Topic: Special Products and Factors Subtopic: Rules on finding factors of polynomials


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1 Quarter I: Special Products and Factors and Quadratic Equations Topic: Special Products and Factors Subtopic: Rules on finding factors of polynomials Time Frame: 20 days Time Frame: 3 days Content Standard: The learner demonstrates understanding of the key concepts on the rules applied to finding factors. Stage 1 Performance Standard: The learner formulates reallife problems involving factors and analyze these using a variety of strategies with utmost accuracy. Essential Understanding(s): Patterns in finding factors of polynomials to facilitate the analysis of real life situations. Essential Question(s): Why do we need to study the rules in factoring? How are patterns used to analyze reallife problems involving factoring of polynomials? The learner will know: rules on finding common monomial factors. factoring trinomials. factoring perfect square trinomials factoring difference of two squares factoring sum or difference of two cubes factoring by grouping apply the concept on rules on factoring in the analysis of real life situation Product or Performance Task: Problems formulated are real life. involve factors of algebraic expressions. are analyzed through The learner will be able to: explore the rules in finding common factors and search for patterns. factor trinomials. factor perfect square trinomials factor difference of two squares factor sum or difference of two cubes factor by grouping apply the concept on rules on factoring in the analysis of real life situations. Stage 2 Evidence at the level of understanding The learner should be able to demonstrate understanding of factoring using the six (6) facets of understanding: Explanation Discuss how factors are found using different methods. Evidence at the level of performance Performance assessment of problems formulated based on the following suggested criteria: reallife situations
2 appropriate and accurate representations Clear Coherent Thorough Interpretation State the process of finding special products of two integers and relate it to the patterns in getting the product of two binomials. Creative Illustrative Meaningful Application Pose situations in real life involving special products and analyze them. Appropriate Authentic Practical Perspective Compare and contrast the different ways of finding special products. Credible Critical Realistic Empathy Describe the difficulties one can experience without knowing the process of getting special products. Open Responsive & Sensitive situations involve factors situations are analyzed through appropriate representations Tools: Rubrics for assessment of problems formulated and analyzed.
3 Self Knowledge Assess how one can give the best representation to a situation involving special products. Receptive Reflective Relevant Stage 3 Teaching  Learning Sequence: The learners are expected to master the following concepts: 1. Factoring polynomials whose terms have a common monomial factor 2. Factoring trinomials which are product of two binomials. 3. Factoring perfect square trinomials and difference of two squares 4. Applications of the concept of factoring polynomials in the analysis of reallife situations ( Note: Present this activity using manila paper, activity cards, overhead projector, power point presentation, etc.) Introduction Consider the figures at the right. What do you think are the measures of the sides of the rectangle? of the square? ac ad bc bd a 2 ab ab b 2
4 Try this opening activity: A. Consider the rectangular paper on the right side. The area of each part is given. 1. Give the possible lengths of the sides for each part. Explain. 2. Express each area as a factor of the possible sides. 3. If the area of the big rectangle is 616 cm 2, what are the possible measures of the sides? 4. What pattern did you discover? B. Consider each of the following: What are the possible sides if the new areas are as follows: A 36 cm 2 C 96 cm 2 B D 160 cm cm 2 E F 48 cm cm 2 A 36 x 2 cm 2 D 160 x 4 y 6 cm 2 B 132 x 2 y 2 cm 2 E 48 x 8 y 4 cm 2 C 96 x 2 y 4 cm 2 F 144x 2 y 2 cm 2 1. Explore ( Group Work ) Note: You may assign two groups work on the same material. At this stage, the learners are expected to: 1. explore factors that form product of two binomials. 2. identify the factors of two binomials. 3. master the process of finding factors of trinomials.
5 Let us have another piece of rectangular paper similar to the opening activity. This time, let us apply manipulatives to discover the factors of two binomials. Follow strictly the directions written in your activity sheet. Activity 1: ( Groups I &II ) : A. 1. Cut papers similar to the figures on the right. 2. Get the total area of the 6 pieces. 3. Fit the pieces in such a way that they form just one big rectangle ( with no holes and no pieces overlapping). 4. Does the total area change? Why? Why not? x 2 x x x 5. Now, write the measures of the sides of the big rectangle formed. 6. Compare the area of the big rectangle with that of the combined areas of the 6 pieces. 7. Generalize your observation. 1 1 a) x 2 x x x b) x 2 x 2 x 2 x x x 1 1 B. 1. Using the algebra tiles above repeat the process in ( A ).
6 2. Relate the process of finding the factors of an integer to the patterns in getting the factors of a trinomial. Activity 2: ( Group III &IV ) : Figures I and II are two congruent squares. One side of the square in Figure I is ( x+2) units, while the area of the square in Figure II is (x+2)(x+2) sq units. (x + 2) ( x+2)(x+2) A. 1. What is the area of the square with side (x + 2) in factored form? Figure I Figure I 2. What is the side of the square in Figure II if its area is (x +2)(x + 2)? 3. What can you say about (x+2) 2 and (x+2)(x+2)? 4. Now, let us do some manipulative activities. Consider again the tiles used in the previous activity. Form the rectangles using the algebra tiles for the following trinomials. a. x 2 + 4x + 4 b. x 2 + 2x + 1 c. x 2 6x + 9 d. x 2 + 2xy + y 2 4. What do you notice about the four rectangles? Describe and compare their sides. 5. Make a generalization of what you observed. B. 1. Using the procedure learned in (A), determine if the following trinomials follow the same pattern. a. x 2 + 8x + 16 b. x 2 + 5x + 10 c. x 2 4x + 4 d. x 2 10x Relate the process of finding factors of numbers which are perfect squares to the pattern in getting the factors of perfect square trinomials. Activity 3: ( Group V & VI) : 10 cm 1. Using a cartolina, cut a square of side 10 cm 2. Cut off a small square of size 2 cm by 2 cm from one of the corners of the big square paper. 3. Label the side of the big square with 10 cm and (10 2) cm. 2cm
7 4. Cut the remaining paper as shown 5. Rearrange the remaining parts to form a new rectangle. 6. Find the area of a) the original square b) the small square c) new rectangle 7. After cutting out the small square, what is the area of the remaining part of the big square? 8. Make a generalization of what you have discovered. B. 1. Repeat the process in ( A ) using x meters and y meters as the lengths, instead of 10 cm and 2 cm, respectively. 2. Relate the process of finding factors of the difference of two squares using integers and the factors of the difference of two squares using variables. 2. Firm Up A. Answer the following questions: 1. Consider Activity I. a. Using the sides of each part of the square, how will you represent the side of the whole square paper? Explain. b. Write the procedure in getting the common monomial factor of ( 4x +2y ), c. Using the steps in (b), factor the following: i 3x + 9 iv. 6a + 9a + 12ab ii. 8cd  18 cd 2 v. 3mn + 5 mn 2 iii. 7x 221x 3 2. Consider Activity 1 under the Explore. a. How are the sides of a rectangle expressed using the algebra tiles? b. Write the procedure in getting the factors of trinomials which are product of two binomials. x 2 + 3x + 2, x 23x + 8 and 3x 2 + 3x + 2 c. Using the steps in (b), answer the following: What are the dimensions of each of the rectangles if the following trinomials represent their product? i. x 2 + 5x + 6 iii. x 2 x  6 v. x 2 9x + 20 ii. 3x 2 + 3x + 2 iv. x 2 + 7x Consider Activity 2 under Explore.
8 a. How is the side of the square obtained given its area? b. Write the procedure in getting the factors of perfect square trinomial x 2 + 4x + 4? x 2 + 2xy+ y 2? c. Explain how to distinguish perfect square trinomials from other trinomials. Describe the factors of perfect square trinomials. Use the steps in (b) to find the factors of the following: i. x 2 + 6x + 9 iii. p p + 36 v. 16x xy + 9y 2 ii. m m + 25 iv. 9x x Use algebra tiles to verify the given equation: 2 2 a. 4x + 12xy + 9y = ( 2x + 3y) 2 b. 4x 2 4xy + y 2 = (2x y) 2 5. Consider Activity 3 under Explore. a. How do you represent the area of the remaining parts? b. Write the procedure in getting factors of the difference of two squares: x 2 y 2 c. Use the procedure in (b) to factor the following: i. x 2 9 iii. 4x v. 9x 2 4 ii. 9x 2 36 iv. 36x Compare the different ways of factoring. 7. If ( x + y)(x 2 xy + y 2 ) = x 3 + y 3 and ( x y)(x 2 + xy + y 2 ) = x 3 y 3, what are the factors of x 3 + y 3 and x 3 y 3? Use the above ideas in getting the factors of the following: a. a 3 27 c. x e. 125m 3 n 3 8 b. 8x 3 + y 3 d. 64x y 3 3. Deepen Analyze the following problems: 1. If the area of a square garden is (x 2 + 8x + 16) sq units, is it possible to find the measure of its side? Why? Or why not? How?
9 2. If the area of a garden is (3x 2 + 8x + 4) sq units, is it possible to find the measure of the side? Is the garden a square? Why? Or why not? 3. Is x 2 y 2 = (x y) 2? Explain why or why not. If you do not understand the rules on factoring, what do you think will happen? 4. What is the difference of the squares of 105 and 5? What is the square of their difference? 5. Explain or illustrate how you can find the difference of the squares of two terms using 105 and Transfer A. Solve: A box with no top is to be made from an 8 inch by 6 inch piece of metal by cutting identical squares from each corner and turning up the sides. The volume of the box is modeled by the polynomial (4x 328x 2 +48x). Find the dimensions of the box. B. 1. Using models, present a problem solving plan that will make use of the concept of factoring. 2. Write a journal about the rules on finding factors of polynomials. Send it to your friend who is studying in a different school through a letter, , etc. Submit a copy of the journal.
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