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# Instrumentation in mathematics

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### Instrumentation in mathematics

1. 1. INSTRUMENTATION IN MATHEMATICS Compilation of PowerPoint Presentations
2. 2. INSTRUCTIONAL MATERIALS IN TEACHING FRACTIONS
3. 3. INSTRUCTIONAL MATERIAL’S NAME: TILE BOARD By Rizaldy A. Castro
4. 4. OBJECTIVES • To show how fractions looks like. • To show similar fractions or fractions with the same denominators. • To show dissimilar fractions or fractions with not the same denominators. • To show how to add and subtract fractions simpler.
5. 5. MATERIALS • Tiles in set – 5, 6, 7, 8, 9 and10. • Tile board
6. 6. HOW TO USE • The tile board is divided into two. • The upper division will represent the numerator while the lower division will represent the denominator. The denominator is based on what set of tiles will be placed, either 6,7,8,9 or 10.
7. 7. SHOWING HOW FRACTIONS LOOKS LIKE • Example 1, to show 5 over 6 or 5/6, put the two six-divided sheets on the two divisions. Place 5 tiles out from the set of 6 tiles on the upper division, which will be the numerator, and place another set of 6 tiles fitted on the lower division, which will be the denominator. • Example 2, to show 8 over 9 or 8/9, put the two six-divided sheets on the two divisions. Place 8 tiles out from the set of 9 tiles on the upper division, which is the numerator, and place another set of 9 tiles fitted on the lower division, which will be the denominator.
8. 8. SHOWING SIMILAR FRACTIONS OR FRACTIONS WITH THE SAME DENOMINATOR • Example 1, to show 7/8 and 3/8, put first the two eight-divided sheets on the two divisions. Then place 7 tiles out from the set of 8 tiles on the upper division, and 3 tiles out from another set of 8 tiles. The filled parts on the two eight-divided sheets will be the numerators. And the numbers of the division of the sheets, which is 8, will be the denominator.
9. 9. • Example 2, to show 9/10 and 5/10, put first the two ten-divided sheets on the two divisions. Then place 9 tiles out from the set of 10 tiles on the upper division, and 5 tiles out from another set of 10 tiles. The filled parts on the two ten- divided sheets, which are 9 and 5, will be the numerators. And the numbers of the division of the sheets, which is 10, will be the denominators.
10. 10. SHOWING DISSIMILAR FRACTIONS OR FRACTIONS WITH NOT THE SAME DENOMINATORS • Example 1, to show 4/9 and 4/5, put first the nine-divided sheet on the upper division and the five-divided sheet on the lower division. Then place 4 tiles out from the set of 9 tiles on the upper division, and place also 4 tiles out from the set of 5 tiles on the lower division. The filled parts on each sheet represent the numerators. On the other hand, the numbers of the division of the sheets, represent the denominator. • Example 2, to show 2/7 and 2/6, put first the seven-divided sheet on the upper division and the six-divided sheet on the lower division. Then place 2 tiles out from the set of 7 tiles on the upper division, and place also 2 tiles out from the set of 6 tiles on the lower division. The filled parts on each sheet represent the numerators. On the other hand, the numbers of the division of the sheets, represent the denominator.
11. 11. SHOWING HOW TO ADD SIMILAR FRACTIONS • Example 1, to show how to add 3/8 and 4/8, first, put the two eight- divided sheets on the two divisions. Place 3 tiles out from the set of 8 tiles on the upper division then place also 4 tiles out from another set of 8 tiles on the lower division. Afterwards, place the tiles from the upper division to the lower division. Count how many tile you have in all, which is 7. Then the answer is 7/8.
12. 12. • Example 2, to show how to add 7/10 and 2/10, first, put the two ten-divided sheets on the two divisions. Place 7 tiles out from the set of 10 tiles on the upper division then place also 2 tiles out from another set of 10 tiles on the lower division. Afterwards, place the tiles from the upper division, which is 7 tiles, to the lower division. Count how many tile you have in all, which is 9. Then the answer is 9/10.
13. 13. SHOWING HOW TO SUBTRACT SIMILAR FRACTIONS • Example 1, to show how to subtract 5/9 from 7/9, first, put the two nine- divided sheets on the two divisions. Place 7 tiles out from the set of 9 tiles on the upper division then place also 5 tiles out from another set of 9 tiles on the lower division. Tiles must start from left to right. Afterwards, remove those tiles from the lower division together with the corresponding or aligned tiles from the upper division. The number of tiles left is the answer. So, in this case, the number of tiles left is 2. Therefore, the answer is 2/9. • Example 2, to show how to subtract 3/6 from 6/6, first, put the two six-divided sheets on the two divisions. Place 6 tiles out from the set of 6 tiles on the upper division then place also 3 tiles out from another set of 6 tiles on the lower division. Tiles must start from left to right. Afterwards, remove those tiles from the lower division together with the corresponding or aligned tiles from the upper division. The number of tiles left is the answer. So, in this case, the number of tiles left is 3. Therefore, the answer is 3/6.
14. 14. • *Note: Answers of fractions to be added or subtracted are only proper fractions.
15. 15. INSTRUCTIONAL MATERIAL’S NAME: FRACTION BLOCKS By Carina Yadao Ancheta , Chezan Marie Brillo and Roselyn Udani
16. 16. OBJECTIVES The student will be able to: • To represent fractions • To add and subtract fractions
17. 17. REPRESENTATION OF FRACTIONS 1 whole 1 /2 1 /4 1 /8
18. 18. ADDITION OF FRACTIONS 1 /2 1 /2 + = 1 whole
19. 19. SUBTRACTION OF FRACTION 1 /2 1 /4 - = 1 /4
20. 20. FRACTION BLOCKS
21. 21. OBJECTIVES • •This fraction blocks is used for adding and subtracting fractions.
22. 22. HOW TO USE This 12’’ by 6’’ box represent as a whole and the smaller boxes represents the fractions.
23. 23. 1/2 1/3 1/4 1/6 1/12
24. 24. ADDITION In adding, just let the indicated fractions put in the fraction box and combined them. Put the first fraction as it labeled to the bigger box and combined the second box. The answer is seen on the labeled box.
25. 25. EXAMPLE + = 1/4 1/2 3/4
26. 26. SUBTRACTION The first fraction should be larger than the second fraction. In subtracting, let the first fraction be at the back and the second fraction should be in front of the first fraction. The fraction should be labeled to what is indicated and the second fraction should be placed in front of the first fraction, it must be end to end. The answer will be the remaining parts of the first fraction.
27. 27. EXAMPLE - = 1/2 1/4 1/4
28. 28. LIMITATIONS
29. 29. 1. Students will add and subtract fractions. 2. Students will develop strategies for adding and subtracting fractions using number lines. OBJECTIVES
30. 30. In adding fractions you just get the made representation of the 2 given fractions and insert it where the number lines is located and you can see the answer on the number lines. In subtracting fractions you just get the made representation of the 2 given fractions and insert first the highest fraction and put the lower fraction under the highest fraction and you can see the answer on the number lines. HOW TO USE
31. 31. Example 1.
32. 32. Example 2.
33. 33. By Angeline P. Bumatay
35. 35. OBJECTIVES • The student will be able to: a)Define units, common denominator, simplify and lowest terms. b)Recognize that only the numerators should be added, not the denominators. c)Describe the procedure for adding fractions with like denominators.
36. 36. HOW TO USE • Put chips in the pockets. Make sure that the denominators are the same and put the yellow chips as your denominator in the lower pocket. And the pink chips will be your numerators representation and put it into the upper pocket. And count the chips in the upper pocket and that is your numerator as well your denominator. If the numerator and denominator are the same just put 1 in the answer.
37. 37. EXAMPLES ½ + ½ = 2/2 2/5 + 1/5 = 3/5
38. 38. INSTRUCTIONAL MATERIALS IN TEACHING INTEGERS AND EQUATIONS
39. 39. ISTRUCTIONAL MATERIAL’S NAME: BOARD CHIPS By Rizaldy A. Castro
40. 40. BOARD CHIPS
41. 41. BOARD CHIPS
42. 42. OBJECTIVES • To represent positive and negative integers. • To show how to add and subtract integers simpler.
43. 43. MATERIALS • Cork board • Double-sided chips • Push pins
44. 44. HOW TO USE • First, place the cork board on a wall.
45. 45. CHIPS REPRESENTATION • The chip has two sides – a blue one and a red one. The blue side of the chip represents the positive integer while the red one represents the negative integer.
46. 46. CHIPS REPRESENTATION • Moreover, a pair of red and blue chip represents zero value, meaning, this pair is disregarded or not counted.
47. 47. ADDING INTEGERS • Positive. To add positive integers, let say positive 4 plus positive 2, simply pin 4 positive chips and 2 positive chips on the board and then combine them. Count how many chips you have. In this situation, you have 6 positive chips. Therefore, the answer is +6. + =
48. 48. • Positive and Negative. To add positive and negative integers, let say positive 3 plus negative 2, simply pin 3 positive chips and 2 negative chips on the board then simply combine them. First, count how many pairs of zero values you have then disregard them. Count how many chips remain. In this situation, you only have 1 positive chip remained. Therefore, the answer is +1. *The sign of the answer depends on what color of chip/s was/were left on the board. + = =
49. 49. • Negative. To add two negative integers, let say negative 5 plus negative 8, simply pin 5 negative chips and 8 negative chips on the board then simply combine them. Count how many negative chips you have in all on the board. In this situation, you have 13 negative chips. Therefore, the answer is -13.
50. 50. SUBTRACTING INTEGERS • *In subtracting integers, first change the sign of the subtrahend then proceed to addition. • Positive. Let say, positive 3 minus positive 10. Positive 10 will become negative 10. Just flip the chips and pin it again. Then we will add negative 10 to positive 3. Positive 3 and negative 10, when combined, we have 3 pairs of zero values, and what were left are 7 red chips. Therefore, the answer is negative 7.
51. 51. • Positive and Negative. Let say positive 6 minus negative 8. Negative 8 will become positive 8. Now, add positive 6 and positive 8. The answer is positive 14.
52. 52. • Negative. Let say negative 5 minus negative 3. Negative 3 will become positive 3. Then add negative 5 and negative 3. We can observe that we have 3 pairs of zero values, therefore, they are disregarded. Count how many chips was left. We have 2 negative chips left, so the answer is negative 2.
53. 53. INSTRUCTIONAL MATERIAL’S NAME: NUMBER LINE By Carina Yadao Ancheta
54. 54. OBJECTIVES Students will be able • To add subtract multiply and divide integers. • To classify real numbers • To round numbers
55. 55. THE YELLOW COLOR REPRESENTS A NEGATIVE NUMBER AND THE RED COLOR REPRESENTS A POSITIVE NUMBER.
56. 56. ADDITION OF INTEGERS • When you add integers we move the adjustment to the right Example: 6 + 3 ?
57. 57. • In equation: -5+ x = 2 this equation simply asking you ―What number do you add to -5 to get 2? • Count the number of units to the right therefore x is equal to 7.
58. 58. SUBTRACTION OF INTEGERS • When you add integers we move the adjustment to the left. Example: 9 - 4?
59. 59. • In equation: 5+ x = -2 this equation simply asking you ―What number do you subtract to 5 to get -2? • Count the number of units to the right therefore x is equal to -7.
60. 60. MULTIPLICATION OF INTEGERS N x M =? N = what number you will add M= how many times you will add Example: 3x 4 = 3 + 3 +3 + 3 = 12
61. 61. - 2 x 3 = -2 + -2 + -2 = -6 -2 x-2 = get the opposite numbers -2=2 and -2 =2 2 x 2 = 2+ 2 = 4
62. 62. DIVISION OF INTEGERS Example: 12/4 = this expression are telling you to divide12 by 4. How many 4’s going to 12? Answer: 3
63. 63. CLASSIFYING REAL NUMBERS
64. 64. NATURAL NUMBERS The numbers we count in our fingers 1, 2, 3, 4, 5……..to positive infinity.
65. 65. WHOLE NUMBERS All natural numbers including zero because makes them whole.
66. 66. INTEGERS Bringing the negative numbers within the set of integers there is the natural numbers and whole numbers.
67. 67. RATIONAL NUMBERS In between the markings on the number line those are the fractional values.
68. 68. ROUNDING OF NUMBERSLabel the tens Round to the nearest ten • 14 = it’s closer to the ten • 4 = it’s closer to 0
69. 69. • -16= it’s closer to -20 • 15 = Notice that its 5 closer to both sides = once you go to the middle you move to the right 15 = 20
70. 70. INSTRUCTIONAL MATERIAL’S NAME: CARTESIAN COORDINATE PLANE OR GRAPHING BOARD By Chezan Marie D. Brillo
71. 71. This Cartesian Coordinate Plane or Graphing Board will be used in the solving systems of linear equations. Students will be able to present the graph systems of linear equations in two variables. They will be able to identify if the graph is inconsistent, dependent and independent systems of linear equations.
72. 72. There are three possible answers in solving systems of linear equations in two variables. DEPENDENT INCONSISTENTINDEPENDENT
73. 73. EXAMPLE y = x + 1 y = -2x + 1 We can easily solve the equation if we let the x and y be zero. y = x + 1 y= 0, x = -1 x = 0, y = 1 y = -2x + 1 y = 0, x = ½ x = 0, y =1
74. 74. This would be the graph of y = x + 1 and y + -2x +1. y = -2x + 1y = x + 1 (0, 1) is the solution of the equation The graphs has a dependent solution (0, 1).
75. 75. INSTRUCTIONAL MATERIAL’S NAME: ALGEBRA BALANCE MODEL: HANDS – ON EQUATIONS By Roselyn T. Udani
76. 76. OBJECTIVES • The students will be able to learn: • how to evaluate an expression and how to combine like terms • how to balance algebra equations (using the subtraction property of equality) • the ability to solve one and two-step algebra equations • To solve equations with unknowns variables on both sides.
77. 77. HOW TO USE Students use a white pawn for the unknown (for the 'x') and cubes for the constants. White cubes for positive integer and black cubes for negative integer. From level II on, they use a black pawn for (-x), also called a "star", and denoted with x. The equations are modeled on the balance so that the left side of the equation goes on the left side of the balance, and similarly for the right side.
78. 78. Students are instructed about "legal moves" with which to "play" with the equations until they arrive to the solution. The legal moves of course correspond to the regular principles used in algebra. For example: • In level I, students are told they can remove the same number of pawns from both sides, and the scale will still balance. • In level II, students are instructed that a pawn and a "star" (the black pawn) are opposites, canceling each other. (Pawn corresponds to x, and star to -x).
79. 79. • In level III, students are taught about a "convenient zero"— essentially adding (x + (-x)) to one of the sides, after which it is possible to remove pawns or stars from both sides, whichever the need might be.
80. 80. Example
81. 81. INSTRUCTIONAL MATERIAL’S NAME: BAR MODEL By Angeline Bumatay
82. 82. BAR MODEL
83. 83. OBJECTIVES: To be able the students learn to represent simple and multi-step word problems by drawing. To enable students to solve difficult math problems and learn how to think symbolically.
84. 84. HOW TO USE In the model area, the bars are provided at the outset, and the student must drag them into position. Question marks are used to indicate what is unknown. The arrangement and labelling of the bars and lines help students understand what they know and what they need to find out.
85. 85. 8 20
86. 86. The first arrangement, by including 2 of the smaller number in the model, allows us to see that two of the smaller number, added to the difference (8), will give us the sum of the two numbers (20). After the model is set up, it functions as a bridge to algebra. A blue block can be labelled b, and from there, we can write equations to express what is shown in the model and solve for a.
87. 87. EXAMPLE: A man sold 230 balloons at a fun fair in the morning. He sold another 86 balloons in the evening. How many balloons did he sell in all? 230 86 X
88. 88. 230 86 316 *The old man sell 316 balloons in all.
89. 89. INSTRUCTIONAL MATERIALS IN TEACHING GEOMETRY
90. 90. THE SURFACE AREA AND THE VOLUME OF PYRAMIDS, PRISMS, CYLIND ERS AND CONES
91. 91. OBJECTIVES • 1. To Solve for the surface area of a given solid figures. • 2. To Compute for the volume of the given solid figures. • 3. To illustrate the relationship between the volume of prism & pyramid, and cone & cylinder.
92. 92. SURFACE AREA is the area that describes the material that will be used to cover a geometric solid. When we determine the surface areas of a geometric solid we take the sum of the area for each geometric form within the solid. VOLUME is a measure of how much a figure can hold and is measured in cubic units. The volume tells us something about the capacity of a figure.
93. 93. A PRISM is a solid figure that has two parallel congruent sides that are called bases that are connected by the lateral faces that are parallelograms. There are both rectangular and triangular prisms.
94. 94. • To find the surface area of a prism (or any other geometric solid) Measure each side of the prism Find the area of the base and its lateral faces. Add the areas of each geometric form. • To find the volume of a prism (it doesn't matter if it is rectangular or triangular) we multiply the area of the base, called the base area B, by the height h. V= BH where B = Area of the base H= Height of the prism
95. 95. A PYRAMID consists of three or four triangular lateral surfaces and a three or four sided surface, respectively, at its base. When we calculate the surface area of the pyramid below we take the sum of the areas of the 4 triangles area and the base square. The height of a triangle within a pyramid is called the slant height.
96. 96. • To find the surface area of a prism (or any other geometric solid) Measure each side of the prism Find the area of the base and its lateral faces. Add the areas of each geometric form. • The volume of a pyramid is one third of the volume of a prism. V= 1/3 BH where B = Area of the base H= Height of the prism.
97. 97. A CYLINDER is a tube and is composed of two parallel congruent circles and a rectangle which base is the circumference of the circle.
98. 98. • To find the Surface Area of the Cylinder The circumference of a circle +The area of one circle + The area of the rectangle. To find the volume of a cylinder we multiply the base area (which is a circle) and the height h. Where: ∏ = 3.14 (constant) r = the radius half of the diameter then square it h = height of the cylinder
99. 99. CONE. The base of a cone is a circle and that is easy to see. The lateral surface of a cone is a parallelogram with a base that is half the circumference of the cone and with the slant height as the height. This can be a little bit trickier to see, but if you cut the lateral surface of the cone into sections and lay them next to each other it's easily seen.
100. 100.  The surface area of a cone is thus the sum of the areas of the base and the lateral surface: • Area of the base The area of one circle Area of the lateral surface where l is the slant height  The volume of a cone is one third of the volume of a cylinder.
101. 101. PROVING THAT PYRAMID AND CONE ARE ONE THIRD THE VOLUME OF A PRISM AND CYLINDER
102. 102. CONES AND CYLINDERS with the same base area and height have a unique relationship. The same relationship exists between pyramids and prisms with the same base area and height. In the following activity you will look for a relationship between these shapes.
103. 103. I. CONES AND CYLINDERS • Fill the cone with sand and pour into the cylinder. • Repeat until the cylinder is filled to the top. Since it takes 3 cones to fill 1 cylinder, the volume of a cone is 1/3 the volume of a cylinder (see figure below).
104. 104. TO FIND THE VOLUME OF A CONE, FIND THE AREA OF THE BASE (THE CIRCLE), MULTIPLY BY THE HEIGHT AND THEN DIVIDE BY 3. V = BH B = AREA OF THE BASE 3 H = HEIGHT OF THE CONE
105. 105. II. PRISMS AND PYRAMIDS Does the same relationship exist between the square prism and the square pyramid with the same base area and height? • Fill the pyramid with sand and pour into the prism. • Repeat until the prism is filled to the top. • Since it takes 3 pyramids to fill 1 prism, the volume of a pyramid is 1/3 the volume of a prism (see figure below).
106. 106. TO FIND THE VOLUME OF A PYRAMID, FIND THE AREA OF THE BASE, MULTIPLY BY THE HEIGHT AND THEN DIVIDE BY THREE. V = BH B= AREA OF THE BASE 3 H= HEIGHT OF THE PYRAMID
107. 107. INSTRUCTIONAL MATERIAL’S NAME: TANGRAMS By Roselyn T. Udani and Angeline P. Bumatay
108. 108. TANGRAMS • Tangrams are an ancient Chinese Mathematical Puzzle. There are 7 pieces in a Tangram Puzzle (5 triangles and 2 quadrilaterals), and the idea is to make different shapes using ALL SEVEN PIECES.
109. 109. OBJECTIVES 1) To form a specific shape (given only in outline or silhouette) using all seven pieces, which may do not overlap. • 2) To find for the perimeter and the area of the formed polygons.
110. 110. RULES 1) Use all seven tans 2) All pieces must touched 3) And non can be overlap
111. 111. EXAMPLE 1
112. 112. EXAMPLE 2
113. 113. INSTRUCTIONAL MATERIAL’S NAME: CIRCULAR ANGLE CLOCK By Chezan Marie D. Brillo and Rachel B. Aranes
114. 114. CIRCULAR ANGLE CLOCK By: Chezan Marie D. Brillo
115. 115. OBJECTIVES • This circular angle clock will be used in demonstrating angles. The following objectives will be obtained in discussing angles. • a. To identify and present what angle is and it’s parts. • b. To show the students what are the kinds of an angles with their measurements. • c. To determine the other kinds of angles by adding two angles.
116. 116. HOW TO USE • Circular Angle clock is composed of 0 degrees to 360 degrees with two hands which the smallest hand represents the initial side and the longer hand represents the terminal side and the vertex that connects the two hands. • For angles there are five kinds the acute, right, obtuse, straight and reflex.
117. 117. EXAMPLES: Right Angle Reflex AngleStraight Angle Obtuse AngleAcute Angle
118. 118. OTHER KINDS OF ANGLES Complementary Angle Supplementary Angle Adjacent Angle
119. 119. INSTRUCTIONAL MATERIALS IN TEACHING STATISTICS- PROBABILITY SNAKE AND LADDER Rizaldy A. Castro - Carina Y. Ancheta - Roselyn T. Udani Chezan Marie D. Brillo – Angeline P. Bumatay
120. 120. OBJECTIVES • To perform a game snake and ladder that shows the probability of how often a certain part of the roulette will be chosen.
121. 121. MATERIALS • Snake and Ladder Board • Roulette • Colored Chips
122. 122. HOW TO USE • The following are the rules and regulations of the game: • Each player will get his own chip. The players’ chips must be different from one another. • Each player will start from zero as the starting point. • The roulette has two circles – the big one and the small one. The big circle is for the steps on how far the player will move upward while the small circle is on how far the player will move downward. • If the player is still at 10 and below, he must not spin yet the small circle.
123. 123. • Whenever a player stops on a number where there is a ladder, he must follow the ladder upward until where it stops but whenever a player stops on a number where there is a snake’s mouth, he must follow the snake downward until where it stops. • On the big circle, whenever a player chooses ‘Twice’ as he spins it, he must make another spin on that big circle to determine what number will he twice as his steps upward. Afterwards, he will move to the small circle for his steps downward. • Whenever a player is at 91-99, he will no longer spin the small circle. But whenever he is near the finishing point, which is 100, let say 98 and he chooses 5, he must move 2 steps upward and move 3 steps downward to make that 5 steps he has chosen. Or in other words, his excess moves will be his downward moves.