2. G OVT. OF T AMILNADU D EPARTMENT OF S CHOOL E DUCATION B RIDGE C OURSE 2011-2012 C LASS VII-M ATHS

3. M ATHEMATICIANS

4. P YTHAGORAS 569 B.C. – 475 B.C. Greece First pure mathematician 5 beliefs Secret society Pythagorean theorem

5. A RISTOTLE 384 B.C. – 322 B.C. Greece Philosopher Studied mathematics in relation to science

6. E UCLID 325 B.C. – 265 B.C. Greece Wrote The Elements Geometry today

7. A L -K HWARIZMI 780 A.D.-850 A.D. Baghdad (in Iraq) 1 st book on Algebra Algebra Natural Number Equation

8. N UMBERS

9. P ERFECT S QUARE Squares of whole numbers

10. S QUARE ROOT One of two equal factors of a number. If a squared equals b then a is the square root of b. The square root of 144 is 12 because 12 squared is 144.

11. R ADICAL S IGN The sign used to represent a nonnegative square root.

12. I RRATIONAL N UMBER Numbers that cannot be expressed as a/b.

13. R EAL N UMBER Irrational numbers together with rational numbers to form the set of all numbers.

14. N ATURAL N UMBERS 1, 2, 3, 4...

15. W HOLE N UMBERS 0, 1, 2, 3, 4, 5...

16. I NTEGERS The whole numbers and their opposites....-3, -2, -1, 0, 1, 2, 3...

17. R ATIONAL N UMBERS Any number that can be expressed in the form a/b where a and b are integers and b is not equal to 0.

18. P RIME N UMBERS Some Simple Tips and Reminders

19. D EFINITIONS Even Numbers – Any number that can be divided by 2 Odd Numbers – Any number that cannot be divided by 2 Composite Number – An integer that can be divided by at least one other number (a factor) other than itself

20. D EFINITIONS Prime Number – An integer whose only factors are 1 and itself. Eg. 3,5,7…. Factor – a number that can divide another number without a remainder. Prime Factors – an expression of numbers that divides another integer without a remainder where all the factors are prime. Composite Number – An integer which is having more than two factors. Eg 2,4,6,8,9…..

21. C OMPOSITE N UMBERS 6 24 15

22. P RIME N UMBERS 7 17 29

23. T HAT ’ S I T ! A prime number cannot be divided by anything other than 1 and itself.

24. I NTEGERS

25. The set of whole numbers and their opposites

26. F UNDAMENTAL O PERATIONS ON I NTEGERS

27. A DDING I NTEGERS

28. R ULE O NE : If the signs are the same you add and take the sign along! Ex: 4+7 = 11 -5+(-3) = -8

29. R ULE T WO : If the signs are different you subtract and take the sign of the larger! Ex: -5+3 = -2 6+(-3) = 3

30. S UBTRACTING I NTEGERS

31. S ING YOUR SONG “Same sign, add and keep Different sign subtract Take the sign of the higher Then it’ll be exact!”

32. L ET ’ S P RACTICE -6-3 = 12-5 -1-5

33. M ULTIPLYING AND D IVIDING I NTEGERS

34. R ULE O NE : If the signs are the same, the answer is positive! Ex: -6 x -3 = 18 6 x 3 = 18

35. R ULE T WO If the signs are different the answer is negative! Ex: -6 x 3 = -18 6 x -3 = -18

36. L ET ’ S P RACTICE -8 x -8 -5 x 4 6x8 -8 / -8 -20 / 4 48/8

37. D ECIMALS Place Value with Decimals

38. H OW DO I KNOW WHAT KIND OF DECIMAL IT IS ? The name of a decimal is determined by the number of places to the right of the decimal point Number of PlacesDecimal NameExample 1tenths0.7 2hundredths0.05 3thousandths0.016

39. W HAT ARE MIXED DECIMALS ? Mixed decimals are numbers with both whole numbers and decimals The name of a whole number is determined by the number of places to the l eft of the decimal point ◦I◦In the number 128.765, 1 is in the hundreds place, 2 is in the tens place, 8 is in the ones place, 7 is in the tenths place, 6 is in the hundredths place, and 5 is in the thousandths place

40. R ATIO Compare the number of boys to girls in the class.

41. The number of boys = The number of girls = If we compare boys to girls we get ___ boys to _____ girls.

42. W HAT DO WE CALL A COMPARISON BETWEEN TWO OR MORE QUANTITIES ? RATIO We just found the RATIO of boys to girls. Is the ratio of girls to boys the same ? No, when writing a ratio, ORDER matters.

43. H OW MANY BASKETBALLS TO FOOTBALLS ARE THERE ? For every 4 basketballs there are 6 footballs. The ratio is 4 to 6.

44. W HAT ARE SOME OTHER WAYS WE CAN WRITE THE RATIO OF BASKETBALL TO FOOTBALLS ? 4 to 6 4 : 6 4 6 First quantity to Second quantity First quantity : Second quantity First quantity divided by the second quantity (as a fraction). Every ratio can be written in 3 ways: Careful!! Order matters in a ratio. 4 to 6 Is NOT the same as 6 to 4

45. E QUIVALENT R ATIOS CAN BE FORMED BY MULTIPLYING THE RATIO BY ANY NUMBER. For example, the ratio 2 : 3 can also be written as ◦ 4 : 6 (multiply original ratio by by 2) ◦ 6 : 9 (multiply original ratio by by 3) ◦ 8 : 12 (multiply original ratio by by 4) The ratio 2 : 3 can be expressed as 2x to 3x (multiply the original ratio by any number x)

46. C OMPOUND R ATIOS A ratio that compares more than 2 quantities is called a compound ratio. Example: A cake recipe says the ratio of cups of milk, sugar, and batter are 1:2:4. This means that there is one cup of milk for every two cups of sugar and four cups of batter.

47. A REA F ORMULAS

48. R ECTANGLE What is the area formula?

49. R ECTANGLE What is the area formula? lb

50. R ECTANGLE What is the area formula? lb What other shape has 4 right angles?

51. R ECTANGLE What is the area formula? lb What other shape has 4 right angles?Square!

52. R ECTANGLE What is the area formula? lb What other shape has 4 right angles? Square! Can we use the same area formula?

53. R ECTANGLE What is the area formula? bh What other shape has 4 right angles? Square! Can we use the same area formula? Yes

54. P RACTICE ! Rectangle Square 10m 17m 14cm

55. T RIANGLE So then what happens if we cut a rectangle in half? What shape is made?

56. T RIANGLE So then what happens if we cut a rectangle in half? What shape is made? 2 Triangles

57. T RIANGLE So then what happens if we cut a rectangle in half? What shape is made? 2 Triangles So then what happens to the formula?

58. T RIANGLE So then what happens if we cut a rectangle in half? What shape is made? 2 Triangles So then what happens to the formula?

59. T RIANGLE So then what happens if we cut a rectangle in half? What shape is made? 2 Triangles So then what happens to the formula? bh

60. T RIANGLE So then what happens if we cut a rectangle in half? What shape is made? 2 Triangles So then what happens to the formula? bh 2

61. P RACTICE ! Triangle 5 ft 14 ft

62. P ARALLELOGRAM Let’s look at a parallelogram.

63. P ARALLELOGRAM Let’s look at a parallelogram. What happens if we slice off the slanted parts on the ends?

64. P ARALLELOGRAM Let’s look at a parallelogram. What happens if we slice off the slanted parts on the ends?

65. P ARALLELOGRAM Let’s look at a parallelogram. What happens if we slice off the slanted parts on the ends?

66. P ARALLELOGRAM Let’s look at a parallelogram. What happens if we slice off the slanted parts on the ends?

67. P ARALLELOGRAM Let’s look at a parallelogram. What happens if we slice off the slanted parts on the ends?

68. P ARALLELOGRAM Let’s look at a parallelogram. What happens if we slice off the slanted parts on the ends?

69. P ARALLELOGRAM Let’s look at a parallelogram. What happens if we slice off the slanted parts on the ends?

70. P ARALLELOGRAM Let’s look at a parallelogram. What happens if we slice off the slanted parts on the ends?

71. P ARALLELOGRAM Let’s look at a parallelogram. What happens if we slice off the slanted parts on the ends?

72. P ARALLELOGRAM Let’s look at a parallelogram. What happens if we slice off the slanted parts on the ends?

73. P ARALLELOGRAM Let’s look at a parallelogram. What happens if we slice off the slanted parts on the ends? What will the area formula be now that it is a rectangle?

74. P ARALLELOGRAM Let’s look at a parallelogram. What happens if we slice off the slanted parts on the ends? What will the area formula be now that it is a rectangle? bh

75. A NSWERS 10.8 cm 2 27 in 2 Parallelogram Rhombus 3 in 9 in 4 cm 2.7 cm

76. A NSWERS 10.8 cm 2 27 in 2 Parallelogram Rhombus 3 in 9 in 4 cm 2.7 cm

77. C ONVERSIONS OF U NITS 1cm 2 = 100mm 2 1m 2 = 10 000cm 2 1m 2 = 1 000 000mm 2

78. L ENGTH, A REA, AND V OLUME

79. A POINT IS AN OBJECT THAT HAS NO DIMENSION. No Length No Width No Height

80. A point is an object that has no dimension. No Length No Width No Height A true point cannot be seen. It is simply a location.

81. L ENGTH IS A MEASURE OF ONE DIMENSION.

82. Length is a measure of one dimension. 1234 cm

83. This line segment has a length of 4.6 cm. 1234 cm

84. Y OU CANNOT MEASURE THE LENGTH OF A LINE OR A RAY.

85. Lines and rays have infinite length.

86. P ERIMETER IS AN EXAMPLE OF LENGTH.

87. The perimeter (length around the shape) is 20 inches. 3 in 3 in + 7 in + 3 in + 7 in = 20 in 7 in 3 in 7 in

88. 3 in 7 in

89. The length of this line segment is the perimeter of the rectangle. 3 in7 in3 in7 in 20 inches in length

90. A REA IS A MEASURE OF 2 DIMENSIONS, LENGTH & WIDTH. 4 cm 6 cm

91. Area is a measure of 2 dimensions, length & width. 4 cm 6 cm

92. Area is the number of squares inside of the shape. 4 cm 6 cm

93. Area is the number of squares inside of the shape. 4 cm 6 cm This rectangle has 24 squares inside of it. The area of the rectangle is 24 square centimeters.

94. A SQUARE CENTIMETER IS A SQUARE THAT MEASURES ONE CENTIMETER ON EACH SIDE. 1234 cm

96. What did the acorn say when he grew up? Points Lines Planes Circles Polygons Congruency Similarity

97. P OINT A position in space, has no size only location D B N D, B and N represent points

98. L INE Continues without end in opposite directions A B AB represents a line

99. P LANE A flat surface that extends in four directions

100. S EGMENT Part of a line made up of 2 points and all the points of the line between the 2 points D E DE represents a segment

101. R AY Part of a line consisting of one endpoint and all the points of the line on one side of the endpoint F G FG represents a RAY

102. C LASSIFYING T RIANGLES

103. T WO W AYS TO C LASSIFY T RIANGLES By Their Sides By Their Angles

104. C LASSIFYING T RIANGLES B Y T HEIR S IDES Scalene Isosceles Equilateral

105. S CALENE T RIANGLES No sides are the same length

106. I SOSCELES T RIANGLES At least two sides are the same length

107. C LASSIFYING T RIANGLES B Y T HEIR A NGLES Acute Right Obtuse

108. A CUTE T RIANGLES Acute triangles have three acute angles

109. R IGHT T RIANGLES Right triangles have one right angle

110. O BTUSE T RIANGLES Obtuse triangles have one obtuse angle

111. C LASSIFY THIS TRIANGLE. Right Scalene

112. C LASSIFY THIS TRIANGLE. Obtuse Isosceles

113. C LASSIFY THIS TRIANGLE. Acute Scalene

114. C LASSIFY THIS TRIANGLE. Acute Isosceles

115. C LASSIFY THIS TRIANGLE. Obtuse Scalene

116. C LASSIFY THIS TRIANGLE. Right Isosceles

117. T RANSLATIONS

118. D EFINITIONS : Translation: A transformation that “slides” a shape to another location.

119. T RANSLATIONS : You “slide” a shape up, down, right, left or all the above. Notation: (x, y) ( x + 2, y - 3)

120. x y Pre-image A (-2, 4) B (-3, 2) C (-1, 1) Image A’ (3, 4) B’ (2, 2) C’ (4, 1) Transformation (x, y) (x + 5, y + 0) A B C A’ B’ C’

121. x y Pre-image A (-2, 4) B (-3, 2) C (-1, 1) Image A’ (-5, 4) B’ (-6, 2) C’ (-4, 1) Transformation (x, y) (x - 3, y + 0) A B C A’ B’ C’

122. x y Image A’ (-2, -1) B’ (-3, -3) C’ (-1, -4) Transformation (x, y) (x + 0, y - 5) A B C A’ B’ C’ Pre-image A (-2, 4) B (-3, 2) C (-1, 1)

123. x y Pre-image A (-2, 4) B (-3, 2) C (-1, 1) Image A’ (-2, 8) B’ (-3, 6) C’ (-1, 5) Transformation (x, y) (x + 0, y + 4) A B C A’ B’ C’

124. x y Pre-image A (-2, 4) B (-3, 2) C (-1, 1) Image A’ (1, 0) B’ (0, -2) C’ (2, -3) Transformation (x, y) (x + 3, y - 4) A B C A’ B’ C’

125. x y Pre-image A (-2, 4) B (-3, 2) C (-1, 1) Image A’ (3, 6) B’ (2, 4) C’ (4, 3) Transformation (x, y) (x + 5, y + 2) A B C A’ B’ C’

126. x y Pre-image A (-2, 4) B (-3, 2) C (-1, 1) Image A’ (-6, -1) B’ (-7, -3) C’ (-5, -4) Transformation (x, y) (x - 4, y - 5) A B C A’ B’ C’

127. x y Pre-image A (-2, 4) B (-3, 2) C (-1, 1) Image A’ (-4, 7) B’ (-5, 5) C’ (-3, 4) Transformation (x, y) (x - 2, y + 3) A B C A’ B’ C’

128. A LGEBRA

129. D EFINITIONS Variable – A variable is a letter or symbol that represents a number (unknown quantity). 8 + n = 12

130. D EFINITIONS A variable can use any letter of the alphabet. n + 5 x – 7 w - 25

131. D EFINITIONS Algebraic expression – a group of numbers, symbols, and variables that express an operation or a series of operations. m + 8 r – 3

132. D EFINITIONS Evaluate an algebraic expression – To find the value of an algebraic expression by substituting numbers for variables. m + 8m = 22 + 8 = 10 r – 3r = 55 – 3 = 2

133. D EFINITIONS Simplify – Combine like terms and complete all operations m = 2 m + 8 + m 2 m + 8 (2 x 2) + 84 + 8 = 12

134. W RITE A LGEBRAIC E XPRESSIONS FOR T HESE W ORD P HRASES Ten more than a number A number decrease by 5 6 less than a number A number increased by 8 The sum of a number & 9 4 more than a number n + 10 w - 5 x - 6 n + 8 n + 9 y + 4

135. W RITE A LGEBRAIC E XPRESSIONS FOR T HESE W ORD P HRASES A number s plus 2 A number decrease by 1 31 less than a number A number b increased by 7 The sum of a number & 6 9 more than a number s + 2 k - 1 x - 31 b + 7 n + 6 z + 9

136. E VALUATE EACH ALGEBRAIC EXPRESSION WHEN X = 10 x + 8 x + 49 x + x x - x x - 7 42 - x 18 59 20 0 3 32

137. C OMPLETE T HIS T ABLE nn - 3 5 10 21 32 2 7 18 29

138. C OMPLETE T HIS T ABLE xx + 6 5 10 21 32 11 16 27 38

139. P ROBLEM N UMBER O NE : E XAMPLE How many moves to make a square from these toothpicks?

140. P ROBLEM N UMBER O NE : E XAMPLE M OVE 1

143. P ROBLEM N UMBER O NE : E XAMPLE M OVE 2

145. P ROBLEM N UMBER O NE : E XAMPLE D ONE IN 2 TOTAL MOVES !!! N OW YOU TRY NUMBER 2!

146. Problem Number 2 How many moves to make the fish swim the opposite way?

147. Problem Number 2 Move One

148. Problem Number 2 Move One

149. Problem Number 2 Move One

150. Problem Number 2 Move Two

151. Problem Number 2 Move Two

152. Problem Number 2 Move Two

153. Problem Number 2 Move Three

154. Problem Number 2 Move Three

155. Problem Number 2 Move Three

156. Problem Number 2 Done in three moves!!!