Teaching & Learning Plans Quadratic Equations Junior Certificate Syllabus The Teaching & Learning Plans are structured as follows: Aims outline what the lesson, or series of lessons, hopes to achieve. Prior Knowledge points to relevant knowledge students may already have and also to knowledge which may be necessary in order to support them in accessing this new topic. Learning Outcomes outline what a student will be able to do, know and understand having completed the topic. Relationship to Syllabus refers to the relevant section of either the Junior and/or Leaving Certificate Syllabus. Resources Required lists the resources which will be needed in the teaching and learning of a particular topic. Introducing the topic (in some plans only) outlines an approach to introducing the topic. Lesson Interaction is set out under four sub-headings: i. Student Learning Tasks – Teacher Input: This section focuses on possible lines of inquiry and gives details of the key student tasks and teacher questions which move the lesson forward. ii. Student Activities – Possible Responses: Gives details of possible student reactions and responses and possible misconceptions students may have. iii. Teacher’s Support and Actions: Gives details of teacher actions designed to support and scaffold student learning. iv. Assessing the Learning: Suggests questions a teacher might ask to evaluate whether the goals/learning outcomes are being/have been achieved. This evaluation will inform and direct the teaching and learning activities of the next class(es). Student Activities linked to the lesson(s) are provided at the end of each plan. Teaching & Learning Plan: Junior Certificate Syllabus Aim • To enable students recognise quadratic equations • To enable students use algebra, graphs and tables to solve quadratic equations • To enable students form a quadratic equation to represent a given problem • To enable higher-level students form quadratic equations from their roots Prior Knowledge Students have prior knowledge of: • Simple equations • Natural numbers, integers and fractions • Manipulation of fractions • Finding the factors of x2 + bx + c where b, c ∈ Z • Finding the factors of ax2+ bx + c where a, b, c ∈ Q, x ∈ R (Higher Level) • Patterns • Basic algebra • Simple indices • Finding the factors of x2 − a2 Learning Outcomes As a result of studying this topic, students will be able to: • understand what is meant by a quadratic equation • recognise a quadratic equation as an equation having as many as two solutions that can be written as ax2 + bx + c = 0 • solve quadratic equations • represent a word problem as a quadratic equation and solve the relevant problem • form a quadratic equation given its roots © Project Maths Development Team 2011 www.projectmaths.ie 1 Teaching & Learning Plan: Quadratic Equations Catering for Learner Diversity In class, the needs of all students, whatever their level of ability level, are equally important. In daily classroom teaching, teachers can cater for different abilities by providing students with different activities and assignments graded according to levels of difficulty so that students can work on exercises that match their progress in learning. Less able students, may engage with the activities in a relatively straightforward way while the more able students should engage in more open–ended and challenging activities. In interacting with the whole class, teachers can make adjustments to suit the needs of students. For example, more challenging material similar to that contained in Question 11 in Section A: Student Activity 1 can be provided to students where appropriate. Apart from whole-class teaching, teachers can utilise pair and group work to encourage peer interaction and to facilitate discussion. The use of different grouping arrangements in these lessons should help ensure that the needs of all students are met and that students are encouraged to verbalise their mathematics openly and to share their learning. Relationship to Junior Certificate Syllabus Topic Number Description of topic Students learn about 4.7 Equations Using a variety of and problem solving inequalities strategies to solve equations and inequalities. They identify the necessary information, represent problems mathematically, making correct use of symbols, words, diagrams, tables and graphs. © Project Maths Development Team 2011 Learning outcomes Students should be able to –– solve quadratic equations of the form x2 + bx + c where b, c ∈ Z and x2 + bx + c is factorisable ax2+ bx + c where a, b, c ∈ Q, x ∈ R (HL only) –– form quadratic equations given whole number roots (HL only) –– solve simple problems leading to quadratic equations www.projectmaths.ie 2 Teaching & Learning Plan: Quadratic Equations Lesson Interaction Student Learning Tasks: Teacher Input Student Activities: Possible Responses Teacher’s Support and Actions Assessing the Learning Section A: To solve quadratic equations of the form (x-a)(x-b) = 0 algebraically »» What is meant by finding the solution of the equation 4x + 6 = 14?» • We find a value for x that makes the statement (that 4x + 6 = 14) true » »» Why is x =1 not a solution of 4x + 6 = 14 • Because 4(1) + 6 ≠ 14 » »» How do we find the solution to 4x + 6 = 14? »» Remember to check your answer. »» Write 4x + 6 = 14 on the board.» »» Ask students to write the solution on the board. Teacher Reflections »» Do students remember what is meant by solving an equation?» »» Can students see why » x =1 is not a solution?» • Students write the solution in their copies. -6 -6 4x + 6 = 14 4x = 8 ÷4 ÷4 x=2 »» Do students know how to check their answer? or 4x + 6 = 14 4x + 6 - 6 = 14 -6 4x = 8 x=2 4 (2) + 6 = 14 True »» »» »» »» »» What is 4 x 0? What is 5 x 0? What is 0 x 5? What is 0 x n? What is 0 x 0?» »» When something is multiplied by 0 what is the answer? © Project Maths Development Team 2011 • • • • • 0 0 0 0 0» »» Write the questions and solutions on the board. 4x0=0 5x0=0 0x5=0 0xn=0 0x0=0 »» Do students understand that multiplication by zero gives zero? • 0 www.projectmaths.ie KEY: » next step • student answer/response 3 Teaching & Learning Plan: Quadratic Equations Student Learning Tasks: Teacher Input »» If xy = 0 what do we know about x and or y? Student Activities: Possible Responses • x = 0 or y = 0 or both are equal »» Compare» (x - 3)(x - 4) = 0, with xy = 0 and what information can we arrive at?» • to 0.» »» If x - 3 = 0, what does this • x = 3 tell us about x? » x=4 »» How do we check if » x = 3 is a solution to (x - 3)(x - 4) = 0? • Insert x = 3 into the equation and check if we get 0.» » »» How do we check if » x = 4 is a solution to (x - 3)(x - 4) = 0? • Insert x = 4 into the equation and check if we get 0. »» Write in your copies in words what x = 3 or x= 4 means in the context of (x - 3)(x - 4) = 0. • Students write in words in their copies what this means and discuss with the student beside them. »» When an equation is written in the form» (x - a) (x - b) = 0, what are the solutions? • © Project Maths Development Team 2011 board.» x - 3 = 0 or x - 4 = 0 or both are »» Allow students to equal to 0.» » » » »» If x - 4 = 0, what does this • tell us about y? Teacher’s Support and Actions »» Write xy = 0 on the x = a or x = b www.projectmaths.ie discuss and compare answers.» Assessing the Learning »» Can students find the solution to (x - 3) (x - 4) = 0? Teacher Reflections »» Do students understand what is meant by the solution to this equation? »» Write on the board:» xy = 0 (x - 3) (x - 4) = 0 x - 3 = 0 or x - 4 = 0 x = 3 or x = 4 »» Write on the board:» (3 - 3) (x - 4) = 0 0(x - 4) = 0 0 = 0» »» Write on the board:» (x - 3) (4 - 4) = 0 (x - 3) 0 = 0 0=0 »» Can students write in words what x = 3 and x = 4 means? »» Write on the board:» (x - a)(x - b) = 0 x-a=0x-b=0 x=a x=b »» Can students generalise the solution to (x - a)(x - b) = 0? KEY: » next step • student answer/response 4 Teaching & Learning Plan: Quadratic Equations Student Learning Tasks: Teacher Student Activities: Possible Input Responses »» Solve (x - 1)(x - 3) = 0 • (x - 1) (x - 3) = 0 » (x - 1) = 0 or (x - 3) = 0 » x = 1 or x = 3. » » » » » Teacher’s Support and Actions Assessing the Learning »» Write the solution on the board. »» Can students detect that if an equation is of the form (x − a) (x−b) = 0, then x = a and x = b are both solutions? »» Distribute Section A: Student Activity 1 »» Are students able to solve equations of the form » (x - a) (x - b) = 0? (1 - 1) (x - 3) = 0 (1 - 1) (x - 3) = 0 0 (x - 3) = 0 0 = 0» »» How do we check that these are solutions?» » » Teacher Reflections • (x - 1) (3 - 3) = 0 (x - 1) 0 = 0 0 = 0» True» »» Is it sufficient to state x = 3 is a • No, both x = 3 and x = 1 are solution to» solutions. (x - 1) (x - 3) = 0? »» Answer questions 1 to 11 on Section A: Student Activity 1. »» If students are unable to make the jump from » (x - a) (x - b) = 0 to (x + a) (x + b) = 0. »» If students are having difficulty, allow them to talk through their work so that misconceptions can be identified and addressed. © Project Maths Development Team 2011 www.projectmaths.ie KEY: » next step • student answer/response 5 Teaching & Learning Plan: Quadratic Equations Student Learning Tasks: Teacher Student Activities: Possible Input Responses »» How does the equation x (x - 5) = 0 • It is the same.» » » differ from the equation» (x - 0)(x - 5) = 0? »» Why are they the same?» » • Because x and x - 0 are the same, i.e. x = x - 0. »» Hence what is the solution? • »» What are the solutions of » x (x - 6) = 0? • »» Answer questions 12-16 on Section A: Student Activity 1. © Project Maths Development Team 2011 www.projectmaths.ie x = 0 or x = 5 x = 0 and x = 6 Teacher’s Support and Actions Assessing the Learning »» Write on the board:» x (x - 5) = 0 and (x - 0) (x - 5) = 0 x = 0 and x - 5 = 0 x = 0 or 5 0 (x - 6) = 0 True x (6 - 6) = 0 True Teacher Reflections »» Write on the board:» x (x - 6) = 0 x = 0 and x - 6 = 0 x = 0 or 6 0 (x - 6)= 0 True x (6 - 6) = 0 True »» Do students recognise that the solutions to » x(x-a) = 0 are x= 0 and x = a? »» On completion of questions 12-16 those students who need a challenge can be encouraged to do question 17 on Section A Student Activity 1. »» Can students solve » x (x - a) = 0? KEY: » next step • student answer/response 6 Teaching & Learning Plan: Quadratic Equations Student Learning Tasks: Teacher Input Student Activities: Possible Responses Teacher’s Support and Actions Assessing the Learning Section B: To solve quadratic equations of the form (x - a) (x - b) = 0 making use of tables and graphs »» Equations of the form » ax2 + bx + c = 0 are given a special name, they are called Quadratic Equations. »» Give me an example of a Quadratic Equation. »» Is (x - 1) (x - 2) = 0 a Quadratic Equation? »» Why is (x - 1) (x - 2) = 0 a Quadratic Equation? »» Write ax2 + bx + c = 0 and Quadratic Equations on the board.» • 2x2 + 3x + 5 = 0 Note: The Quadratic Equations that the students list do not • Yes» have to be factorisable ones at the moment and special cases » • Because when you multiply it ax2 + bx = 0 2 out you get x - 3x + 2 = 0, which ax2 + c = 0 is in the format ax2 + bx + c = 0. will be dealt with later. »» What does it mean to solve the equation (x - 1) (x - 2) = 0? • Find values for x that makes this statement true.» »» What is meant by the roots of an equation?» » • The roots of an equation are the values of x that make the equation true.» • (x - 1) (x - 2 ) = 0 x - 1 = 0 or x - 2 = 0 x = 1 or x = 2 (1 - 1) (x - 2) = 0 True (x - 1) (2 - 2) = 0 True »» So what are the roots of » (x - 1) (x - 2) = 0 • © Project Maths Development Team 2011 »» Do students recognise a Quadratic Equation?» »» Do students understand that quadratic equations can be written in the form» (x - a) (x - b)=0? »» Can students solve the quadratic equations of the form » (x - a) (x - b) = 0? »» Do students understand that finding the roots of an equation and solving an equation are the same thing? »» It is true that finding the roots • Yes» of an equation and solving the » equation mean the same thing?» » »» Solve the equation» (x - 1) (x - 2) = 0 using algebra. » » » Teacher Reflections x = 1 and x = 2. www.projectmaths.ie KEY: » next step • student answer/response 7 Teaching & Learning Plan: Quadratic Equations Student Learning Tasks: Teacher Input Student Activities: Possible Responses »» Copy the table on the board into your exercise book and complete it.» » » » » » » » »» For what values of x did (x - 1)(x - 2) = 0? • x (x - 1) (x - 2) -2 12 -1 6 0 2 1 0 2 0 3 2 x = 1 and x = 2 Teacher’s Support and Actions Assessing the Learning »» Display the following table on the board: »» Are students able to complete the table, read from the table the values of x when (x - 1) (x - 2) = 0 and hence solve the equation? x (x - 1) (x - 2) -2 -1 0 1 Teacher Reflections 2 3 » »» What name is given to the value(s) of x • Solution(s) or roots. that make(s) an equation true? »» Draw a graph of the information in the table, letting y = (x - 1) (x - 2). »» For what values of x did the graph cut the x axis? »» What is meant by the solution of an equation?» »» What was the value of y =(x - 1)(x - 2) when x = 1 and x = 2? © Project Maths Development Team 2011 www.projectmaths.ie »» Draw the graph on the board • »» Are students able to draw the graph and read from the graph the values of x when » (x - 1) (x - 2) = 0? x = 1 and x = 2 » • The equation is true for the value of x. • 0 KEY: » next step • student answer/response 8 Teaching & Learning Plan: Quadratic Equations Student Learning Tasks: Teacher Input Student Activities: Possible Responses Teacher’s Support and Actions Assessing the Learning »» When using algebra to solve, what were the values of x for which (x - 1) (x - 2) = 0 called? • Solutions, roots» » » »» Write the student's answers on the board. »» How can we get the solution by looking at the table?» • When (x - 1) (x - 2) has the value zero.» »» How can we get the solution by looking at the graph? • Where it cuts the x axis or where the y value is zero »» Do students understand that: • solving the equation using algebra • finding the value of x when the equation equals zero in the table • finding where the graph of the function cuts the x axis are all methods of finding the solution to the equation? »» Complete the exercises in Section B: Student Activity 2. © Project Maths Development Team 2011 www.projectmaths.ie Teacher Reflections »» Distribute Section B. Student Activity 2. KEY: » next step • student answer/response 9 Teaching & Learning Plan: Quadratic Equations Student Learning Tasks: Teacher Input Student Activities: Possible Responses Teacher’s Support and Actions Assessing the Learning Section C: To solve quadratic equations of the form x2 + bx + c = 0 that are factorable »» What does it mean to find the factors of a number?» • Rewriting the number as a product »» Write on the board:» of two or more numbers.» x2 + 3x + 2 = 0 a =1, b =3, c =2 »» What does it mean to find the • It means to rearrange an algebraic Guide number =1 x 2 = 2» factors of x2 + 3x + 2? expression so that it is a product of » its prime factors.» »» Write on the board 1 x 2 = 2 and -1x - 2 = 2. »» What is the Guide number of this • 2» equation?» » »» Write on the board:» x2 + 3x + 2 = 0 »» How did you get this Guide • Multiplied 1 x 2 because x2 + 2x + 1x + 2 = 0 number?» comparing this equation to the » form ax2 + bx + c = 0, a = 1 and c »» Write on the board» » = 2.» x2 + 1x + 2x + 2 = 0 »» What are the factors of 2?» • 1 and 2 or -1 and -2» »» Which pair shall we use?» » »» Why would I not use the pair -1 and -2?» » »» Ask students if this looks familiar to any other type of factorising they have done before?» »» Could I have written 1x plus 2x instead of 2x and 1x? © Project Maths Development Team 2011 »» Allow students to compare their work.» • 1 and 2 because added together 1x + 2x gives us +3x. »» Ask two students to come to the board and write • They would give you -3x which down their solution to» is not a term in the original x2 + 2x + 1x + 2 = 0 equation.» and» x2 + 1x + 2x + 2 = 0 • This is Factorising by Grouping» » » Teacher Reflections »» Do students understand the Guide number method of finding the factors of a quadratic?» »» Can students find the factors of a simple equation?» »» Do students understand why they should use 1 and 2?» »» Can students connect this to their prior knowledge of Factorising by Grouping? • Students work on factorising each. www.projectmaths.ie KEY: » next step • student answer/response 10 Teaching & Learning Plan: Quadratic Equations Student Learning Tasks: Teacher Student Activities: Input Possible Responses »» I want you to investigate this for yourselves by Factorising by grouping each of these» x2 + 2x + 1x + 2 = 0 x2 + 1x + 2x + 2 = 0 » Teacher’s Support and Actions Assessing the Learning • One student writes on board:» x2 + 2x + 1x + 2 = 0 x(x + 2) + 1(x + 2) = 0 (x + 1) (x + 2) = 0 x = -1 or x = -2 Teacher Reflections »» Why does the first solution • Another student writes have (x+2) in each bracket and on the board:» the second solution have (x+1) x2 + 1x + 2x + 2 = 0 in each bracket?» x(x + 1) + 2(x + 1) (x + 2) (x + 1) = 0 »» How would you check that x = -2 or x = -1 both are correct? • (x + 1) (x + 3) = 0 x + 1 = 0 or x + 3 = 0 x = -1 or x = -3 (-1 + 1) (x + 3) = 0 True (x + 1) (-3 + 3) = 0 True »» Solve the equation:» x2 + 4x + 3 = 0 »» Answer questions 1 - 8 on Section C: Student Activity 3. »» Write the solution on the board. »» Are students able to factorise the equation and hence solve it? »» Distribute Section C: Student Activity 3. »» Circulate and see what answers the students are giving and address any misconceptions.» »» Can students factorise an expression of the form and solve an equation of the form» x2 + bx + c = 0? »» Ask individual students to write their answers on the board. © Project Maths Development Team 2011 www.projectmaths.ie KEY: » next step • student answer/response 11 Teaching & Learning Plan: Quadratic Equations Student Learning Tasks: Teacher Input »» The width of a rectangle is 5cm greater than its length. Could you write this in terms of x? Student Activities: Possible Responses • x (x + 5) » » » Teacher’s Support and Actions Assessing the Learning »» Get the students to draw a diagram of a suitable rectangle.» »» Are students extending their knowledge of quadratic equation?» » » » » » » » » » » » » » » »» Allow students to adopt an explorative approach here before giving the procedure.» »» If we know the area is equal to 36cm2, write the information we know about this rectangle as an equation.» • A = (length) (width)» x (x + 5) = 36 » » » »» Solve this equation.» » » » » » • »» What is the length and width of the rectangle?» » • The length is equal to 4cm and width is equal to 9cm.» »» Do students understand why x = -9 was a spurious solution?» »» Is it sufficient to leave this question as x = 4? • No. You must bring it back to the context of the question. »» Do students understand that saying x = 4 is not a sufficient answer to the question, but that it must be bought back into context of the question? © Project Maths Development Team 2011 x2 + 5x - 36 = 0 (x + 9) (x - 4) = 0 x+9=0x-4=0 x = -9 or x = 4 (-9 + 9) (x - 4) = 0 True (x + 9) (-4 - 4) = 0 True www.projectmaths.ie »» Ask a student to write the solution on the board explaining what they are doing in each step.» »» Challenge students to explain why x= -9 is rejected. KEY: » next step • student answer/response Teacher Reflections 12 Teaching & Learning Plan: Quadratic Equations Student Learning Tasks: Teacher Input Student Activities: Possible Responses »» Complete the remaining exercises on Section C: Student Activity 3. Teacher’s Support and Actions Assessing the Learning »» Teacher may select a number of these questions or get students to complete this activity sheet for homework.» »» Can students answer the problems posed in the student activity? Teacher Reflections »» Ask individual students to do questions on the board when the class has done some of the work. Students should explain their reasoning in each step. © Project Maths Development Team 2011 www.projectmaths.ie KEY: » next step • student answer/response 13 Teaching & Learning Plan: Quadratic Equations Student Learning Tasks: Teacher Input Student Activities: Possible Responses Teacher’s Support and Actions Assessing the Learning Section D: To solve quadratic equations of the form ax2 + bx + c that are factorable (Higher Level only) Teacher Reflections »» Solve the equation» • 2x2 + 5x + 2 = 0 »» Ask a student to come to the board and »» Do students understand that » 2 2x + 5x + 2 = 0. explain how they factorised this. 2x2 + 5x + 2 = (2x + 1) (x + 2)? Guide number is: 4 [coefficient 2 2x + 4x +1x + 2 = 0 of x2 and 2 the constant] 2x (x + 2) + 1 (x + 2) = 0 2 x 2 = 4» (2x + 1) (x + 2) = 0 Factors of 4 are 4 x 1 or 2 x 2» 2x + 1 = 0 or x + 2 = 0 x = -1/2 or x = -2 • Use 4 x 1 because this gives or you a sum of 5» 2x2 + 1x +4x + 2 = 0 » x (2x + 1) + 2 (2x + 1) = 0 2x2 + 4x +1x + 2 = 0 (x + 2) (2x + 1) = 0 2x (x + 2) + 1 (x + 2) = 0 x + 2 = 0 or 2x + 1 = 0 (2x + 1) (x + 2) = 0 x = -2 or x = -1/2 2x + 1 = 0 or x + 2 = 0 x = -1/2 or x = -2 »» Find the factors of » • (3x + 1)(x + 1) 3x2 + 4x +1 and » hence solve » 3x + 1 = 0 x + 1 = 0 2 3x + 4x +1 = 0 3x = -1 x = -1 x= -⅓ x = -1 »» Answer questions contained in Section D: Student Activity 4. »» Circulate and see what answers the students are giving and address any misconceptions. »» Distribute Section D: Student Activity 4. »» Can students factorise expressions of the form» »» Circulate and check the students’ work ax2 + bx + c and hence solve ensuring that all students can complete equations of the form» the task.» ax2 + bx + c = 0? »» Ask individual students to do questions on the board when some of the work is done using the algorithm, table and graph. © Project Maths Development Team 2011 www.projectmaths.ie »» Can students solve equations of the form ax2 + bx + c = 0 by algorithm, table and graphically? KEY: » next step • student answer/response 14 Teaching & Learning Plan: Quadratic Equations Student Learning Tasks: Teacher Input Student Activities: Possible Responses »» Are x2 + 6x + 5 = 0 and x2 + 6x = 0 quadratic equations? Explain why.» » • Yes because the highest power of the unknown in these examples is 2.» »» Give a general definition of a quadratic equation?» • »» Are ax2 + c = 0? ax2 + bx = 0 ax2 = 0 quadratic equations? • Yes, because the highest power of the unknown is still 2.» » »» Do you have a quadratic equation if a = 0? If not what is it called? • No, because the highest power is no longer 2. This is called a linear equation. Teacher’s Support and Actions Assessing the Learning Section E: To solve quadratic equations of the form x2 - a2 = 0 »» What would the general form of the equation look like if» (i) b = 0, (ii) c = 0, (iii) a = 0, (iv) b = 0 and c = 0? »» Do students understand that quadratic equations all take the form» ax2 + bx + c = 0 and b and/or c can be zero ax2 + bx + c = 0. » • • • • »» Do students understand the difference between a quadratic and a linear equation? ax2 + c = 0 ax2 + bx = 0 bx + c = 0 ax2 = 0 »» So is x2 - 16 a quadratic expression? » • Yes, because the highest power »» Give students time to of the unknown is 2.» find the factors.» »» What are the factors of x2 - 16? • (x + 4) (x - 4) »» What was this called?» • The difference of two squares.» »» What is the solution of x2 - 16 = 0? • »» How can we prove these values are the factors of x2 - 16? • (-4)2 - 16 = 0 True © Project Maths Development Team 2011 www.projectmaths.ie x = -4 and x = 4 Teacher Reflections »» Ask a student to write the expression and its factors on the board explaining his/her reasoning. KEY: » next step »» Can students get the factors of » x2 - b2 = 0 and hence solve equations of the form » x2 - b2 = 0? • student answer/response 15 Teaching & Learning Plan: Quadratic Equations Student Learning Tasks: Teacher Input Student Activities: Possible Responses Teacher’s Support and Actions Assessing the Learning »» How can you use the following diagrams of two squares to graphically show that x2 - y2 = (x + y) (x - y)? • Area of the small square is y2 and the area of the big square is x2. »» Draw the squares on the board. »» Can students see how this is a physical representation of » x2 - y2 = (x + y) (x - y)? • Area of the unshaded region is x2 - y2. y 2 x2 • The area of the unshaded region is also x(x - y) + y(x - y). x-y y2 x Teacher Reflections x-y y2 y y • Hence x2 - y2 = x(x - y) + y(x - y). • Hence x2 - y2 = (x + y) (x - y)? »» Complete Section E: Student Activity 5. © Project Maths Development Team 2011 www.projectmaths.ie »» Distribute Section E: Student »» If students are Activity 5. having difficulties, allow them to »» Circulate the room and talk through their address any misconceptions. work.This will help them identify their misunderstandings and misconceptions. Difficulties identified can then be addressed. KEY: » next step • student answer/response 16 Teaching & Learning Plan: Quadratic Equations Student Learning Tasks: Teacher Input Student Activities: Possible Responses Teacher’s Support and Actions Assessing the Learning Section F: To solve quadratic equations using the formula (Higher Level only) »» Is x2 + 3x + 1 a quadratic equation? » »» Can you find the factors of» x2 + 3x + 1? • Yes, because the highest power of the unknown is 2.» • No »» Mathematicians use the formula » » 2 » to find the solutions to quadratic equations when they are unable to find factors. It is worth noting however that the formula can be used with all quadratic equation.» »» We will now try it for » x2 + 3x + 2 = 0, which we already know has x = - 2 and x= - 1 as its solutions. »» Give students time to consider if they can get the factors and justify their answers. »» Do students see that not all quadratics are factorable? »» Write x2 + 3x + 1 = 0 on the board.» »» It is not always possible to solve quadratic equations through the use of factors, but there are alternative methods to solve them.» »» Comparing x2 + 3x + 1 = 0 to the general form of a quadratic equation » ax2 + bx + c = 0. What are the values of a, b and c? »» Write the expression on the board.» Teacher Reflections »» Write a = 1, b = 3 and c = 1 on the board. • a = 1, b = 3 and c = 1 »» Write x2 + 3x + 2 = 0 and (x + 2) (x + 1) = 0 x = -2 and x = -1 on the board. »» Can students complete the formula? »» Write the formula on the board.» » » » » x = -1 or x = -2 © Project Maths Development Team 2011 www.projectmaths.ie KEY: » next step • student answer/response 17 Teaching & Learning Plan: Quadratic Equations Student Learning Tasks: Teacher Input »» Solve x2 + 3x + 1 = 0 using the formula. Student Activities: Possible Responses • x = -2.816 and x = 0.382 Teacher’s Support and Actions Assessing the Learning »» Challenge students to find the factors on their own before going through the algorithm.» Teacher Reflections »» Write on the board» » » » » x = -2.618 or x = -0.382 »» Complete the exercises in Section F: Student Activity 6. © Project Maths Development Team 2011 www.projectmaths.ie »» Distribute Section F: Student Activity 6. KEY: » next step »» Can students use the formula to solve quadratic equations? • student answer/response 18 Teaching & Learning Plan: Quadratic Equations Student Learning Tasks: Teacher Input Student Activities: Possible Responses Teacher’s Support and Assessing the Learning Actions Section G: To solve quadratic equations given in rational form • 3x2 - 2x = 0 x (3x - 2) = 0 x = 0 or x = 2/3 »» Simplify » » » »» Get individual students to write the solution on the board and explain their work. »» Can students convert equations given in simple fraction form to the form» ax2 + bx + c = 0? »» Get an individual student to write the solution on the board and explain their reasoning. »» Can students simplify the equation and then solve it? »» Now solve the equation. »» Write x + 3 = 10/x in the form ax2 + bx + c = 0 x + 3 = 10/x x2 + 3x -10 = 0 • Multiplied both sides by x. »» How did you do this?» »» Hence solve x + 3 = 10/x • (x + 5) (x - 2) = 0 x = -5 or x = 2 (-5) 2 + 3 (-5) -10 = 0 True» (2) 2 + 3 (2) -10 = 0 True »» Attempt the questions on Section G: Student Activity 7. © Project Maths Development Team 2011 • www.projectmaths.ie Teacher Reflections »» Distribute Section G: »» Can students correctly Student Activity 7. answer the questions on the activity sheet? »» Circulate the room and address any misconceptions. KEY: » next step • student answer/response 19 Teaching & Learning Plan: Quadratic Equations Student Learning Tasks: Teacher Input Student Activities: Possible Teacher’s Support and Assessing the Responses Actions Learning Section H: To form quadratics given whole number roots (Higher Level Only) »» What are the factors of x2 + 6x + 8? • (x + 4) (x + 2) »» What is the solution of x2 + 6x + 8 = 0? • »» If we were told the equation had roots » x = -4 and x = -2, how could we get an equation? »» Given that an equation has roots 2 and 5, write the equation in the form (x - s) (x - t) = 0. »» Then write it in the form ax2 + bx + c = 0. x = -4 and x = -2 • (x - (-4)) (x- (-2)) = 0 (x + 4) (x + 2) = 0 x2 + 4x + 2x + 8 = 0 x2 + 6x + 8 = 0 • (x -2) (x -5) = 0 » » x2 -2x -5x + 10 = 0 x2 -7x + 10 = 0 »» Given that an equation has roots 1 and -3, write • (x -1) (x + 3) = 0 the equation in the form ax2 + bx + c = 0. x2 - 1x + 3x - 3 = 0 x2 + 2x -3 = 0 »» Challenge the students to find the equation for themselves.» »» Get individual students to write the solution on the board and explain their work. »» Write students’ responses on the board. »» Given the roots, can students write the equation in the form » ax2 + bx + c = 0? • »» What do you notice about the following equations:» x2 + 2x - 8 = 0 2x2 + 4x - 16 = 0 8 - 2x - x2 = 0? »» Now complete Section H: Student Activity 8. © Project Maths Development Team 2011 www.projectmaths.ie • They are the same equations but are arranged in a different order. Teacher Reflections »» Write students’ responses on the board. »» Write each equation »» Can students on the board. recognise that the equations are the same? »» Distribute Section H: Student Activity 8. KEY: » next step • student answer/response 20 Teaching & Learning Plan: Quadratic Equations Section A: Student Activity 1 Note:It is always good practice to check solutions. The roots of a quadratic equation are the elements of its solution set. For example if x = 1, x = 2 are the root ⇒ {1, 2} = solution set. The roots of a quadratic equation are another name for its solution set. 1. If xy = 0, what value must either x or y or both have? 2. Write in your own words what solving an equation means. 3. Solve the following equations: a. (x - 1) (x - 2) = 0 b. (x - 4) (x - 5) = 0 c. (x - 3) (x - 5) = 0 d. (x - 2) (x - 5) = 0 4. What values of x make the following statements true: a. (x - 2) (x - 5) = 0 b. (x - 4) (x + 5) = 0 c. (x - 2) (x + 4) = 0 5. Find the roots of (x - 4) (x + 5) = 0. 6. Solve the equation (x - 3) (x + 2) = 0. Hence state what the roots of (x - 3) (x + 2) = 0 are. 7. Find a positive value for x that makes the statement (x - 4) (x + 2) = 0 true. 8. Solve the following equations: a. x (x - 1) = 0 b. x (x - 2) = 0 c. x (x + 4) = 0 9. a. These students each made at least one error, explain the error(s) in each case: Student C Student B Student A (x - 8) (x - 9) = 0 (x - 7) (x - 9) = 0 (x + 5) (x + 9) = 0 x-8=0 x-9=0 x-7=0 x-9=0 x+5=0 x+9=0 x = -8 x=9 x = -7 x = -9 x=5 x=9 b. Solve each equation correctly showing all the steps clearly. 10.If x = 5 is a solution to the equation (x - 4) (x - b) = 0, what is the value of b? 11.Is x = 3 a solution to the equation (x - 3) (x - 2) = 2. Explain your reasoning. Solve this equation. © Project Maths Development Team 2011 www.projectmaths.ie 21 Teaching & Learning Plan: Quadratic Equations Section B: Student Activity 2 (continued) 1. a. Compete the following table: x −4 −3 −2 −1 0 1 2 x+2 x+1 y = f (x) = (x + 2) (x + 1) b. From the table above determine the values of x for which the equation is equal to 0. c. Solve the equation (x + 2) (x + 1) = 0 by algebra. d. What do you notice about the answer you got to parts b. and c. in this question? e. Draw a graph of the data represented in the above table. f. Where does the graph cut the x axis? What is the value of f (x) = (x + 2)(x + 1) at the points where the graph cuts the x axis? g. Can you describe three methods of finding the solution to (x + 2) (x + 1) = 0. 2. Solve the equation (x - 1) (x - 4) = 0 a) by table, b) by graph and c) algebraically. 3. Write the equation represented in this table in the form (x−a)(x−b)=0. x −2 −1 0 1 2 3 © Project Maths Development Team 2011 f(x) 20 12 6 2 0 0 www.projectmaths.ie 22 Teaching & Learning Plan: Quadratic Equations Section B: Student Activity 2 4. The graph of a quadratic function f (x) = ax2 + bx + c is represented by the curve in the diagram below. Find the roots of the equation f (x) = 0 and so identify the function. 5. The graph of a quadratic function f (x) = ax2 + bx + c is represented by the curve in the diagram below. Find the roots of the equation f (x) = 0 and so identify the function. 6. Where will the graphs of the following functions cut the x axis? a. f (x) = (x - 7) (x - 8) b. f (x) = (x + 7) (x + 8) c. f (x) = (x - 7) (x + 8) d. f (x) = (x + 7) (x - 8) 7. For what values of x does (x - 7) (x - 8) = 0? © Project Maths Development Team 2011 www.projectmaths.ie 23 Teaching & Learning Plan: Quadratic Equations Section C: Student Activity 3 Note: It is always good practice to check solutions. It is recommended you use the Guide number method to find the factors. 1. Solve the following equations: a. x2 + 6x + 8 = 0 b. x2 + 5x + 4 = 0 c. x2 - 9x + 8 = 0 d. x2 - 6x + 8 = 0 2. Solve the equations: a. x2 + 2x = 3 b. x (x - 1) = 0 c. x (x - 1) = 6 3. Are the following two equations different x (x - 1) = 0 and x (x - 1) = 6? Explain. 4. When a particular Natural number is added to its square the result is 12. Write an equation to represent this and solve the equation. Are both solutions realistic? Explain. 5. A number is 3 greater than another number. The product of the numbers is 28. Write an equation to represent this and hence find two sets of numbers that satisfy this problem. 6. The area of a garden is 50cm2. The width of the garden is 5cm less than the breadth. Represent this as an equation. Solve the equation. Use this information to find the dimensions of the garden. 7. A garden with an area of 99m2 has length xm. Its width is 2m longer than its length. Write its area in term of x. Solve the equation to find the length and width of the garden. 8. The product of two consecutive positive numbers is 110. Represent this as an algebraic equation and solve the equation to find the numbers. 9. Use Pythagoras theorem to generate an equation to represent the information in the diagram below. Solve this equation to find x. © Project Maths Development Team 2011 www.projectmaths.ie 24 Teaching & Learning Plan: Quadratic Equations Section C: Student Activity 3 (continued) 10.One number is 2 greater than another number. When these two numbers are multiplied together the result is 99. Represent this problem as an equation and solve the equation. 11.Examine these students’ work and spot the error(s) in each case and solve the equation fully: Student A Student B Student C x2 - 6x - 7 = 0 (x + 7) (x + 1) = 0 x = -7 x = -1 x2 - 6x - 7 = 0 (x - 7) (x + 1) = 0 x=7 x=1 x2 - 6x - 7 = 0 (x - 1) (x + 7) = 0 x=1 x = -7 12. a. Complete a table for x2 - 2x + 1 for integer values between -2 and 2. b. Draw the graph of x2 - 2x + 1 for values of x between -2 and 2. Where does this graph cut the x axis? c. Factorise x2 - 2x + 1 and solve x2 - 2x + 1 = 0. d. What do you notice about the values you got for parts a), b) and c)? 13. a. Complete a table for x2 + 3x + 2 for integer values between -3 and 3. b. Draw the graph of x2+3x+2 for values of x between -3 and 3. Where does this graph cut the x axis? c. Factorise x2+3x +2 and solve x2+3x +2=0. d. What do you notice about the values you got for parts a), b) and c)? © Project Maths Development Team 2011 www.projectmaths.ie 25 Teaching & Learning Plan: Quadratic Equations Section D: Student Activity 4 Note: It is always good practice to check solutions. It is recommended you use the Guide number method to find the factors. 1. Find the factors of 2x2 + 7x + 3. Hence solve 2x2 + 7x + 3 = 0. 2. Find the factors of 3x2 + 4x + 1. Hence solve 3x2 + 4x + 1 = 0. 3. Find the factors of 4x2 + 4x + 1. Hence solve 4x2 + 4x + 1 = 0. 4. Find the factors of 3x2 - 4x + 1. Hence solve 3x2 - 4x + 1 = 0. 5. Find the factors of 2x2 + x - 3. Hence solve 2x2 + x - 3 = 0. 6. a. Is 2x2 + x = 0 a quadratic equation? Explain your reasoning. b. Find the factors of 2x2 + x. Hence solve 2x2 + x = 0. 7. Factorise 4x2 - 1x2 + 9. Hence solve 4x2 - 1x2 + 9 = 0. 8. Twice a certain number plus four times the same number less one is 0. Find the numbers. 9. a. Complete the following table and using your results, suggest solutions to 2x2 + 3x + 1 = 0. x 2x2 3x 1 -2 1 -1 1 0 1 1 1 2 1 2x2 + 3x + 1 b. Using the information in the table above, draw a graph of f(x) = 2x2 + 3x + 1, hence solve the equation. c. Did your results for a. agree with your results in b? 10.Find the function represented by the curve in the diagram opposite in the form f (x) = ax2 + bx + c = 0. Then solve the equation. © Project Maths Development Team 2011 www.projectmaths.ie 26 Teaching & Learning Plan: Quadratic Equations Section E: Student Activity 5 1. Factorise: a. x2 - 25 b. x2 - 49 2. Solve: a. x2 - 36 = 0 b. 36 - x2 = 0 c. n2 - 625 = 0 d. x2 - x = 0 e. 3x2 - 4x = 0 f. 3x2 = 11x g. (a + 3)2 - 25 = 0 3. Think of a number, square it, and subtract 64. If the answer is 0, find the number(s). 4. Given an equation of the form x2 - b = 0, write the solutions to this equation in terms of b. 5. Solve the equation x2 - 16 = 0 graphically. Did you get the results you expected? Explain your answer. 6. Calculate: a. 992 - 1012 b. 1032 - 972 Higher Level Only 7. Solve the following equations: a. 4x2 - 36 = 0 b. 4x2 - 9 = 0 c. 4x2 - 25 = 0 d. 16x2 - 9 = 0 8. A man has a square garden of side 20m. He builds a pen for his dog in one corner. If the area of the remaining part of his garden is 144m2, find the dimensions of the dog’s pen. © Project Maths Development Team 2011 www.projectmaths.ie 27 Teaching & Learning Plan: Quadratic Equations Section F: Student Activity 6 1. Using the formula solve the following equations: a. x2 + 5x + 4 = 0 b. x2 + 4x + 3 = 0 c. x2 + 4x -3 = 0 d. x2 - 4x + 3 = 0 e. x2 - 4x -3 = 0 f. x2 - 4 = 0 g. x2 + 3x = 4 h. x2 + 2x -3 = 0 2. Solve the equation x2 -5x - 2 = 0. Write the roots in the form a±√b. 3. Given 2+√3 as a solution to the equation ax2 + bx + c = 0 find the other solution. 4. When using the quadratic formula to solve an equation and you know x = 3 is a solution, does that mean that x = -3 is definitely the other solution? Explain your reasoning with examples. 5. a. Solve the equation x2 + x2 + 1 = 0 by using a: i. Table. ii.Graph. iii.Factors. iv.Formula. b. Did you get the same solutions using all four methods? © Project Maths Development Team 2011 www.projectmaths.ie 28 Teaching & Learning Plan: Quadratic Equations Section G: Student Activity 7 Solve the following equations: 2. Square a number add 9, divide the result by 5. The result is equal to twice the number. Write an equation to represent this and solve the equation. 3. A prize is divided equally among five people. If the same prize money is divided among six people each prize winner would get €2 less than previously. Write an equation to represent this and solve the equation. © Project Maths Development Team 2011 www.projectmaths.ie 29 Teaching & Learning Plan: Quadratic Equations Section G: Student Activity 7 Higher Level Only 1. If the roots of the following quadratic equations are as follows write the equations in the form (x - s) (x - t) = 0: a. 2 and 3 b. 2 and -3 c. -2 and -3 d. -2 and 3 e. 2 and ¾ 2. What is the relationship between the roots of a quadratic equation and where the graph of the same quadratic cuts the x axis? 3. The area of the triangle below is 8cm2. Find the length of the base and the height of the triangle. 4. A field is in the shape of a square with side equal to x metres. One side of the field is shortened by 5 metres and the other side is lengthened by 6 metres. If it is known the area of the new field is 126m2, write an equation that represents the area of the new field. Solve the equation and find the dimensions of the original square field. ©http://www.examinations.ie/archive/exampapers/2009/JC003ALP100EV.pdf 5. a. Draw the graph of f (x) = 3x2 - 10x + 8 for values between -3 and 3. Where does this graph cut the x axis? b. Factorise and solve 3x2 - 10x + 8 = 0 c. What do you notice about the values you got for part a and part b? 6. Find the factors of 2x2 - 5x +3. Hence solve 2x2 - 5x + 3 = 0 © Project Maths Development Team 2011 www.projectmaths.ie 30

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