UNIT TWO POLYNOMIALS MATH 421A 22 HOURS Revised May 2, 00 38
UNIT 2: POLYNOMIALS Previous Knowledge: With the implementation of APEF Mathematics at the intermediate level, students should be able to: - Grade 7 - addition and subtraction of first degree terms - Grade 7 - use algebra tiles to add and subtract polynomials and multiply a polynomial by a scalar - Grade 9 - factor algebraic expressions with common monomial factors - Grade 7 - solve simple linear equations - Grade 8 - solve and verify linear equations algebraically - Grade 8 - create and solve problems using linear equations - Grade 9 - solve and verify linear equations algebraically - Grade 9 - demonstrate an understanding of and apply the exponent laws for integral exponents Overview: - a brief review of the exponent laws - classification and operations on polynomials - factoring polynomials - solving linear equations - solving literal equations 39
SCO: By the end of grade 10 students will be expected to: A6 apply properties of numbers when operating upon and expressing equations Elaboration- Instructional Strategies/Suggestions Exponent Laws (1.7) Review exponent laws including introduction to zero and negative exponents. When developing zero exponent law, show students that ( x 2 ) ( x 0 ) = x 2. Then ask students to fill in the blank in the following: ( x 2 )( ) = x 2. Students will recognize that the blank must be 1 and thus x 0 = 1. Another method to illustrate this could be : You may also wish to expand the left side of this equation and reduce. Students should see that anything divided by itself equals one so therefore x 0 = 1. B23 understand the relationship that that exists between the arithmetic operations and the operations on exponents or logarithms B30 understand and use zero and negative exponents (fractional exponents optional) The same principle applies when working with negative exponents. Using the exponent laws, we know that This is an excellent opportunity to use TI-83 to explore its use in working with exponents. For example when a student evaluates 3-2 they would enter 3^-2 (use the gray minus sign)into the calculator the answer is 0.11111... This can be changed to common fraction form by pressing math Frac enter enter. The display on the screen shows 1/9 as the equivalent fraction. Note: Students may want to simplify numerical base problems like this: 2 3 2 2 = 4 5. The students should explore problems like this to convince themselves that the law does apply here. Optional Rational exponents may be explored. 40
Worthwhile Tasks for Instruction and/or Assessment Exponent Laws (1.7) Journal Write out all of the exponent laws. For each law, include an example problem and its solution. Suggested Resources Exponent Laws Mathpower 10 p. 33 #1-21,35-43 51-81 odd 85,86 Technology Using the TI-83, solve the following in common fraction form: a) (1/2) 3 b) (-2) -4 c) (3) -3 d) (-2/3) 3 Journal Explore problems such as: 2 2 2 4 and justify why the answer is 2 6 and not 4 6. Application Calculate the volume of air in a tennis ball container with diameter of 8 cm and a height of 24 cm, assuming the container has 3 tennis balls in it. Rational Exponent - Optional Mathpower 10 p.34-38 41
SCO: By the end of grade 10 students will be expected to: B1 model(with concrete materials and pictorial representations) and express the relationship between arithmetic operations and operations on algebraic expressions and equations B3 use concrete materials, pictorial representations and algebraic symbolism to perform operations on polynomials Elaboration - Instructional Strategies/Suggestions Polynomials (3.1)(3.3)(3.4) a) Classify by number of terms and degree: < A mathematical expression can be named by the number of terms which it contains. (ie. monomial, binomial, trinomial, polynomial) < An expression can also be classified by its degree. The degree of a monomial is determined by the sum of the exponents of the variable bases. The degree of a polynomial is determined by the highest degree term. A constant term is considered to have a degree of 0. (Note: The degree cannot be determined until the expression is simplified.) b) Operations on Polynomials: i) Review the concept of like terms and combining like terms through addition and subtraction. Algebra tiles should be used to demonstrate like terms. Students will have been previously exposed to algebra tiles. ii) Review multiplication of polynomials. This topic will rely on the students knowledge of the exponent laws. Again, algebra tiles should be used. Multiplication will include product of two trinomials and a binomial cubed but not beyond these problems. Note: Division of monomials has been previously covered under exponent laws. Invite students to discover the use of algebra tiles in multiplication of polynomials. An example is: (2x + 1)(x! 2) 42
Worthwhile Tasks for Instruction and/or Assessment Polynomials (3.1)(3.3)(3.4) a) Classify by number of terms and degree Journal/Communication A student has missed today s class. You are responsible to explain the degree of a monomial or a polynomial and the names of polynomials according to the number of terms. Write a paragraph explaining today s lesson and include examples for each of the following polynomials: a) binomial, degree three b) trinomial, degree four c) four terms, degree six d) monomial, degree five b) Operations on Polynomials Activity Like Term Memory Game Make cards with monomials on them - be sure that each card has a like term. The number of cards that you need will depend on the size of your class and the number of games necessary (a minimum of 20 cards per game is suggested). Students can play in pairs or small groups. Each student takes a turn flipping over two cards. If they turn over two like terms then they will pick up the cards, if not the cards will be turned back over. The winner is the student with the most pairs! Find cards in the Appendix. Manipulatives Have students model each of the following problems using algebra tiles and then state the product. 1. 2x (x + 1) 2. (x + 2)(x - 1) 3. (4x - 1) (x + 2) 4. ( x + 6) (2x + 1) An example with 2 variables is: (x + 1)(y!2) = xy!2x + y!2 Suggested Resources Polynomials a) Classify by number of terms and degree Mathpower 10, p. 102 # 1-8 b) Operations on Polynomials Mathpower, p. 102 #27-51 odd Problem Solving Strategies Math Power 10 p.105 #1-3,16-19 Mathpower 10, p.108 #44-54 even #59-65 odd #79-83 odd Applications p.109 # 87,88,90,92 p.112 #9-19 odd #31-37 odd #43-47 Applications p.113 #58-60 43
SCO: By the end of grade 10 students will be expected to: B3 use concrete materials, pictorial representations and algebraic symbolism to perform operations on polynomials Elaboration -Instructional Strategies/Suggestions Factoring Polynomials (3.6)(3.8) Note: Algebra tiles should be used to assist students in understanding each of the following types of factoring methods. a) Greatest Common Factor - Remove a numerical and/or algebraic expression from any polynomial. This topic will be a review from grade 9. Examples: 2x(x - 3) C21 expand and factor polynomial expressions using perimeter and area models x (a + b) + y (a + b) = (a + b)(x + y) b) Factor by Grouping in Pairs Students should realize that this is directly related to greatest common factoring. When given a four term polynomial, grouping the terms in pairs will often allow the removal of a common factor from each pair, leaving a common binomial factor. Examples: ac - ad + bc - bd = a(c - d) + b(c - d) = (c - d) (a + b) 2x 3-2x 2 y + xy - y 2 = 2x 2 ( x - y) + y(x - y) = (x - y) (2x 2 + y) c) Trinomials of the form ax 2 +bx + c, where a = 1 This topic should be introduced with algebra tiles. Making rectangles with tiles will reveal the factors of the expression as the dimensions of each side of the figure. Refer to your algebra tile manual for instructions and examples. Mathpower 10, p.98-99 is a great introduction to tiles and p. 125 shows one example of factoring this type of trinomial with tiles. Math 10, p. 362 gives an example for factoring using tiles. Students should now be capable of factoring mentally. This will involve finding the two numbers whose product is equal to the last term of the trinomial and whose sum is equal to the middle term. Invite students to construct a rectangle representing x 2-5x - 6 44
Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Factoring Polynomials (3.6)(3.8) a) Greatest Common Factoring Pencil/Paper Write a trinomial with different numerical coefficients that has a greatest common factor of 2xy. Write your trinomial in both unfactored and factored form. b) Factor by Grouping in Pairs Use algebra tiles to formulate and construct models of the following expression. State the dimensions of the rectangle formed. (x 2 + x) + (xy + y) c) Trinomials of the form ax 2 +bx + c, where a = 1 Journal Use pictorial representations of each expression to construct rectangles and determine their dimensions. 1) x 2 + 7x + 6 2) y 2 + 6y + 9 3) x 2-2x - 3 How would you determine the first term in each factor? Generate criteria that seems to describe the pattern between the last terms in the factors and the last term in the trinomial? Examine and discuss the pattern between the last term of the factors and the middle term of the trinomial? Explain why m 2 + 9m + 6 can t be factored. Write three other trinomials that don t have integral factors. Pencil/Paper The trinomial x 2 + kx + 24 has eight different pairs of binomial factors. Investigate the possible values for k. Manipulatives Place the tiles to represent x 2 + 5x + 4 on the overhead projector and invite a volunteer to arrange these tiles into a rectangle. What are the dimensions of the rectangle? Students should recognize that these are the factors of the trinomial. Factoring Polynomials a) Greatest Common Factoring Mathpower10, p. 120 #11-29 odd 37-41 odd b) Factor by Grouping in Pairs Mathpower 10, p. 120 # 30-36 c) Trinomials ax 2 + bx + c, a = 1 Mathpower 10, p. 127 #19-55 odd, 64 Problem Solving Strategies Math Power 10 p.115 #1,3,5 45
SCO: By the end of grade 10 students will be expected to: Elaboration - Instructional Strategies/Suggestions d) Trinomials of the form ax 2 +bx + c, where a > 1 (3.9) Again, algebra tiles should be used to show how to factor these trinomials concretely. Challenge students to create a rectangle using tiles for 2x 2-5x + 3 B3 use concrete materials, pictorial representations and algebraic symbolism to perform operations on polynomials C21 expand and factor polynomial expressions using perimeter and area models After students are comfortable with the tiles, you can move to symbolic factoring. There are at least two different approaches to this: 1) Breaking up the Middle Term(decomposition) - This process is similar to factoring ax 2 +bx + c, where a = 1. In factoring ax 2 +bx + c, where a > 1, follow these steps: < Find the product of a and c. < Find the factors of a and c that add or subtract to get the value of b. < Use these factors to replace the middle term to obtain a four termed expression. < Factor by grouping in pairs Example: 6x 2 + 13x - 5 ac =!30 6x 2-2x + 15x - 5 b = 13 2x(3x - 1) + 5(3x - 1) the two factors are chosen from (2x + 5)(3x - 1)!1, 30 1,!30!2, 15 2,!15!3, 10 3,!10!5, 6 5,!6 46
Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Factoring Polynomials (3.9) d) Trinomials of the form ax 2 +bx + c, where a > 1 Manipulatives Encourage students to model a trinomial such as 2x 2 + 5x + 2 with algebra tiles. By finding the dimensions of the rectangle, the students will discover the factors of the trinomial. d) Trinomials ax 2 + bx + c, a >1 Mathpower 10, p. 130 #13-49 odd Applications p.131 #51,53 Pencil/Paper Construct a pictorial rectangle for each trinomial and state their dimensions. a) 2x 2-11x + 15 b) 4x 2-17x - 15 Factor by Guess and Check: c) 2x 2 + 13x + 15 d) 2x 2-11x + 12 c) 5x 2-7x - 6 d) 5x 2 + 19x - 4 Problem Solving Strategies Math Power 10 p.123 #1(a),5,6,9 Factor the following by breaking up the middle term(decomposition): a) 6x 2 + 11x + 3 b) 3x 2-5x - 2 47
SCO: By the end of grade 10 students will be expected to: C21 expand and factor polynomial expressions using perimeter and area models Elaboration - Instructional Strategies/Suggestions 2) Guess and Check - Trinomials of the form ax 2 +bx + c, where a > 1, can be factored by looking at all combinations of binomial factors that when expanded will contain the first term and the last term of the trinomial. One of these pairs of factors will also produce the middle term when expanded. This can be easy if there are only a few factors but can be very time consuming for problems with a number of possible factors. Factor the following by Guess and Check: 2x 2 + x! 6 All of the following factors will give the first and last terms but only one pair will also give the middle term. (2x + 1) (x! 6) (2x! 1 ) (x + 6) (2x + 3) (x! 2) (2x! 3) (x + 2) The correct pair is (2x! 3) (x + 2). Verify by multiplication or using algebra tiles. e) Difference of Squares (3.10) Introduce this pattern for factoring by using algebra tiles. The students will need to complete the rectangle by adding zeros and then they will be able to determine the factors by finding the dimensions. Students should be able to recognize which factoring questions are a difference of squares by looking for two perfect squares that are being subtracted. f) Perfect Square Trinomials To identify these special products, the trinomial must meet the following criteria: < Are the first and last terms perfect squares and the sign of the last term positive? < Is the middle term equal to twice the product of the square roots of the first and last terms? Therefore the factors are two identical binomials (a binomial squared). Again, these can be explored through the use of algebra tiles. 48
Worthwhile Tasks for Instruction and/or Assessment Suggested Resources e) Difference of Squares (3.10) Activity Have students draw a table with binomial conjugates in the first column, their expanded form in the second column and the simplified expression in the third column. Use these binomials for the first column: e) Difference of Squares Mathpower 10, p. 133 # 1-11 odd After the students have completed their charts, have them answer the following questions: What do you notice about each expression in the third column? Observe and describe the pattern developed in the third column. f) Perfect Square Trinomial Journal/Communication Explain how you can recognize a perfect square trinomial. Activity Make a three column chart for expanding binomial squares. In the first column, write the two identical factors. In the second column, use FOIL to determine the four terms. In the third column, write the simplified form. After the chart is completed, examine the columns and look for a pattern. Can you find any pattern for multiplying a binomial squared? How would this help you in factoring a perfect square trinomial? f) Perfect Square Trinomials Mathpower 10, p. 133 #13-25 odd p.133 #29-43 odd 49
SCO: By the end of grade 10 students will be expected to: B1 model(with concrete materials and pictorial representations) and express the relationship between arithmetic operations and operations on algebraic expressions and equations Elaboration - Instructional Strategies/Suggestions Solving Linear Equations (p.194) In order for students to be able to solve quadratic equations, they must first review solving linear equations. This will involve both one-step and multi-step equations. Students must understand the concept of keeping the equation balanced by performing the same operation to each side of the equation. This can be easily demonstrated through the use of algebra tiles. Mathpower 10 gives a pictorial explanation of using tiles to solve linear equations on p. 176-177. Your algebra tile manual will also contain examples of these equations. {page numbers to be included when available.} B7 understand the relationships that exist between arithmetic operations used when solving equations and inequalities add!x and!1 to both sides Solving Literal Equations (4.9) This topic will assist students later in the course when they are required to solve an equation for y in order to graph manually or with the TI-83. The process is the same as solving linear equations except the students will be working with more than one variable. Example: Solve the following equation for t : I = p r t I = p r t p r p r I = t p r 50
Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Solving Linear Equations (p.194) Research/Presentation The air temperature drops by 1 0 C for every 100m increase in altitude. If the air temperature at sea level is 20 0 C, a) write an equation describing the situation b) What would the temperature be at the top of the CN Tower? Do some research to find the tower s height. c) What would be the temperature outside a jet at an altitude of 11,000m? Solving Linear Equations These have already been done in unit one but it wouldn t hurt to do a few more. Technology Solve some of the problems on p.194 Green #1 with the TI- 83. Ex. 3x + 2 = 11 Press 2 nd calc 5:intersect and press enter 3 times We can see that at an x value of 3" 3x + 2 equals 11. Solving Literal Equations (4.9) Pencil/Paper/Research Do some reading on the life of a famous mathematician or scientist and present a short history of their life along with a formula that they developed. Initiate a class discussion on how that formula can be re-arranged various ways. Research Research a famous mathematician and bring out some interesting features about their life. Also discuss in the paper the person s major contributions to mathematics. Solving Literal Equations Mathpower 10, p. 192-193 http://www.alph0.clarku.edu/~djoyce mathhist/chronolgy.html http://forum.swarthmore.edu/dr.math Problem Solving Strategies Math Power 10 p.143 #1,5 51