Page 1. A Five Day Exploration of Polynomials Using Algebra Tiles and the TI-83 Plus Graphing Calculator Pre- Algebra Grade Seven By: Cory Savard
Page. Overall Unit Objectives Upon completion of this unit, students will be able to do the following things: 1. Both identify and classify polynomials and find their degree. Standards met: NCTM: #1 (Mathematics as problem solving), # (Mathematics as communication), #3 (Mathematics as reasoning), #4 (Mathematical connections), and #9 (Algebra). NYS: #1 (Mathematical Analysis, part 1a).. Evaluate polynomials. Standards met: NCTM: #1 4, 9 NYS: #3 (Mathematical Reasoning). 3. Use algebra tiles to represent polynomials. Standards met: NCTM: #1 4, 9 NYS: #6 (Models). 4. Add, subtract and multiply polynomials using algebra tiles. Standards met: NCTM: #1 4, 9 NYS: #1 (Analysis). 5. Use the TI-83 Plus graphing calculator to graph polynomials. Standards met: NCTM: #1 4, 9
Page 3. NYS: # (Information Systems) 6. Identify the degree of a polynomial based on the shape of its graph, and identify the shape of a graph based on the degree of its polynomial equation. Standards met: NCTM: #1 4, 9 NYS: # (Information Systems).
Page 4. Resources Used 1. Pre-Algebra: An Integrated Transition To Algebra and Geometry, by Price, Rath, Leschensky, Malloy and Alban; Copyright 1997 by Glencoe/The McGraw Hill Companies, Inc. Material used includes Chapter 14 (pp. 706-730).. Curriculum and Evaluation Standards for School Mathematics, Copyright 1989 by The National Council of Teachers of Mathematics, Inc.
Page 5. Materials and Equipment Needed 1. A class set of homemade algebra tiles.. One set of algebra tiles for use with the overhead projector. 3. An overhead projector. 4. A class set of TI-83 plus graphing calculators, as well as an overhead projector kit to display the calculator screen on the overhead. 5. Handouts #1 9 6. Answer keys for handouts #1-4, 6-9. 7. Bookwork answer keys #1-5. 8. Pre-Algebra: An Integrated Transition To Algebra and Geometry, by Price, Rath, Leschensky, Malloy and Alban; Copyright 1997 by Glencoe/The McGraw Hill Companies, Inc.
Page 6. Polynomials: A Unit Overview 1. Introducing Polynomials a. Monomials, binomials and trinomials b. The constant c. Finding the degree d. Evaluating polynomials. Adding Polynomials with algebra tiles a. Introduce algebra tiles i. Explain that each tile has a value based in its area. ii. Explain negative tiles iii. Practice modeling polynomials with the tiles. b. Explain some characteristics of polynomials. i. The coefficient ii. Like terms c. Begin adding polynomials with the tiles. i. Explain zero pairs ii. Practice problems 3. Subtracting polynomials with algebra tiles a. Explain that subtracting is simply adding the additive inverse. i. Distribution ii. Additive inverse and how to find it
Page 7. b. Practice finding the additive inverse c. Practice subtracting polynomials 4. Multiplying Polynomials with algebra tiles a. Using rectangles b. Multiplying a polynomial by a monomial i. Distribution ii. Commutativity c. Practice problems 5. More multiplying with algebra tiles a. Multiplying binomials b. Combining like terms c. Introducing word problems d. Practice problems
Page 8. Day #1 Introducing Polynomials Concept: Identifying, classifying, finding the degree of and evaluating polynomials. Objectives: By the end of the day students should be able to identify monomials, binomials and trinomials. Students should also be able to find the degree of a monomial, binomial, trinomial or constant. Finally, students should be able to evaluate polynomials when given a value for the variable. Materials: One copy of Handout #1 per group of five students, Handout #1 answer key, one copy of Handout # for each student, Handout # answer key, Bookwork answer key #1. Opening Activity: Introduce the terms monomial, binomial, trinomial, polynomial and constant. Developmental Activity: 1. Identifying monomials, binomials, trinomials and the degree of each.. Evaluating polynomials for a given variable value. Closing Activity: Pose questions to verify student understanding in the form of a relay.
Page 9. Day #1 Teacher Notes Opening Activity: Have students recall the definitions of the terms monomial, binomial, trinomial, polynomial and constant. Use words in which the prefixes are familiar, such as bicycle and tricycle, to help the students understand the definitions. Developmental Activity: Ask the students to identify each of the following expressions as a monomial, binomial or trinomial, or not a polynomial at all. 1. x. x 3 3. x + 4 Answer: binomial Answer: monomial + 4x + 4 Answer: trinomial 4. x + x 1 Answer: not a polynomial Introduce the term constant and ask the students to identify the constant term of each of the following: 1. x + 1 Answer: 1. x + 3x + 5 Answer: 5 Introduce the term degree and explain how to find the degree of a monomial, binomial, trinomial or constant. Ask students to find the degree of the following: (Recall that the exponent of x is 1.) 1. 3x Answer: 1. xy Answer: 3 Inform students that the degree of a constant term is zero.
Page 10. Explain to students how to find the degree of a polynomial. Ask the students to find the degree of the following polynomial by finding the degree of each monomial within the polynomial: x 3 + xy + 5 Answer: 3 x = Degree 3 xy = Degree 5 = Degree 0, Since 5 is a constant 3 x + xy + 5 = Degree 3, Since 3 is the highest monomial degree in the polynomial. Ask the students to find the degree of the polynomial x y + x + y 3. Answer: 4 Show students how to evaluate polynomials for given variable value/values. Solve the following equation with the students: x + xy + 1 when x=3 and y=. 1. Plug in 3 for x and for y.. Solve, obtaining 16. Ask the students to evaluate xy + y + 3 for x=7 and y=5. Answer: 63. Closing Activity: Conduct a relay with the students. Separate the students into rows of five, if they are not already in such rows. Cut up one copy of handout #1 per row of students and distribute question #1 to the first student in each row, question # to the second student in each row, etc. Student #1 will solve his/her question and pass that answer on to student #, who will use the answer from student #1 to solve his/her question, and so on. There should be no talking allowed. Upon completion of the last
Page 11. question student #5 should bring the final answer to the teacher for confirmation or denial. If correct the teacher says yes and that row has successfully completed the task. If incorrect the teacher says no and that row must uncover where the mistake was made, at which point they may be permitted to talk it over. Homework Assignment: From Pre-Algebra: An Integrated Transition To Algebra and Geometry, by Price, Rath, Leschensky, Malloy and Alban; Copyright 1997 by Glencoe/The McGraw Hill Companies, Inc. Page 707 / 1-4 Page 708 / 5, 1, 16, 18, 7, 36, 45, 55 Also, Handout #
Page 1. Handout #1 #1. Find the degree of the polynomial 3x + xy + 1. #. Evaluate the polynomial x + 4x + 3 when x=(the # you received). #3. Evaluate the polynomial 5x + 4 when x=(the # you received). #4. Evaluate x when x=(the # you received). #5. Evaluate the polynomial x 7 when x=(the # you received).
Page 13. Handout #1 Answer Key #1. Find the degree of the polynomial 3x Answer: 3 + xy + 1. #. Evaluate the polynomial x + 4x + 3 when x=(the # you received). Answer: 4 #3. Evaluate the polynomial 5x + 4 when x=(the # you received). Answer: 144 #4. Evaluate x when x=(the # you received). Answer: 1 #5. Evaluate the polynomial x 7 when x=(the # you received). Answer: 17
Page 14. Handout # NAME DATE Here is a strategy for finding the degree of a monomial: *Recall that the degree of a monomial is the sum of the exponents of its variables. Also remember that x = x 1.* Monomial Variables Exponents Degree x x 1 1 m m 5 3 4 z 5 y z, y 4, 5 4 + 5 = 9 xy 3 z x, y, z 1,, 3 1 + + 3 = 6 Complete the following chart: Monomial Variables Exponents Degree xy x yz 4 3xy x mn 3
Page 15. Handout # Answer Key NAME DATE Here is a strategy for finding the degree of a monomial: *Recall that the degree of a monomial is the sum of the exponents of its variables. Also remember that x = x 1.* Monomial Variables Exponents Degree x x 1 1 m m 5 3 4 z 5 y z, y 4, 5 4 + 5 = 9 xy 3 z x, y, z 1,, 3 1 + + 3 = 6 Complete the following chart: Monomial Variables Exponents Degree xy x, y 1, 1 1 + 1 = x yz 4 x, y, z, 1, 4 + 1 + 4 = 7 3xy x, y 1, 1 1 +1 = x x 1 1 mn 3 m, n 1, 3 1 + 3 = 4
Page 16. Bookwork Answer Key #1 Page 707 / 1-4 Page 708 / 5, 1, 16, 18, 7, 36, 45, 55, 56 1. No because there is a variable in the denominator.. Sample answers: x + 1, x + 3, x 3 - x. These are binomials because they each the sum or difference of two monomials. 3. Of a monomial: sum the degrees of every variable in the term. Of a polynomial: Find the degree of each term in the expression and take the highest one. 4. Learn the meaning of the prefixes. For example, mono- means one, bi- means two, tri- means three and poly- means more than one. The prefixes represent the number of terms in the expression. 5. yes; monomial 1. Answer: 4 16. Answer: -60 18. yes; monomial 7. yes; monomial 36. Answer: 3 45. Answer: -18 55. a. xy + x + y + z b. 56. a. 5 b. 104,000
Page 17. Day # Adding Polynomials With Algebra Tiles Concept: Modeling and adding polynomials with algebra tiles. Objectives: By the end of the day students should understand how tiles are based on area, be able to represent polynomials with the tiles, add polynomials using the tiles, and understand the concept of zero pairs. Materials: One set of overhead projector compatible algebra tiles, one overhead projector, a class set of homemade algebra tiles, two copies of handout #3 for each student in the class, the handout #3 answer key, bookwork answer key #, and calculator work answer key #1. Opening Activity: Introduce algebra tiles and explain how each tile has a specific value represented by its area. Developmental Activity: 1. Use tiles to represent monomials, followed by polynomials.. Look at tile representations of polynomials and identify each as a monomial, binomial or trinomial. 3. Introduce the concept of adding like terms. 4. Add polynomials involving negative tiles and demonstrate an understanding of the concept of zero pairs. Closing Activity: Pose questions to verify student understanding in the form of a group activity.
Page 18. Day # Teacher Notes Opening Activity: Introduce the algebra tiles to the students. Explain to them how each tile has a specific value represented by its area. Tile Dimensions Area Color 1 by 1 (1)(1)= 1 Yellow 1 by X (1)(X)= X Green X by X (X)(X)= X Blue Negative tiles are of the same dimensions as the above tiles, only they are red in color. Developmental Activity: 1. A. Have the students represent the monomials x, 3x and 4 with algebra tiles. Answer: x :
Page 19. 3x: 4: Together these tiles represent the polynomial x + 3x + 4. B. Have the students model the polynomial x + 3x 3 with the tiles. Recall that the negative tiles are red. Answer: x + 3x 3. Ask the students to identify whether x + x + 1 is a monomial, binomial or trinomial based only on its tile representation, which is as follows: Answer: Trinomial Explanation: The fact that we use tiles of three different areas to represent this polynomial leads us to the conclusion that this polynomial is a trinomial.
Page 0. 3. Explain to the students what like terms are and how to add them. These like terms can be added as follows: = 3x Ask students to write the equation of the following tile representation: Answer: x + 4x + 3 4. A. Ask students to add the following polynomials using their knowledge of like terms: x Answer: + x + 1 and x + 3x +.
Page 1. Group like terms: Add them together and come to the conclusion of 3x + 4x + 3. B. 1. Explain to the students the concept of zero pairs. = 0 = 0 = 0 Explain to students that zero pairs can be removed from the group without changing the value of the polynomial the tiles represent.. Have students add the following polynomials and remove any zero pairs using the three steps outlined below: x + x + 1 and -x - x 1.
Page. Step #1: Group all tiles together, like areas next to each other. Answer: Step #: Remove any zero pairs. Answer: that are left. Step #3: Write the equation of the resulting polynomial represented by the tiles Answer: x + x Closing Activity: Instruct each student to write the equations of any three polynomials, labeling them 1, and 3. Pair off the students with the person sitting next to them. Should there be a student without a partner, the teacher shall work with that student. Pass out to each student two copies of Handout #3 (one for this activity and the other for the homework). The pairs shall take and add their first two polynomials labeled 1, add their second two polynomials labeled, and add their last two polynomials labeled 3, filling in the handout as they go. The tile representations should be drawn on the handout. The
Page 3. teacher should walk around the room and look at the student s work to be sure that they have a clear understanding of how to develop and add polynomials. Homework: From Pre-Algebra: An Integrated Transition To Algebra and Geometry, by Price, Rath, Leschensky, Malloy and Alban; Copyright 1997 by Glencoe/The McGraw Hill Companies, Inc.: Page 713 / 1-3, 4-6, and 13. Instructions for 4-6: Use Handout #3 to carry out the three steps for adding polynomials as covered in the lesson. Draw the tile representations for each step. Also, using your TI-83 plus, graph each of the following sets of polynomials, then graph their sum. What can you say about the graph of the sum when compared to the graphs of the original two polynomials? 1. (x + 1) and (x + 3). (-x ) and (3x 1) 3. (x - x 1) and (x + x + 4)
Page 4. Calculator Work Answer Key #1 1. (x + 1) (x + 3) Sum: (3x + 4). (-x ) (3x 1) Sum: (x 3)
Page 5. 3. (x - x 1) (x + x + 4) Sum: (3x + x +3) If we add two lines, the resulting sum is also a line. If we add two curves, the resulting sum is also a curve.
Page 6. Handout #3 NAME DATE Polynomials to be summed: Step #1: Step #: Step #3: Polynomials to be summed: Step #1: Step #: Step #3: Polynomials to be summed: Step #1: Step #: Step #3:
Page 7. Bookwork Answer Key # Page 713 / 1-3, 4-6, 13 1. Answer: x. Answer: and x, 7x and 3x, and 1 and 5. a. Group like terms; (5x + x ) + (6x + 3x) + (4+1) b. Remove any zero pairs; There are none in this case c. Write the equation of the resulting polynomial; 7x + 9x + 5 3. Caroline is correct because the variables are the same in both monomials, only in a different order. The answers to 4 6 are on the Handout #3 Answer Sheet. 13. Perimeter of rectangle = 6x + 4 units
Page 8. Handout #3 Answer Key / Bookwork Answers to #4 - #6 #4. Polynomials to be summed: (3x 1. Group like terms: - x + 1) and (x + 5x 3). Remove zero pairs: 3. Write the equation of the resulting polynomial: 4x + 3x #5. Polynomials to be summed: (-x 1. Group like terms: + x 5) and (x - 3x + )
Page 9.. Remove zero pairs: 3. Write the equation of the resulting polynomial: -x - x 3 #6. Polynomials to be summed: (x - 3x + 6) and (x + 5x 4) and (x + x +1) 1. Group like terms:
. Remove zero pairs Page 30.
Page 31. 3. Write the equation of the resulting polynomial: 4x + 3x + 3
Page 3. Day #3 Subtracting Polynomials with Algebra Tiles Concept: Understanding that subtracting a polynomial is simply adding the negation of that polynomial, or the additive inverse. Also, distribution. Objectives: By the end of the day students should be able to find the additive inverse of a polynomial as well as subtract polynomials. Materials: A class set of homemade algebra tiles as well as an overhead set and an overhead projector, bookwork answer key #3, handout #4, handout #4 answer key, and calculator work answer key #. Opening Activity: Show students the take away model for subtracting polynomials with algebra tiles. Practice with this model. Developmental Activity: 1. Introduce the additive inverse and how to find it.. Teach students how to distribute. 3. Subtract polynomials with algebra tiles. Closing Activity: Pose questions to verify student understanding in the form of a relay.
Page 33. Day #3 Teacher Notes Opening Activity: Begin with the take away model for subtracting polynomials with algebra tiles. Ask the students to make a model of the polynomial 4x + 4x + 3. Answer: Now ask the students to subtract 3x + x + 1 from the model by taking away three x -tiles, two x-tiles and one unit tile. Answer: The result is the polynomial x + x +.
Page 34. Now ask the students to subtract x Answer: + x + 1 from 3x + 4x +. 1. Model 3x + 4x +. Take away x + x + 1 3. Write the equation of the resulting polynomial: x + 3x + 1. Developmental Activity: Explain to the students that subtracting one polynomial from another is simply adding the additive inverse of one polynomial to the other. Explain that the additive inverse of a polynomial is found by multiplying that polynomial by negative one. Have the students practice finding the additive inverse of each of the following monomials: 1. 3xy Answer: -(3xy) = -3xy. x Answer: -(x ) = -x
Page 35. Now ask the students to find the additive inverse of the polynomial 4x + 3x +. To do this we multiply the entire polynomial by negative one and distribute the negative. Distribution of the negative changes the sign of each term in the polynomial, positive to negative and negative to positive. Answer: -1(4x + 3x + ) = -4x + 3x + Ask the students to find the additive inverse of each of the following polynomials on their own: 1. -x + 5x + 4 Answer: x - 5x 4. -x - 3x + 4 Answer: x + 3x 4 Explain to the students that adding a polynomial to its additive inverse yields a result of zero. Examples: -4xy + 4xy = 0 (x + 3) + (-x 3) = 0 Ask the students to subtract the following polynomials: 1. (x + x + 5) (x - 3x + ) Step #1: Find the additive inverse of (x - 3x + ) Answer: -x + 3x Step #: Add the additive inverse to x + x + 5 Answer: (x + x + 5) + (-x + 3x ) = -x + 5x + 3. (3x y + y + 4) (x y y + ) Step #1: Find the additive inverse of (x y y + ) Answer: -x y + y
Page 36. Step #: Add the additive inverse to (3x y + y + 4) Answer: (3x y + y + 4) + (-x y + y ) = x y + 3y + Closing Activity: Have the students participate in a relay activity in order to demonstrate their understanding of the material. Separate the students into rows of five, if they are not already in such rows. Cut up one copy of handout #4 per row of students and distribute question #1 to the first student in each row, question # to the second student in each row, etc. Student #1 will solve his/her question and pass that answer on to student #, who will use the answer from student #1 to solve his/her question, and so on. There should be no talking allowed. Upon completion of the last question student #5 should bring the final answer to the teacher for confirmation or denial. If correct the teacher says yes and that row has successfully completed the task. If incorrect the teacher says no and that row must uncover where the mistake was made, at which point they may be permitted to talk it over. Homework: From Pre-Algebra: An Integrated Transition To Algebra and Geometry, by Price, Rath, Leschensky, Malloy and Alban; Copyright 1997 by Glencoe/The McGraw Hill Companies, Inc. Page 717 / 1-4, 3-33 odd, 36-40, 4 Also, ask students to graph the following pairs of polynomials. What can they say about the graph of their difference in comparison to the graphs of the original two polynomials? 1. (x + 1) and (x + 3); for the difference subtract (x + 3) from (x + 1).
Page 37.. (-x ) and (3x 1); for the difference subtract (3x 1) from (-x ). 3. (x - x 1) and (x + x + 4); for the difference, subtract (x + x + 4) from (x - x 1).
Page 38. Calculator Work Answer Key # 1. (x + 1) (x + 3) Difference: (-x ). (-x ) (3x 1) Difference: (-4x 1)
Page 39. 3. (x - x 1) (x + x + 4) Difference: (x - 3x - 5) If we subtract two lines the resulting difference is also a line. If we subtract two curves the resulting difference is also a curve.
Page 40. Bookwork Answer Key #3 Page 717 / 1-4, 3-33 odd, 36-40, 4 1. Addition and subtraction are inverse operations.. Multiply the polynomial by negative one. The additive inverse of 3x - 5x + 7 is -3x + 5x 7. 3. Zero 4. One possible answer: ( 7m 3. x 3 5. x 8 7. 3x + 5y - m + 6) (4m + 3m + ) 9. -x 31. m + 6x 7-7m + 10 33. 15m - m + 10 36. 7x 37. 8y 38. 4a + 6x + 3 + 4y 5 + 4ab b 39. 6x 3-4x y + 13xy 40. A= x + 3x 5 B= x - x + 3 4. 7x + 5
Page 41. Handout #4 #1. Subtract (x + x + 1) from (5x + 7x + 9). #. Distribute 1(The polynomial you received). #3. Subtract (The polynomial you received) from 3x + x + 1. #4. Subtract (x + 4x + 7) from (The polynomial you received). #5. Find the additive inverse of (The polynomial you received).
Page 4. Handout #4 Answer Key #1. Subtract (x Answer: 3x + x + 1) from (5x + 6x + 8 + 7x + 9). #. Distribute 1(The polynomial you received). Answer: -3x - 6x - 8 #3. Subtract (The polynomial you received) from 3x + x + 1. Answer: 6x + 8x + 9 #4. Subtract (x Answer: 5x + 4x + 7) from (The polynomial you received). + 4x + #5. Find the additive inverse of (The polynomial you received). Answer: -5x - 4x
Page 43. Day #4 Multiplying with Algebra Tiles Concept: Using the area of a rectangle to represent the product of two polynomials. Also, understanding distribution and commutativity. Objectives: By the end of the day students should understand the concepts of distribution and commutativity and be able to apply these concepts to the multiplication of a monomial and a polynomial. Students should also understand that the product of two polynomials can be represented as the area of a rectangle. Materials: A class set of homemade algebra tiles and one set of overhead tiles, one overhead projector, handout #5 and bookwork answer key #4. Opening Activity: Demonstrate how the product of polynomials can be represented as the area of a rectangle. We begin with a monomial and a binomial. Developmental Activity: Include a discussion on distribution and commutativity, and practice multiplying monomials by binomials. Closing Activity: Play a review game involving the overhead projector and algebra tiles in which the students are split into groups of about 5.
Page 44. Day #4 Teacher Notes Opening Activity: Show students how to use the area of a rectangle to represent the product of two polynomials. One polynomial represents the length and the other polynomial represents the width of the rectangle. Demonstrate for the students how to find the product of x and x + 1, using handout #5 as a guide. We want to match the dimensions of length and width to fill in the rectangle with tiles. The tile that goes in the outlined region must have a length of x and a width of x.
Page 45. The tile that goes in this outlined region must have a length of x and a width of 1. We add the tiles in the resulting rectangle to get the product of the two polynomials: Answer: x (x + 1) = x + x Now ask the students to find the product of x and (x + 3). Answer: x + 6x
Page 46. Developmental Activity: In order to understand the multiplication of a monomial by a polynomial the students must understand distribution and commutativity. Ask the students how they would calculate (x + 1). Answer: x + Explain to the students that this is much like distributing the negative one when we found the additive inverse, only this time we multiplied each term of the polynomial by two instead of negative one. Ask the students to evaluate x(x + ) and (x + )x. Answer: x + x and x + x, respectively. This demonstrates the property of multiplication known as commutativity, which means that it does not make a difference in which order you perform the operation, in this case multiplication. The results will be the same. Ask the students to evaluate the following: a. 4x(5 + x) Answer: 0x + 4x
Page 47. b. (6x + 3) 3 Answer: 9 + 18x c. x(7x + 3) Answer: -14x - 6x d. 3(6x + 3) Answer: 18x + 9 Now ask the students if any two of the four above solutions are commutative and to justify their answers. Answer: Yes, b and d are commutative because we are multiplying the same terms, just in a different order, and we obtain the same result. Now have the students practice multiplying monomials and polynomials, using handout #5 as a guide. Remind students that thanks to commutativity it does not matter which polynomial they choose to be the length and which one they choose to be the width. 1. Evaluate 3x(x + ). Answer: 3x + 6x. Evaluate x (x + 3).
Page 48. Answer: -x - 3x Closing Activity: Begin by dividing the class into groups of four or five students. Instruct each group to nominate a leader who will be responsible for going up to the overhead projector and carrying out designated tasks. What we will do is introduce a monomial and a polynomial to one group at a time, at which point the group leader will multiply the two together using the overhead tiles. The group leader will only be allowed to carry out the multiplication as suggested by his/her group members. Once the group thinks that they have reached the solution they will present it to the teacher for verification. Once the group has achieved the desired solution we move on to the next group and repeat the process. Groups that get stuck should be allowed to struggle for a bit before they will be permitted to receive assistance first from other classmates and, as a last resort, from the teacher. Sample monomial / polynomial pairs to use include: X (x + 1) Answer: x + x 3x(5 x) Answer: 15x 3x -x(-9 4x) Answer: 18x + 8x
Page 49. 6 (7 + 3x) Answer: 4 + 18x 4x (-x + 4) Answer: -4x + 16-5 (x 1) Answer: -10x + 5 x (8x + 3) Homework: Answer: 8x + 3x From Pre-Algebra: An Integrated Transition To Algebra and Geometry, by Price, Rath, Leschensky, Malloy and Alban; Copyright 1997 by Glencoe/The McGraw Hill Companies, Inc. Page #76 77 / 1-3, 6-31, 35, 36
Page 50. Handout #5 NAME DATE
Page 51. Bookwork Answer Key #4 Page 76-77/ 1, 3, 6-31, 35, 36 1. Distribute the x (multiply both the x and the 1 by x). 3. Multiplication is commutative. 6. 18y + 7y - 1y 3 7. 4c 4 + 8c 3-40c 8. 15x 9. -1x 5 5 + 40x 3 + 48x 4 + 60x - 4x 30. 10x 4-8x 3 + 1x - 18x 31. -x 5 + x 4-3x 3 + 5x 35. a. 1x + 8x + 1 b. 96 in 36. 3s - 3s
Page 5. Day #5 More Multiplying with Algebra Tiles Concept: Multiplying two binomials and representing their product as the area of a rectangle. Also, introducing word problems. Objectives: By the end of the day students should be able to multiply two binomials and solve related word problems. Materials: A class set of homemade algebra tiles, one set of overhead projector tiles, one overhead projector, handout #5, handout #6, handout #6 answer key, bookwork answer key #5, and computer work answer key #3. Opening Activity: Demonstrate for the students how to multiply two binomials. Have them follow along with their tiles and handout #5. Developmental Activity: Introduce a word problem to the students and teach them how to go about solving it. Closing Activity: In an effort to pull together the material learned throughout the week, we will conduct a review relay.
Page 53. Day #5 Teacher Notes Opening Activity: Explain to the students that two binomials are multiplied much the same way a monomial and a binomial are multiplied; by constructing a rectangle and finding its area. Ask the students to multiply (x + ) and (x + 1). Instruct them to use handout #5 as a guide. Answer: x + 3x + Now ask the students to multiply (x + 3) and (x + 4). Answer: x + 10x + 1
Page 54. Developmental Activity: Introduce a word problem to the students and teach them how to go about solving it. Introduce Example # from page 79 of the text, which states: A rectangular garden is five feet longer than twice its width. It has a sidewalk three feet wide on two of its sides. The area of the sidewalk is 13 square feet. Find the dimensions of the garden. *Ask the students to refer to the picture in the text* Instruct the students to go about solving the problem as follows: 1. Let x = the width of the garden. Let x + s = the length of the garden 3. Let x + 3 = the width of the garden and the sidewalk 4. Let x + 8 = length of the garden and the sidewalk 5. Let 13 = the area of the sidewalk (x + 3)(x + 8) x(x + 5) = 13 x + 6x + 8x + 4 - x - 5x = 13 (count tiles in picture in text) (x - x ) + (6x + 8x 5x) + 4 = 13 9x + 4 = 13 9x = 189 x = 1 Solve the length and width of the garden (#1 and # from above list). We now have: 1 = width of the garden
Page 55. x + 5 = (1) + 5 = 47 = the length of the garden Closing Activity: In an effort to link all of the key material learned in the past week we should conduct a review relay. Separate the students into rows of five. Cut up one copy of handout #6 per row and distribute question #1 to the first student in each row, question # to the second student in each row, and so on. Student #1 will solve his/her question and pass that answer on to student #, who will use that answer to answer his/her question, and so on. There should be no talking allowed. Upon completion of the last question, student #5 should bring the final answer to the teacher for confirmation or denial. If correct the teacher says yes and that row has completed the task. If incorrect the teacher says no and the students in that row must find where the mistake was made. This tests the students understanding of all of the important material learned in the past week. Homework: From Pre-Algebra: An Integrated Transition To Algebra and Geometry, by Price, Rath, Leschensky, Malloy and Alban; Copyright 1997 by Glencoe/The McGraw Hill Companies, Inc. Page #730 / 1, 17-33 odd Also, ask the students to graph the following pairs of polynomials and the graph of their product. What can they say about the graph of the product when compared to the graphs of the original two polynomials? 1. (x + 1) and (x + 3). (-x ) and (3x 1)
Page 56. Computer Work Answer Key #3 1. (x + 1) (x + 3) Product: (x + 5x + 3). (-x ) (3x 1) Product: (-3x - 5x + ) When two lines are multiplied, the result is a curve.
Page 57. Handout #6 #1. Multiply: (x + 3) (x + 5) #. Add (4x + 7x + 6) to the polynomial you received. #3. Subtract (x + 15x + x y - 3) from the polynomial you received. #4. Find the degree of the polynomial you received. #5. Evaluate (3x - 4x + 5) when x = the number you received.
Page 58. Handout #6 Answer Key #1. Multiply: (x + 3) (x + 5) Answer: x + 13x + 15 #. Add (4x Answer: 6x + 7x + 6) to the polynomial you received. + 0x + 1 #3. Subtract (x + 15x + x y - 3) from the polynomial you received. Answer: 5x + 5x x y + 4 #4. Find the degree of the polynomial you received. Answer: 3 #5. Evaluate (3x - 4x + 5) when x = the number you received. Answer: 7 1 + 5 = 0
Page 59. Handout #5 NAME DATE
Page 60. Bookwork Answer Key #5 Page #730 / 1, 17-33 odd 1. Sample answer: x + 7x + 3 17. x 19. x 1. x + 7x + 1-4x 1-5x 36 3. x - 5x - 1 5. x + 6x + 9 7. 10x 9. 18x - 11x - 6-3x + 8 1 31. a. (x + )(x 1) b. x + x - 33. a. 14 in. and 7 in. b. 378 in c. 464 in
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