Grade 7 new module Math

ESPERANZA NATIONAL HIGH SCHOOL
Esperanza, Sultan Kudarat
A Simplified WORKBOOK in Grade 7 Mathematics aligned with K to 12 Basic Education
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FIRST QUARTER: NUMBERS AND NUMBER SENSE
Lesson 1: SETS: AN INTRODUCTION
A set is a well – defined group of objects that share a common characteristic.
Examples of well – defined sets:
1. The set of vowels in English Alphabet.
2. The set of barangay officials of Poblacion, Esperanza.
Examples of not well – defined sets:
1. The set of very large numbers.
2. The set of all rich people in Esperanza.
Two Methods of Describing Sets
1. Listing or Roster Method – all elements of the sets are listed
Example: Set B contains the set of vowels in English Alphabet is written as:
B = { a, e, i, o, u }
2. Rule Method – the elements of a set is written with the use of a descriptor
Example: B = { x|x is a vowel in English Alphabet }
Exercises:
A. Determine whether the following sets is well-defined or not. Write your answer
on the space provided after each statement.
1. The set of Grade 7 teachers in Esperanza National High School.
____________
2. The set of beautiful girls in the campus. ____________
3. The set of Mathematics books in the library. ____________
4. The set of colors in the rainbow. ____________
5. The set of flowers. ____________
B. Express the following sets in Roster Method and Rule Method.
1. Set of counting numbers from 1 to 7.
Roster Method: _______________________________________________
Rule Method: ________________________________________________
2. Set of consonants in English Alphabet.
Roster Method: _______________________________________________
Rule Method: ________________________________________________
3. Set of even numbers from 112 to 138.
Roster Method: _______________________________________________
Rule Method: ________________________________________________

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4. Set of primary colors.
Roster Method: _______________________________________________
Rule Method: ________________________________________________
5. Set of odd numbers from 10 to 25.
Roster Method: _______________________________________________
Rule Method: ________________________________________________
 Elements refer to the objects or members of the set.
Example: A = { b, a, c, k }, b, a, c, and k are the elements of set A
 The set F is a subset of set A, written as F  A, if all elements of F are also
elements of A. The F is a proper subset of A, written as F  A, if F does not
contain all elements of A.
Example: A = { b, a, c, k } and B = { c, a, b }. Since c, a and b which are elements
of B are also elements of A, then we can say that B is a subset of A, or in
symbols, B  A.
 The universal set, denoted by U, is the set that contains all objects under
consideration.
 Two sets are said to be disjoint sets if they have no element in common.
 The null set, denoted by Ø or { }, is an empty set.
 The cardinality of a set A, denoted by n(A), is the number of elements contained
in A.
Example: If A = { b, a, c, k }, then n(A) = 4.
Note: 1. Every set is a subset of itself.
2. Null set Is a subset of any set.
Exercises:
A. Determine the following based on the given sets below.
A = { m, o, n, k , e, y, s }
B = { m, o, n, e, y }
C = { k, e, y, s }
D = { n, o }
1. The elements of set A.
______________________________________________________________
2. Describe B in relation to A.
______________________________________________________________

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3. Describe C in relation to D.
______________________________________________________________
4. The cardinality of set A.
______________________________________________________________
5. The cardinality of set B.
______________________________________________________________
6. The cardinality of set C.
______________________________________________________________
7. The cardinality of set D.
______________________________________________________________
8. The subsets of set D.
______________________________________________________________
9. Set B is the universal set. True or False?
______________________________________________________________
10.Null set is a subset of set A. True or False?
______________________________________________________________
 Power Set
The set of all subsets of a given set is called power set. The power set of set A is
denoted by P(A), and the order of P(A) is 2n(A), where n(A) is the cardinality of set A.
Example:
1. A = { h, o, p, e }
Solution:
n(A)= 4, then 2n(A) = 24 = 16 subsets
P(A) = { Ø, {h}, {o}, {p}, {e}, {h,o}, {h,p}, {h,e}, {o,p}, {o,e}, {p,e},
{h,o,p}, {h,o,e}, {h,e,p}, {o,p,e}, {h,o,p,e} }
Exercises:
A. Find the power set of the following sets.
1. E = { g, o, d }
2. N = { l, o, v, e }

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3. H = { c, a, r, e }
4. S = { u, s }
Lesson 2.1: Union and Intersection of Sets
Let A and B be sets. The union of the sets A and B, denoted by A ᴜ B, is the set
that contains those elements that are either in A or in B, or in both.
A ᴜ B = {x l x is in A or x is in B}
Example: A = { 1, 2, 3, 4 }, B = { 3, 4, 5, 6 }
A ᴜ B = { 1, 2, 3, 4, 5, 6 }
Let A and B be sets. The intersection of the sets A and B, denoted by A ∩ B, is
the set containing those elements in both A and B.
A ∩ B = {x l x is in A and x is in B}
Example: A = { 1, 2, 3, 4 }, B = { 3, 4, 5, 6 }
A ∩ B = { 3, 4 }

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Exercises:
A. Do the following exercises. Write your answers on the spaces provided:
X = { 0, 1, 2, 3, 4 }
Y = { 0, 2, 4, 6, 8 }
Z = { 1, 3, 5, 7, 9 }
Given the sets above, determine the elements and cardinality of:
1. X ᴜ Y
2. X ᴜ Z
3. Y ᴜ Z
4. X ∩ Y
5. X ∩ Z
6. Y ∩ X
7. X ᴜ Y ᴜ Z
8. Y ∩ X ∩ Z
9. (X ᴜ Y) ∩ Z
10.X ᴜ (Y ∩ Z)
Lesson 2.2: Difference of Two Sets
Let A and B be sets. The difference of the sets A and B, denoted by A - B, is the
set of elements in A which are not in B.

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Example: A = { 1, 2, 3, 4 }, B = { 3, 4, 5, 6 }
A – B = { 1, 2 }
B – A = { 5, 6 }
Exercises:
A. Given the sets below, determine the elements and cardinality. Write your answer
on the space provided.
X = { 0, 1, 2, 3, 4 }
Y = { 0, 2, 4, 6, 8 }
Z = { 1, 3, 5, 7, 9 }
1. X – Y
2. Y – Z
3. Z – X
4. (X ᴜ Y) – Z
5. (Y ∩ Z) – X
Lesson 2.3: Complement of a Set
The complement of a set A, written as A’, is the set of all elements found in the
universal set, U, that are not found in set A. The cardinality n (A’) is given by
n (A’) = n (U) –n (A).
Examples:
1. Let U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, and A = {0, 2, 4, 6, 8}.
Then the elements of A’ are the elements from U that are not found in A.
Therefore, A’ = {1, 3, 5, 7, 9} and n (A’) = 5

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2. Let U = {1, 2, 3, 4, 5}, A = {2, 4} and B = {1, 5}. Then
A’= {1, 3, 5}, B’= {2, 3, 4}
A’ ᴜ B’ = {1, 2, 3, 4, 5} = U
3. Let U = {1, 2, 3, 4, 5, 6, 7, 8}, A = {1, 2, 3, 4} and B = {3, 4, 7, 8}. Then
A’ = {5, 6, 7, 8}
B’ = {1, 2, 5, 6}
A’ ᴜ B’ = {5, 6}
4. Let U = {1, 3, 5, 7, 9}, A = {5, 7, 9} and B = {1, 5, 7, 9}. Then
A ∩ B = {5, 7, 9}
(A ∩ B)’= {1, 3}
Exercises:
A. Shown in the table are names of students of a high school class by sets according
to the definition of each set.
A
Like Singing
B
Like Dancing
C
Like Acting
D
Don’t Like Any
Mary
June
Anne
Luke
Ben
Ben
Sid
Love
Grace
Anne
Ben
Mary
Lori
Meg
Tef
Cris
Zac
After the survey has been completed, find the following sets.
1. U =
2. A ᴜ B’ =
3. A’ ᴜ C =
4. (B ᴜ D)’
5. A’ ∩ B =
6. A’ ∩ D’ =
7. (B ∩ C)’ =

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Lesson 2.4: Cartesian Product
The Cartesian product A×B between two sets A and B is the set of all possible
ordered pairs with first element from A and second element from B:
A×B = { (x, y) : x∈A and y∈B }.
Example: A = { b, i, g }, B = { l, i, f, e }
A×B = { (b,l), (b,i), (b,f), (b,e), (i,l), (i,i), (i,f), (i,e), (g,l), (g,i), (g,f), (g,e) }
B×A = { (l,b), (l,i), (l,g), (i,b), (i,i), (i,g), (f,b), (f,i), (f,g), (e,b), (e,i), (e,g) }
Exercises:
Find the Cartesian products of the following sets.
A = { m, a, k, e }
B = { s, i, t }
C = { f, a, i, r }
D = { s, k, y }
1. A×B =
2. A×C =
3. B×C =
4. D×B =
5. A×D =
Lesson 3: Problems Involving Sets
Venn diagrams can be used to solve word problems involving union and
intersection of sets.
Example:
A group of 25 high school students were asked whether they use either Facebook
or Twitter or both. Fifteen of these students use Facebook and twelve use Twitter.
a. How many use Facebook only?
b. How many use Twitter only?
c. How many use both social networking sites?
Solution:
Let S1= set of students who use Facebook only
S2= set of students who use both social networking sites
S3= set of students who use Twitter only

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Finding the elements in each region:
n(S1) + n( S2) + n(S3) = 25
n(S1) + n( S2) = 15
_________________________
n(S3) = 10
n(S1) + n(S2) + n(S3) = 25
n(S2) + n(S3) = 12
__________________________
n(S1) = 13
n(S1) + n( S2) + n(S3) = 25
13 + n( S2) + 10 = 25
n( S2) + 23 = 25
n( S2) = 25 - 23
n( S2) = 2
The number of elements in each region is shown below:

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Exercises:
1. Among the 70 kids in Barangay Poblacion, 53 like eating in Hide Out while 42
like eating in Burger Bucks. How many like eating both in Hide Out and in
Burger Bucks? in Hide Out only? in Burger Bucks only?
2. The following diagram shows how all the Grade 7 students of Esperanza
National High School go to school. Write your answer on the space provided.
1. How many students ride in a car, townace and tricycle going to their school? _______
2. How many students ride in both a car and a townace? _______
3. How many students ride in both a car and tricycle? _______
4. How many students ride in both a townace and the tricycle? _______
5. How many students go to school in a car only? ______
in a townace only? _______
in a tricycle only? ______
walking _______
6. How many Grade 7 students of Esperanza National High School are there? ______

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Lesson 4: Absolute Value of a Number in Number Line
Number Line – is best described as a straight line which is extended in both
directions as illustrated by arrowheads. A number line consists of three elements:
a. set of positive numbers, and is located to the right of zero.
b. set of negative numbers, and is located to the left of zero; and
c. Zero.
The absolute value of a number, denoted "| |" is the distance of the number from
zero.
Examples:
1. | -4| = 4
2. | +7| = 7
3. - | -10| = -(10) = -10
Two integers that are the same distance from zero in opposite directions are
called opposites.
Examples:
1. The opposite of -4 is +4.
2. The opposite of +7 is -7.
Exercises:
A. Simplify the following.
1. │7.04 │=
2. │0 │=
3. -│-29│=
4.-│15 + 6 │=
5. │-22│ =

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B. List at least two integers that can replace N such that.
1. │N │= 4
2. │N │< 3
3. │N │> 5
4. │N │≤ 9
5. 0<│N │< 3
C. Answer the following.
1. Insert the correct relation symbol (>, =, <): │-7 │__│-4 │.
2. If │x - 7│= 5, what are the possible values of x?
3. If │x │= 2/7, what are the possible values of x?
4. Evaluate the expression, │x + y │ -│y -x │, if x = 4 and y = 7.
5. A submarine navigates at a depth of 50 meters below sea level while
exactly above it; an aircraft flies at an altitude of 185 meters. What is the
distance between the two carriers?

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Lesson 5.1: Fundamental Operations on Integers: Addition of Integers
Addition of integers with the same sign
If the integers have the same sign, just add the numbers.
Examples:
1. 5 + 12 = 17
2. (-13) + (-26) = -39
Addition of integers with the different signs
If the integers have different signs, get the difference of the of the two numbers
and copy the sign of the larger number to the result.
Examples:
1. -7 + 22 = 7 – (+22)
= 15
2. (-45) + (+32) = - 45 – (+32 )
= -13
Addition of integers with more than two addends
In adding integers with more than two addends, just combine like signs then add
following the rule of addition.
Examples:
1. (- 4) + 8 + 32 + (-21) = - ( 4 + 21 ) + (8 + 32)
= - 25 + 40
= 15
2. 55 + (-15) + (+2) + (-16) + (-31) = -(15 + 16 +31) + (55 + 2)
= - 62 + 57
= -5

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Exercises:
A. Find the sum.
Who was the first English mathematician who first used the modern symbol of
equality in 1557?
(To get the answer, compute the sums of the given below. Write the letter of the
problem corresponding to the answer found in each box at the bottom).
A 20 + 95
C (30) + (-20)
R 65 + 75
B 38 + (-15)
D (110) + (-75)
O (-120) + (-35)
O 45 + (-20)
T (16) + (-38)
R (165) + (-85)
R (-65) + (-20)
R (-65) + (-40)
E 47 + 98
E (78) + (-15)
E (-75) + (20)
ANSWER: __________________________________
B. Add the following:
1. (-18) + (-11) + (3)
2. (-9) + (-19) + (-6)
3. (-4) + (125) + (-15)
4. (50) + (-13) + (-12)
5. (-100) + (48) + (49)
C. Solve the following problems:
1. Mrs. Garcia charged P4,752.00 worth of groceries on her credit card.
Find
her balance after she made a payment of P3,530.00.
2. In a game, Team ENHS lost 10 yards in one play but gained 14 yards in
the next play. What was the actual yardage gain of the team?

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3. A vendor gained P60.00 on the first day; lost P38.00 on the second day,
and gained P59.00 on the third day. How much profit did the vendor gain in 3
days?
4. Richard had PhP3280 in his checking account at the beginning of the month.
He wrote checks for PhP550, PhP2200, and PhP800. He then made a deposit
of PhP2000. If at any time during the month the account is overdrawn, a
PhP200 service charge is deducted. What was Richard’s balance at the end of
the month?
Lesson 5.2: Fundamental Operations on Integers: Subtraction of Integers
In subtracting integers, just change the sign of the subtrahend then add following
the rules of addition.
Examples:
1. 12 – 2 = 10
2. 2 – 12 = 2 + (-12) = -10
3. (-12) – (-2) = (-12) + (+2) = -10
4. 12 – (-2) = 12 + 2 = 14
5. -12 – 2 = -12 + (-2) = -14
Exercises:
A. Find the difference.
1. -84 – (-31) =
2. 24 – (-71) =
3. 54 – 78 =
4. -19 – (-116) =
5. 264 – 981 =
6. 386 – (-617) =
7. 379 – (-187) =

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8. -6768 – (-427) =
9. -8762 – 1235 =
10. 948 – (-672) =
B. Solve the following problems.
1. Gian deposited P63,400.00 in her account and withdrew P20,650.00 after
a week. How much of his money was left in the bank?
2. Two trains start at the same station at the same time. Train A travels
82km/h, while train B travels 72km/h. If the two trains travel in opposite
directions, how far apart will they be after an hour? If the two trains travel in
the same direction, how far apart will they be in two hours?
3. During the Christmas season, the Supreme Student Government was able
to solicit 2,656 grocery items and was able to distribute 2,498 to one
barangay. If this group decided to distribute 1,601 grocery items to the next
barangay, how many more grocery items did they need to solicit?
Lesson 5.3: Fundamental Operations on Integers: Multiplication of Integers
In multiplying integers, find the product of their absolute values.
1. If the integers have the same signs, their product is positive.
2. If the integers have different signs their product is negative.
Examples:
1. (+10)(+3) = +30
2. (-11)(-4) = +44
3. (-6)(+3) = - 18
4. (+2)(-31) = -61

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Exercises:
A. Find the product
1. (5)(12) =
2. (-8)(4) =
3. (-5)(3)(2) =
4. (-7)(4)(-2) =
5. (3)(8)(-2) =
6. (9)(-8)(-9) =
7. (-3)(-4)(-6) =
8. (-5)(-7)(3) =
9. (-9)(14)(-2) =
10.(-12)(-3)(-10) =
1. Lara has twenty P5 coins in her coin purse. If her niece took 5 of the coins
how much has been taken away?
2. Luke can type 45 words per minute, how many words can Mark type in 30
minutes?
3. Give an arithmetic equation which will solve the following
a. The messenger came and delivered 6 checks worth PhP40 each. Are
you richer or poorer? By how much?
b. The messenger came and took away 3 checks worth PhP110 each. Are
you richer or poorer? By how much?
c. The messenger came and delivered 12 bills for PhP76 each. Are you
richer or poorer? By how much?

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Lesson 5.4: Fundamental Operations on Integers: Division of Integers
The quotient of two integers with the same signs is a positive integer and the
quotient of two integers having unlike signs is a negative integer.
Examples:
1. 24 ÷ 4 = 6
2. (-36) ÷ (-3) = 12
3. (-72) ÷ 8 = - 9
4. 90 ÷ (-9) = -10
Exercises:
A. Find the quotient.
1. 40 ÷ 5 =
2. 54 ÷ 6 =
3. (-88) ÷ (-11) =
4. (-50) ÷ (-10) =
5. (-64) ÷ (-4) =
6. (-51) ÷ (-3) =
7. (-49) ÷ (7) =
8. 125 ÷ (-25) =
9. (-81) ÷ 9 =
10.94 ÷ (-94) =
1. Oliveros’ store earned P8750 a week, how much is her average earning in
a day?
2. Rich worked in a factory and earned P7875.00 for 15 days. How much is
his earning in a day?
3. There are 336 oranges in 12 baskets. How many oranges are there in 3
baskets?

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4. A teacher has to divide 280 pieces of graphing paper equally among his 35
students. How many pieces of graphing paper will each student receive?
5. A father has 976 square meters lot, he has to divide it among his 4
children. What is the share of each child?
Lesson 6: Properties of the Operations on Integers
1. Closure Property
Two integers that are added and multiplied remain as integers. The set of
integers is closed under addition and multiplication.
a, b I, then a+b  I
a, b I, then a•b  I
Examples:
1. 2, 3 I, then 2 + 3  I
2. 4,5 I, then (4)(5) I
2. Commutative Property
Changing the order of two numbers that are either being added or
multiplied does not change the value.
a + b = b + a
ab = ba
Examples:
1. 2 + 3 = 3 + 2
2. (2)(3) = (3)(2)
3. Associative Property
Changing the grouping of numbers that are either being added or multiplied
does not change its value.
(a + b) + c = a + (b + c)
(ab) c = a (bc)
Examples:
1. (2 + 3) + 4 = 2 + (3 + 4)
2. (2 x 3) x 4 = 2 x (3 x 4)

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4. Distributive Property
When two numbers have been added / subtracted and then multiplied by a
factor, the result will be the same when each number is multiplied by the factor
and the products are then added / subtracted.
a(b + c) = ab + ac
Example: 2(3 + 4) = (2)(3) + (2)(4)
5. Identity Property
Additive Identity - states that the sum of any number and 0 is the given
number. Zero, “0” is the additive identity.
a + 0 = a
Examples:
1. 4 + 0 = 4
2. -10 + 0 = -10
Multiplicative Identity - states that the product of any number and 1 is the
given number, a • 1 = a. One, “1” is the multiplicative identity.
a • 1 = a
Examples:
1. 4 x 1 = 4
2. -10 x 1 = -10
6. Inverse Property
Additive Inverse - states that the sum of any number and its additive
inverse, is zero. The additive inverse of the number a is - a.
a + (-a) = 0
Examples:
1. 4 + (-4) = 0
2. -10 + (10) = 0
Multiplicative Inverse - states that the product of any number and its
multiplicative inverse or reciprocal, is 1. The multiplicative inverse of the number 𝑎
is
1
𝑎
.
𝑎∙
1
𝑎
= 1
Examples:
1. 4 +
1
4
= 1
2. -10 + (−
1
10
) = 1

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A. Identify the property used in each statement. Write your answer on the space
provided.
1. 0 + (-7) = -7 ________________________________
2. 6 (3 -5) = 6(3) - 6(5) ________________________________
3. (-8) + (-7) = (-7) + (-8) ________________________________
4. 1 x (-13) = -13 ________________________________
5. -4 x - = 1 ________________________________
6. 2 x (4 x 7) = (2 x 4) x 7 ________________________________
7. 11 + (-11) = 0 ________________________________
8. 3(5) = 5(3) ________________________________
9. 1 x (-) ________________________________
10. (-3)( + 9) = (-3)(5) + (-3)(9) ________________________________
B. Rewrite the following expressions using the given property.
1. 12a –5a Distributive Property _____________________
2. (7a)b Associative Property _____________________
3. 8 + 5 Commutative Property _____________________
4. -4(1) Identity Property _____________________
5. 25 + (-25) Inverse Property _____________________
Lesson 7: Rational Numbers in the Number Line
The word rational is derived from the word “ratio” which means quotient. Rational
numbers are numbers which can be written as a quotient of two integers
𝑎
𝑏
,where b ≠ 0.
Examples:
1. 4 =
1
4
2. 0.07 =
7
100
3.
3
4
4. 2.5 =
25
10
We can locate rational numbers on the real number line.
Examples:
1. Locate ½ on the number line.
a. Since 0 < ½ < 1, plot 0 and 1 on the number line.

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b. Get the midpoint of the segment from 0 to 1. The midpoint now
corresponds to ½
2. Locate
5
7
on the number line.
3. Locate −
8
3
on the number line.
Exercises:
A. Locate and plot the following on a number line.
1. −
11
3
2. 3.07
3.
3
5
4. 11
5. -0.02
6. 8
8
9
7. 0
8. 6.6
9. −
6
7
10. 1.5
B. Name 10 rational numbers that are greater than -1 but less than 1 and arrange
them from least to greatest on the real number line.

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C. Name one rational number X that satisfies the descriptions below:
1. −10 ≤ 𝑥 ≤ −9
2.
1
10
≤ 𝑥 ≤
1
2
3. 3 ≤ 𝑥 ≤ 𝜋
Lesson 8.1: Forms of Rational Numbers and Addition and Subtraction of Rational
Numbers
Changing Fraction Form to Decimal Form
To change a fraction form to decimal form, you need only to divide the numerator
by the denominator.
Examples:
1.
5
8
= 0.625 2.
1
3
= 0.333 …
Note: 0.333… can be written using vinculum, 0.3.
Changing Decimal Form to Fraction Form
To change a terminating decimal to fraction, express the decimal part of the
numbers as a fractional part of a power of 10.
Examples:
1. 0.7 =
7
10
2. 0.12 =
12
100
=
3
25
3. −1.625 = −1
625
1000
= −
1625
1000
= −
13
8

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To change a repeating and nonterminating decimal to fraction, follow the steps
below:
Examples:
1. Change 0.2 to its fraction form.
Solution: Let r = 0.222... (Since there is only 1 repeated digit,
10r = 2.222... multiply the first equation by 10)
Then subtract the first equation from the second equation and obtain
9r = 2.0
r =
2
9
Therefore, 0.2 =
2
9
.
2. Change 1.35 to its fraction form.
Solution: Let r = 1.353535...
100r = 135.353535...
Then subtract the first equation from the second equation and obtain
99r = 134 (Since there are 2 repeated digits, multiply the first
r =
134
99
equation by 100. In general, if there are n repeated
digits, multiply the first equation by 10n
.)
Therefore, 1.35 =
134
99
.
Exercises:
A. Convert these decimals to fraction form. Express the answers in the lowest
term.
1. 0.45
2. 0.11
3. 0.8181…

25
4. -0.02
5. 5.333...
Addition and Subtraction of Fractions
To add or subtract fractions with the same denominator, simply add or
subtract the numerators and copy the common denominator. Thus,
𝑎
𝑏
+
𝑐
𝑏
=
𝑎+𝑐
𝑏
and
𝑎
𝑏
−
𝑐
𝑏
=
𝑎−𝑐
𝑏
where 𝑏 ≠ 0.
Examples:
1.
5
7
+
4
7
=
5+4
7
=
9
7
2.
7
11
−
9
11
=
7−9
11
= −
2
11
3.
6
15
+
4
15
−
8
15
=
6+4−8
15
=
2
15
To add or subtract fractions with the different denominators, find the least
common denominator(LCD) of the given fractions.
Examples:
1.
1
4
+
2
3
Solution:
The LCD of
1
4
and
2
3
is 12.
1
4
+
2
3
=
3
12
+
8
12
=
11
12
.
Exercises:
A. Perform the indicated operations and express your answer in simplest form.
1.
2
9
+
3
9
+
1
9
2.
4
5
+
3
5
+
16
5
3.
7
10
+
2
5
4.
16
24
−
3
12
5. 2
5
12
−
1
3

26
6. 9
1
4
+
1
7
7. 3
1
4
+ 6
2
3
8. 9
2
7
− 4
5
7
9.
1
12
−
7
9
10.12
5
6
− 8
5
9
11.
1
4
+
2
3
−
1
2
12.11 − 4
5
11
13.
7
12
+
5
9
−
3
4
14.
9
20
+
5
8
+
4
5
15.7
2
8
+ 6
3
2
+ 5
1
4
B. Solve each problem.
1. Irish and Chilby are comparing their heights. If Irish’s height is 120
3
4
cm
and Chilby’s height is 96
1
3
cm. What is the difference in their heights?
2. Dhanela bought 6
3
4
meters of silk, 3
1
2
meters of satin and 8
2
5
meters of
velvet. How many meters of cloth did she buy?
3. Seline needs 11
1
4
kg. of meat to serve 55 guests, If she has4
1
2
kg of
chicken, a 9
2
5
kg of pork, and kg of beef, is there enough meat for 55
guests?

27
4. Mr. Delna has liters of gasoline in his car. He wants to travel far so he
added 16 liters more. How many liters of gasoline is in the tank?
5. After boiling, the liters of water was reduced to 9 liters. How much water
has evaporated?
Addition and Subtraction of Decimals
Arrange the decimal numbers in a column such that the decimal points are
aligned, then add or subtract as with whole numbers.
Examples:
1. 23.76 + 36.7
Solution:
23.76
+ 36.7_
60.46
2. 98.21 – 53.10
Solution:
98.21
– 53.10_
45.11
Exercises:
A. Perform the indicated operation.
1. 1,902 + 21.36 + 8.7
2. 700 – 678.891
3. 45.08 + 9.2 + 30.545
4. 7.3 – 5.182
5. 900 + 676.34 + 78.003
6. 51.005 – 21.4591
7. 0.77 + 0.9768 + 0.05301

28
8. (2.45 + 7.89) – 4.56
9. 5.44 – 4.97
10.(10 – 5.891) + 7.99
B. Solve the following problems:
1. Pia had P8500 for shopping money. When she got home, she had P232.75 in her
pocket. How much did she spend for shopping?
2. Clint contributed P79.25, while Josh and Julia gave P66.25 each for their gift to
Teacher Jen. How much were they able to gather altogether?
3. Ily said, “I’m thinking of a number N. If I subtract 11.34 from N, the difference is
1.34.” What was Ily’s number?
4. Ana said, “I’m thinking of a number N. If I increase my number by 66.2, the sum is
24.62.” What was Ana’s number?
5. Kera ran the 100-meter race in 136.46 seconds. Mel ran faster by 16.7 seconds.
What was Mel’s time for the 100-meter dash?
Lesson 8.2: Multiplication and Division of Rational Numbers
Multiplication and Division of Fractions
To multiply rational numbers in fraction form simply multiply the numerators and
multiply the denominators. In symbol,
𝑎
𝑏
×
𝑐
𝑑
=
𝑎𝑐
𝑏𝑑
where 𝑏 ≠ 0 and 𝑑 ≠ 0.
Examples:
1.
4
5
×
2
7
=
8
35
2. −
10
13
×
1
2
= −
10
26
To divide rational numbers in fraction form, you take the reciprocal of the second
fraction (called the divisor) and multiply it by the first fraction. In symbol,
𝑎
𝑏
÷
𝑐
𝑑
=
𝑎
𝑏
×
𝑑
𝑐
=
𝑎𝑑
𝑏𝑐
where 𝑏 ≠ 0, 𝑐 ≠ 0 and 𝑑 ≠ 0.
Examples:

29
1.
3
7
÷
4
9
=
3
7
×
9
4
=
27
28
2. −
2
5
÷
1
6
= −
2
5
×
6
1
= −
12
5
Exercises:
A. Find the products. Express in lowest terms (i.e. the numerator and denominators
do not have a common factor except 1). Mixed numbers are acceptable as well.
1.
2
3
∙
5
6
2. 8 ∙
3
4
3.
2
5
∙
4
20
4. 4
2
3
∙ 11
5
6
5. −
9
20
∙
20
27
6. 5
2
3
∙ 3
1
2
7.
3
5
∙
2
15
8.
2
3
∙
5
6
∙
1
2
9. −
2
5
∙
5
11
∙
11
12
10.
7
9
∙ (−
5
6
) ∙
3
7
B. Find the quotient.
1. 21 ÷
5
6
2.
3
4
÷ (−
7
12
)
3.
12
13
÷
13
12
4. −
4
7
÷
6
14
5.
8
15
÷
12
25
6. (−
10
14
) ÷ (−
5
6
)

30
7.
9
16
÷
3
4
÷
1
6
8. −
2
9
÷
11
15
9.
15
6
÷
2
3
÷
5
8
10.
2
3
÷
5
6
C. Solve the following:
1. Kath spent hours doing her assignment. DJ did his assignment for times as
many hours as Kath did. How many hours did DJ spend doing his assignment?
3. How many fourths are there in six - fifths?
4. Honey donated of her monthly allowance to the Marawi survivors. If her
monthly allowance is P6500, how much did she donate?
5. The enrolment for this school year is 5340. If
1
6
are Grade 8 and
1
4
are Grade
10, how many are Grade 7 or Grade 9?
6. At the end of the day, a store had
2
5
of a cake leftover. The five employees
each took home the same amount of leftover cake. How much of the cake did
each employee take home?
Multiplication and Division of Decimal Form
In multiplying decimals, just multiply the numbers the way we do with integers.
The product has the same number of decimal places as the total of decimal places in the
factors.
Example: 2.46 × 1.5
Solution:

31
2.46 (2 decimal places)
× 1.5_ (1 decimal place)
1230
246_
3.690 (3 decimal places)
In dividing decimals, convert the decimals into integers then divide.
Example: 3.65 ÷ 2.5
1.
3.65
2.5
×
100
100
=
365
250
= 1.46
Exercises:
A. Perform the indicated operation
1. 3.5 ÷ 2
2. 27.3 x 2.5
3. 78 x 0.4
4. 9.7 x 4.1
5. 9.6 x 13
6. 3.415 ÷ 2.5
7. 3.24 ÷ 0.5
8. 53.61 x 1.02
9. 1.248 ÷ 0.024
10. 1948.324 ÷ 5.96
Lesson 9: Principal Roots and Irrational Numbers
The principal nth
root of a positive number is the positive nth
root. The principal nth
root of a negative number is the negative nth
root if n is odd. If n is even and the number
is negative, the principal nth
root is not defined. The notation for the principal nth
root of a
number b is √ 𝑏
𝑛
. In this expression, n is the index and b is the radicand. The nth
roots
are also called radicals.
Examples:
1. √4 = 2
2. √−1000
3
= −10
3. √−100
4
is not defined

32
To determine whether a principal root is a rational or irrational number, determine
if the radicand is a perfect nth
power of a number. If it is, then the root is rational.
Otherwise, it is irrational.
Examples:
1. √9 = 3 rational
2. √8 irrational since 8 is not a perfect square
Approximating the Square Root of a Number
The square root of a number can be approximated by looking for consecutive
integers between which the square root lies. A series of estimations can also be made to
approximate values up to a certain number of decimal places.
Example: √40
The principal root √40 is between 6 and 7, principal roots of the two perfect
squares 36 and 49, respectively. Now, take the square of 6.5, midway between 6 and
7. Computing,(6.5)2
= 42.25. Since 42.25 > 40 then √40 is closer to 6 than to 7. Now,
compute for the squares of numbers between 6 and 6.5:
(6.1)2
= 37.21, (6.2)2
= 38.44, (6.3)2
= 39.69, (6.4)2
= 40.96.
Since 40 is close to 39.69 than to 40.96 is approximately 6.3.
√40 is between 6 and 7, principal roots of 36 and 49. Since 40 is closer to 36 than
to 49, √40 is closer to 6. Plot√40 closer to 6.
Exercises:
A. Tell whether the principal roots of each number is rational or irrational.
1. √81
2. √144
3. √0.02
4. √24
5. √1000
6. √22.5
7. √47
8. √289
9. √600
10.√0.36

33
B. Between which two consecutive integers does the square root lie?
1. √88
2. √800
3. √244
4. √444
5. √47
6. √91
7. √2046
8. √905
9. √1799
10.√100000
C. Estimate each square root to the nearest tenth and plot on a number line.
1. √51
2. √73
3. √14
4. √55
5. √137
6. √240
7. √6
8. √87
9. √39
10.√102
Lesson 10: Subsets of Real Numbers
Real Numbers – are any of the numbers from the preceding subsets. They can be
found on the real number line. The union of rational numbers and irrational numbers is
the set of real numbers.
Rational Numbers – are numbers that can be expressed as a quotient of two
integers. The integer a is the numerator while the integer b, which cannot be 0 is the
denominator. This set includes fractions and some decimal numbers.
Irrational Numbers – are numbers that cannot be expressed as a quotient of two
integers. Every irrational number may be represented by a decimal that neither repeats
nor terminates.

34
Integers – are the result of the union of the set of whole numbers and the negative of
counting numbers.
Whole Numbers – are numbers consisting of the set of natural or counting numbers
and zero.
Natural Counting Numbers – are the numbers we use in counting things, that is
{1,2,3,4, …}. The three dots, called ellipses, indicate that the pattern continues
indefinitely.
Exercises:
A. Determine the subset of real numbers to which each number belongs. Use a
tick mark (√) to answer.
NUMBER WHOLE
NUMBER
INTEGER RATIONAL IRRATIONAL
1. -19
2. 34.74
3.
𝟐
𝟕
4. √ 𝟖𝟏
5. √ 𝟏𝟑
6. -0.125
7. −√ 𝟔𝟒
8. e
9. -45.34
10.-1.3535…
B. Based on the stated information, show the relationships among natural or
counting numbers, whole numbers, integers, rational numbers, irrational
numbers and real numbers using the Venn diagram below.

35
1. Are all rational numbers whole numbers?
2. Are all real numbers rational numbers?
3. Are −
1
3
and −
2
5
negative integers?
4. How is a rational number different from an irrational number?
5. How do natural numbers differ from whole numbers?
C. Complete the details in the Hierarchy Chart of the Set of Real Numbers.

36
Lesson 11: Significant and Digits and the Scientific Notation
Rules for Determining Significant Digits
A. All digits that are not zeros are significant.
Examples:
1. 2781 has 4 significant digits
2. 82.973 has 5 significant digits
B. Zeros may or may not be significant. Furthermore,
1. Zeros appearing between nonzero digits are significant.
Examples:
2. Zeros appearing in front of nonzero digits are not significant.
Examples:
2. 0.0000009 has 1 significant digit
3. Zeros at the end of a number and to the right of a decimal are significant digits.
Zeros between nonzero digits and significant zeros are also significant.
Examples:
4. Zeros at the end of a number but to the left of a decimal may or may not be
significant. If such a zero has been measured or is the first estimated digit, it is
significant. On the other hand, if the zero has not been measured or estimated
but is just a place holder it is not significant. A decimal placed after the zeros
indicates that they are significant.
Examples:
2. 560000. has 6 significant digits
Significant Figures in Calculations
1. When multiplying or dividing measured quantities, round the answer to as many
significant figures in the answer as there are in the measurement with the least
number of significant figures.
2. When adding or subtracting measured quantities, round the answer to the same
number of decimal places as there are in the measurement with the least number
of decimal places.

37
Examples:
a. 3.0 x 20.536 = 61.608
Answer: 61 since the least number of significant digits is 2, coming from 3.0
b. 3.0 + 20.536 = 23.536
Answer: 23.5 since the addend with the least number of decimal places is 3.0
Writing a Number in Scientific Notation
1. Move the decimal point to the right or left until after the first significant digit and
copy the significant digits to the right of the first digit. If the number is a whole
number and has no decimal point, place a decimal point after the first significant
digit and copy the significant digits to its right.
For example, 300 000 000 has 1 significant digit, which is 3. Place a decimal point
after 3.0
The first significant digit in 0.000 000 089 is 8 and so place a decimal point after
8, (8.9).
2. Multiply the adjusted number in step 1 by a power of 10, the exponent of which is
the number of digits that the decimal point moved, positive if moved to the left and
negative if moved to the right.
For example, 300 000 000 is written as 3.0 x 108 because the decimal point was
moved past 8 places.0.0000 089 is written as 8.9 x 10-8 because the decimal
point was moved 8 places to the right past the first significant digit 8.
Exercises:
A. Determine the number of significant digits in the following measurements. Rewrite
the numbers with at least 5 digits in scientific notation.
1. 0.0000056 L
2. 8207 mm
3. 4003 kg
4. 0.83500 kg
5. 350 m
6. 50.800 km
7. 4113.000 cm
8. 0.0010003 m
9. 700.0 mL
10.8 000 L
B. a. Round off the following quantities to the specified number of significant figures.
1. 5 487 129 m to three significant figures
2. 0.013 479 265 mL to six significant figures

38
3. 31 947.972 cm to four significant figures
4. 192.6739 m to five significant figures
5. 786.9164 cm to two significant figures
b. Rewrite the answers in (a) using the scientific notation.
1.
2.
3.
4.
5.
C. Write the answers to the correct number of significant figures
1. 4.5 X 6.3 ÷ 7.22
2. 5.567 X 3.0001 ÷ 3.45
3. (37 X 43) ÷ (4.2 X 6.0)
4. (112 X 20) ÷ (30 X 63)
5. 47.0 ÷ 2.2

39
Prepared By:
EDEN JOEY T. PAGHUBASAN
Teacher III, Math Department Head
Validated By:
SAMSUDIN N. ABDULLAH, Ph.D. HENRY T. LEGASTE, Ed.D.
Regional Learning Resources Evaluator Regional Learning Resources Evaluator
Recommending Approval:
SHERYL L. OSANO
Education Program Supervisor
LRMDS
OFELIA C. BETON, Ed.D.
Education Program Supervisor
Mathematics
Approved:
RAPHAEL C. FONTANILLA, Ph.D., CESO V
Schools Division Superintendent

40
References:

41
1. Mathematics Grade 7 Learner’s Module and Teacher’s Guide. Department of
Education – Instructional Materials Council Secretariat (DepEd-IMCS), 5TH
Floor. Mabini Building, DepEd Complex , Meralco Avenue, Pasig City,
Philippines 1600
2. DILAO, S. J. Intermediate Algebra.(2003). JTW Corporation, 1281 Gregorio
Araneta Avenue, Quezon City, Philippines
3. ESCAÑO, R.S. (2005). Intermediate Algebra Workbook. Vicarish Publication &
Trading, Inc., 1946-A, F. Torres St., Corner Diamante Ext., Sta. Ana, Manila
Philippines

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