PASIG CATHOLIC COLLEGE Grade School Department School

Transcription

PASIG CATHOLIC COLLEGE
Grade School Department
School Year 2015-2016
MATHEMATICS 5
THIRD QUARTER
Activity Sheet No. 1
TYPE OF ACTIVITY: Concept Development
TOPIC
LEARNING OBJECTIVES
: Place Value and Value of Decimals
: Identify the place value and value of each decimal digit.
Read and write decimals .
REFERENCE
: Soaring 21st Century Mathematics 2nd Edition, pp. 84-87
Manuel T. Kotah, et al.
Math for Life 5 Revised Edition, p. 106 - 109
Adelaida Celeridad-Wright & Adela C. Villamayor
CONCEPT NOTES
:
A decimal is a number less than one that represents units divided equally into powers of
ten of which is separated by a decimal point from the whole number.
Whole
Numbers
Ones
1
4
Decimal
Point
Recall: Place Value System
.
Decimals
Tenths
Hundredths
Thousandths
1/10
6
1/100
7
1/1000
2
Ten
thousandths
1/10 000
5
Hundred
thousandths
1/100 000
3
Example:
1.)
In
1.)
2.)
3.)
4.)
4.67253
reading decimals the standard way, follow these steps:
Read the whole numbers as it is,
Read the decimal point as “and”,
Read the decimal digits like that of a whole number,
State the place value of the last decimal digit.
Therefore, 4.672 is read as four and sixty-seven thousand two hundred fifty-three
hundred thousandths.
In cases where the whole number is zero (0), steps 1 and 2 should be omitted.
2.)
0.5913 is read as five thousand nine hundred thirteen ten thousandths.
Note: To emphasize the decimal point in numbers less than one, a zero should be written as the
digit in the ones place.
MATHEMATICS 5
THIRD QUARTER
TYPE OF ACTIVITY: Computational Skills
TOPIC
LEARNING OBJECTIVES
REFERENCE
AUTHOR/S
CONCEPT NOTES
:
:
:
:
:
Fractions to Decimals
Rename fractions as decimals.
Math for Life 5 Revised Edition, p. 172
A fraction can be changed to decimal easily if the denominator is a power of ten. However, not
all fractions has a denominator which is a power of ten.
Case 1: If the denominator of the simplest form is a factor of any power of ten, change the
fraction to an equivalent fraction with a denominator of a power of ten of which it is a factor.
Simplifying the given fraction first may also help.
1.)
2.)
!
!
!
!
à
à
!
!
!
!
×
!"
!"
=
!"
𝟏𝟎𝟎
!
!"
!
𝟏𝟎
× =
(seventy-five hundredths). Therefore,
=2
!
!
!
(two and five tenths). Therefore,
!"
!
!
= 0.75
= 2.5
[Hint: A fraction can be changed to an equivalent fraction if the denominator of its simplest form
is a power of 2 or a power of 5.]
Case 2: If the fraction cannot be changed to an equivalent fraction that has a denominator of
any power of ten, divide the numerator by the denominator.
3.)
!
!
Note: If the divisor is larger than the dividend, affix a decimal to the dividend without changing
its value. Thus, affix zero (0) every time a number can’t be divided. Then, divide the dividend
like whole numbers. Secure the ones digit with zero (0) to emphasize that the number is less
than one. This also proves that any proper fraction is always less than one. 0.44
9 4.00
- 36
40
- 36
,
Notice that the digits in the
quotient are repeating and
seems to be endless when
divided continuously. This is an
example of a repeating and nonterminating decimal.
MATHEMATICS 5
THIRD QUARTER
TYPE OF ACTIVITY: Mathematical Investigation
TOPIC
LEARNING OBJECTIVES
CONCEPT NOTES
: Changing Fraction to Decimals
: Hypothesize how a fraction is changed to decimal with denominators
9, 99, 999...
:
Decimals are classified into 4 kinds:
Non-repeating and Terminating Decimal – the digits do not repeat or occur in a pattern and
has a definite number of place value.
Example: 0.129, 0.36162, 1.0986
Repeating and Terminating Decimal – the digits are repeated and has a definite number of
place values.
Example: 0.555, 0.767676, 0.234234
Non-repeating and Non-Terminating Decimal – the digits do not repeat and has infinite
number of place values.
Example: the value of pi (π) = 3.14159265358979323846264338327...
Repeating and Non-Terminating Decimal – the digits do not repeat and has infinite number
of place values.
Example: 0.33333333... = 0.3
(the bar on top of the digit
0.84848484… = 0.84
signifies that the digits under it
are being repeated in a pattern) PASIG CATHOLIC COLLEGE
MATHEMATICS 5
THIRD QUARTER
TOPIC
LEARNING OBJECTIVES
REFERENCE
AUTHOR/S
CONCEPT NOTES
:
:
:
:
:
Decimals to Fractions
Rename decimals as fractions in its simplest form.
Math for Life 5 Revised Edition, p. 171 - 172
Since a fraction can be changed to a decimal, then a decimal can be changed back to fraction.
To change a decimal to fraction, write the fraction the way you read its decimal equivalent.
Whenever possible, reduce the fraction to its simplest form.
Example:
1.)
0.45 (read as forty-five hundredths similar to
0.45 =
𝟒𝟓
!"
!""
)
𝟏𝟎𝟎
Simplify the fraction if necessary by dividing the numerator and denominator by their GCF;
!" ÷ !
!
𝟗
= . Therefore, 0.45 = .
!"" ÷ !
!"
𝟐𝟎
2.)
4.056 (read as four and fifty-six thousandths similar to
4.056 =
𝟒
𝟓𝟔
4
56
)
1000
𝟏𝟎𝟎𝟎
Simplify the fraction if necessary by dividing the numerator and denominator by their GCF;
!" ÷ !
!
𝟕
=
. Therefore, 4.056 = 4
.
!""" ÷ !
!"#
𝟏𝟐𝟓
MATHEMATICS 5
THIRD QUARTER
TOPIC
LEARNING OBJECTIVES
REFERENCE
: Rounding Off Decimals
: Round off decimals to the indicated palce value.
: Math for Life 5 Revised Edition, pp. 114 - 116
CONCEPT NOTES
:
To round off decimals,
Find your digit,
Look right next door,
Five or more, add one more,
Four or less, just ignore.
Example:
(locate the digit being rounded)
(look at the digit to the right of the place being rounded
to.)
(if the digit is 5 or more, add 1 to the digit being rounded
remove the digits to its right)
(if the digit is 4 or less, let the digit remain as it is and
remove the digits to its right.)
Round off 34.9673 to the nearest hundredths
6 is the digit to be rounded.
Since the digit to its right is greater than 4, then we add 1 to 6.
34.9683 ≈ 34.97
MATHEMATICS 5
THIRD QUARTER
TOPIC
LEARNING OBJECTIVES
REFERENCE
: Comparing and Ordering Decimals
: Compare and arrange decimals in descending and ascending order.
: Soaring 21st Century Mathematics 5, p. 99
Mauel T. Kotah, et al.
Math for Life 5 Revised Edition, pp. 121 - 123
CONCEPT NOTES
:
To compare decimals, compare the digits in the same place value beginning at the left
until you find a difference. The number with a greater digit in the same place value is the
greater number, otherwise, the lesser number.
Example: 8.43 ___ 8.49
a.
Compare the whole numbers.
8.43 ___ 8.49
same value
b.
Compare the digits in the tenths place
8.43 ___ 8.49
same value
c.
Compare the digits in the hundredths place
8.43 ___ 8.49
3 is less than 9
Therefore, 8.43 < 8.49
In arranging decimals, compare all the decimals in a given set. To arrange in ascending
order means to start with least to greatest. To arrange in descending order means to start with
greatest to least.
Example: 0.845, 0.019, 0.892, 0.91
Tip: Arrange the numbers horizontally to easily compare each digit with the same place value.
0.845
0.845
0.845 3rd
0.019
0.019
0.019 4th
0.892
0.892
0.892 2nd
0.91
0.911st
0.91 1st
descending order: 0.91, 0.892, 0.845, 0.019 ascending order: 0.019, 0.845, 0.892, 0.91
MATHEMATICS 5
THIRD QUARTER
TOPIC
LEARNING OBJECTIVES
REFERENCE
: Addition and Subtraction of Decimals
: Perform addition and subtraction of decimals.
: Soaring 21st Century Mathematics 5, p. 99
CONCEPT NOTES
:
When adding or subtracting decimals, the alignment of the digits according to its place
value must be observed. To do this, simply align the decimal points and the place values will
follow. Annex zeroes to the right as place holders if needed. Then, add or subtract in the same
way as whole numbers.
Example
What is the sum of 8.529 and 7.95?
1
8.529
+ 7.95
8.529
+ 7.950
9
8.529
+ 7.950
79
1
8.529
8.529
+ 7.950
+ 7.950
.479
16.479
What is 9.526 subtracted form 9.8?
791
9.8
– 9.526
9.800
– 9.526
4
791
9.800
– 9.526
74
791
9.800
– 9.526
.274
791
9.800
– 9.526
0.274
<
MATHEMATICS 5
THIRD QUARTER
TOPIC
LEARNING OBJECTIVES
REFERENCE
: Multiplication of Decimals
: Multiply decimals by a whole number or another decimal.
: Soaring 21st Century Mathematics 5, pp. 112-114
Math for Life 5 Revised Edition, p. 150-151
:
CONCEPT NOTES
To find the product of decimals, multiply the numbers in the same way as we multiply
whole numbers. The number of decimal places must be equal to the total number of decimals
places in the factors.
Example:
What is the product of 13.42 and 3.5?
<
1 1
1 1
12 1
12 1
12 1
13.42
× 3.5
6710
13.42
× 3.5
6710
4026
13.42
× 3.5
5036
4026
45.296
What is the product of 2.518 and 3.12?
1
<
1
2.518
× 3.12
5036
1
1
2.518
× 3.12
5036
2518
1
2
1
1
2.518
× 3.12
5036
2518
7554
There is a total of 3 decimal places in the factors. Therefore, same number of decimal places there must be in the product. 2.518
× 3.12
5036
2518
7554
7.85616
There is a total of 5 decimal places in the factors. Therefore, same number of decimal places there must be in the product. PASIG CATHOLIC COLLEGE
MATHEMATICS 5
THIRD QUARTER
TOPIC
LEARNING OBJECTIVES
REFERENCE
: Division of Decimals
: Divide decimals by whole numbers and vice versa.
Divide decimals by another decimal.
: Soaring 21st Century Mathematics 5, pp. 119 - 122
CONCEPT NOTES
:
To divide decimals;
a. Change the divisor to a whole number by moving the decimal point to the right as
many places needed to make it a whole number. (If the divisor is a whole number, there is no
need to move the decimal point.)
b. Do likewise with the decimal point in the dividend. Move the decimal point as many
places as you moved in the divisor.
c. Secure the place of the decimal point in the quotient directly above the decimal point in
the dividend.
d. Divide the numbers in the same way we divide whole numbers.
Example:
1.) Divide: 3.43 ÷ 0.7
.
0.7 3.43
07 34.3
2.) Divide: 750 ÷ 0.25
.
0.25 750
25 75000.
- 75
The decimal point in a whole number is placed after the ones digit. 4.9
7 34.3
- 28
63
- 63
0
3000
25 75000.
-
00
00
00
00
00
00
0
The decimal point is not necessary if the number does not include a decimal, thus, if it is a whole number. PASIG CATHOLIC COLLEGE
MATHEMATICS 5
THIRD QUARTER
TYPE OF ACTIVITY: Problem Solving
TOPIC
LEARNING OBJECTIVES
REFERENCE
AUTHOR/S
: Solving Word Problems involving Decimals
: Analyze and solve word problems involving fundamental operations
on decimals.
: Math for Life 5 Revised Edition, p. 178
: Adelaida Celeridad-Wright & Adela C. Villamayor
CONCEPT NOTES
:
In solving word problems, one must carefully understand what the problem is asking.
Some word problems includes hidden questions that the problem solver must figure out in order
to answer what is asked.
Example:
Francis takes a 12-month loan for a motorcycle that costs Php75, 723.75. He paid a down
payment of Php30, 000. How much should he still pay monthly?
Understand
a.
What are the given information in the problem?
Php75, 723.75 – cost of the motorcycle
Php30,000.00 – down payment
b.
What is asked in the problem? How much is his monthly payment?
Plan
c.
What operations will you use to answer the problem?
subtraction, multiplication and division
Solve
d.
What is the solution to the problem?
Php30, 000.00
?
Php75, 723.75
To know the amount of the remaining balance, subtract the amount of down payment
from the actual price of the motorcycle:
Php75, 723.75
- Php30, 000.00
Php45, 723.75
To know the amount of the monthly payment, divide the remaining balance by 12
months:
Php45, 723.75 ÷ 12 months = Php3, 810.3125
Look Back: Php3, 810.3125 × 12 months = Php45, 723.7500
MATHEMATICS 5
THIRD QUARTER
TYPE OF ACTIVITY: Concept Development / Computational Skills
TOPIC
LEARNING OBJECTIVES
REFERENCE
: Ratio
: Compare two quantities in the form of ratio.
Express ratios in several forms and in simplest terms.
: Soaring 21st Century Mathematics 2nd Edition,
pp. 242-243
Manuel T. Kotah, et al.
CONCEPT NOTES
:
Ratio is a comparison of quantities. The ratio of a to b is commonly written as a : b. It
may also be written as a fraction
(e.g. 2:3 or
!
!
!
!
however, it should be read as a ratio and not as a fraction
is read as 2 is to 3 and not two-thirds.
Example:
In a Catholic bible, there are 27 books in the New Testament and 46 books in the Old
Testament. What is the ratio of books in the New Testament to the books in the Old Testament.?
The ratio is 27 : 46 or
!"
!"
.
Ratios may also be written in simplest form by dividing both terms by their GCF. These
ratios, the higher term and its lowest term are called equivalent ratios.
Example:
21 : 14 = 3 : 2
24 : 30 = 4 : 5
35 : 65 = 7 : 13
MATHEMATICS 5
THIRD QUARTER
TYPE OF ACTIVITY: Concept Development / Computational Skills
TOPIC
LEARNING OBJECTIVES
REFERENCE
AUTHOR
CONCEPT NOTES
: Rate
: Compute for the unit rate of a given comparison.
pp. 246
: Manuel T. Kotah, et al.
:
Rate is a comparison of two quantities of different units.
Example: The car travels at 60 kilometers per hour while the motorcycle travels at 40
kilometers per hour.
Kilometer is a unit of distance while hour is a unit of time. Since two different units were used to
compare the two quantities, the rate is being describe in the situation. The rate of distance over
time is called speed.
Rate can be written as 60km per hour or 60km/hour.
A rate that is simplified to a per-unit form is called unit rate. Unit rate is easily identified if the
denominator is 1
Example: Luke can type 90 words in 3 minutes. On an average, how many words can he type
every minute?
Rate
à 90 words per 3 minutes or
90 words/3 minutes
To compute for the unite rate, divide the numerator by the denominator then copy the units of
each term respectively.
Unit rate à 30 words/minute
MATHEMATICS 5
THIRD QUARTER
TOPIC
LEARNING OBJECTIVES
: Proportion
: Determine if two ratios make a proportion.
Find the missing term in a proportion.
REFERENCE
pp. 252 - 254
AUTHOR
: Manuel T. Kotah, et al.
CONCEPT NOTES
:
Proportion is a statement of two equal ratios.
a:b=c:d
Example
a.)
b.)
1 : 2 = 12 : 24
21 : 14 = 6 : 4
In a proportion, the inner terms are called the means and the outer terms are called the
extremes.
extremes
a:b=c:d
means
The law of proportion states that the product of the means is equal to the product of the
extremes.
axd=bxc
a.)
24
1 : 2 = 12 : 24
24
b.)
84
21 : 14 = 6 : 4
84
If the product of the means is not equal to the product of the extremes, then the two ratios do
not make a proportion.
MATHEMATICS 5
THIRD QUARTER
TYPE OF ACTIVITY: Problem Solving
TOPIC
LEARNING OBJECTIVES
REFERENCE
: Solving Word Problems Involving Proportion
: Analyze and solve word problems that involve proportion.
p. 254, Manuel T. Kotah, et al.
CONCEPT NOTES
:
In solving problems involving proportion, the terms must correspond accordingly. Identify
what object is being described by each quantity so as not to interchange the means by the
extremes and vice versa.
Using an approach can make the given problem clearer and easier to solve. Here are some
examples of word problems that involve proportion.
Example #1
In a library, there are 25 books for every two rows in the shelf. How many books are there
if there are 8 rows in a shelf?
Write a proportion based on the problem.
25 : 2
=
N:8
books : rows books : rows
25 × 8 = 200
25 : 2 = N : 8
2 × N = 200
What should be the value of N to make 2 × N = 200?
2 × N = 200
N = 200 ÷ 2
N = 100
Therefore, there are 100 books in a shelf of 8 rows.
MATHEMATICS 5
FOURTH QUARTER
TOPIC
: Percent
Changing Percent to Fraction and vice versa
LEARNING OBJECTIVES : Define percent in relation to fraction.
Express fractions as percents and vice versa.
REFERENCE
: Soaring 21st Century Mathematics
Kotah, Manuel T., et al.
CONCEPT NOTES
:
Percent means per hundred or hundredths. (‘per’ means ‘for every’ and ‘cent’ means
‘hundred’). The symbol used for percent is %.
Example
In a 10 by 10 grid, there are 100 boxes.
30% of the grid is shaded with blue,
34% of the grid is shaded with red,
30% of the grid is shaded with yellow,
6% of the grid is unshaded.
1.) In a group of 100 students, 24 of them like strawberries, 18
students like blueberries and 39 like raspberries. That means; 24% of
the group likes strawberries, 18% of the group likes blueberries
and 39% of the group likes raspberries.
Take note that in the first example, the shaded region can be expressed as fractions. Thus, a
fraction can be expressed as percent and vice versa.
To change a percent to fraction, drop the percent symbol (%) and put 100 as the
denominator. Reduce if necessary.
30% =
24% =
!"
!""
!"
!""
=
=
!
!"
!
!"
,
,
34% =
39% =
!"
!""
!"
!""
=
!"
!"
,
6% =
, 18% =
!"
!""
!
!""
=
=
!
!"
!
!"
,
To change a fraction to percent, divide the numerator by the denominator. Multiply the
quotient by 100 and put the percent symbol (%).
24
= 24 ÷ 100 = 0.24
100
0.24 × 100 = 24 à 24%
MATHEMATICS 5
FOURTH QUARTER
TOPIC
LEARNING OBJECTIVES
REFERENCE
: Changing Percent to Decimal and vice versa
: Express decimals as percent and vice versa
Math for Life 5 Revised Edition
Wright, Amelia C. and Villamayor, Adelia C.
CONCEPT NOTES
: To change a percent to a decimal, drop the % symbol and move the
decimal point two places to the left.
35% = 0.35
Example:
1.)
2.)
3.)
4.)
5.)
45% = 0.45
37% = 0.37
99% = 0.99
100% = 1.00
125% = 1.25
To change a decimal to a percent, move the decimal point two places
to the right then affix the % symbol.
0.98 = 98%
Example:
1.)
2.)
3.)
4.)
5.)
0.33 = 33%
0.52 = 52%
0.07 = 7%
0.7 = 70%
0.093 = 9.3%
MATHEMATICS 5
FOURTH QUARTER
TOPIC
LEARNING OBJECTIVE
REFERENCE
CONCEPT NOTES
: Percentage, Base and Rate
: Identify the elements of percent in a give problem.
Math for Life 5 Revised Edition
:
Percentage is part of the base determined by the rate.
Base is the whole on which the rate operates.
Rate is the number of hundredths part taken and is written with
percent symbol (%).
MATHEMATICS 5
FOURTH QUARTER
TOPIC
LEARNING OBJECTIVES
REFERENCE
: Finding Percentage
: Solve for the percentage in a worded problem.
: Math for Life 5 Revised Edition, 218-219
CONCEPT NOTES
:
In finding the percentage, base or rate from a given problem, we use the PBR
Triangle in obtaining the formula for each situation.
To find the percentage, multiply the base and the rate.
P B ×
R P = B × R.
Example 1 What is 30% of 500?
R = 30%
B = 500
P=?
P = 500 × 30% (change percent to decimal)
P = 500 × 0.3
P = 150
Example 2
15% of the fruits in the basket are apples. If there are 60 fruits in the
basket, how many are apples?
R = 15%
B = 60
P=?
P = 60 × 15% (change percent to decimal)
P = 60 × 0.15
P = 9, therefore, there are 9 apples in the basket.
MATHEMATICS 5
FOURTH QUARTER
TOPIC
LEARNING OBJECTIVES
REFERENCE
: Finding the Base
: Solve for the base in a worded problem.
CONCEPT NOTES
:
To find the base, divide the percentage by the rate.
B=P÷R
Example 1: 24 is 60% of what number?
P = 24
R = 60%
B=?
B=P÷R
B = 24 ÷ 60% (change percent to decimal)
B = 24 ÷ 0.6
B = 40
P B R Example 2: 20% of the class are non-Catholics. If there are 7 non-Catholics, how
many students are there in the class?
P=7
R = 20%
B=?
B=P÷R
B = 7 ÷ 20% (change percent to decimal)
B = 7 ÷ 0.2
B = 35, therefore there are 35 students in the class.
MATHEMATICS 5
FOURTH QUARTER
TOPIC
LEARNING OBJECTIVES
REFERENCE
: Finding the Rate
: Solve for the rate in a worded problem.
CONCEPT NOTES
:
To find the rate, divide the given percentage by the base or rewrite the
number as a fraction.
R=P÷B
P B R Example 1: 56 is what percent of 80?
P = 56
B = 80
B=?
R=P÷B
R = 56 ÷ 80
R = 0.7 (change decimal to percent)
R = 70%
Example 2: Czarina’s score in her exam is 38. If there are 40 items in the
exam, what is the rate of her score?
P = 38
B = 40
B=?
R=P÷B
R = 38 ÷ 40
R = 0.95 (change decimal to percent)
R = 95%

PASIG CATHOLIC COLLEGE Grade School Department School

Transcription

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