Multiples, Common Multiples and Lowest Common Multiples

Resourses:

http://www.mathgoodies.com/Lessons/vol3/lcm.html

http://webmath.com/intlcm.html

http://www.mathsteacher.com.au/year8/ch01_arithmetic/03_mult/mult.htm

Multiples
The multiples of a number are its products with the natural numbers 1, 2, 3, 4, 5, ....
Example 1
1 x 8 = 8
2 x 8 =16
3 x 8 =24
4 x 8 =32
5 x 8 =40
So, the multiples of 8 are 8, 16, 24, 32, 40 and so on.
Note:
The multiples of a number are obtained by multiplying the number by each of the natural numbers.

Example 2
Write down the first five multiples of 9.
Solution:
1 x 9 = 9
2 x 9 =18
3 x 9 =27
4 x 9 =36
5 x 9 =45
The multiples of 9 are obtained by multiplying 9 with the natural numbers 1, 2, 3, 4, 5 …
So, the first five multiples of 9 are 9, 18, 27, 36 and 45.

Common Multiples
Common multiples are multiples that are common to two or more numbers.

Example 3
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 4
Find the common multiples of 3 and 4.
Solution:
Multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, …Multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, …
So, the common multiples of 3 and 4 are 12, 24, 36, …

Lowest Common Multiple
The lowest common multiple (LCM) of two or more numbers is the smallest common multiple.
Example 5
Multiples of 8 are 8, 16, 24, 32, …
Multiples of 6 are 6, 12, 18, 24, …
The LCM of 6 and 8 is 24
.
Example 6
Find the lowest common multiple of 2 and 5.
List the multiples of 5 and stop when you find a multiple of 2.
Multiples of 5 are 5, 10, …
Multiples of 2 are 2, 4, 6, 8, 10, …
The LCM of 2 and 5 is 10

Write Words on Calculators?

This is a fun activity to do when you are bored in your Math class!

Calculator Words. Type 200 Words Using A Simple Calculator - These bloopers are hilarious
CALCULATOR WORDS
Use a calculator to find these words by doing the following calculations and then turning your calculator upside down.
a. 3357 -2223 to get a place you wouldn’t go to. (………………..…….)
b. 300 000 + 18 830 to get a girl’s name. (…………………..….)
c. 85 423 + 294 496 to get something you might do if you’re happy or embarrassed about something. (…………………..….)
d. 6411 − 897 to get a noise you wouldn’t want to hear while bushwalking. (………..…………….)
e. 63 552 ÷ 64 to get something you eat. (…………………..….)
f. 203 × 15 to get something you wear. (……………..……….)
g. 52 043 ÷ 71 to get an animal. (………………..…….)
h. 52 360 ÷ 17 to get a musical instrument. (…..………………….)
i. 4417 x 8 to get some animals. (………………..…….)
j. 923 x 5 to get something you need your breath to do. (……..……………….)
k. 23 x 19 + 7301 to get something you often hear at school. (……..……………….)
l. 888 888 ÷ 2 – 65 638 to get something you shouldn’t do when you eat. (……..……..………..)
m. 4536 ÷ 81 + 261 to get something you should never do. (……………….……..)
n. 237 023 x 2 ÷ 421 x 2 + 4853 to get something you find in the garden. (……….……………..)
o. 6716 ÷ 73 x 125 136 ÷ 16 – 642 187 to get something you may find on the beach. (………………….…..)
p. 7 + 700 + 7000 + 10 000 + 300 000 + 5 000 000 to get something that you shouldn’t eat too much of. (……..……………....)

The following videos show more examples of the application of PEMDAS

COMBINED OPERATIONS
If the expression consists of parenthesis, exponents, +, –, × and ÷, then the operations MUST be performed in the following order.
Always work on the calculations within parenthesis first if any.
Next, calculate the exponents.
Then, carry out multiplication or division, working from left to right.
Lastly, do addition or subtraction, working from left to right.

The order to perform combined operations is called the PEMDAS rule.
Note: A common mnemonic for PEMDAS is Please Excuse My Dear Aunt Sally.
Example:
Evaluate 10 ÷ 2 + 12 ÷ 2 × 3
Using the PEMDAS rule, we need to evaluate the division and multiplication before subtraction and addition. It is recommended that you put in parenthesis to remind yourself the order of operation.
Solution:
10 ÷ 2 + 12 ÷ 2 × 3
= ( 10 ÷ 2) + (12 ÷ 2 × 3)
= 5 + 18
= 23

HAPPY NEW YEAR.

WELCOME BACK TO SCHOOL.

........................................................................

WHOLE NUMBERS
Divisibility Tests

Divisibility by 2
A whole number is divisible by 2 if the digit in its units position is even, (either 0, 2, 4, 6, or 8).
Examples:
The number 84 is divisible by 2 since the digit in the units position is 4, which is even.The number 333336 is divisible by 2 since the digit in the units position is 6, which is even.The number 1297000 is divisible by 2 since the digit in the units position is 0, which is even.

Divisibility by 3
A whole number is divisible by 3 if the sum of all its digits is divisible by 3.
Examples:
The number 177 is divisible by three, since the sum of its digits is 15, which is divisible by 3.The number 8882151 is divisible by three, since the sum of its digits is 33, which is divisible by 3.The number 162345 is divisible by three, since the sum of its digits is 21, which is divisible by 3.
If a number is not divisible by 3, the remainder when it is divided by 3 is the same as the remainder when the sum of its digits is divided by 3.
Examples:
The number 3248 is not divisible by 3, since the sum of its digits is 17, which is not divisible by 3. When 3248 is divided by 3, the remainder is 2, since when 17, the sum of its digits, is divided by three, the remainder is 2.
The number 172345 is not divisible by 3, since the sum of its digits is 22, which is not divisible by 3. When 172345 is divided by 3, the remainder is 1, since when 22, the sum of its digits, is divided by three, the remainder is 1.

Divisibility by 4
A whole number is divisible by 4 if the number formed by the last two digits is divisible by 4.
Examples:
The number 3124 is divisible by 4 since the number formed by its last two digits, 24, is divisible by 4.The number 1333336 is divisible by 4 since the number formed by its last two digits, 36, is divisible by 4.The number 1297000 is divisible by 4 since the number formed by its last two digits, 0, is divisible by 4.
If a number is not divisible by 4, the remainder when the number is divided by 4 is the same as the remainder when the last two digits are divided by 4.
Example:
The number 172345 is not divisible by 4, since the number formed by its last two digits, 45, is not divisible by 4. When 172345 is divided by 4, the remainder is 1, since when 45 is divided by 4, the remainder is 1.

Divisibility by 5
A whole number is divisible by 5 if the digit in its units position is 0 or 5.
Examples:
The number 95 is divisible by 5 since the last digit is 5.The number 343370 is divisible by 5 since the last digit is 0. The number 129700195 is divisible by 5 since the last digit is 5.
If a number is not divisible by 5, the remainder when it is divided by 5 is the same as the remainder when the last digit is divided by 5.
Example:
The number 145632 is not divisible by 5, since the last digit is 2. When 145632 is divided by 5, the remainder is 2, since 2 divided by 5 is 0 with a remainder of 2.
The number 7332899 is not divisible by 5, since the last digit is 9. When 7332899 is divided by 5, the remainder is 4, since 9 divided by 5 is 1 with a remainder of 4.

Divisibility by 6
A number is divisible by 6 if it is divisible by 2 and divisible by 3. We can use each of the divisibility tests to check if a number is divisible by 6: its units digit is even and the sum of its digits is divisible by 3.
Examples:
The number 714558 is divisible by 6, since its units digit is even, and the sum of its digits is 30, which is divisible by 3. The number 297663 is not divisible by 6, since its units digit is not even.The number 367942 is not divisible by 6, since it is not divisible by 3. The sum of its digits is 31, which is not divisible by 3, so the number 367942 is not divisible by 3.

Divisibility by 8
A whole number is divisible by 8 if the number formed by the last three digits is divisible by 8.
Examples:
The number 88863024 is divisible by 8 since the number formed by its last three digits, 24, is divisible by 8.The number 17723000 is divisible by 8 since the number formed by its last three digits, 0, is divisible by 8.The number 339122483984 is divisible by 8 since the number formed by its last three digits, 984, is divisible by 8.
If a number is not divisible by 8, the remainder when the number is divided by 8 is the same as the remainder when the last three digits are divided by 8.
Example:
The number 172045 is not divisible by 8, since the number formed by its last three digits, 45, is not divisible by 8. When 172345 is divided by 8, the remainder is 5, since when 45 is divided by 8, the remainder is 5.

Divisibility by 9
A whole number is divisible by 9 if the sum of all its digits is divisible by 9.
Examples:
The number 1737 is divisible by nine, since the sum of its digits is 18, which is divisible by 9.The number 8882451 is divisible by nine, since the sum of its digits is 36, which is divisible by 9.The number 762345 is divisible by nine, since the sum of its digits is 27, which is divisible by 9.
If a number is not divisible by 9, the remainder when it is divided by 9 is the same as the remainder when the sum of its digits is divided by 9.
Examples:
The number 3248 is not divisible by 9, since the sum of its digits is 17, which is not divisible by 9. When 3248 is divided by 9, the remainder is 8, since when 17, the sum of its digits, is divided by 9, the remainder is 8.
The number 172345 is not divisible by 9, since the sum of its digits is 22, which is not divisible by 9. When 172345 is divided by 9, the remainder is 4, since when 22, the sum of its digits, is divided by 9, the remainder is 4.

Divisibility by 10
A whole number is divisible by 10 if the digit in its units position is 0.
Examples:
The number 1229570 is divisible by 10 since the last digit is 0.The number 676767000 is divisible by 10 since the last digit is 0.The number 129700190 is divisible by 10 since the last digit is 0.
If a number is not divisible by 10, the remainder when it is divided by 10 is the same as the units digit.
Examples:
The number 145632 is not divisible by 10, since the last digit is 2. When 145632 is divided by 10, the remainder is 2, since the units digit is 2.The number 7332899 is not divisible by 10, since the last digit is 9. When 7332899 is divided by 10, the remainder is 4, since the units digit is 9.

Divisibility by 11
Starting with the units digit, add every other digit and remember this number. Form a new number by adding the digits that remain. If the difference between these two numbers is divisible by 11, then the original number is divisible by 11.
Examples:
Is the number 824472 divisible by 11? Starting with the units digit, add every other number:2 + 4 + 2 = 8. Then add the remaining numbers: 7 + 4 + 8 = 19. Since the difference between these two sums is 11, which is divisible by 11, 824472 is divisible by 11.
Is the number 49137 divisible by 11? Starting with the units digit, add every other number:7 + 1 + 4 = 12. Then add the remaining numbers: 3 + 9 = 12. Since the difference between these two sums is 0, which is divisible by 11, 49137 is divisible by 11.
Is the number 16370706 divisible by 11? Starting with the units digit, add every other number:6 + 7 + 7 + 6 = 26. Then add the remaining numbers: 0 + 0 + 3 + 1=4. Since the difference between these two sums is 22, which is divisible by 11, 16370706 is divisible by 11.

Divisibility by 12
A number is divisible by 12 if it is divisible by 4 and divisible by 3. We can use each of the divisibility tests to check if a number is divisible by 12: its last two digits are divisible by 4 and the sum of its digits is divisible by 3.
Examples:
The number 724560 is divisible by 12, since the number formed by its last two digits, 60, is divisible by 4, and the sum of its digits is 30, which is divisible by 3.The number 36297414 is not divisible by 12, since the number formed by its last two digits, 14, is not divisible by 4.The number 367744 is not divisible by 12, since it is not divisible by 3. The sum of its digits is 29, which is not divisible by 3, so the number 367942 is not divisible by 3.

Divisibility by 15
A number is divisible by 15 if it is divisible by 3 and divisible by 5. We can use each of the divisibility tests to check if a number is divisible by 15: its units digit is 0 or 5, and the sum of its digits is divisible by 3.
Example:
The number 7145580 is divisible by 15, since its units digit is even, and the sum of its digits is 30, which is divisible by 3.

Divisibility by 16
A whole number is divisible by 16 if the number formed by the last four digits is divisible by 16.
Examples:
The number 898630032 is divisible by 16 since the number formed by its last four digits, 32, is divisible by 16.The number 1772300000 is divisible by 16 since the number formed by its last four digits, 0, is divisible by 16.The number 339122481296 is divisible by 16 since the number formed by its last four digits, 1296, is divisible by 16.
If a number is not divisible by 16, the remainder when the number is divided by 16 is the same as the remainder when the last four digits are divided by 16.
Example:
The number 172411045 is not divisible by 16, since the number formed by its last four digits, 1045, is not divisible by 16. When 172411045 is divided by 16, the remainder is 5, since when 1045 is divided by 16, the remainder is 5.

Divisibility by 18
A number is divisible by 18 if it is divisible by 2 and divisible by 9. We can use each of the divisibility tests to check if a number is divisible by 18: its units digit is even and the sum of its digits is divisible by 9.
Examples:
The number 7145586 is divisible by 18, since its units digit is even, and the sum of its digits is 36, which is divisible by 9. The number 2976633 is not divisible by 18, since its units digit is not even.The number 367942 is not divisible by 18, since it is not divisible by 9. The sum of its digits is 31, which is not divisible by 9, so the number 367942 is not divisible by 9.

Divisibility by 20
A number is divisible by 20 if its units digit is 0, and its tens digit is even. In other words, the last two digits form one of the numbers 0, 20, 40, 60, or 80.
Examples:
The number 3351002760 is divisible by 20, since the number formed by its last two digits is 60.The number 802199730000 is divisible by 20, since the number formed by its last two digits is 0.

Divisibility by 22
A number is divisible by 22 if it is divisible by the numbers 2 and 11. We can use each of the divisibility tests to check if a number is divisible by 22: its units digit is even, and the difference between the sums of every other digit is divisible by 11.
Example:
Is the number 117524 divisible by 22? The units digit is even, so it is divisible by 2. The two sums of every other digit are 4 + 5 + 1 = 10 and 2 + 7 + 1 = 10, which have a difference of 0. Since 0 is divisible by 11, 117524 is divisible by 11. Thus, 117524 is divisible by 22, since it is divisible by both 2 and 11.

Divisibility by 25
A number is divisible by 25 if the number formed by the last two digits is any of 0, 25, 50, or 75 (the number formed by its last two digits is divisible by 25).
Examples:
The number 73224050 is divisible by 25, since its last two digits form the number 50.The number 1008922200 is divisible by 25, since its last two digits form the number 0.

Solid Geometry

Solid geometry is concerned with three-dimensional shapes. Some examples of three-dimensional shapes are cubes, rectangular solids, prisms, cylinders, spheres, cones and pyramids.

Cubes
A cube is a three-dimensional figure with six matching square sides.

The figure above shows a cube.

The dotted lines indicate edges hidden from your view.

If s is the length of one of its sides,

the Volume of the cube = s x s x s.

Since the cube has six square-shape sides,

the Surface area of a cube = 6 x s x s

Cuboid

In a cuboid, the length, width and height may be of different lengths.

The volume of the above cuboid would be the product of the length, width and height that is
Volume of rectangular solid = lwh
Surface area of rectangular solid = 2(lw + wh + lh)

Prisms
A prism is a solid that has two congruent parallel bases that are polygons. The polygons form the bases of the prism and the length of the edge joining the two bases is called the height.

A rectangular solid is a prism with a rectangle-shaped base.
The volume of a prism is given by the product of the area of its base and its height.

Cylinders
A cylinder is a solid with two congruent circles joined by a curved surface.
In the above figure, the radius of the circular base is r and the height is h.

The volume of the cylinder is the area of the base × height.

Spheres
A sphere is a solid with all its points the same distance from the center.

Cones

A circular cone has a circular base, which is connected by a curved surface to its vertex. A cone is called a right circular cone, if the line from the vertex of the cone to the center of its base is perpendicular to the base.

Pyramids
A pyramid is a solid with a polygon base and connected by triangular faces to its vertex. A pyramid is a regular pyramid if its base is a regular polygon and the triangular faces are all congruent isosceles triangles.

Symmetry / Reflection

• A plane figure is symmetrical about a line if it is divided into two identical (coincident) parts by that line. The line is called its line (or axis) of symmetry.
• A plane figure is symmetrical about a point if every line segment joining two points of the figure and passing through the point is bisected at that point. The point is called its point (or center) of symmetry.
• A plane figure has a rotational symmetry if on rotation through some angle ( 180°) about a point it looks the same as it did in its starting position.
• If A° ( 180°) is the smallest angle through which a figure can be rotated and still looks the same, then it has a rotational symmetry of order 360/A.
• The reflection (or image) of a point P in a line AB is a point P' such that AB is the perpendicular bisector of the line segment PP'.
  To find the reflection (or image) of a point P in a line ABFrom P, draw PM perpendicular to AB and produce PM to P' such that MP'= MP, then P' is the reflection (or image) of P in the line AB.
• The reflection of the point P (x, y) in the x-axis is the point P'(x, -y).
• The reflection of the point P (x, y) in the y-axis is the point P'(-x, y).
• If a point P (x, y) is rotated through 180° (clockwise or anti-clockwise) about the origin to the point P', then co-ordinates of P' are (-x,-y).
• If a point P (x, y) is rotated through 90° clockwise about the origin to the point P', then co-ordinates of P' are (y,-x).
• If a point P (x, y) is rotated through 90° anti-clockwise about the origin to the point P', then co-ordinates of P' are (-y,x).

Brain Teaser

CIRCLES

• A circle is the set of all those points, say P, in a plane, each of which is at a constant distance from a fixed point in that plane.
  The fixed point is called the center and the constant distance is called the radius.
  All radii of a circle are equal.

• A line segment joining any two points of a circle is called a chord of the circle.
  A chord of a circle passing through its center is called a diameter of the circle

• Length of diameter = 2 × radius

Exercise.
1. Fill in the blanks with correct word(s) to make the statement true:

(i) Radius of a circle is one-half of its.....

(ii) A radius of a circle is a line segment with one end point at..... and the other end on.....

(iii) A chord of a circle is a line segment with its end points....

(iv) A diameter of a circle is a chord that..... the center of the circle.

(v) All radii of a circle are.....

2. State which of the following statements are true and which are false:

(i) A line segment with its end-points lying on a circle is called a diameter of the circle.

(ii) Diameter is the longest chord of the circle.

(iii) The end-points of a diameter of a circle divide the circle into two parts; each part is called a semi-circle.

(iv) A diameter of a circle divides the circular region into two parts; each part is called a semi-circular region.

(v) The diameters of a circle are concurrent. The center of the circle is the point common to all diameters.

(vi) Every circle has unique center and it lies inside the circle.

(vii) Every circle has unique diameter.

Maths Challenge 2

7th July- 17th July

Each letter stands for a digit from 0 to 9. No two digits have the same letter code and each letter corresponds to a distinct digit. Discover the code so that the problems become true.

1. T H R E E + T H R E E + F O U R = E L E V E N

2. A B C D E x 4 = E D C B A

3. WRONG + WRONG = R I G H T

Preparation For Ujian Cermin Diri 2

Loci In Two Dimension
EXPRESS NOTES

1. The locus of a point is the path traced out by the moving point under given geometrical condition (or conditions). Alternatively, the locus is the set of all those points which satisfy the given geometrical condition (or conditions).

2. TYPES OF LOCUS
The locus of a point, which is equidistant from two fixed points, is the perpendicular bisector of the straight line joining the two fixed points.
The locus of a point, which is equidistant from two intersecting straight lines, consists of a pair of straight lines which bisect the angles between the two given lines.

EXERCISE

1. Draw and describe the locus in each of the following cases:

(i) The locus of points at a distance 2 cm from a fixed line.

(ii) The locus of points (in a plane) at a constant distance 2 cm from a fixed point in the plane.

(iii) The locus of points (in space) at a constant distance 2 cm from a fixed point.

(iv) The locus of centers of all circles passing through two fixed points.

(v) The locus of a point in the rhombus ABCD which is equidistant from the sides AB and AD.

(vi) The locus of a point in the rhombus which is equidistant from the points A and C.

(vii) The locus of center of a circle of varying radius and touching two arms of ABC.

(viii) The locus of center of a circle of varying radius and touching a fixed circle, center O, at a fixed point A on it.

(ix) The locus of center of a circle of radius 2 cm and touching a fixed circle of radius 3 cm with center O.

2. A, B and C are three collinear points.

(i) Construct the locus of point equidistant from A and B.

(ii) Construct the locus of point equidistant from B and C.

(iii) Is it possible to locate a point equidistant from A, B and C.

3. Take PQ = 6 cm. Construct

(i) the locus of points equidistant from P and Q.

(ii) the locus of points 5 cm from P.Mark the two points which lie on both loci and measure the distance between them.

4. In a triangle ABC, the right bisectors of AB and BC meet at P.

(i) Assign the special name to the point P.

(ii) Prove that PA = PB = PC.

(iii) If ABC = 90°, find the exact location of the point P with respect to the side AC.

5. Construct a triangle PQR with PQ = 5 cm, QR = 4 cm and RP = 3·6 cm. Find by construction a point P which is equidistant from the three vertices P, Q and R.

6. Construct a triangle ABC in which BC = 3·8 cm, CA = 4 cm and AB = 5·1 cm. Find by construction a point P which is equidistant from BC and AB, and also equidistant from B and C.

7. Construct a triangle ABC, in which AC = BC = 5 cm and AB = 6 cm. Locate the points equidistant from AB, AC and at a distance 2 cm from BC. Measure the distance between the two points obtained.

8. Construct a triangle ABC, in which AC = BC = 5 cm and AB = 6 cm. Locate the points equidistant from B, C and at a distance 2 cm from A. Measure the distance between the two points obtained.

9. Construct ABC = 75°. Mark a point P equidistant from AB and BC such that its distance from another line DE is 2·3 cm.

10. Draw two intersecting straight lines to include an angle of 135°. Also locate points which are equidistant from these lines and also 1·8 cm away from their point of intersection. How many such points exist?

11. AB and CD are two intersecting st. lines. Locate points which are at distances 2·5 cm and 1·8 cm from AB and CD respectively. How many such points are there?

12. Without using set square or protractor, construct the quadrilateral ABCD in which BAD = 45°, AD = AB = 6 cm, BC = 3·6 cm and CD = 5 cm.

(i) Measure BCD.

(ii) Locate the point P on BD which is equidistant from BC and CD.

13. Without using set square or protractor, construct rhombus ABCD with sides of length 4 cm and diagonal AC of length 5 cm. Measure ABC. Find the point P on AD such that PB = PC. Measure the length AP.

14. ABCD is a rhombus with side 4 cm and ABC = 120°.

Construct the locus of points inside the rhombus

(i) equidistant from A and C.

(ii) equidistant from B and D.

(iii) equidistant from A and B.

15. Draw a line segment AB of length 6 cm, M is mid point of AB. Construct:

(i) the locus of points 3 cm from AB

(ii) the locus of points 5 cm from M.

Mark two points P and Q on the same side of AB satisfying the above loci. Measure the distance between P and Q.

16. AB is a fixed line. Draw and describe the locus of the center of a circle of radius 2·5 cm and touching the line AB.

17. If the diagonals of a quadrilateral bisect each other, prove that the quadrilateral is a rhombus.

18. Using ruler and compasses only, construct a quadrilateral ABCD in which AB = 6 cm, BC = 5 cm, B = 60°, AD = 5 cm and D is equidistant from AB and BC. Measure CD.

19. Use ruler and compasses only for this question. Draw a circle of radius 4 cm and mark two chords AB and AC of the circle of length 6 cm and 5 cm respectively.

(i) Construct the locus of points, inside the circle, that are equidistant from A and C. Prove your construction.

(ii) Construct the locus of points, inside the circle, that are equidistant from AB and AC.

20. Use ruler and compasses only for the following question:Construct triangle BCP, where CB = 5 cm, BP = 4 cm, PBC = 45°.

Complete the rectangle ABCD such that

(i) P is equidistant from AB and BC, and

(ii) P is equidistant from C and D.Measure and write down the length of AB.

Answers

1. (i) A pair of straight lines parallel to the given line.

(ii) A circle with fixed point as center and radius 2 cm.

(iii) A sphere with fixed point as center and radius 2 cm.

(iv) Perpendicular bisector of the line segment joining given points.

(v) Diagonal AC of the rhombus ABCD.

(vi) Diagonal BD of the rhombus ABCD.

vii) The bisector of ABC.

(viii) The straight line passing through O and A.

(ix) Concentric circle of radius 1 cm if circles touch internally; and concentric circle of radius 5 cm if circles touch externally.

2. (iii) No

3. 8 cm

4. (i) circumcenter

(iii) Mid-point of AC

7. 4·1 cm (app.)

8. 3·4 cm (app.)

10. Four

11. Four

12. (i) 65°.

13. 78°; 1·2 cm

14. (i) Diagonal BD (ii) diagonal AC (iii) right bisector of AB

15. (i) Pair of straight lines parallel to AB and at distance 3 units on either side of AB (ii) Circle with center M and radius 5 cm; 8 cm

16. Pair of straight lines parallel to AB and at distance 2·5 cm on either side of AB18. 5·25 cm approximately

19. (i) The diameter of the circle right bisecting AC (ii) The segment of the circle bisecting BAC.

20. 5·5 cm (app.)

Cartesian Coordinate
Revision Exercise

1 . Diagram 1 shows a Cartesian plane.

Diagram 1
Based on the Cartesian plane in diagram 1;

a. state the coordinates of A, B and C

b. find the distance from A to the y-axis

c. find the midpoint of AC

d. if the midpoint of BD is (-2,0), what is the coordinate of D

2. In diagram 2, PQ = QR.

Diagram 2
Find

a) the length of PQ.

b) the coordinates of point R.

c) the midpoint between points Q and R.

3. In diagram 3, points A, B and D are three of the vertices of a rectangle.

Diagram 3

State the coordinates of the fourth vertex of the rectangle.

Coordinate System



Distance Between Two Points



Finding The Midpoint

Math Challenge 1
27th June - 3rd July
Three challenges to set you thinking: How many can you solve...........?

No. 1: "A Perfect Match"

In the diagram below, 11 matches make 3 squares:
Your challenge is to move 3 matches to show 2 squares.

No. 2: "Nenek’s age"

Mamat asked his nenek how old she was. Rather than giving him a straight answer, she replied: "I have 6 children, and there are 4 years between each one and the next. I had my first child (Pak Long Husin) when I was 19. Now the youngest one (Mak Su Jamilah) is 19 herself. That's all I'm telling you!". How old is Mamat's nenek?

No. 3: "Peas Galore"

At a school fete people were asked to guess how many peas there were in a jar. No one guessed correctly, but the nearest guesses were 163, 169, 178 and 182. One of the numbers was one out, one was three out, one was ten out and the other sixteen out.
How many peas were there in the jar?

e-mail your answers to your teacher before 4th July 2009

Baking By Numbers
Not all people are chefs, but we are all eaters. Most of us need to learn how to follow a recipe at some point. To create dishes with good flavor, consistency, and texture, the various ingredients must have a kind of relationship to one another. For instance, to make cookies that both look and taste like cookies, you need to make sure you use the right amount of each ingredient. Add too much flour and your cookies will be solid as rocks. Add too much salt and they'll taste terrible.

Ratios: Relationships between quantities
That ingredients have relationships to each other in a recipe is an important concept in cooking. It's also an important math concept. In math, this relationship between 2 quantities is called a ratio. If a recipe calls for 1 egg and 2 cups of flour, the relationship of eggs to cups of flour is 1 to 2. In mathematical language, that relationship can be written in two ways:
1/2 or 1:2
Both of these express the ratio of eggs to cups of flour: 1 to 2. If you mistakenly alter that ratio, the results may not be edible.

Working with proportion
All recipes are written to serve a certain number of people or yield a certain amount of food. You might come across a cookie recipe that makes 2 dozen cookies, for example. What if you only want 1 dozen cookies? What if you want 4 dozen cookies? Understanding how to increase or decrease the yield without spoiling the ratio of ingredients is a valuable skill for any cook.

Let's say you have a mouth-watering cookie recipe:

1 cup flour

1/2 tsp. baking soda

1/2 tsp. salt

1/2 cup butter

1/3 cup brown sugar

1/3 cup sugar1 egg

1/2 tsp. vanilla1 cup chocolate chips

This recipe will yield 3 dozen cookies. If you want to make 9 dozen cookies, you'll have to increase the amount of each ingredient listed in the recipe. You'll also need to make sure that the relationship between the ingredients stays the same. To do this, you'll need to understand proportion. A proportion exists when you have 2 equal ratios, such as 2:4 and 4:8. Two unequal ratios, such as 3:16 and 1:3, don't result in a proportion. The ratios must be equal.

Going back to the cookie recipe, how will you calculate how much more of each ingredient you'll need if you want to make 9 dozen cookies instead of 3 dozen? How many cups of flour will you need? How many eggs? You'll need to set up a proportion to make sure you get the ratios right.
Start by figuring out how much flour you will need if you want to make 9 dozen cookies. When you're done, you can calculate the other ingredients. You'll set up the proportion like this:

(1 cup flour /x cups flour ) x 3 dozen/9 dozen

You would read this proportion as "1 cup of flour is to 3 dozen as X cups of flour is to 9 dozen." To figure out what x is (or how many cups of flour you'll need in the new recipe), you'll multiply the numbers like this:

x times 3 = 1 times 9,

Therefore 3x = 9

Now all you have to do is find out the value of x. To do that, divide both sides of the equation by 3. The result is X = 3. To extend the recipe to make 9 dozen cookies, you will need 3 cups of flour. What if you had to make 12 dozen cookies? Four dozen? Seven-and-a-half dozen? You'd set up the proportion just as you did above, regardless of how much you wanted to increase the recipe.
What if your recipe has metric measurements? Find out more about the metric system in

"Meters and Liters: Converting to the Metric System of Measurements."

Source:http://www.learner.org/interactives/dailymath/cooking.html

All About Patterns
Mathematics has been called 'science of patterns'. Recognizing and describing patterns and using patterns are important mathematical skills.

A Bee Tree
Although a female honeybee has two parents, a male honeybee's ancestors reveals an interesting pattern of numbers.

1 bee

1 parent

2 grandparent

3 great-grandparent

5 greatt-great-grandparent

8 great-graet-great-grandparent

The numbers of bees in the generation: 1,1,2,3,5,8.......form a famous list of numbers knowns as the Fibonacci sequence. This number sequence has fascinated peoples for centuries because it appears so often in nature.

You can find numbers from the Fibonacci sequence in the patterns of leaves on plants, in the arrangement of scales pineapples and in the spirals of nautilus shells

The surface of a honeycomb is made up of a pattern of hexagons that fit together with no overlaps

About Fibonacci
Leonardo Fibonacci was born in Pisa, Italy, around 1175. His father was Guilielmo Bonacci, a secretary of the Republic of Pisa.His father was also a customs officer for the North African city of Bugia. Some time after 1192. Bonacci brought his son with him to Bugia.Guilielmo wanted for Leonardo to become a merchant and so arranged for his instruction in calculational techniques, escpecially those involving the Hindu - Arabic numerals which had not yet been introduced into Europe.Since Fibonacci was the son of a merchant, he was able go travel freely all over the Byzantine Empire. Merchants at the time were immuned, so they were allowed to move about freely. This allowed him to visit many of the area's centers of trade. While he was there, he was able to learn both the mathematics of the scholars and the calculating schemes in popular use, at the time.

MATHS IN HOME DECORATING

What does math have to do with home decorating? Most home decorators need to work within a budget. But in order to figure out what you'll spend, you first have to know what you need. How will you know how many rolls of wallpaper to buy if you don't calculate how much wall space you have to cover? Understanding some basic geometry can help you stick to your budget.

The word geometry literally means "to measure the Earth." Geometry is the branch of math that is concerned with studying area, distance, volume, and other properties of shapes and lines. If you need to know the distance between two points, the volume of water in a pool, the angle of a tennis serve, or how much wallpaper it will take to cover a wall, geometry holds the answers.

Figuring area: Squares and rectangles

Imagine you're planning to buy new carpeting for your home. You're going to put down carpeting in the living room, bedroom, and hallway, but not in the bathroom. You could try to guess at how much carpet you might need to cover these rooms, but you're better off figuring out exactly what you need. To determine how much carpet you'll need, you'll use this simple formula:

Area = Length x Width

This formula is used to determine the area of a rectangle or square. In the floor plan below, all of the floor space (as well as the walls and ceilings) is made up of squares or rectangles, so this formula will work for figuring the area you need to carpet.

Start by figuring the total area of the floor plan. When you're done, you can deduct the area of the bathroom, since you don't want to carpet that room. To figure out the total area of the floor plan, you'll need to know the total length and width. The total length of the floor plan shown above is 12 feet plus 10 feet, or 22 feet. The total width is 7 feet plus 5 feet, or 12 feet. Plug these numbers into your equation to get the total area of the floor plan:

A = 22feet x 12 feet = 264 sq feet

The total area of your floor plan is 264 square feet. Now you need to figure out the area of the bathroom so you can deduct it from the total area. The bathroom is 7 feet long and 5 feet wide, so it has an area of 35 square feet. Deducting the area of the bathroom from the total area (264 minus 35) leaves you with 229 square feet to carpet.

Figuring Area : Circles

Calculating how much carpet you'll need is a fairly simple task if your home has only square or rectangular rooms. But what if you have a circular alcove at the end of one room? How do you figure the area of a circle? Use this formula:

A = pi x r²

In English, this formula means "area equals pi times the radius squared." A circle's radius is one half of its diameter, or one half of what you get if you measure all the way across its widest part. "Squaring" something means you multiply it by itself. Pi is a number that roughly equals 3.14159.

If your living room has a semi-circular alcove as shown in the floor plan above, you'll need to use this additional equation to figure out its area. To figure the radius of your alcove, the number you'll need to plug into the equation, you'll divide its diameter in half. Its diameter is the same as the width of the living room: 12 feet. Half of that is its radius: 6 feet.
Let's plug in the numbers:

A = 3.14159 x (6 feet x 6 feet)

A = 113 square feet (rounded to the closest square foot)

If your alcove were a complete circle, it would have an area of 113 square feet. Because it's a half circle, its area is half of that, or 56.5 square feet. Adding 56.5 square feet to the rest of your floor plan's area of 229 square feet gives you the total area you want to carpet: 285.5 square feet. Using geometry, you can buy exactly the amount of carpet you need.

http://www.learner.org/interactives/dailymath/decorating.html