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.