Jian earned P100 for selling junk. He gave his mother P80 and put P20 in his coin bank. The document explores percent relationships through models and examples. It examines converting between fractions, decimals, and percentages. Formulas are derived for finding the circumference of a circle based on observations of circles with different diameters. Patterns in number sequences are investigated to formulate rules for determining subsequent terms.

1. LESSON 1

2. Visualizing Percent and its Relationship to Fractions, Ratios, and Decimal
Numbers Using Models.
Explore and Discover!
JianearnedP100 for selling junk. He gave his mother P 80
and
put P 20 in his coin bank. What percent of his earnings did
he give to
his mother? What percent did he save?
Let us draw the model.

3. .80
5 00.4
4 0
0
0
0
Another way:
80% =
100
80
÷
20
20
=
5
4
The relationship of percent to decimal. To change percent to decimal move the decimal point 2
places to the left and drop the % sign.
80% = 0.80
B. Compare the shaded part in each grid.
100
25
25:100 0.25 25%
Fraction Ratio Decimal Percent
Study these.
4
1
=
4
1
x
25
25
=
100
25
’ so
4
1
=
100
25
= 0.25 = 25%
5
2
=
5
2
x
20
20
=
100
40
, so
5
2
=
100
40
= 0.40 = 40%
To change from percent to fraction, change percent to fraction with a denominator of
100, and then simplify it.
To change percent to decimal, drop the % symbol and move the decimal point 2 places
to the left.
To change decimal to percent, simply move the decimal point 2 places to the right, then
annex the symbol % to the number.

4. LESSON 2

5. Formulating the Rule in Finding the Next Term in a Sequence
Explore and Discover!
1
2
3
4 5
6
7 8 9
10
11 12 13 14
15
16 17 18 19 20
21
22 23 24 25 26 27
28
29 30
31 32 33 34 35
36
37 38 39 40
41 42 43 44
45
46 47 48 49 50
A.
1, 3, 6, 10, 15, 21, 28, 36, 45
Look for the pattern in the
sequence of encircled
numbers
The encircled numbers form a number sequence. A number sequence is a list of numbers
in which successive terms follow a rule or pattern. Each number in s sequence is called
a term.
Looking at the pattern of the encircled number, to find the 2nd
term add 2, for the
3rd
term add 3, for the 4th
term add 4, for the 5th
term add 5, for the 6th
term add 6, for the
7th
term add 7, for the 8th
term add 8, and for the 9th
term add 9. Therefore. the rule in the
sequence is (+2, +3, +4, +5, +6, +7, +8, +9).
By studying the sequence of numbers, we can find the rule governing the terms.
The rule can tell us what number will come next in the sequence.

6. B. Find the missing terms in the following situations below:
3 2 9 4 45 16 315 96
Can you find the pattern or sequence used?
The numbers inside the pentagon are multiplied by consecutive odd numbers 3, 5, 7.
Starting with 3 x 3 = 9, then 9 x 5 = 45, then 45 x 7 = 315, so the missing number in the
last pentagon is 2 835 ( 315 x 9 = 2 835 ).
The series of numbers inside the hexagon uses even numbers as factors. So, the
missing number inside the last hexagon is 768 ( 96 x 8 = 768 ).

7. LESSON 3

8. Deriving a Formula in Finding the Circumference of a Circle
Explore and Discover!
Get an empty milk can. (The top of the can represents a circle.) Wind a strip
of paper once around the can. The length of the strip is the distance
around the circle.
The distance around a circle is called circumference. Is there a relationship
between the circumference and the length of a diameter of a circle?
Look at the length of a diameter of each circle below. Each circle has been
rolled once to find its circumference. Study the table.

9. Start
A
B
1 cm
2 cm
4 cm
C

10. Circle
Length of Diameter
(d)
Estimated Circumference
(C)
Ratio of C to d
A 1 cm about 3 or about 3.16 = 3.16
B 2 cm about 6 or about 6.33 = 3.165
C 4 cm about 12 or about 12.5 = 3.125
What is the average ratio of the circumference to the length of the diameter?
From this point on, we will use the words “diameter” and “radius” to mean not only the line
segments but also the length of the line segments.