Instrumentation in mathematics

INSTRUMENTATION IN
MATHEMATICS
Compilation of PowerPoint Presentations

INSTRUCTIONAL MATERIALS
IN TEACHING FRACTIONS

INSTRUCTIONAL MATERIAL’S
NAME:
TILE BOARD
By Rizaldy A. Castro

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.

MATERIALS
• Tiles in set – 5, 6, 7, 8, 9 and10.
• Tile board

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.

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.

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.

• 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.

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.

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.

• 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.

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.

• *Note: Answers of fractions to be added or
subtracted are only proper fractions.

NAME:
FRACTION BLOCKS
By Carina Yadao Ancheta , Chezan Marie Brillo and Roselyn Udani

OBJECTIVES
The student will be able to:
• To represent fractions
• To add and subtract fractions

REPRESENTATION OF FRACTIONS
1 whole 1 /2 1 /4 1 /8

ADDITION OF FRACTIONS
1 /2 1 /2
+ =
1 whole

SUBTRACTION OF FRACTION
1 /2 1 /4
- =
1 /4

OBJECTIVES
•
•This fraction blocks is used for
adding and subtracting fractions.

HOW TO USE
This 12’’ by 6’’ box represent as
a whole and the smaller boxes
represents the fractions.

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.

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.

1. Students will add and subtract fractions.
2. Students will develop strategies for adding and subtracting
fractions using number lines.
OBJECTIVES

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

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.

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.

EXAMPLES
½ + ½ = 2/2
2/5 + 1/5 = 3/5

IN TEACHING INTEGERS
AND EQUATIONS

ISTRUCTIONAL MATERIAL’S
NAME:
BOARD CHIPS
By Rizaldy A. Castro

OBJECTIVES
• To represent positive and negative integers.
• To show how to add and subtract integers simpler.

MATERIALS
• Cork board
• Double-sided chips
• Push pins

HOW TO USE
• First, place the cork board on a wall.

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.

CHIPS REPRESENTATION
• Moreover, a pair of red and
blue chip represents zero value,
meaning, this pair is disregarded
or not counted.

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.
+ =

• 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.
+ = =

• 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.

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.

• 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.

• 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.

NAME:
NUMBER LINE
By Carina Yadao Ancheta

OBJECTIVES
Students will be able
• To add subtract multiply and divide integers.
• To classify real numbers
• To round numbers

THE YELLOW COLOR REPRESENTS A NEGATIVE
NUMBER AND THE
RED COLOR REPRESENTS A POSITIVE NUMBER.

ADDITION OF INTEGERS
• When you add integers we move the adjustment to the right
Example: 6 + 3 ?

• 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.

SUBTRACTION OF INTEGERS
• When you add integers we move the adjustment to the left.
Example: 9 - 4?

• 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.

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

- 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

DIVISION OF INTEGERS
Example:
12/4 = this expression are telling you to divide12 by 4.
How many 4’s going to 12?
Answer: 3

NATURAL NUMBERS
The numbers we count in our fingers 1, 2, 3, 4, 5……..to positive infinity.

WHOLE NUMBERS
All natural numbers including zero because makes them whole.

INTEGERS
Bringing the negative numbers within the set of integers there is the natural
numbers and whole numbers.

RATIONAL NUMBERS
In between the markings on the number line those are the fractional values.

ROUNDING OF NUMBERSLabel the tens
Round to the nearest ten
• 14 = it’s closer to the ten
• 4 = it’s closer to 0

• -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

NAME:
CARTESIAN COORDINATE
PLANE OR GRAPHING BOARD
By Chezan Marie D. Brillo

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.

There are three possible answers in solving systems of
linear equations in two variables.
DEPENDENT INCONSISTENTINDEPENDENT

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

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).

INSTRUCTIONAL MATERIAL’S NAME:
ALGEBRA BALANCE MODEL:
HANDS – ON EQUATIONS
By Roselyn T. Udani

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.

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.

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).

• 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.

NAME:
BAR MODEL
By Angeline Bumatay

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.

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.

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.

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

230 86
316
*The old man sell 316 balloons in all.

IN TEACHING GEOMETRY

THE SURFACE AREA AND
THE VOLUME OF
PYRAMIDS, PRISMS, CYLIND
ERS AND CONES

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.

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.

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.

• 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

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.

• 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.

A CYLINDER is a tube and is composed of
two parallel congruent circles and a
rectangle which base is the
circumference of the circle.

• 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

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.

 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.

PROVING THAT PYRAMID AND
CONE ARE ONE THIRD THE
VOLUME OF A PRISM AND
CYLINDER

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.

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).

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

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).

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

NAME:
TANGRAMS
By Roselyn T. Udani and Angeline P. Bumatay

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.

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.

RULES
1) Use all seven tans
2) All pieces must touched
3) And non can be overlap

NAME:
CIRCULAR ANGLE CLOCK
By Chezan Marie D. Brillo and Rachel B. Aranes

CIRCULAR
ANGLE CLOCK
By: Chezan Marie D. Brillo

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.

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.

EXAMPLES:
Right Angle
Reflex AngleStraight Angle
Obtuse AngleAcute Angle

OTHER KINDS OF ANGLES
Complementary Angle Supplementary Angle Adjacent Angle

IN TEACHING STATISTICS-
PROBABILITY
SNAKE AND LADDER
Rizaldy A. Castro - Carina Y. Ancheta - Roselyn T. Udani
Chezan Marie D. Brillo – Angeline P. Bumatay

OBJECTIVES
• To perform a game snake and ladder that shows the
probability of how often a certain part of the roulette will be
chosen.

MATERIALS
• Snake and Ladder Board
• Roulette
• Colored Chips

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.

• 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.

Instrumentation in mathematics

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