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Lesson Design Subject Area: Algebra Benchmark Period III Grade Level: 8-12 Duration of Lesson: 5 – 6 days (1 day second degree with coefficient of 1, 1-2 days second degree with coefficient greater than 1, 1 day perfect square trinomials and difference of two squares, and 1 day common factors) Standard(s): 11.0: Apply basic factoring techniques to second and simple third degree polynomials. These techniques include finding a common factor for all terms in a polynomial, recognizing the differences of two squares, and recognizing perfect squares of binomials. Big Ideas involved in the lesson: Using basic factoring technique to factor second degree polynomials with a leading coefficient of one. Use basic factoring techniques to factor second degree polynomials with a leading coefficient greater than one. Recognize the difference of two squares and perfect squares of binomials. As a result of this lesson students will: Know: Vocabulary: x-intercept, roots, zeros, solutions, like terms, coefficient, degree, constant term, exponents, variables, monomial, binomial, trinomial, polynomial, factor, common factor, factorization, quadratic expression and equation, linear expression and equation, leading coefficient, degree of a polynomial, cubic expression, prime, composite, perfect square, prime polynomial, area model Standard form of a quadratic polynomial: ax2+bx+c, a 0. Factored form: a polynomial is in factored form if it is written as the product of two or more linear factors, such as, for a quadratic, (ax+b)(cx+d) where a 0 and c 0. Special products of polynomials o Difference of two squares pattern a2-b2 = (a+b)(a-b) o Square of a binomial pattern a2 2ab+b2 = (ab)2. Discriminant expression: b2 - 4ac. Understand: Factoring a quadratic trinomial means to express it as the product of two linear expressions. Factoring a quadratic expression is accomplished by using the converse of the distributive property. In an area model, when two linear factors are multiplied, the product represents the area of a rectangle whose length and width are the linear factors. All quadratic equations are not factorable over the set of rational numbers. The relation between an equation in mathematics and a sentence in English, (symbolically and verbally), in particular when the situation represents a quadratic function Be Able To Do: Factor quadratic trinomial by using the greatest common factor and the distributive property. Factor a quadratic trinomial by grouping. Factor a quadratic expression by recognizing special product forms such as the difference of two squares and the square of a binomial pattern. Find the dimensions of a rectangle or a square, given its area as a factorable quadratic polynomial. Translate a word problem that can best be represented by a quadratic model into a mathematical model, apply the model and determine a solution or solutions to the problem. 1 Lesson Design Check and verify solutions to quadratic equations, whether within the context of a word problem or not, for accuracy and reasonableness. Assessments: Formative: CFU Questions: What will be evidence of student DWA, classroom assessments, knowledge, understanding & ability? ABWA Embedded in lesson Summative: CST Lesson Plan Anticipatory Set: Use with lessons for Factoring with a coefficient of 1 and greater than 1 a. T. focuses students Use area model to multiply two polynomials. b. T. states objectives Go to file for Anticipatory Set. Show three different models of multiplication of c. T. establishes purpose of binomials. the lesson CFU d. T. activates prior Is there another way that we could use to solve this problem? knowledge What property did you use to solve this problem? Would you get the same answer if you solved it using a different method? How do the three different models compare? Which explains the multiplication best to you? Why? Use with lessons for factoring perfect square trinomials and difference of two squares binomials. Use PowerPoint 11.0 Factoring slide 1 Use with lesson for factoring with common factors Use PowerPoint 11.0 Factoring slide 9 2 Factoring x2 + bx + c; b > 0 and c > 0 LO: Factoring trinomials with a leading coefficient of 1 where b > 0 and c > 0. Give algebra tiles and x2 + 5x +6 to students. Ask students to use the algebra tiles to form a rectangle. Have student demonstrate on board their answer. -1 * 6 = -6 -1 +6 = 5 1 * -6 = -6 1 + -6 = -5 -2 + 3 = +1 -2 * 3 = -6 x2 5x x + 2 Instruction: a. Provide information Explain concepts State definitions Provide exs. Model b. Check for Understanding Pose key questions Ask students to explain concepts, definitions, attributes in their own words Have students discriminate between examples and nonexamples Encourage students generate their own examples Use participation x + 3 6 Lesson Design CFU: What is the dimension of the rectangle? (x + 2) (x + 3) Label each rectangle in the term of area. What is the total area of this rectangle? How do you arrive this answer? Can we place (x + 2) on the horizontal line and (x + 3) on the vertical line? Why or why not? If you switch the length and width of the rectangle, what property are you applying? Now students will be given the following problems to form rectangles using algebra tiles. x2+6x + 9 x2 + 7x + 6 x2 + 3x +1 With the dimensions of each rectangle create a table. Length Width 2 x + 5x + 6 x+2 x+3 x2 + 6x + 9 x+3 x+3 2 x + 7x + 6 x+1 x+6 x2 + 3x + 2 x+2 x+1 CFU: Look at each expression and its dimensions. How are they related? For the expression x2 + 5x + 6, to prove that (2)(3) = 6 (product) and 2 + 3 = 5 (Sum), using area model to show solution. Draw an area model, x2 goes to the top left box, the constant +6 goes to the bottom right box, and the second term in the middle of the box. x2 5x 6 x For the factors of x2 are x times x. For the factors of 6, starting with 1. 1 * 6 = 6, 1 + 6 = 7. These are not correct factors because we need the sum to equal 5. Find the other factors of six: 2 * 3 = 6 and 2 + 3 = 5. These are correct factors because the product equals 6 and the sum equals 5. x + 2 product of 6 sum of 5 x2 2x 1*6=6 1 + 6 =7 5x 2*3=6 2+3=5 6 + 3 3x 3 Lesson Design So the factors for x2 + 5x + 6 are (x+2) and (x+3). Continue working on completing the rest of expressions in the table. For x2 + bx+ c and using the area model, the first term goes in the top left box, the constant term, c, goes in the bottom right box, and the second term, bx, in the center of the box. The sum of the two factors equals b and the product of these two factors must also equal c. CFU Does the order matter when all the terms are positive? Why? Factoring x2 + bx + c ; b < 0 and c > 0 LO: Factoring trinomials with a leading coefficient of 1 where b < 0 and c > 0. Give algebra tiles and x2 - 5x + 6 to students. Ask students to use the algebra tiles to form a rectangle. Have student demonstrate on board their answer. x2 x -5x -3 +6 x -2 CFU: What is the dimension of the rectangle? (x - 2) (x - 3) Label each rectangle in the term of area. Why this section is (-3x) instead of (3x)? Why (-2x) instead of (2x)? Why is the constant tern (6), not (-6)? What is the total area of this rectangle? How do you arrive this answer? Can we place (x - 2) on the horizontal line and (x - 3) on the vertical line? Why or why not? If you switch the length and width of the rectangle, what property are you applying? Students will be given the following problems to form rectangles using algebra tiles. x2 – 5x + 4 x2 –4x + 4 x2 – 8x + 7 x2 – 7x +12 4 Lesson Design With the dimensions of each rectangle create a table. Length Width 2 x –5x + 4 x –4 x-1 x2 –4x + 4 x–2 x-2 2 x –8x + 7 x –1 x –7 2 x –7x + 12 x –4 x -3 Look at each expression and its dimensions. How are they related? For the expression x2 –5x + 6, to prove that (-2)(-3) = 6 and – 2 + -3 = 5, using area model to show solution. Draw an area model, x2 goes to the top left box, the constant +6 goes to the bottom right box, and the second term –5x in the middle of the box. x2 -5x +6 With a T-table we need to determine the factors for +6 where the product is +6 and the sum is negative –5. Product of +6 Sum of -5 x 2*3=6 2+3=5 3 1 + 6 =7 - 1 * 6= 6 x -2 x2 -2x -5x - 3s 6 -2 * -3 =6 -2 + -3 = -5 So the factors for x2 –5x +6 are ( x –2) and (x –3). Continuing working on completing the rest of the expression in the table. Factoring x2 + bx + c ; b < 0 and c < 0 LO: Factoring trinomials with a leading coefficient of 1 where b < 0 and c < 0. x2 –5x -6 x2 5 -5x -6 Lesson Design Students will probably place the ones unit with the –x’s. A negative multiplied by a negative is not negative. Ask student what can they do to place the negative units. (zero pairs) CFU: What are the differences of this expression (x2 – 5x –6) from the previous expressions we had been working with (x2 + 5x + 5) and (x2 –5x +6)? Can you create a rectangle using only these tiles? Why or why not? If you create a rectangle with only these tiles, does your model match our expression? (x2 – 5x –6) Which section does not match? (The section does not match is the constant, the product of –2 and –3 is 6 not –6 as indicated in our expression) We can’t change the constant value of the expression, so what can we change in order to form a rectangle? Teacher’s note: Students might not think about creating zero pairs. Teacher may remind them the warm-up exercises they did today. What would happen if we add just one zero pair? Is that enough to create a rectangle for x2 –5x –6? What is the dimension of the rectangle? (x - 6) (x +1) Label each rectangle in the term of area. Why this section is (-6x) instead of (6x)? Why (x) instead of (-x)? Why is the area of the constant tern (-6), not (6)? What is the total area of this rectangle? How do you arrive this answer? Can we place (x - 6) on the vertical line and (x +1) on the horizontal line? Why or why not? If you switch the length and width of the rectangle, what property are you applying? x -6 x +1 x2 – 3x –10 x2 –13x –14 x2 – 6x –7 Length ( x – 5) ( x – 14) ( x – 7) -6 x2 – 3x –10 –13x –14 x2 – 6x –7 x x2 x -6x x2 -5x +1 6 +1x -6 Width ( x + 2) (x +1) (x +1) Lesson Design Create a T-table, we need to determine the factors for –6 and the sum is -5 Product of -6 Sum of -5 -1 * 6= -6 1 + -6 = -5 1 * -6 = -6 1 + -6 = -5 So the factors for x2 –5x – 6 are ( x +1) ( x – 6) Factoring x2 + bx + c; b > 0 and c < 0 LO: Factoring trinomials with a leading coefficient of 1 where b > 0 and c < 0. x2 + x - 6 +x +3 x2 x -6 x -2 x2 + 2x – 8 x2 + 8x – 20 x2 + 3x –10 Length ( x – 2) ( x – 2) ( x – 2) x2 + 2x – 8 x2 + 8x – 20 x2 + 3x –10 x -3 x - x2 3x Width ( x +4) ( x + 10) ( x + 5) +1 Product of -6 -2 -2x 7 -6 Sum of +1 Lesson Design So the factors for x2 + x – 6 are ( x –2 ) ( x + 3). CFU: Can you make a model of the rectangle using the given quantity (x2 + x –6) of algebra tiles? What do we need to do in order to make a rectangular model? How many zero pair do we need in this case? How does this change the original expression? Why won’t this change the value of the original expression? What property are we applying? (Identity property of addition) What is the dimension of the rectangle? (x - 2) (x +3) Label each rectangle in the term of area. Why this section is (3x) instead of (-3x)? Why (-2x) instead of (2x)? Why is the constant tern (-6), not (6)? What is the total area of this rectangle? How do you arrive this answer? Can we place (x - 2) on the horizontal line and (x + 3) on the vertical line? Why or why not? If you switch the length and width of the rectangle, what property are you applying? Factoring perfect square trinomials and difference of squares binomials Use Riverdeep Course 2, Module: Powers and Polynomials, Unit: Factoring Polynomials, Tutorial 3: Special Cases Screens 1 and 2 only CFU questions attachment 11.0 Special Cases Riverdeep PowerPoint 11.0 Factoring slides 2-3 (Difference of squares) and 5-6 (Perfect square trinomials) Factoring using a common factor PowerPoint11.0 Factoring slides 10-13 and 15-18 odds (common factor) Use Riverdeep Course 2, Module: Powers and Polynomials, Unit: Factoring Polynomials, Tutorial 3: Special Cases Screens 3 only CFU questions attachment 11.0 Special Cases Riverdeep Guided Practice: a. Initiate practice activities under direct teacher supervision – T. works problem stepby-step along w/students at the same time b. Elicit overt responses 8 Factor with coefficient of one. x2 + 4x + 3 x2 + 8x + 7 x2 + 7x +6 Students can work independently or with a partner. Answer can be written in notes, on whiteboards, and be verbally expressed. CFU (change numbers for each expression) What thing that I want to do to factor this trinomial? (draw area model) Lesson Design from students that demonstrate behavior in objectives c. T. slowly releases student to do more work on their own (semi-independent) d. Check for understanding that students were correct at each step e. Provide specific knowledge of results f. Provide close monitoring What opportunities will students have to read, write, listen & speak about mathematics? Closure: a. Students prove that they know how to do the work b. T. verifies that students can describe the what and why of the work c. Have each student perform behavior Where is x2 located on the area model? 3? 4x? What are the factors of x2? What is the next step? (making a T-chart to find the product of 3 and then sum of 4) What are the products of 3? If you add them up, what will it equal to? What are the factors of the trinomial x2 + 4x + 3? Factoring perfect square trinomials and difference of squares binomials PowerPoint 11.0 Factoring slides 4 (Difference of squares) and 7 (Perfect square trinomials) Factoring using a common factor PowerPoint11.0 Factoring slides 14 and 15-18 even (common factor) Pair share, whiteboards, writing in notes with summary, draw with algebra tiles and explain each step Use with lessons for Factoring with a coefficient of 1 and greater than 1 Draw the algebra diagram and explain in detail how you found the factors for x2 + 7x + 10. Draw the algebra diagram and explain in detail how you found the factors for x 2 - 5x + 6. Draw the algebra diagram and explain in detail how you found the factors for x2 5x 6. Draw the algebra diagram and explain in detail how you found the factors for x 2 + x - 6. Use with lessons for perfect squares trinomials and differences of squares binomials. Use PowerPoint 11.0 Factoring slide 8 Independent Practice: a. Have students continue to practice on their own b. Students do work by themselves with 80% accuracy c. Provide effective, timely feedback Resources: materials needed to complete the lesson 9 Use with lessons for factoring with common factors Use PowerPoint 11.0 Factoring slide 19 Use with lessons for Factoring with a coefficient of 1 and greater than 1 Independent practice day 1, 2, 3, and 4 alg 11 ip coeff greater 1 Use with lessons for perfect squares trinomials and differences of squares binomials, and factoring with common factors St11 ind prac perfect diff wksht Algebra tiles, whiteboard, markers, graphic paper, line paper, rulers, pencils, color pencils worksheet sheets for independent practice . Lesson Design 10