Example 3 Find the lowest common multiple of 6 and 9. Solution:

List the multiples of 9 and stop when you find a multiple of 6. Multiples of 9 are 9, 18, … Multiples of 6 are 6, 12, 18, …

Example 4 Find the lowest common multiple of 5, 6 and 8. Solution:

List the multiples of 8 and stop when you find a multiple of both 5 and 6. Multiples of 8 are 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112, 120, …

Stop at 120 as it is a multiple of both 5 and 6. So, the LCM of 5, 6 and 8 is 120.

Factors A whole number that divides exactly into another whole number is called a factor of that number.

So, 4 is a factor of 20 as it divides exactly into 20.

So, 5 is a factor of 20 as it divides exactly into 20. Note:

If a number can be expressed as a product of two whole numbers, then the whole numbers are called factors of that number.

So, the factors of 20 are 1, 2, 4, 5, 10 and 20. Example 5 List all the factors of 42. Solution:

So, the factors of 42 are 1, 2, 3, 6, 7, 14, 21 and 42. Note:

Example 6

Is 7 a factor 15? Solution:

Common Factors Factors that are common to two or more numbers are said to be common factors.

Example 7 Find the common factors of 10 and 30. Solution:

So, the common factors of 10 and 30 are 1, 2, 5 and 10. Example 8 Find the common factors of 26 and 39. Solution:

So, the common factors of 26 and 39 are 1 and 13.

Highest Common Factor The largest common factor of two or more numbers is called the highest common factor (HCF).

Setting out:

Often, we set out the solution as follows:

Example 9 Find the highest common factor of 14 and 28. Solution:

Prime Numbers If a number has only two different factors, 1 and itself, then the number is said to be a prime number.

7 is a prime number since it has only two different factors. Note:

But 1 is not a prime number since it does not have two different factors.

Composite Numbers A number that has more than two factors is called a composite number.

So, 14 is a composite number as it has more than two factors. Note:

1 is considered neither a prime number nor a composite number. Example 10 State which of the following numbers are a prime: a. 46 b. 19 Solution:

a. 46 is not a prime because 46 = 2 × 23. b. 19 is a prime since it has only two different factors, 1 and 19. Example 11 Express 210 as a product of prime numbers. Solution:

Note:

We try the prime numbers in order of their magnitude. Example 12 Express 90 as a product of prime numbers. Solution:

Alternatively, we can use a factor tree to express 90 as a product of prime numbers as illustrated below.

Percentages A fraction that is written out of one hundred is called a percentage. The symbol used for per cent is %.

Note:

From the preceding discussion, we notice that:

A percentage is the numerator of a fraction with a denominator of 100.

Changing a Percentage to a Fraction To convert a percentage to a fraction, write it as a fraction with a denominator of 100 and then

simplify the fraction if possible. Example 1 Express 65% as a fraction. Solution:

Mixed Number Percentage

To convert a mixed number percentage to a fraction, change the mixed number to an improper fraction, write the percentage as a fraction with a denominator of 100 and simplify the fraction if possible. Example 2

Solution:

Changing a Percentage to a Decimal To convert a percentage to a decimal, first write the percentage as a fraction out of 100 and then move the decimal point two places to the left. Example 3

Express 15% as a decimal. Solution:

Example 4

Solution:

Changing a Decimal or Fraction to a Percentage Example 5 Express 0.15 as a percentage. Solution:

From this we can state that:

To change a number into a percentage, multiply the number by 100%. Then simplify. Example 6

Solution:

Finding a Percentage of a Quantity To find a certain percentage of a given quantity, we multiply it by the corresponding fraction. Example 7 Find 20% of 45. Solution:

Example 8

Solution:

Expressing a Quantity as a Percentage of another Quantity To express one quantity as a percentage of another, make sure that both quantities are expressed in the same units. Write the given quantity as a fraction of the total and multiply it by 100%. Then simplify. Example 9

I obtained 30 marks out of 40 in a test. Convert this test mark into a percentage. Solution:

Example 10 What percentage of $4 is 32¢? Solution:

Note:

To express one quantity as a percentage of another, we must have both quantities in the same units.

Squaring the square From Wikipedia, the free encyclopedia (Redirected from Perfect square dissection) Jump to: navigation, search

A square with sides equal to a unit length multiplied by an integer is called an integral square. Squaring the square is the problem of tiling one integral square using only other integral squares. Squaring the square is a trivial task unless additional conditions are set. The most studied restriction is the "perfect" squared square, where all contained squares are of different size (see below). •

Perfect squared squares

Smith diagram of a rectangle

A "perfect" squared square is a square such that each of the smaller squares has a different size. The name was coined in humorous analogy with squaring the circle. It is first recorded as being studied by R. L. Brooks, C. A. B. Smith, A. H. Stone, and W. T. Tutte, at Cambridge University. They transformed the square tiling into an equivalent electrical circuit — they called it "Smith diagram" — by considering the squares as resistors that connected to their neighbors at their top and bottom edges, and then applied Kirchhoff's circuit laws and circuit decomposition techniques to that circuit. The first perfect squared square was found by Roland Sprague in 1939. If we take such a tiling and enlarge it so that the formerly smallest tile now has the size of the square S we started out from, then we see that we obtain from this a tiling of the plane with integral squares, each having a different size. It was an unsolved problem for many years whether the plane can be tiled with a set of integral tiles such that each natural number is used exactly once as size of a square tile. This can in fact be done: see [1] Martin Gardner has published an extensive [2] article written by W. T. Tutte about the early history of squaring the square.

Lowest-order perfect squared square

Simple squared squares A "simple" squared square is one where no subset of the squares forms a rectangle or square, otherwise it is "compound". The smallest simple perfect squared square was discovered by A. J. W. Duijvestijn using a computer search. His tiling uses 21 squares, and has been proved to be minimal. The smallest perfect compound squared square was discovered by T.H. Willcocks and has 24 squares.

Mrs. Perkins's quilt When the constraint of all the squares being different sizes is relaxed, a squared square such that the side lengths of the smaller squares do not have a common divisor larger than 1 is called a "Mrs. Perkins's quilt". In other words, the greatest common divisor of all the smaller side lengths should be 1. The Mrs. Perkins's quilt problem is to find a Mrs. Perkins's quilt with the fewest pieces for a given n × n square. [edit] External links •

Mrs. Perkins's Quilt on MathWorld

Cubing the cube Cubing the cube is the analogue in three dimensions of squaring the square: that is, given a cube C, the problem of dividing it into finitely many smaller cubes, no two congruent. Unlike the case of squaring the square, a hard but solvable problem, cubing the cube is impossible. This can be shown by a relatively simple argument. Consider a hypothetical cubed cube. The bottom face of this cube is a squared square; lift off the rest of the cube, so you have a square region of the plane covered with a collection of cubes Consider the smallest cube in this collection, with side c. Since the smallest square of a squared square cannot be on its edge, its neighbours will all tower over it, meaning that there isn't space to put a cube of side larger than c on top of it. Since the construction is a cubed cube, you're not allowed to use a cube of side equal to c; so only smaller cubes may stand upon S. This means that the top face of S must be a squared square, and the argument continues by infinite descent. Thus it is not possible to dissect a cube into finitely many smaller cubes of different sizes.

BAHAGIAN SEKOLAH KEMENTERIAN PENDIDIKAN MALAYSIA PARAS 5, BLOK J (SELATAN) PUSAT BANDAR DAMANSARA 50604 KUALA LUMPUR Diilustrasi kembali oleh Ahmad Faris bin Johan, Unit ICT, Bahagian Sekolah, Kementerian Pelajaran Malaysia

Tel : 03-2556900 Fax : 03-2562389

Ruj. Tuan : Ruj. Kami : KP(BS)8591/Jld.VIII / (83) Tarikh : 6 April 1995 Semua Pengarah Pendidikan Negeri Y.Bhg. Datuk/Tuan,

SURAT PEKELILING IKHTISAS BIL. 1/1995 : Keselamatan Diri Pelajar Semasa Pengajaran Pendidikan Jasmani dan Kesihatan Serta Kegiatan Kokurikulum dan Sukan Di Dalam dan Di Luar Kawasan Sekolah Sebagaimana Y.Bhg. Datuk/Tuan sedia maklum bahawa keselamatan semasa pengajaran Pendidikan Jasmani dan Kesihatan, kegiatan kokurikulum dan sukan adalah sentiasa diutamakan. Walau bagaimanapun kemalangan atau kecederaan mungkin boleh berlaku tanpa diduga. 2. Tujuan surat pekeliling ini adalah untuk memperingatkan semua guru Pendidikan Jasmani dan Sukan supaya berwaspada terhadap kemungkinan-kemungkinan yang boleh menyebabkan berlakunya kejadian yang tidak diingini ke atas diri pelajar serta mengambil langkah-langkah tertentu untuk mengelakkannya. Pada setiap masa, keselamatan pelajar semestinya menjadi pertimbangan utama. 3. Sebagai panduan, berikut adalah di antara beberapa langkah yang boleh dilaksanakan :3.1 Guru Pendidikan Jasmani dan Kesihatan 3.1.1 Menjaga keselamatan pelajar dengan rapi semasa di dalam atau di luar bilik darjah. 3.1.2 Bertanggungjawab untuk mengeluar, mengguna dan menyimpan alatalat sukan, walaupun boleh dibantu oleh pelajar. 3.2 Guru Kokurikulum dan Sukan 3.2.1 Guru-guru mestilah mengawasi dan peka terhadap keselamatan pelajar-pelajar yang melibatkan diri dalam aktiviti-aktiviti kokurikulum dan sukan. 3.2.2 Guru-guru hendaklah mempastikan semua alat sukan berada dalam keadaan baik dan selamat sebelum digunakan. 3.3 Pelajar 3.3.1 Semua pelajar mestilah mematuhi setiap arahan guru dengan sepenuhnya semasa mengikuti pelajaran Pendidikan Jasmani dan Kesihatan atau gerakerja kokurikulum dan sukan. Diilustrasi kembali oleh Ahmad Faris bin Johan, Unit ICT, Bahagian Sekolah, Kementerian Pelajaran Malaysia

3.3.2 Pelajar-pelajar yang tidak dapat mengikuti pelajaran Pendidikan Jasmani dan Kesihatan hendaklah berada bersama guru. 3.4 Stor Sukan 3.4.1 Stor sukan mestilah kemas dan tersusun. 3.4.2 Alat-alat sukan mestilah diletakkan pada tempat-tempat yang mudah diambil untuk kegunaan.

3.4.3 Setiap alat yang dikeluarkan dari stor sukan mestilah direkodkan dalam buku ’keluar dan masuk alat-alat sukan’. 3.5 Cuaca/Musim 3.5.1 Guru hendaklah menggunakan budi bicara supaya keadaan cuaca/musim tidak membahayakan pelajar sewaktu melaksanakan aktiviti. 3.6 Laporan Kemalangan atau Kecederaan 3.6.1 Sebarang kemalangan atau kecederaan yang berlaku hendaklah disiasat dan dilaporkan kepada pengetua/guru besar dengan segera. 3.7 Skim Perlindungan Diri 3.7.1 Sekolah digalakkan mengadakan skim perlindungan diri untuk semua pelajar. 3.8 Penyelenggaraan Peralatan Sukan 3.8.1 Ketua Panitia Pendidikan Jasmani dan Kesihatan/Setiausaha Sukan hendaklah memeriksa dan bertanggungjawab menyelenggara stor dan alat-alat sukan dari masa ke masa untuk mempastikan semua peralatan sukan dalam keadaan baik. 4. Y.Bhg. Datuk/Tuan adalah diminta menyebarkan kandungan surat ini kepada semua sekolah-sekolah di negeri Y.Bhg. Datuk/Tuan. Sekian, terima kasih.

“BERKHIDMAT UNTUK NEGARA” “CINTAILAH BAHASA KITA” (DATUK HAJI ABDUL TALIB BIN MD ZIN) Pengarah Bahagian Sekolah Kementerian Pendidikan Malaysia Diilustrasi kembali oleh Ahmad Faris bin Johan, Unit ICT, Bahagian Sekolah, Kementerian Pelajaran Malaysia

s.k 1. Y.B. Datuk Amar Dr. Sulaiman Haji Daud Menteri Pendidikan Malaysia 2. Y.B. Dr. Leo Michael Toyad Timbalan Menteri Pendidikan Malaysia 3. Y.B. Dr. Fong Chan Onn Timbalan Menteri Pendidikan Malaysia 4. Ketua Setiausaha Kementerian Pendidikan Malaysia 5. Ketua Pengarah Pendidikan Kementerian Pendidikan Malaysia 6. Timbalan Ketua Setiausaha I Kementerian Pendidikan Malaysia 7. Timbalan Ketua Pengarah Pendidikan I Kementerian Pendidikan Malaysia 8. Timbalan Ketua Setiausaha II Kementerian Pendidikan Malaysia 9. Timbalan Ketua Pengarah Pendidikan II Kementerian Pendidikan Malaysia 10. Semua Ketua Bahagian Kementerian Pendidikan Malaysia 11. Ketua Unit Perhubungan Awam

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