Grade 10 - Academic - Quadratic Relations - Brendon May's E
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Big Ideas and Essential Questions Big Ideas: - Students should be able to expand and factor quadratic expressions using a variety of tools. - Students should be able to graph the base function y=x2 as well as all transformations of the form: y = a (x – h)2 +k - Students should be able to solve quadratic equations using different strategies. - Students should be able to model real-life problems using quadratic functions and use that model in order to determine/calculate variables of interest. Essential Questions: - What is a quadratic relation? - Which strategies can we utilize in order to factor quadratic expressions? - Can we develop a formula to solve the real roots for any given quadratic function? - How do the variables: a, h, and k affect the graph of the quadratic function y = a (x- h)2 + k? - How can we model real life problems using quadratic functions to solve variables of interest? Students will know… - How to factor/expand polynomials, specifically quadratics - How to solve the roots of polynomials - How to graph any give quadratic function - How to determine the max/min of a quadratic function - How to solve real world problems through the use of quadratics Students will do…. - A real life investigation during their culminating task, representing the height of a soccer ball using quadratic functions. - Several cooperative learning activities in which students will have to investigate different theorems. - Investigations on the different transformations of y=x2 - Presentations of their work and will justify their conclusions Cross-curricular and/or real world connections: - Students will use quadratic relations to model real world problems. - Students will use quadratic graphing to represent single variable quadratic functions found in our everyday life - Students will discover applications of their theoretical math knowledge, specifically finding the roots and points of intersection, and max/min of functions. - Students will investigate curved graphs for the first time, and will be able to finally study real life phenomenon which cannot be described using linear functions. Considerations for how instruction, assessment and evaluation will be differentiated to achieve a goal of “Learning for All” - Calculators can be used for students who wish to use it. - Placing students into groups and practicing cooperative leaning will help foster educational and social growth within the students - Scaffolding within the zone of proximal development will be practiced by the teacher whenever possible - Unit Plan Outline Learning Outcomes (Curriculum expectations – overall and specific) Lesson 1 Lesson 2 Lesson 3 Assessment activities (Diagnostic, formative & summative) Minds on Activity: assessment as learning and assessment for learning as students recall what they already know in expansion and use these skills to apply them to a new concept of expansion. Action: assessment for learning students are applying the rules expansion of perfect squares (a new concept that builds off what is already known). Learning activities Helpful Resources In Class: Practice worksheets, interactive SMARTboard lesson, cooperative learning. Appendix 1, Appendix 3. Overall Expectation Covered: Solving quadratic equations. Specific Expectation Covered: Factor polynomial expressions involving different common factors, trinomials and differences of squares. Quiz: assessment for learning – quiz on expansion will assess the students’ learning to date on the concepts of expansion. Action: assessment for learning students are applying the rules expansion of perfect squares (a new concept that builds off what is already known). Assessment as learning: students are learning a brand new concept – factoring. Zone of proximal development is used as they will be using their knowledge from the expansion lessons and building off that to apply to the factoring lessons. Some may find this difficult. In Class: Interactive SMARTboard lesson, volunteering opportunities, review factoring checklist. Overall Expectation Covered: Solving quadratic equations Specific Expectation Covered: Factor polynomial expressions involving different common factors, trinomials and differences of squares Action: assessment for learning students are applying the rules expansion of difference squares and what is known about general factoring (both are old concepts) to build on this new concept of factoring differences of squares. Assessment as learning: students are learning a new type of factoring. They will be using their knowledge from to build off it. Some may find this difficult. In Class: Review worksheet, guided problem solving, interactive SMARTboard lesson. Overall Expectation Covered: Solving quadratic equations. Specific Expectation Covered: Expand and simplify second-degree polynomial expressions. Outside Class: Concept practice skill drill, asssigned homework problems. Appendix 4. Outside Class: Concept practice skill drill, asssigned homework problems. Outside Class: Concept practice skill drill, asssigned homework problems. Appendix 3, Appendix 5. Unit Plan Outline Cont'd Lesson 7 Lesson 8 Learning Outcomes (Curriculum expectations – overall and specific) Overall Expectations: i) Solve quadratic equations and interpret the solutions with respect to the corresponding relations. Specific Expectations: i) Factor polynomial expressions involving common factors, trinomials, and differences of squares. ii) Determine, through investigation, and describe the connection between the factors of a quadratic expression and the x-intercepts of the graph of the corresponding quadratic relation, expressed in the form a(x-r)(x-s). iii) Solve quadratic equations that have real roots, using a variety of methods. Overall Expectations: i) Solve quadratic equations and interpret the solutions with respect to the corresponding relations. Specific Expectations: i) Factor polynomial expressions involving common factors, trinomials, and differences of squares. ii) Explore the algebraic development of the quadratic formula. iii) Solve quadratic equations that have real roots, using a variety of methods. Assessment activities (Diagnostic, formative & summative) Diagnostic Assessment: During the Minds On phase of the lesson, this will give the teacher a good idea of their students’ previous understandings of how one would solve the x-intercepts of a single variable function. Learning activities Helpful Resources In Class: - Guided generalized instruction, active participation during lesson, peer sharing/instruction, independent practice on specific cases. Khan Academy Video Searches: Formative Assessment: The students will be given a small 15 minute quiz at the end of class to gauge whether or not they fully grasped the newly acquired concepts. The results will determine whether more time is needed on this topic before moving forward to the quadratic formula. Outside Class: - Follow-up reinforcement exercises as well as reference to Khan Academy’s videos (optional). Diagnostic Assessment: The Minds-On activity will give the teacher a good idea of where the students are at in terms of dealing with general equations, and whether or not they are comfortable manipulating generalized variables. In Class: - Guided generalized instruction, active participation during lesson, peer sharing/instruction, independent practice on specific cases, cooperative learning. Formative Assessment: While the students are in their groups trying to deduce the quadratic formula, it will be a good opportunity for the teacher to assess whether or not the students can complete the square on a generalized function. Completing the square is a concept that was taught much earlier in the unit, so they should have those skills. Also reviewing the assigned homework the next day will gauge the class’s understanding. Outside Class: - Follow-up reinforcement exercises as well as reference to Khan Academy’s videos (optional). 1. “Solving a quadratic equation by factoring” 2. “Solving a quadratic by factoring” Khan Academy Video Searches: 1. “How to use the quadratic formula” 2. “Example 1: Using the quadratic formula” 3. “Example 2: Using the quadratic formula” 4. “Example 3: Using the quadratic formula” Quadratic Functions – Lesson 8: Solving Quadratic Equations (Quadratic Formula) Quadratic – Lesson 1: Perfect Squares Expansion 70 minute lesson Functions Overall Expectation Covered: 70 minute lesson Grade 10 Grade 10 Materials Materials Overall Expectation Covered: Solving quadratic equations i) Solve quadratic equations and interpret the solutions with respect to the corresponding relations. Black/White/SMART Specific Expectation Covered: Covered: Expand and simplify second-degree polynomial board SmartBoard Specific Expectation i) Factor polynomial expressions involving common factors, trinomials, and differences of squares. Chart paper expressions ii) Explore the algebraic development of the quadratic formula. Markers iii) Solve quadratic equations that have real roots, using a variety of methods. Academic Objective: expand perfect square polynomials Academic SolveHave quadratic equations using the quadratic SocialObjective: Objective: students work together and formula. practice mutual respect to | Assessmentto(A) and DI build a safe, warm and fun classroom environment learn in. (D) Opportunities Before the students get to class, place the following question on the board: Diagnostic Minds-On… Given the standard quadratic ax2 + bx + c = 0, can you isolate x? (Hint: Try completing the Assessment: The Assessment (A) and DI (D) Opportunities 10 minutes square). Give the students about 5 minutes to see if they can isolate x. Minds-On activity Minds On… AfterOver a fewthe minutes, class if they can explain doing such a feat may be favorable. past ask fewthelessons, students havewhy learned a variety of ways to expand. will give the teacher a Activity: Probe the class asking familiar them how many would in anytricks specifictocase of them ax 2 + goodMinds idea ofonwhere They are by already with unknowns the FOILthere rules andbesome help assessment c = 0. (Provide them with a distinct example if they are unable to see it i.e. 2x 2 + 3x + 2 = the students are at as in 10 minutes bx +expand successfully. Have students work on theisolating expansion worksheet learning andwith 0) They should be able to see that x is the only unknown. Thus, x would give us a (see terms of dealing appendix) to get their mathematical juices flowing. assessment generalized formula for determining the roots of any quadratic. general equations,for and learning students Ask the class why it may be favorable to have such a formula. Scaffold their learning by asking whether or notasthey recall what they if every quadratic can be factored using product/sum. The answer is definitely no, provide a comfortable Action! Introduce the lesson on perfect squares expansion ο building off of the other are already know in counterexample to illustrate this (i.e x2 +10x + 1). 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Give each group a piece of chart paper and a Action! for learning formula, it will bestudents a marker. Make sure that the groups know to work out the math on scrap paper before transferring applying for the goodare opportunity 35 minutes theirWhat answeristoathe chart paper. The groups will be given 35 minutes to do their best in perfect square? rules expansion the teacher to assess of formulating the quadratic formula and transferring their best efforts onto the provided chart (a + b)2 / (a - b)2 ο Using the smart board, demonstrate how these are just perfect squares whether or not the (a paper. new can concept that they are expanded out. 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The groups should record their work/ thought process on chart paper using 2π thehave teacher, should those While this activity thethe teacher shouldtechnique walk aroundto anddisplay observe and the groups, markers. Asisaunderway, class, use Bansho view the reinforced by peer skills. Also reviewing providing themthe withstudents guided instruction if necessary. answers have come up with. group work, further the assigned reinforced through homework the next Ask the class if there are any groups that would like to volunteer to come up and show their Consolidate day explanations will gauge the in the After a solid poster (correct answer and work-process) has been identified, work. It is very possible that none of the groups will have completely generated the quadratic Debrief Bansho activity. class’s understanding. encourage to record their notes theytocan refer back to it formula, which is students fine. 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Since thoseadditional groups who in this Debrief can’tknowledge move any further there will be a quiz on expansion the following class, do a quick review on subject area. without specific all the important points of expansion and the various topics that have been direction. 10 minutes Concept Practice Skill Drill Concept Reinforcement covered in expansion. When the rules are followed and students are meticulous, expansion is foolproof arithmetic. Analyze expansion rules and the basic steps of foiling. Home Activity or Further Classroom Consolidation Assign some homework questions from the textbook based on the perfect squares expansion. Encourage students to study for the upcoming quiz on expansion. Home Activity or Further Classroom Consolidation Assign 5-6 homework problems that the students can do while at home to practice using the quadratic formula to solve the roots of a given function. These questions should be followed by 2-3 more where the question simply states, “Solve for x in the optimal manner.” From there the students will have to determine whether using the quadratic formula, or solving by factoring is more favorable. Note 2: Students should have the right to pass, theyplan should Did as your feel include as though they are activities in a that safe are learning environment. • visual • kinesthetic Students may also refer to Khan Academy and run the searches provided in the Unit Outline. Quadratic Functions – Lesson 2: Factoring Quadratic Trinomials 70 minute lesson Overall Expectation Covered: Solving quadratic equations Specific Expectation Covered: Factor polynomial expressions involving different common factors, trinomials and differences of squares Grade 10 Materials SmartBoard Academic Objective: Factor quadratic trinomials Assessment (A) and DI (D) Opportunities Minds On… 30-40 minutes 1. 2. 3. Over the past few lessons, students have learned all of the necessary types of expansion and the rules that accompany them. Students will be given a short quiz on expansion (assessment of learning). Quiz: (2x + 3) (2x - 3) (x2 + 4)2 -2 (x + 3y) (x – 3)2 Students should not use calculators but they can use scrap paper or any kinaesthetic materials that may help them solve their answer. Although this is a short quiz that should not take more than 30 minutes, allow students all of the time they need to complete the quiz. Action! 25 minutes Introduce factoring: What is factoring? Why do we factor? How is factoring related to expansion? Reminder of trinomial versus quadratic trinomial Introduce the lesson on factoring quadratic trinomials ο building off of the expansion concepts and what students already know about trinomials. Draw students’ attention to their learning goal and success criteria. Learning Goal: Factoring Quadratic Trinomials Work through the example on the SmartBoard using the four-step rule process. Ensure students copy down these rules in their notebooks so that they can refer back to them as they practice factoring ο it will become second nature as they practice and master it. x2 + 4x + 4 (x +/- ____) (x +/- ____) = x2 + 4x + 4 There will not be enough time for students to work in groups or pairs today (as too much time has already been dedicated to the quiz). Work through more examples together as a class, and have students volunteer to come up to the SmartBoard and interact in the lesson. x2 + 2x + 1 Consolidate Debrief 5 minutes Concept Practice Skill Drill x2 + 6x + 9 x2 + x – 12 Factoring is a difficult process that involves a lot of practice. If it doesn’t make sense initially, stick with it because it will make sense eventually if you dedicate yourself to it and work at it. Review the four step rule process on how to find the factors. Trying to get it into this form (x +/- ____) (x +/- ____) . Home Activity or Further Classroom Consolidation Assign some homework questions from the textbook based on the factoring quadratic trinomials. Help to reinforce learning. Quiz: assessment of learning – quiz on expansion will assess the students’ learning to date on the concepts of expansion. Action: assessment for learning students are applying the rules expansion of perfect squares (a new concept that builds off what is already known). Assessment as learning: students are learning a brand new concept – factoring. Zone of proximal development is used as they will be using their knowledge from the expansion lessons and building off that to apply to the factoring lessons. Some may find this difficult. Explain the concept of factoring in multiple ways through examples. The step-by-step rule process is something that should be memorized in order to find the correct factors every time. Understanding will come through plenty of practice. Did your plan include activities that are • visual Quadratic Functions – Lesson 3: Differences of Squares 70 minute lesson Grade 10 Overall Expectation Covered: Solving quadratic equations Specific Expectation Covered: Factor polynomial expressions involving different common factors, trinomials and differences of squares Materials SmartBoard Academic Objective: Factor differences of squares Social Objective: Have students work together and practice mutual respect to build a safe, warm and fun classroom environment to learn in. Assessment (A) and DI (D) Opportunities Minds On… 10 minutes To begin this lesson, I would have a mini minds-on worksheet for students to work on independently to get their factoring juices flowing from the class prior to this one. The worksheet would serve as both a mini review / reminder of factoring quadratic polynomials before moving on to factoring difference of squares. Action! 50 minutes Introduce factoring of difference of squares: Draw students’ attention to their learning goal and success criteria. Learning Goal: Factoring Differences of Squares Remind them of the quadratic trinomial - ax2 + bx + c Think of difference of square problems as trinomials where b = 0; ax2 + c Introduce the rules of the concept before working through an example. Encourage students to record these rules in their notebook. 1. 2. Both terms of the binomial must be perfect squares (you can find the square root of each) ο square root of x2 is x; square root of y2 is y When factored, the second terms of the binomials must have different signs (e.g. [x +y][x – y]) x 2 − π¦ 2 = (π₯ + π¦)(π₯ − π¦) To factor a difference of squares, we must find a number or variable that when multiplied by itself it equals the term we began with. Work through the following example as a class: (x 2 − 16) Work through the following non-example with the class: (x 2 − 14) Have students work in pairs to come up with more examples of difference of square factoring. Share these examples with the class and work through them on the SmartBoard. Encourage students to record these examples in their notebooks so that they can refer back to them in the future while doing homework or studying for an assessment. Consolidate Debrief 10 minutes Concept Practice Skill Drill Factoring is a difficult process that involves a lot of practice. Need to incorporate the concepts learned in expansion (expansion of difference of squares specifically in this case). Make a list of ways to successfully expand and factor. Remind the students of the rules of factoring and the rules of factoring for difference of squares. Note the similarities and how students can identify a difference of squares problem versus a normal quadratic trinomial. Home Activity or Further Classroom Consolidation Assign some homework questions from the textbook based on the factoring difference of squares. Encourage students to look over their expansion of difference of squares notes before working on their factoring difference of squares homework. Homework will help to reinforce learning. Quadratic Culminating Task Action: assessment for learning students are applying the rules expansion of difference squares and what is known about general factoring (both are old concepts) to build on this new concept of factoring differences of squares. Assessment as learning: students are learning a new type of factoring. They will be using their knowledge from to build off it. Some may find this difficult. Differentiated Learning: Explain the concept of factoring in multiple ways through examples and nonexamples. Students also have the opportunity to learn through working with their peers. The step-by-step rule process is something that should be memorized in order to find the correct factors every time. Understanding will come through plenty of practice. Did your plan include activities that are • visual Quadratic Functions – Lesson 7: Solving Quadratic Equations (Factoring) 70 minute lesson Academic Objective: Solve quadratic equations by factoring. | Assessment (A) and DI (D) Opportunities When the students walk into the classroom, leave these two questions on the board. Ask the students to take a few minutes to pondering the following questions. 1. Given any function, f(x), how does one find the x-intercepts? Justify your answer. 2. Suppose ab = 0, what can we conclude about either a or b? Justify your answer. Minds-On… 30 minutes After about 10 minutes, ask the class if any of the students have any answers to these questions. If there are students with answers, invite them to the board to write their answers and attempt to justify it. Whether or not the students who present are correct, begin by explaining #1. Show visually that for any function, the y-value of the x-intercept will always be 0. Thus, to solve for the xintercept, all must one do is set f(x) = 0. In the particular quadratic case, we have ax 2 + bx + c = 0. For #2, explain that ab = 0 iff a= 0 or b = 0. This is easily shown by attempting to isolate either a or b. Ask the class if anyone can see how we can use this theorem to solve for the x-intercepts of a given quadratic function. (If no one responds, probe the students by asking them if they know how to express a quadratic expression as a product of two factors.) Pose the following question to the class: Suppose f(x) = ax2 + bx + c = a(x-r)(x-s) = 0. Using the previously identified theorem, determine the roots of f(x). Have students come up and share their answers with the class until one gets it correct. (Note: Be sure to formally define roots as the x-intercepts of a function). Now, pose the class with a unique case; ask them to determine the x-intercepts of f(x) = 3x2 + 8x + 4. Give the students a few minutes to write an answer in their notebook, then ask some volunteers to come up to share and justify their answer with the class. Give the students one more specific case and ask them to determine the roots of f(x) = -x2 – 7x + 8. Once again, give the students a few minutes to come up with a solution in their notebook. Then, invite volunteers to show and justify their answer to the classroom. Urge the students to ensure they have the correct solution in their notebook. This lesson should be very straightforward seeing as the factoring component is something that has been taught to them earlier in the unit. It is a skill they should have already mastered by now. Action! 25 minutes Consolidate Debrief 15 minutes 1. 2. Concept Reinforcement Grade 10 Overall Expectation Covered: Materials i) Solve quadratic equations and interpret the solutions with respect to the corresponding relations. Black/White/SMART Specific Expectation Covered: board i) Factor polynomial expressions involving common factors, trinomials, and differences of squares. ii) Determine, through investigation, and describe the connection between the factors of a quadratic expression and the x-intercepts of the graph of the corresponding quadratic relation, expressed in the form a(x-r)(x-s). iii) Solve quadratic equations that have real roots, using a variety of methods. The students will take a quick 2 question quiz to be collected by the teacher as they exit the classroom. The teacher may write the 2 questions on the board, and the students may write their solutions on a piece of lined paper. Quiz: Solve the roots of the following functions: f(x) = -4x2 - 6x + 4 g(x) = 4x2 + 16 Home Activity or Further Classroom Consolidation Assign 5-6 homework problems that the students can do while at home to reinforce their newly acquired mathematical knowledge. Diagnostic Assessment: During the Minds-On phase of the lesson, this will give the teacher a good idea of their students’ past understanding of how one would solve the xintercepts of a single variable function. Formative Assessment: The students will be given a small 15 minute quiz at the end of class to gauge whether or not they fully grasped the newly acquired concepts. The results will determine whether more time is needed on this topic before moving forward to the quadratic formula. Note 1: The use of a calculator throughout the course of this lesson and the quiz are permissible as an accommodation to those who feel they need it. Note 2: Students should have the right to pass, as they should feel as though they are in a safe learning environment. Students may also refer to Khan Academy and run the searches provided in the Unit Outline. The following is the culminating task designed specifically for the Grade 10 Unit on quadratic relations of the form y = ax2 + bx + c. Necessary Equipment: i. ii. iii. Access to a Computer Soccer Ball Video Camera Instructions: 1. Place the students into small groups of 3 or 4. 2. The students will be given a soccer ball and a video camera to use. 3. One student will kick a soccer ball up as high as they can. While this is happening, another student will be videotaping the kick, being sure to keep both the student and the soccer ball in the shot the whole time. (Note: The camera should be stationary the whole time! Also, be sure to begin filming before the kick is done!) It is recommended that the camera man stand at a distance to ensure both the ball and the kicker remain in the shot at all times. If necessary, ask the kicker to not kick the ball so high to ensure both can be kept in the same shot. 4. The students will then be asked to create a graph showing the height of the ball (in meters) with respect to the amount of time elapsed in the video (in seconds). This will be done by determining the following: i. At what time in the video was the ball kicked? (First root) ii. At what time in the video did the ball hit the ground? (Second root) iii. At approximately what time did the ball reach the height of the kicker? (Co-ordinate [t, h(t)] for determining the stretch factor of the function. Note: It is necessary for the group to know the kicker’s height in meters.) 5. After determining the function describing the height of the ball, the students will be asked to graph that function, along with all of the transformations from the base function y = x2. 6. The students will then be asked to determine the time at which the ball hit its maximum height using their generated function. They will then compare this calculated time with the approximate time shown in the video. 7. Using their function, the students will determine the maximum height of the ball in meters. 8. The groups will share their findings with the classroom, and explain how they derived their function, the time in which the ball hit the maximum height, and the maximum height of the ball. They will also share how accurate their calculated results were to that of the actual video. (Note: If the students do not have their own video cameras, the teacher can video tape each of the kicks, and send the video to each of the respective groups.) Student Exemplar: (See below for the rubric.) Rubric for Culminating Activity: Level 1 Presentation (Communication) The group presented the content in a disorganize fashion; hard to follow/understand. Graphs (Communication) Not all of the graphs were included, several mistakes were made. Algebra: Expansion/Factoring (Knowledge & Understanding) None of the algebra was executed in a rigorous, organized fashion, with many errors. Problem Solving Strategy/Rationale (Thinking & Application) There was no clear logical strategy, and it was either unjustified or the justification was incorrect. Level 2 Level 3 Level 4 The group presented the material in a way that was somewhat clear and understandable. Most of the graphs were included, and were plotted with a few mistakes. Some of the algebra was executed in a rigorous, organized fashion, with some errors. The strategy used was applied somewhat rigorously with some justification. The group presented the content in a way that was understandable and clear. The group presented the content in a very rigorous and understandable manner. The graphs were all included, and were plotted correctly for the most part. The graphs were all included, plotted correctly with no errors. Most of the algebra was executed in a rigorous, organized fashion, with few errors. The strategy used was applied rigorously and was justified. All of the algebra was executed in a rigorous, organized fashion, with no errors. The strategy used was the optimal strategy for solving the problem and it was applied rigorously. Appendices Appendix 1: EXPANSION MINDS-ON 1. Expand: (a) 6(x+3) (b) 2(3x − 4) (c) −3(x + y) (d) 5(m−4) (e) x(x+y) (f) y(y2 − 2) 2. Expand. Remove the brackets. Collect like terms (a) 2(m+4)+3(m+6) (b) 4(t−2)−3(t+1) (c) 7(m−3)−2(m−4) (d) −4(x+1)+3(x+2) Appendix 2: FACTORING MINDS-ON 1. Factor completely: (a) (b) x2 + 4x + 4 (c) x2 – x – 6 (d) Appendix 3: LESSON 1: Expansion of Perfect Squares To begin this lesson, I would have a mini minds-on worksheet for students to work on independently to get their expansion juices flowing from the class prior to this one. See Appendix for Expansion Worksheet. The worksheet would serve as both a mini review / reminder of expanding different types of quadratic relations before moving on. Why don’t you give these a try and see if you recall anything about the rules of expansion. Feel free to work with a partner. Overall Expectation Covered: Solving quadratic equations Specific Expectation Covered: Expand and simplify second-degree polynomial expressions Learning Goal: Expand perfect square polynomials Success Criteria: – – – – FOIL – algebraic expansion for binomials (learned previously) Be thorough Use brackets Check work At this point, students are already familiar with the quadratic relation and the notion of expanding. In this lesson, they will simply be learning a new tool for their expansion toolkit. What is a perfect square? – A perfect square is a number made by squaring a whole number o 4 is a perfect square (22 = 4) o 9 is a perfect square (32 = 9) o 16 is a perfect square (42 = 16) So what…? – – A perfect square relation is a simplified trinomial We are trying to identify the relationship between a trinomial that we will discover in our final step, and the perfect square. The perfect square is something that is easy to work with and it will help to simplify our lives in the future. Perfect Squares ο Trinomialsο (a + b)2 = (a + b) (a + b) = aa + ab + ab + bb = a2 + 2ab + b2 (a - b)2 = (a - b) (a – b) = aa – ab – ab = bb = a2 - 2ab + b2 Rule: (a + b)2 = a2 + 2ab + b2 As a class, we will work through the following examples. Expand the following: (x + 7)2 (x - 3)2 (x + 5)2 Consolidation/ review of expansion of perfect squares ο quiz in the next class See the Appendix for a copy of an Expansion Quiz Review the main points and the rules of expansion for perfect squares. Since there will be a quiz on expansion the following class, do a quick review on all the important points of expansion and the various topics that have been covered in expansion. When the rules are followed and students are meticulous, expansion is foolproof arithmetic. Analyze expansion rules and the basic steps of foiling. Appendix 4: LESSON 2: Introduction to Factoring No minds-on activity – QUIZ on expansion Over the past three lessons we have learned about polynomial expressions and how to expand polynomials to solve for variables. Students will be given a quiz to work on for the first half of the lesson. This quiz will be followed by an introduction to factoring. Now we will be learning a bit about factoring. What is factoring? – – Process of finding the factors E.g. the factors of the number 6 are (2, 3; -2,-3; 1,6; -1,-6) o When multiplied together, each pair equals 6 Why do we factor quadratic equations? – It will allow us to solve quadratic functions, draw and analyze them Factoring Polynomial Expressions involving: ο· ο· Quadratic trinomials (Lesson 2) Difference of squares (Lesson 3) LESSON 2: Factoring Quadratic Trinomials: Overall Expectation Covered: Solving quadratic equations Specific Expectation Covered: Factor polynomial expressions involving different common factors, trinomials and differences of squares Learning Goal: Factoring Quadratic Trinomials Success Criteria: Use the rules established Identify how expansion and factoring are interrelated Recall common factors Record all work (don’t skip steps) – – Reminder: trinomial is a polynomial with three terms (ax2 + bx + c) Quadratic trinomial is a polynomial with three terms with one part of the trinomial that has a exponential degree of two Example: x2 + 4x + 4 Non-example: x3 + 6x2 + 12 This is not a quadratic trinomial because there is an exponent greater than 2 Factor: x2 + 4x + 4 (x +/- ____) (x +/- ____) = x2 + 4x + 4 We must find numbers and variables that when multiply/expanded back out, they will be equivalent to our starting formula. 1. Identify a, b and c a=1 b=4 c=4 2. List all the factors of ‘c’ 1,4 -1,-4 2,2 -2,-2 3. Which pair of factors sum to ‘b’? 2,2 ο 2 + 2 = 4 4. Substitute this factor pair into two binomials (x + 2) (x+2) Check ο Expand (x + 2) (x + 2) = x2 + 2x +2x + 4 Gather like terms ο x2 + 4x + 4 Try the following with the class; have students come up to the SmartBoard and work them through out loud. The class should support the student at the board but allow them to work through the problem without people shouting out answers. x2 + 2x + 1 x2 + 6x + 9 x2 + x – 12 After students have worked in pairs for 5-10 minutes on the problems, I would draw their attention back to the front and take up these problems as a class. I would have volunteers work through them on the board and go over areas where the students may have gotten a bit lost. I will simultaneously be doing the consolidation and reviewing what factoring is and why it is important. Homework will also be assigned to reinforce learning from this lesson. Appendix 5: LESSON 3: Factoring Difference of Squares To begin this lesson, I would have a mini minds-on worksheet for students to work on independently to get their factoring juices flowing from the class prior to this one. The worksheet would serve as both a mini review / reminder of factoring quadratic polynomials before moving on to factoring difference of squares. Overall Expectation Covered: Solving quadratic equations Specific Expectation Covered: Factor polynomial expressions involving different common factors, trinomials and differences of squares Learning Goal: Factoring Difference of Squares Success Criteria: Use the rules established Identify how expansion and factoring are interrelated Recall common factors Record all work (don’t skip steps) ax2 + bx + c – Think of difference of square problems as trinomials where b = 0 ax2 + c Difference of Squares 3. Both terms of the binomial must be perfect squares (you can find the square root of each) ο square root of x2 is x; square root of y2 is y 4. When factored, the second terms of the binomials must have different signs (e.g. [x +y][x – y]) x 2 − π¦ 2 = (π₯ + π¦)(π₯ − π¦) To factor a difference of squares, we must find a number or variable that when multiplied by itself it equals the term we began with. Example: (x 2 − 16) x 2 is a perfect square; 16 is a perfect square What times itself = x2? What times itself =16? a=1 b = 0 c = -16 In order to solve, we must find a pair of factors that when multiplied together their product must equal (a β c) and when added together they sum to b. (a β c) = (1 β -16) = -16 b=0 Factors of 16 = 1, -16; -1, 16; -4, 4; 4,-4 The pair that satisfies the condition is either (-4, 4) and (4, -4) ο 4 β -4 = -16 = (a β c) and 4 + (-4) = 0 = b ο -4 β 4 = -16 = (a β c) and -4 + 4 = 0 = b Therefore: (x 2 − 16) = (π₯ + ___)(π₯ − ____) = (π₯ + 4)(π₯ − 4) Check: Expanded out again = x2 -4x +4x - 16 = (x 2 − 16) Non-example: (x 2 − 14) = What times itself = 14? (π₯ + ___)(π₯ − ____) – This is not a difference of squares because 14 is not the square of an integer What about (p2 +16)? This is not a difference of squares because when foiled out, it is a sum of squares. =(p + 4) (p + 4) = (p2 + 4p + 4p + 16) = p2 + 8p + 16 …… ≠ (p2 +16) Work with a partner to come up with 2 more examples of differences of squares? After students have worked in pairs for 5-10 minutes, I would then draw their attention back again as a class to hear about what they have discovered. This would be followed by a consolidation activity, getting the students to identify patterns in the examples they came up with in order to encourage them to identify difference of square problems with ease. Reflection So I decided to format my summative assessment plan (SAP) through the use of a table (as can be seen in my unit plan outline) as I found the fishbone outline a bit confusing. For each of my lessons I was very careful to ensure that all of the expectations highlighted within the achievement chart were properly covered. As you can in my unit plan, for each lesson I outlined the exact expectations that the lesson would cover. I even took the liberty of writing out extremely detailed lesson plans showing exactly how I would go about teaching the specific topics. I am fairly confident that my SAP carefully examines each of the necessary expectations for the quadratic relations strand as they were the central building blocks for my SAP. In terms of destination and discipline, seeing as this is an academic mathematics course, the SAC was designed to be particularly engaging as these students will be pursing university preparatory courses starting the following year. This allowed me to make the level of rigor a bit higher and gave me the flexibility to develop and structure my lessons in such a way that would properly balance theoretical and practical applications of quadratics. I was cognizant though, of the students’ expected background knowledge and was very careful in my design as I needed to ensure I properly considered their grade level and current depth of understanding. It is for this reason that my SAC calls for several diagnostic assessments to allow flexibility to adjust my instruction accordingly. As for the formative assessments, I believe it is pretty self-explanatory. Within my unit plan design I very clearly highlighted all of the assessments that would take place for each individual lesson. Not only that, I also specified within my lesson plans exactly when and how to execute not only the formative assessments, but also the diagnostic and summative ones as well. For more detailed information on what they call for or how to administer them, you can refer to either my unit plan for a brief overview, or the actual lesson itself whereby I detail the assessment in more detail. Finally, the way in which a student’s grade will be determined is quite straight-forward (ah, the beauty of mathematics!). Almost every-other day, there is a formative assessment that will take place, typically in the form of a quiz. This is particularly important to administer in order to track student progress before moving forward onto more complex concepts. For each unit, there is typically 3-4 quizzes before there is a final unit-test and summative task. Typically, the accumulation of all the quiz scores will account for 33.33% of their mark, the unit test will account for 33.33 of their mark, and the summative task will also account for 33.33% as well. That means for each strand, you will typically have at least 6 pieces of evidence to determine the student’s grade. In the grade 10 academic math curriculum, there are six strands, which means before exams, you will have accumulated at least thirty-six pieces of solid evidence that can be used to determine the student’s term mark before exams.
