GRADE 8
DAILY LESSON LOG
Grade Level 8
Learning Area MATHEMATICS
Quarter FIRST
School
Teacher
Teaching Dates and Time
Session 1
Session 2
Session 3
Session 4
I. OBJECTIVES
1. Content Standards The learner demonstrates understanding of key concepts of factors of polynomials, rational algebraic expressions,
linear equations and inequalities in two variables, systems of linear equations and inequalities in two variables and
linear functions.
The learner is able to formulate real-life problems involving factors of polynomials, rational algebraic expressions,
2. Performance
linear equations and inequalities in two variables, systems of linear equations and inequalities in two variables and
Standards
linear functions, and solve these problems accurately using a variety of strategies
Factors completely different Factors completely different Factors completely different Factors completely different
3. Learning
types of polynomials
types of polynomials
types of polynomials
types of polynomials
Competencies /
(polynomials with common (polynomials with common (polynomials with common (polynomials with common
Objectives
monomial factor , difference monomial factor , difference monomial factor , difference monomial factor , difference
of two squares, sum and
of two squares, sum and
of two squares, sum and
of two squares, sum and
difference of two cubes,
diffeence of two cubes,
difference of two cubes,
difference of two cubes,
perfect square trinomials
perfect square trinomials
perfect square trinomials
perfect square trinomials
and general trinomials)
and general trinomials)
and general trinomials)
and general trinomials)
(M8AL-Ia-b-1)
(M8AL-Ia-b-1)
(M8AL-Ia-b-1)
(M8AL-Ia-b-1)
a. Factor polynomials with
common monomial factor.
b. Apply the theorems in
proving inequalities in
triangle.
c. Appreciate the concept
about factoring out the
common factor in
polynomials.
a. Factor the difference of
two squares .
b. Solve equations by
factoring the difference of
two squares.
c. Find pleasures in
working with numbers.
a. Find the factors of the
sum or difference of two
cubes.
b. Completely factor a
polynomial involving the
sum and difference of two
cubes.
c. Find pleasures in working
with numbers.
1. Identify a perfect square
trinomial.
2. Get the square of the
numbers.
3. Factor a perfect square
trinomial
II. CONTENT
Factor of Polynomials
With Common
Monomial Factor(CMF)
Factoring the
Difference of Two
Squares
Factoring a Perfect
Square Trinomial
Factoring the Sum or
Difference of Two
Cubes
III. LEARNING
RESOURCES
A. References
1.
Teacher’s
Guide
pages 29-33
pages 34-35
pages 36-37
pages 38-39
2.
Learner’s
Materials
pages 27-31
pages 32-33
pages 34-35
pages 36-38
3.
Textbook
Intermediate Algebra UBD
pages 22-23
Mathematics Activity
Sourcebook pages 22-23
Mathematics Activity
Sourcebook pages 25- 26
Intermediate Algebra UBD
pages 24-25
4.
http://lmrds.deped.gov.ph.
Additional
Materials from
Learning
Resource (LR)
portal
http://lmrds.deped.gov.ph.
http://lmrds.deped.gov.ph.
http://lmrds.deped.gov.ph.
B. Other Learning
Resources
Grade 8 LCTG by Dep Ed
Cavite Mathematics 2016
laptop, LCD
Grade 8 LCTG by Dep Ed Grade 8 LCTG by Dep Ed Grade 8 LCTG by Dep Ed
Cavite Mathematics 2016 Cavite Mathematics 2016 Cavite Mathematics 2016
laptop, LCD
laptop, LCD
laptop, LCD
IV. PROCEDURES
A. Reviewing previous 1. Asking the common
lesson or presenting physical features/
behavioural traits among
the new lesson
siblings in the family.
SECRET MESSAGE
Find the square roots and
solve the secret message.
4 = ___ 16 = ___
16 = ___ 81 = ___
49 = ___
9 = ___
Purpose Setting Activity
So here are the formulas
that summarize how to
factor the sum and
difference of two cubes.
Find the square of the
following:
1. 1
2. 4
3. 9
6. 36
7. 49
8. 81
2. What are the things
common to each set of
pictures?
81 = ___ 25 = ___
16 = ___ 100 = ___
9 = ___ 36 = ___
121= ___ 16 = ___
25 = ___
9 = ___
144 = ___ 64 = ___
81= ___ 289 = ___
225 = ___ 49 =___
9 = ___ 81 = ___
25= ___ 16 =___
100 = ___ 9 =___
A
16
B
16
C
25
E
299
F
100
G
400
I
36
J
81
K
64
M
144
N
100
O
9
Q
49
R
900
S
121
U
24
V
9
W
81
Y
8
X
9
Study them carefully using
the following diagrams.
D
1000
H
4
L
81
P
64
T
4
X
225
Observations:
•For the “sum” case, the
binomial factor on the right
side of the equation has a
middle sign that is positive.
•In addition to the “sum”
case, the middle sign of the
trinomial factor will always
be opposite the middle sign
of the given problem.
Therefore, it is negative.
4. 16
5. 25
9. a2
10. x4
•For the “difference” case,
the binomial factor on the
right side of the equation
has a middle sign that is
negative.
•In addition to the
“difference” case, the middle
sign of the trinomial factor
will always be opposite the
middle sign of the given
problem. Therefore, it is
positive.
B. Establishing a
purpose for the
lesson
Factoring the common
Factoring the difference of Factoring the sum or Factoring a perfect square
monomial factor is the
two squares is the reverse difference of two cubes is the trinomial is the reverse
reverse process of monomial process of the product of reverse process of product process of square o
to polynomials.
sum and difference of two of binomial and trinomial.
binomial.
a(b + c) = ab + ac
(x + y)(x2 – xy + y2)
(x + y)2 = x2 + 2xy + y2
terms.
(x + y)(x – y) = x2 – y2
= x3 + y3
(x - y)2 = x2 - 2xy + y2
2
2
(x + y)(x + xy + y )
= x3 - y3
C. Presenting examples/ a. Factor xy + xz
Get the CMF, x
instances of the
Divide xy + xz by x
lesson
Quotient: y + z
Thus xy + xz = ( y + z)
b. Factor 5n² + 15n
Get the CMF, 5n
Divide 5n² = 15 n by 5n
Quotient: n + 3
Thus 5n² + 15n
= 5n (n + 3)
Factor 4y2 - 36y6
1: Factor x3 + 27
Study the trinomials and
•There is a common factor
Currently
the their corresponding binomial
of 4y2 that can be factored problem is not written in the factors.
out first in this problem, to form that we want. Each 1. x2 + 10x + 25 = ( x + 5)2
make the problem easier. term must be written as 2. 49x2 – 42 + 9
4y2 (1 - 9y4)
cube, that is, an expression = ( 7x – 3)2
4
•In the factor (1 - 9y ), 1 and raised to a power of 3. The 3. 36 + 20 m + 16m2
9y4 are perfect squares term with variable x is okay = (6 + 4m)2
(their coefficients are perfect but the 27 should be taken 4. 64x2 – 32xy + 4y2
squares and their exponents care of. Obviously we know = (8x – 2y)2
are even numbers). Since that 27 = (3)(3)(3) = 33.
subtraction is occurring
Rewrite the original
between these squares, this problem as sum of two
D. Discussing new
concepts and
practicing new skills
#1
c. Factor 27y² + 9y -18
The CMF is 9
Divide 27y² + 9y -18 by 9
The quotient is 3y² + y -2
Thus 27y² + 9y -18
= 9 ( 3y² + y -2)
expression is the difference cubes, and then simplify. a. Relate the first term in the
of two squares.
Since this is the "sum" case, trinomial to the first term
the binomial factor and in the binomial factors.
trinomial factor will have b. Compare the second
•What times itself will give positive and negative middle term in the trinomial
1?
signs, respectively.
factor and the sum of the
•What times itself will give x3 + 27 = (x)3 + (3)3
product of the inner terms
4
2
2
9y ?
= (x+3)[{x) –(x)(3)+(3) ]
and outer terms of the
•The factors are (1 + 3y2) =(x+3)(x2-3x+9)
binomials.
and (1 - 3y2).
c. Observe the third term in
•Answer:
Example 2: Factor y3 - 8
the trinomial and the
2
2
2
4y (1 + 3y )(1 - 3y ) or
This is a case of product of the second
4y2 (1 - 3y2) (1 + 3y2)
difference of two cubes since terms in the binomials.
the number 8 can be written
as a cube of a number,
where 8 = (2)(2)(2) = 23.
Apply the rule for
difference of two cubes, and
simplify. Since this is the
"difference"
case,
the
binomial factor and trinomial
factor will have negative and
positive
middle
signs,
respectively.
Question : What fruit is the
main product of Tagaytay
City? You will match the
products in Column A with
the factors in Column B to
decode the answer.
Factor each of the following: Factor the following:
1. c² - d²
1. x3 – 8
2. 1 - a²
2. 27x3 + 1
3. ( a + b )² - 4c²
3. x3y6 – 64
4. 16x² - 4
4. m³ + 125
5. a²b² - 144
5. x³ + 343
Supply the missing term to
make a true statement.
1. m2 + 12m + 36
= (m + ___)2
2. 16d2 – 24d + 9
= (4d – ___)2
3. a4b2 – 6abc + 9c2
= (a2b ___)2
4. 9n2 + 30nd + 25d2
= (____ 5d)2
5. 49g2 – 84g +36
= ( ______)2
E. Discussing new
concepts and
practicing new skills
#2
Factor the following
1. a²bc + ab²c + abc²
2. 4m²n² - 4mn³
3. 25a + 25b
4. 3x² + 9xy
5. 2x²y + 12xy
F. Developing mastery Factor the following:
1. 10x + 10y + 10z
(Leads to Formative
2. bx + by + bz
Assessment 3)
3. 3x³ + 6x² + 9x
4. 10x + 5y –20z
5. 7a³ + 14a² + 21
Fill in the blanks to make
the sides of each equation
equivalent.
1. ( _____ ) ( x – 9)
= x² -81
2. ( 20 + 4) ( _____ )
= 20² -4²
3. ( _____ ) (2a +3 )
= 4a² - 9
4. ( 6x²y + 3ab)(6x²y -3ab)
= ( _____ ) - 9a²b²
5. ( 13 + x ) (13 – x)
= _____ - x²
Complete the factoring.
1. t3 - w3
=(t–w)(
2. m3 + n3
=(m+n)(
3. x3 + 8
= (x+2)(
4. y3 - 27
=(y–3)(
5. 8- v3
=(2–v)(
Factorize the following by
taking the difference of
squares:
1. x2 – 100
2. a2 – 4
3. ab2 – 25
4. 36𝑥2 – 81
5. 54𝑥2 – 6y2
Factor each completely.
Factor the following:
a) x ³ + 125
1. 1. x2 – 5x + 25
b) a ³ + 64
2. 2. b2 -10b + 100
c) x ³ – 64
3. 36b2 – 12b + 1
d) u ³ + 8
4. 49p2 – 56p = 16
5. 49k2 – 28kp + 4p2
)
)
)
Factor
the
following
trinomials.
1. x2 + 4x + 4
2. x2 - 18x + 81
3. 4a2 + 4a + 1
4. 25m2 – 30m + 9
5. 9p2 – 56p + 16
)
)
Factor the following
G. Finding practical
1. 16a² + 12a
applications of
concepts and skills in 2. 12am + 6a²m
3. 72x² + 36xy – 27x
daily living
4. 5a³ + a³b
5. 30a + 5ay - 25 az
Factor the following.
1. 100a2 – 25b2
2. 1 – 9a2
3. 81x2 – 1
4. – 64a2 + 169 b2
5. x2 – 144
Directions. Find the cube Complete the perfect square
roots. Then, match each trinomial and factor them.
solution to the numbers at 1. ___ + 16x + 64
the bottom of the page. Write 2. x2 - ___ + 49
the corresponding letter in 3. x2 + 4x + ___
each blank to the question.In 4. x2 + ___ + 9y2
the survey, Best place for 5. ___ + 10k + 25
family picnic in Tagaytay
City?
No 1
2
3
4
27
512
343
216
C
R
G
O
5
6
7
8
1728
8
1
729
P
2
1
1
9
10
11
1331
1000
219
I
C
V
12
0
0
13
64
E
14
125
N
12
11
3
5
9
10
7
8
6
13
4
H. Making
generalizations and
abstractions about
the lesson
I. Evaluating learning
Common Monomial Factor The factors of the difference 1. The sum of the cubes of
of two squares are the sum
two terms is equal to
To factor polynomial with
of the square roots of the
the sum of the two terms
common monomial factor, first and second terms times multiplied by the sum
expressed the given
the difference of their
of the squares of these
polynomial as a product of square roots.
terms minus the product
the common monomial
of these two terms.
*The factors of 𝑎2 − 𝑏2
a³ + b³
=𝑎𝑟𝑒 ( 𝑎 + 𝑏 ) 𝑎𝑛𝑑 ( 𝑎 −𝑏 ).
factor and the quotient
= ( a + b ( a² - ab + b² )
obtained when the given
polynomial is divided by the
2. The difference of the
common monomial factor.
cubes of two terms is
equal to the difference of
the two terms multiplied
by the sum of the
squares of these two
terms plus the product of
these two terms.
a³ - b³
= ( a - b ( a² + ab + b² )
Factor the following:
Factorize the following by Supply the missing
1. 5x + 5y + 5z
taking the difference of
expression.
3.
2. ax + ay + az
squares:
4.
1. 𝑚3 - 27
3. 4x³ + 8x² + 12x
1. x2 – 9
= (m – 3) _________
4. 6x + 18y – 9z
2. a2 – 1
2. 64 + 27𝑛3
5. 3a³ + 6a² + 12
3. ab2 – 16
= ____(16 – 12n + 9𝑛2 )
4. 16𝑥2 – 49
3. _______
5. 54𝑥2 – 6y2
= ( 2p + 5q ) ( 4𝑝2 – 10pq +
25𝑞2 )
4. 𝑥6 + 1000
= _____𝑥4 - 10𝑥2 + 100 )
In factoring a perfect square
trinomial, the following
should be noted:
1. The factors are binomials
with like terms
wherein the terms are the
square roots of the first
and the last terms of the
trinomial.
2. The sign connecting the
terms of the binomial
factors is the same as
the sign of the middle
term of the trinomial.
Factor the following:
1. x2 – 6x + 9
2. b2 -12b + 36
3. 4b2 – 4b + 1
4. 49p2 – 56p = 16
5. 49k2 – 28kp + 4p2
J. Additional activities
for application or
remediation
A. Follow up
Factorize the following by
taking the difference of
Supply the missing term
squares:
1. 3a + 3b = ____ (a + b)
1. x2 – 9
2. bx + by + bz
2. a2 – 1
= _____ (x + y + z)
3. ab2 – 16
3. a²b - ab² = ab (_____
4. 16𝑥2 – 49
4. 4x + 6y = ____(2x + 3y ) 5. 54𝑥2 – 6y2
5. m³ - m = ____(m² - 1)
B. Study Factoring
Polynomials
1. What is a common
monomial factor?
2. How will you factor
polynomial by grouping?
Reference: G8 Mathematics
Learner’s Module pages
45-46
V. REMARKS
VI. REFLECTION
1.
No.of learners who
earned 80% on the
formative assessment
2.
No.of learners who
require additional
5. ________
= ( 6x – 7y ) ( 36𝑥2 + 42xy +
49𝑦2 )
Solve the following:
Complete the perfect square
1. The product of two
trinomial and factor them.
consecutive even
1. ___ + 16x + 64
integers is 528. Find the 2. x2 - ___ + 49
value of each integer.
3. x2 + 4x + ___
4. x2 + ___ + 9y2
5. ___ + 10k + 25
activities for
remediation.
3.
Did the remedial
lessons work? No.of
learners who have
caught up with the
lesson.
4.
No.of learners who
continue to require
remediation
5.
Which of my teaching
strategies worked
well? Why did these
work?
6.
What difficulties did I
encounter which my
principal or supervisor
can help me solve?
7.