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,
difference 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
1. Identify a perfect square
sum or difference of two
trinomial.
cubes.
2. Get the square of the
b. Completely factor a
numbers.
polynomial involving the
3. Factor a perfect square
sum and difference of two trinomial
cubes.
c. Find pleasures in working
with numbers.
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
Intermediate Algebra UBD
Sourcebook pages 25- 26 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.
B. Other Learning
Resources
http://lmrds.deped.gov.ph.
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
1. Asking the common
physical features/
behavioural traits among
siblings in the family.
SECRET MESSAGE
Find the square roots and
solve the secret message.
4 = ___ 16 = ___
16 = ___ 81 = ___
49 = ___ 9 = ___
IV. PROCEDURES
A. Reviewing previous
lesson or presenting
the new lesson
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?
Study them carefully using 4. 16
9. a2
the following diagrams.
5. 25 10. x4
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
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.
•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.
Factoring the common
Factoring the difference of Factoring the sum or Factoring a perfect square
B. Establishing a
monomial
factor
is
the
two squares is the reverse difference of two cubes is trinomial is the reverse
purpose for the lesson
reverse process of monomial process of the product of the reverse process of process of square o
to polynomials.
sum and difference of two product of binomial and binomial.
a(b + c) = ab + ac
(x + y)2 = x2 + 2xy + y2
terms.
trinomial.
2
2
2
2
(x + y)(x – y) = x – y
(x + y)(x – xy + y )
(x - y)2 = x2 - 2xy + y2
= x3 + y3
(x + y)(x2 + xy + y2)
= 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
of 4y2 that can be factored problem is not written in the binomial 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 raised to a power of 3. The 3. 36 + 20 m + 16m2
and 9y4 are perfect squares term with variable x is okay = (6 + 4m)2
(their
coefficients
are but the 27 should be taken 4. 64x2 – 32xy + 4y2
perfect squares and their care of. Obviously we know = (8x – 2y)2
exponents
are
even that 27 = (3)(3)(3) = 33.
numbers).
Since
Rewrite the original a. Relate the first term in
subtraction is occurring problem as sum of two the trinomial to the first
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)
between these squares, cubes, and then simplify. term in the binomial
this expression is the Since this is the "sum" case, factors.
difference of two squares. the binomial factor and b. Compare the second
trinomial factor will have term in the trinomial
positive
and
negative factor and the sum of the
•What times itself will give middle signs, respectively.
product of the inner
1?
x3 + 27 = (x)3 + (3)3
terms and outer terms of
2
2
•What times itself will give = (x+3)[{x) –(x)(3)+(3) ]
the binomials.
9y4 ?
=(x+3)(x2-3x+9)
c. Observe the third term in
•The factors are (1 + 3y2)
the trinomial and the
2
3
and (1 - 3y ).
Example 2: Factor y - 8
product of the second
•Answer:
This is a case of terms in the binomials.
4y2 (1 + 3y2)(1 - 3y2) or
difference of two cubes
4y2 (1 - 3y2) (1 + 3y2)
since 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:
1. c² - d²
2. 1 - a²
3. ( a + b )² - 4c²
4. 16x² - 4
5. a²b² - 144
Factor the following:
1. x3 – 8
2. 27x3 + 1
3. x3y6 – 64
4. m³ + 125
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
(Leads to Formative
Assessment 3)
Factor the following:
1. 10x + 10y + 10z
2. bx + by + bz
3. 3x³ + 6x² + 9x
4. 10x + 5y –20z
5. 7a³ + 14a² + 21
G. Finding practical
Factor the following
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 the following.
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 – 36p + 16
)
)
Directions. Find the cube Complete
the
perfect
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
1.
2.
3.
4.
5.
100a2 – 25b2
1 – 9a2
81x2 – 1
– 64a2 + 169 b2
x2 – 144
roots. Then, match each square trinomial and factor
solution to the numbers at them.
the bottom of the page. 1. ___ + 16x + 64
Write the corresponding 2. x2 - ___ + 49
letter in each blank to the 3. x2 + 4x + ___
question.In
the
survey, 4. x2 + ___ + 9y2
Best place for family picnic 5. ___ + 10k + 25
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
Common Monomial Factor
H. Making
generalizations and
To factor polynomial with
abstractions about the
common monomial factor,
lesson
expressed the given
polynomial as a product of
the common monomial
factor and the quotient
obtained when the given
polynomial is divided by the
common monomial factor.
The factors of the
1. The sum of the cubes of
difference of two squares
two terms is equal to
are the sum of the square
the sum of the two terms
roots of the first and
multiplied by the sum
second terms times the
of the squares of these
difference of their square
terms minus the product
roots.
of these two terms.
a³ + b³
*The factors of 𝑎2 − 𝑏2
= ( a + b ( a² - ab + b² )
=𝑎𝑟𝑒 ( 𝑎 + 𝑏 ) 𝑎𝑛𝑑 ( 𝑎 −𝑏 ).
I. Evaluating learning
Factorize the following by
taking the difference of
squares:
1. x2 – 9
2. a2 – 1
3. ab2 – 16
4. 16𝑥2 – 49
5. 54𝑥2 – 6y2
Factor the following:
1. 5x + 5y + 5z
2. ax + ay + az
3. 4x³ + 8x² + 12x
4. 6x + 18y – 9z
5. 3a³ + 6a² + 12
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
2. The difference of the
terms of the binomial
cubes of two terms is
factors is the same as
equal to the difference of the sign of the middle
the two terms multiplied
term of the trinomial.
by the sum of the
squares of these two
terms plus the product of
these two terms.
a³ - b³
= ( a - b ( a² + ab + b² )
Supply the missing
Factor the following:
expression.
3. 1. x2 – 6x + 9
4. 2. b2 -12b + 36
1. 𝑚3 - 27
3. 4b2 – 4b + 1
= (m – 3) _________
4. 49p2 – 56p = 16
2. 64 + 27𝑛3
5. 49k2 – 28kp + 4p2
= ____(16 – 12n + 9𝑛2 )
3. _______
= ( 2p + 5q ) ( 4𝑝2 – 10pq +
25𝑞2 )
4. 𝑥6 + 1000
= _____𝑥4 - 10𝑥2 + 100 )
J. Additional activities for A. Follow up
application or
Supply the missing term
remediation
1. 3a + 3b = ____ (a + b)
2. bx + by + bz
= _____ (x + y + z)
3. a²b - ab² = ab (_____
4. 4x + 6y = ____(2x + 3y )
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
activities for
Factorize the following by
taking the difference of
squares:
1. x2 – 9
2. a2 – 1
3. ab2 – 16
4. 16𝑥2 – 49
5. 54𝑥2 – 6y2
5. ________
= ( 6x – 7y ) ( 36𝑥2 + 42xy +
49𝑦2 )
Solve the following:
Complete
the
perfect
1. The product of two
square trinomial and factor
consecutive even
them.
integers is 528. Find the 1. ___ + 16x + 64
value of each integer.
2. x2 - ___ + 49
3. x2 + 4x + ___
4. x2 + ___ + 9y2
5. ___ + 10k + 25
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.