STUDENTS REASONING IN QUADRATIC EQUATIONS WITH ONE UNKNOWN


 Georgia Poole
 3 years ago
 Views:
Transcription
1 STUDENTS REASONING IN QUADRATIC EQUATIONS WITH ONE UNKNOWN M. Gözde Didiş, Sinem Baş, A. Kürşat Erbaş Middle East Technical University This study examined 10 th grade students procedures for solving quadratic equations with one unknown. An openended test was designed and administered to 113 students in a high school in Antalya, Turkey. The data were analyzed in terms of the students foci while they were answering the questions. The results revealed that factoring the quadratic equations was challenging to them, particularly when students experienced them in a different structure from what they are used to. Furthermore, although students knew some rules related to solving quadratics, they applied these rules thinking about neither why they did so, nor whether what they were doing was mathematically correct. It was concluded that the students understanding in solving quadratic equations is instrumental (or procedural), rather than relational (or conceptual). Key words: Quadratic equations, instrumental understanding, relational understanding INTRODUCTION For many secondary school students, solving quadratic equations is one of the most conceptually challenging subjects in the curriculum (Vaiyavutjamai, Ellerton, & Clements, 2005). In Turkey, where a national mathematics curriculum for elementary and secondary levels is implemented, the teaching and learning of quadratic equations are introduced through factorization, the quadratic formula, and completing the square by using symbolic algorithms. Of these techniques, students typically prefer factorization when the quadratic is obviously factorable. With this technique, students can solve the quadratic equations quickly without paying attention to their structure and conceptual meaning (Sönnerhed, 2009). However, as Taylor and Mittag (2001) suggest, the factorization technique is only symbolic in its nature. Since students simply memorize the procedures and formulas to solve quadratic equations, they have little understanding of the meaning of quadratic equations, and do not understand what to do and why. This can be described using Skemp s (1976) categorization of mathematical understanding as either instrumental or relational. He simply described instrumental understanding as rules without reasons and relational understanding as knowing both what to do and why (p. 20). Using the language of Skemp, it can be said that students can perform instrumentally to solve the quadratic equations by applying the factorization technique; however, they become deprived of relational understanding. Although quadratic equations take an important role in secondary school algebra curricula around the world, it appears that studies concerning teaching and learning 1
2 quadratic equations are quite scarce in algebra education research (Kieran, 2007; Vaiyavutjamai & Clements, 2006). Therefore, this study was designed to widen the research considering students reasoning when engaging in different types of quadratic equations in one unknown. In particular, this study investigated students processes for solving quadratic equations with one unknown by using the factorization technique. The findings of this study may provide teachers with insight into the reasoning that leads to the common mistakes that students make while solving quadratic equations, and hence guide them in creating a more efficient pedagogical design for teaching how to solve quadratics. Challenges faced by Students in Solving Quadratic Equations According to Kotsopoulos (2007), for many secondary school students, solving quadratic equations is one of the most conceptually challenging aspects in the high school curriculum. She indicated that many students encounter difficulties recalling main multiplication facts, which directly influences their ability to engage in quadratics. And, since the factorization technique of solving quadratic equations requires students to be able to rapidly find factors, factoring simple quadratics (i.e., x 2 +bx+c=0 where b, c R) become a quite challenge, while nonsimple quadratics (i.e., ax 2 +bx+c=0 a, b, c R and a 1) become nearly impossible. Moreover, students encounter crucial difficulties in factoring quadratic equations if they are presented in nonstandard forms. For example, factoring x 2 +3x+1=x+4 is challenging for students, since the equation is not presented in standard form (Kotsopoulos, 2007). Similarly, Bossè and Nandakumar (2005) stated that the factoring techniques for solving quadratic equations are problematic for students. They indicated that students can find factoring the quadratics considerably more complicated when the leading coefficient or constant in the quadratic has many pairs of possible factors. Skemp s (1976) description of instrumental and relational understanding can be used as a framework to discuss the difficulties students have with factoring quadratic equations. While an instrumental understanding of factorizing quadratic equations with one unknown requires memorizing rules for equations presented in particular structures, relational understanding enables students to apply these rules to different structures easily (Reason, 2003). That is, when students have relational understanding, they can transfer knowledge of both what rules (and formulas) worked and why they worked from one situation to another (Skemp, 2002). Lima (2008) found that students may perceive quadratic equations just like they do calculations. Since they focus mostly on the symbols used to perform operations, they may not be aware of the concepts that are involved. Vaiyavutjamai and Clements (2006) explain that students difficulties with quadratic equations arise from the lack of both instrumental and relational understanding of the associated mathematics. They found several misconceptions regarding variables which were obstacles to 2
3 understanding quadratic equations. For example, students thought that the two x s in the equation (x3) (x5) =0 stood for different variables, even though most of them obtained the correct solutions x=3 and x=5. Hence, they concluded that students performance in that context reflect rote learning and a lack of relational understanding. METHODOLOGY Participants and the Instrument The sample of this study consisted of 113 students in four 10 th grade classes, and this study was performed in a high school in Antalya, Turkey during the spring term For the purpose of the study, a questionnaire was formed by the authors since no test to specifically explore students errors and understanding was available. The test questions were carefully selected from secondary mathematics textbooks and from research regarding quadratic equations (e.g., Crouse & Sloyer, 1977). All questions used in this questionnaire were selected to measure the study objective of determining how students determine the roots and solution set of [a] quadratic equation in one unknown. During the selection process, two mathematics educators and a mathematics teacher were consulted about whether the content of the selected questions were consistent with the objective of the test. In light of their suggestions, seven open ended question were determined. Although the format of the all of the questions was openended, they varied in type so as to be consistent with the objective of the study. Questions 1 to 4 were in the standard format in which students were expected to find the solution set of the given quadratic equation. These questions were based on procedural skills, and they were mostly used to detect students procedural abilities in solving quadratic equations in different structures. On the other hand, questions 5 to 7 introduced a mathematical scenario that included both a quadratic equation and a solution belonging to it. In these type of questions, students were expected to determine whether the solutions [belonging to] the equations were correct or not, and to make judgment about their decision. Therefore, in addition to procedural skills, these questions were used to detect students understanding of and reasoning level when dealing with quadratic equations. The mathematics teacher administered the questionnaire during the regular class period and the students were given 30 minutes to complete it. Analysis of Data Initially, the responses given to each question were givens scores of either 1 or 0. A score of 1 was given for answers that were mathematically correct in terms of both solution process and final answer. A score of 0 was given for answers that were either omitted or incorrect in terms of either solution process or final answer. Then, in order to obtain a general view of the students performance, the percentage of correct, 3
4 incorrect and omitted questions were calculated. The aim of this process was descriptive analysis. Afterwards, qualitative data analysis was conducted. The subjects responses were studied in order to provide substantial information about their type of understanding. In this analysis, it was attempted to identify the common mistakes that students made while solving the quadratic equations. Therefore, the incorrect answers for all questions have been analyzed item by item with respect to the students focus when they solved the questions in the test situation. In this process, students types of mistakes were coded by two researchers of this study who worked initially separately. Next, the mistakes were both combined and renamed based on their common features, and then they were classified by two researchers together. Lastly, these mistakes were interpreted in terms of students instrumental understanding and relational understanding. RESULT The first item in the instrument was related to finding the roots of a quadratic equation given in standard form (e.g., ax 2 +bx+c=0 where a, b, c R). Almost all students correctly solved this equation by factorization. In the following questions, quadratic equations were given in different structures (e.g., ax 2 bx=0, c=0). In these types of questions, just 64% of them solved the equation ax 2 bx=0, correctly. When the solution processes of students who made mistakes (36%) were analyzed, it was recognized that their mistakes were based on two different types. Find the solution set of the equation 2 2 = 0. Figure 1: An example of students first type of mistake Find the solution set of the equation 2 2 = 0. Figure 2: An example of students second type of mistake 4
5 Find the solution set of the equation 2 = 12 Figure 3: An example of students mistake when just the form of equation changed. In the first type of wrong solution (see Figure 1), students carried the term 2x from left side to the right, and then simplified the term x in both sides of the equation. Consequently, they ignored one of the roots of the equation, which is 0. In the second type of wrong solution (see Figure 2), students tried to factorize the equation. Here, students perceived the form ax 2 bx=0 just like ax 2 +bx+c=0 and thought 2x as the constant term of the quadratic equations. Even, when just the form of the equation was changed instead of the structure (e.g., ax 2 +bx=c where a, b, c 0), 12% of the students incorrectly solved the quadratic. Because the constant term was in the right side, they didn t perceive that the equation was in standard form (see Figure 3). In this type of solution, they were able to find only one of the roots, 4. Statements Question 5 To solve the equation (x3) (x2) = 0 for real numbers, Ali answered in a single line that: x=3 or x=2 Is this answer correct? If it is correct, how can you show it correctness? Students types of responses with their reasoning I. II. III. IV. Right Since I wrote (x3) (x2) = 0 as x 25x+6=0 and then I factorize to find roots of it. from (x3)=0 and from (x2)=0 x=3 and x=2 Table 1: Common examples of students types of responses with their reasoning for question 5. Although all of the students stated that Ali s answer was correct by choosing either one of the statements I, II, III, and IV, the ways they justified for the correctness of Ali s solution were different. For instance, in statement I, students first transformed the factorized expression into the standard form, and then factorized the expression again in the same way and found the roots by rote. In statement II, students unconsciously applied the null factor law. In statement III, the way of justification for solution was based on substitution method. In all of these three statements, they could not clearly justify the correctness of the solution. In statement IV, students substituted x=3 into (x3) and x=2 into (x2) simultaneously, and concluded that their solution 5 The answer is Right Because (x3)=0 (x2)=0 x=3 x=2 The answer is Right. Since we substitute 3 and 2 into x, the equation is provided. (explanation made only with words) Right. If the x=2 and x=3 are substituted into the equation (33) (22)=0 0.0 = 0
6 were correct since 0 0=0. Namely, they thought that the two x s stood for different numbers. Statements Question 6 A student hands in the following work for the following problem. Solve ; x 214x+24=3 (x12) (x2)=3 (x12) (x2)=3 1 x12=3 x2=1 x=15 x=3 Ç.K= {3, 5} Is the student correct? Explain your answer with its reasons? The answer is Wrong Because, firstly, 3 must carry the left side of the equation and equalize the 0. Then, the other operations must be done. In this way, the equation x 214x+21=0 Students types of responses with their reasoning I. II. III. IV. The answer is Wrong. Because when we substitute 3 and 15 for x, the equation is not provided. Right Since the result is equal to 3, we equate 3 rather than 0 while factoring it. Therefore, the result is true. Students again solve as: x 214x+24=3 (x12) (x2)=3 (x12) (x2)=3 1 x12=3 x2=1 x=15 x=3 (3,1) Wrong Since the equations are separated as (3,1) there is no error when (x12)=3 however, there is error when (x2)=1. It must be (x2)=3 then, x=5. Therefore, the solution will be {5, 15} rather than {3, 15}. Table 2: Common examples of students types of responses with their reasoning for question 6. In statements I and II (see Table 2), students were aware of the error in the solution of the given question. However, to explain the reasons for the mistake, they presented procedural explanations like the responses in statements I, II, III for question 5 (see Table 1). In statement III, students incorrectly stated that the answer was right. Looking at the statement since the result is equal to 3, we equate to 3 rather than 0 while factoring it, it can be said that they wrongly tried to transfer the null factor law to this context. That is, they equated the factors of equation x 214x+24 with the integer factors of 3. In statement IV, students correctly claimed the answer of the question wrong ; however, their explanations were fully erroneous. Similar to statement III, these students tried to apply the null factor law to the equation. Nonetheless, in this case, they only equated the factors to 3 rather than to the factors of 3. In both statements III and IV, students did not check whether the roots they found were appropriate or not. 6
7 Statements Question 7 The solution of the quadratic equation 2 x 2 =3x is given in the following; According to you, is this solution correct or not? Explain your answer with its reasons? Solution: I. step 2 x 2 =3x II. step 2 x x=3 x III. step 2 x =3 IV. step x= 3/2 Ç.K = {3/2} Students types of responses with their reasoning I. II. III. IV. Wrong Because 3x must be carried the left side of the equation and equalized the 0. Then, 2 x 2 =3x 2 x 23x=0 x (2x3)=0 x=0, x=3/2. The answer is Right The solution is right; however, it must be added 0 to the solution set. The answer Right Because when we substitute the value for x, the equation is satisfied. Right 2 x 2 =3x and x 2 is opened. 2 x x=3 x Yes the x is simplified. 2x=3 so x=3/2. Table 3: Common examples of students types of responses with their reasoning for question 7. In statement I (see Table 3), students stated that the answer was correct. They explained an appropriate procedure required for solving the equation. Since they memorized the rule without its reasons, they could only exhibit how the procedure must be worked. In statement II, on the other hand, students were aware that the roots of the equation were 0 and 3/2. However, they did not recognize that when was simultaneously canceled from both sides, the root 0 disappeared. Furthermore, in statement III, the explanation for solution was just based on the substitution method. In statement IV, students incorrectly stated that the answer was right. Like in statement II, students were not aware of the missing root 0 when canceling an in the equation DISCUSSION The results indicate that most of the students used the factorization technique to solve quadratic equations. This result supports Bosse and Nandakumar (2005), who claimed that a large percentage of the students preferred to apply the factorization techniques to find the solutions of quadratic equations. Also, in parallel with the results of Bossé and Nandakumar (2005) and Kotsopoulos (2007), the result of this study revealed that factoring the quadratic equations was challenging when they were presented to students in nonstandard forms and structures. After looking at the examples of students solutions (see Figures 1, 2, and 3), it can be said that the students knew some rules (or procedures) related to solving quadratics. However, they tried to apply these rules thinking about neither why they did so, nor whether if what they were doing was mathematically correct. These results give some clues about students instrumental understanding of solving quadratic equations with one unknown. However, to make an exact judgment about students relational or 7
8 instrumental understanding as Skemp (2002) defined, indepth interviews with individual students are required. Furthermore, results also indicate that students incorrectly tried to transfer some rules from one form of equation to another (e.g., in Figure 2). This can be considered another clue to students instrumental understanding (Reason, 2003). When students were asked to examine a solution process of a quadratic equation and judge whether it was correct (i.e., in questions 5, 6, and 7), the results give additional clues about their reasoning in solving quadratics. In question 5, for example, although most of the students were aware of the correctness of the result, they did not explain the underlying null factor law used to solve the quadratics by factorization. The responses also reveal their misunderstanding of the unknown concept in a quadratic equation (see the statement IV, in question 5), which is consistent with the results of Vaiyavutjamai and Clements (2006). Students were not aware that the two s in the equation represent a specific unknown when dealing with equations in the form (xa) (xb) =0. All of these can be regarded as clues to students instrumental understanding. As stated by Lima (2008), and Vaiyavutjmai and Clements (2006), students knew how to get correct answers but were not aware of what their answers represented. Similar interpretations can be made for the responses of students to question 6. There are two salient points related to their reasoning in explaining the given solution. First, although students were expected to explain the reason(s) why the given solution process was wrong, they could not detect the conceptual errors in the solution. They just presented some rules or procedures to solve the quadratic. Second, as was clear from statements III and IV (see Table 2), due to their lack of conceptual understanding of the null factor law in solving quadratics given in standard form, they wrongly transferred this principle to a quadratic in a nonsuitable form. This can also be a clue for students instrumental understanding. Because when students relationally understand a rule, they can use it in a different context (Reason, 2003). Similar inferences can be made for the students responses to questions 7 where they did not offer any explanation for why canceling s was wrong. In other words, they did not recognize that when x was simultaneously canceled from both sides, the root 0 disappeared. Also, consistent with the results reported by Bossé and Nandakumar s (2005) and Kotsopoulos (2007), although students knew the null factor law, they could not apply it appropriately when the structure of equation was changed. Collectively, all these results reveal that students attempted to solve the quadratic equations as quickly as possible without paying much attention to their structures and conceptual meaning (Sönnerhed, 2009). Although we cannot be sure if their reasoning was based on instrumental or relational understanding without indepth interviews with students, their written answers provide clues to their reasoning, and it can be said that their reasoning underlying solving quadratic equations was based on instrumental understanding. 8
9 Having instrumental understanding does not generally cause trouble for students. It is much easier to obtain and use than relational understanding, just because it requires less knowledge, and with instrumental understanding, students can generally obtain the right answers more quickly. However, it necessities memorizing, and without relational understanding the learning cannot be adapted to new tasks, and students cannot give real reasons for their answers (Skemp, 2002). For that reason, greater attention should be given to how the concept is introduced to reduce the possibility of students learning the subjects/rules/procedures by rote. Any mechanism of solution must allow students to understand the meaning of the process that they apply in order to arrive at the correct answer; otherwise, the mechanism they learn will be a source of error (Blanco & Garrote, 2007). Recommendation As a result of this study, several suggestions can be made to contribute to improving students understanding of quadratic equations. Since factoring the quadratic equations was challenging when they are presented in nonstandard forms and structures, it would be better if teachers introduce various kinds of quadratic equations in different structures rather than just in the standard form. On the other hand, it would be also helpful for students to understand the factorization techniques as relational when teachers clearly emphasize meaning of the null factor rather than presenting it just as rule. In addition, because the students can attribute different meanings to the symbols (Küchemann, 1981), their understanding of the meanings of the algebraic symbols needs to be taken into account. Therefore, if teachers emphasize the meaning of the algebraic symbols, it would also useful for students to understand what the symbols represent in quadratic equations. Moreover, when teachers encourage students to use different techniques while solving quadratic equations, students learning may improve, and they may also gain a conceptually understanding. Similar recommendations can also be found in the related literature (e.g., Bossè & Nandakumar, 2005; Sönnerhed, 2009). Undoubtedly, teachers play an important role in encouraging students to learn relationally. This should be the most important part of teachers pedagogical content knowledge. However, research studies demonstrate a lack of secondary school mathematics teachers pedagogical content knowledge in this respect (Vaiyavutjamai, Ellerton, & Clements, 2005). Indeed, there is a need to research teachers knowledge about students difficulties concerning quadratic equations. REFERENCES Blanco, L. J., & Garrote, M. (2007). Difficulties in learning inequalities in students of the first year of preuniversity education in Spain. Eurasia Journal of Mathematics, Science & Technology Education, 3(3), Bossé, M. J., & Nandakumar, N. R. (2005). The factorability of quadratics: Motivation for more techniques (section A). Teaching Mathematics and its Applications, 24(4),
10 Crouse, J. R., & Sloyer, W. C. (1977). Mathematical questions from the classroom. USA: Prindle, Weber & Schmidt. Kieran, C. (2007). Learning and teaching algebra at the middle school through college levels. In F. Lester (ed.), Second Handbook of Research on Mathematics Teaching and Learning: A project of the National Council of Teachers of Mathematics. Vol II (pp ). Charlotte, NC: Information Age Publishing. Kotsopoulos, D. (2007). Unraveling student challenges with quadratics: A cognitive approach. Australian Mathematics Teacher, 63(2), Küchemann, D. E. (1981). Algebra. In Hart, K., Brown, M. L., Küchemann, D. E., Kerslake, D., Ruddock, G., & McCartney, M. (Eds.), Children's understanding of Mathematics: (pp ). London: John Murray. Lima, R. N. (2008). Procedural embodiment and quadratic equations. Retrieved April 1, 2010, from Reason, M. (2003). Relational, instrumental and creative understanding. Mathematics Teaching, 184, 57. Skemp, R. R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77, Skemp, R. R. (2002). Mathematics in the primary school. London: Routledge Falmer. Sönnerhed, W. W. (2009). Alternative approaches of solving quadratic equations in mathematics teaching: An empirical study of mathematics textbooks and teaching material or Swedish Uppersecondary school. Retrieved April 5, 2010, from Taylor, S. E. & Mittag, K. C. (2001). Seven wonders of the ancient and modern quadratic world. Mathematics Teacher, 94, Vaiyavutjamai, P., Ellerton, N. F., & Clements, M. A. (2005). Students attempts to solve two elementary quadratic equations: A study in three nations. Retrieved April 1, 2010, from Vaiyavutjamai, P., & Clements, M. A. (2006). Effects of classroom instruction on students understanding of quadratic equations. Mathematics Education Research Journal, 18(1),
Grade Level Year Total Points Core Points % At Standard 9 2003 10 5 7 %
Performance Assessment Task Number Towers Grade 9 The task challenges a student to demonstrate understanding of the concepts of algebraic properties and representations. A student must make sense of the
More informationStep 1: Set the equation equal to zero if the function lacks. Step 2: Subtract the constant term from both sides:
In most situations the quadratic equations such as: x 2 + 8x + 5, can be solved (factored) through the quadratic formula if factoring it out seems too hard. However, some of these problems may be solved
More informationHigh School Algebra Reasoning with Equations and Inequalities Solve equations and inequalities in one variable.
Performance Assessment Task Quadratic (2009) Grade 9 The task challenges a student to demonstrate an understanding of quadratic functions in various forms. A student must make sense of the meaning of relations
More informationThe Mathematics School Teachers Should Know
The Mathematics School Teachers Should Know Lisbon, Portugal January 29, 2010 H. Wu *I am grateful to Alexandra AlvesRodrigues for her many contributions that helped shape this document. Do school math
More informationPerformance Assessment Task Which Shape? Grade 3. Common Core State Standards Math  Content Standards
Performance Assessment Task Which Shape? Grade 3 This task challenges a student to use knowledge of geometrical attributes (such as angle size, number of angles, number of sides, and parallel sides) to
More informationRoots of quadratic equations
CHAPTER Roots of quadratic equations Learning objectives After studying this chapter, you should: know the relationships between the sum and product of the roots of a quadratic equation and the coefficients
More informationHigh School Functions Interpreting Functions Understand the concept of a function and use function notation.
Performance Assessment Task Printing Tickets Grade 9 The task challenges a student to demonstrate understanding of the concepts representing and analyzing mathematical situations and structures using algebra.
More informationSolving Quadratic Equations by Factoring
4.7 Solving Quadratic Equations by Factoring 4.7 OBJECTIVE 1. Solve quadratic equations by factoring The factoring techniques you have learned provide us with tools for solving equations that can be written
More informationFACTORING QUADRATICS 8.1.1 and 8.1.2
FACTORING QUADRATICS 8.1.1 and 8.1.2 Chapter 8 introduces students to quadratic equations. These equations can be written in the form of y = ax 2 + bx + c and, when graphed, produce a curve called a parabola.
More informationEquations, Inequalities & Partial Fractions
Contents Equations, Inequalities & Partial Fractions.1 Solving Linear Equations 2.2 Solving Quadratic Equations 1. Solving Polynomial Equations 1.4 Solving Simultaneous Linear Equations 42.5 Solving Inequalities
More informationMathematics Cognitive Domains Framework: TIMSS 2003 Developmental Project Fourth and Eighth Grades
Appendix A Mathematics Cognitive Domains Framework: TIMSS 2003 Developmental Project Fourth and Eighth Grades To respond correctly to TIMSS test items, students need to be familiar with the mathematics
More informationAlgebra 1 Course Information
Course Information Course Description: Students will study patterns, relations, and functions, and focus on the use of mathematical models to understand and analyze quantitative relationships. Through
More informationHow Old Are They? This problem gives you the chance to: form expressions form and solve an equation to solve an age problem. Will is w years old.
How Old Are They? This problem gives you the chance to: form expressions form and solve an equation to solve an age problem Will is w years old. Ben is 3 years older. 1. Write an expression, in terms of
More information5.4 The Quadratic Formula
Section 5.4 The Quadratic Formula 481 5.4 The Quadratic Formula Consider the general quadratic function f(x) = ax + bx + c. In the previous section, we learned that we can find the zeros of this function
More informationis identically equal to x 2 +3x +2
Partial fractions 3.6 Introduction It is often helpful to break down a complicated algebraic fraction into a sum of simpler fractions. 4x+7 For example it can be shown that has the same value as 1 + 3
More informationStudent Performance Q&A:
Student Performance Q&A: AP Calculus AB and Calculus BC FreeResponse Questions The following comments on the freeresponse questions for AP Calculus AB and Calculus BC were written by the Chief Reader,
More informationSOLVING QUADRATIC EQUATIONS  COMPARE THE FACTORING ac METHOD AND THE NEW DIAGONAL SUM METHOD By Nghi H. Nguyen
SOLVING QUADRATIC EQUATIONS  COMPARE THE FACTORING ac METHOD AND THE NEW DIAGONAL SUM METHOD By Nghi H. Nguyen A. GENERALITIES. When a given quadratic equation can be factored, there are 2 best methods
More information3.6. Partial Fractions. Introduction. Prerequisites. Learning Outcomes
Partial Fractions 3.6 Introduction It is often helpful to break down a complicated algebraic fraction into a sum of simpler fractions. For 4x + 7 example it can be shown that x 2 + 3x + 2 has the same
More informationHigh School Algebra Reasoning with Equations and Inequalities Solve systems of equations.
Performance Assessment Task Graphs (2006) Grade 9 This task challenges a student to use knowledge of graphs and their significant features to identify the linear equations for various lines. A student
More informationFactorising quadratics
Factorising quadratics An essential skill in many applications is the ability to factorise quadratic expressions. In this unit you will see that this can be thought of as reversing the process used to
More informationFACTORISATION YEARS. A guide for teachers  Years 9 10 June 2011. The Improving Mathematics Education in Schools (TIMES) Project
9 10 YEARS The Improving Mathematics Education in Schools (TIMES) Project FACTORISATION NUMBER AND ALGEBRA Module 33 A guide for teachers  Years 9 10 June 2011 Factorisation (Number and Algebra : Module
More informationAcademic Success Centre
250) 9606367 Factoring Polynomials Sometimes when we try to solve or simplify an equation or expression involving polynomials the way that it looks can hinder our progress in finding a solution. Factorization
More informationTIMSS 2011 Mathematics Framework. Chapter 1
TIMSS 2011 Mathematics Framework Chapter 1 Chapter 1 TIMSS 2011 Mathematics Framework Overview Students should be educated to recognize mathematics as an immense achievement of humanity, and to appreciate
More informationDRAFT. New York State Testing Program Grade 8 Common Core Mathematics Test. Released Questions with Annotations
DRAFT New York State Testing Program Grade 8 Common Core Mathematics Test Released Questions with Annotations August 2014 Developed and published under contract with the New York State Education Department
More informationUnit 3: Day 2: Factoring Polynomial Expressions
Unit 3: Day : Factoring Polynomial Expressions Minds On: 0 Action: 45 Consolidate:10 Total =75 min Learning Goals: Extend knowledge of factoring to factor cubic and quartic expressions that can be factored
More informationExponents and Radicals
Exponents and Radicals (a + b) 10 Exponents are a very important part of algebra. An exponent is just a convenient way of writing repeated multiplications of the same number. Radicals involve the use of
More informationMATHEMATICS FOR ENGINEERING BASIC ALGEBRA
MATHEMATICS FOR ENGINEERING BASIC ALGEBRA TUTORIAL 3 EQUATIONS This is the one of a series of basic tutorials in mathematics aimed at beginners or anyone wanting to refresh themselves on fundamentals.
More informationYear 9 set 1 Mathematics notes, to accompany the 9H book.
Part 1: Year 9 set 1 Mathematics notes, to accompany the 9H book. equations 1. (p.1), 1.6 (p. 44), 4.6 (p.196) sequences 3. (p.115) Pupils use the Elmwood Press Essential Maths book by David Raymer (9H
More informationG C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
Performance Assessment Task Circle and Squares Grade 10 This task challenges a student to analyze characteristics of 2 dimensional shapes to develop mathematical arguments about geometric relationships.
More information3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes
Solving Polynomial Equations 3.3 Introduction Linear and quadratic equations, dealt within Sections 3.1 and 3.2, are members of a class of equations, called polynomial equations. These have the general
More informationPartial Fractions Examples
Partial Fractions Examples Partial fractions is the name given to a technique of integration that may be used to integrate any ratio of polynomials. A ratio of polynomials is called a rational function.
More informationHIDDEN DIFFERENCES IN TEACHERS APPROACH TO ALGEBRA a comparative case study of two lessons.
HIDDEN DIFFERENCES IN TEACHERS APPROACH TO ALGEBRA a comparative case study of two lessons. Cecilia Kilhamn University of Gothenburg, Sweden Algebra is a multidimensional content of school mathematics
More informationINTRODUCTION FRAME WORK AND PURPOSE OF THE STUDY
STUDENTS PERCEPTIONS ABOUT THE SYMBOLS, LETTERS AND SIGNS IN ALGEBRA AND HOW DO THESE AFFECT THEIR LEARNING OF ALGEBRA: A CASE STUDY IN A GOVERNMENT GIRLS SECONDARY SCHOOL KARACHI Abstract Algebra uses
More informationSuccessful completion of Math 7 or Algebra Readiness along with teacher recommendation.
MODESTO CITY SCHOOLS COURSE OUTLINE COURSE TITLE:... Basic Algebra COURSE NUMBER:... RECOMMENDED GRADE LEVEL:... 811 ABILITY LEVEL:... Basic DURATION:... 1 year CREDIT:... 5.0 per semester MEETS GRADUATION
More informationNSM100 Introduction to Algebra Chapter 5 Notes Factoring
Section 5.1 Greatest Common Factor (GCF) and Factoring by Grouping Greatest Common Factor for a polynomial is the largest monomial that divides (is a factor of) each term of the polynomial. GCF is the
More informationFactoring Quadratic Expressions
Factoring the trinomial ax 2 + bx + c when a = 1 A trinomial in the form x 2 + bx + c can be factored to equal (x + m)(x + n) when the product of m x n equals c and the sum of m + n equals b. (Note: the
More informationA Systematic Approach to Factoring
A Systematic Approach to Factoring Step 1 Count the number of terms. (Remember****Knowing the number of terms will allow you to eliminate unnecessary tools.) Step 2 Is there a greatest common factor? Tool
More informationGCSE MATHEMATICS. 43602H Unit 2: Number and Algebra (Higher) Report on the Examination. Specification 4360 November 2014. Version: 1.
GCSE MATHEMATICS 43602H Unit 2: Number and Algebra (Higher) Report on the Examination Specification 4360 November 2014 Version: 1.0 Further copies of this Report are available from aqa.org.uk Copyright
More informationAlgebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 201213 school year.
This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra
More informationSTUDENTS DIFFICULTIES WITH APPLYING A FAMILIAR FORMULA IN AN UNFAMILIAR CONTEXT
STUDENTS DIFFICULTIES WITH APPLYING A FAMILIAR FORMULA IN AN UNFAMILIAR CONTEXT Maureen Hoch and Tommy Dreyfus Tel Aviv University, Israel This paper discusses problems many tenth grade students have when
More informationUnit 7 Quadratic Relations of the Form y = ax 2 + bx + c
Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c Lesson Outline BIG PICTURE Students will: manipulate algebraic expressions, as needed to understand quadratic relations; identify characteristics
More informationEquations and Inequalities
Rational Equations Overview of Objectives, students should be able to: 1. Solve rational equations with variables in the denominators.. Recognize identities, conditional equations, and inconsistent equations.
More informationALGEBRA I (Common Core)
The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA I (Common Core) Wednesday, August 12, 2015 8:30 to 11:30 a.m. MODEL RESPONSE SET Table of Contents Question 25...................
More informationFactoring Polynomials
Factoring Polynomials Hoste, Miller, Murieka September 12, 2011 1 Factoring In the previous section, we discussed how to determine the product of two or more terms. Consider, for instance, the equations
More informationPerformance Assessment Task Gym Grade 6. Common Core State Standards Math  Content Standards
Performance Assessment Task Gym Grade 6 This task challenges a student to use rules to calculate and compare the costs of memberships. Students must be able to work with the idea of breakeven point to
More informationA Concrete Introduction. to the Abstract Concepts. of Integers and Algebra using Algebra Tiles
A Concrete Introduction to the Abstract Concepts of Integers and Algebra using Algebra Tiles Table of Contents Introduction... 1 page Integers 1: Introduction to Integers... 3 2: Working with Algebra Tiles...
More informationCORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREERREADY FOUNDATIONS IN ALGEBRA
We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREERREADY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical
More informationBalanced Assessment Test Algebra 2008
Balanced Assessment Test Algebra 2008 Core Idea Task Score Representations Expressions This task asks students find algebraic expressions for area and perimeter of parallelograms and trapezoids. Successful
More informationa) x 2 8x = 25 x 2 8x + 16 = (x 4) 2 = 41 x = 4 ± 41 x + 1 = ± 6 e) x 2 = 5 c) 2x 2 + 2x 7 = 0 2x 2 + 2x = 7 x 2 + x = 7 2
Solving Quadratic Equations By Square Root Method Solving Quadratic Equations By Completing The Square Consider the equation x = a, which we now solve: x = a x a = 0 (x a)(x + a) = 0 x a = 0 x + a = 0
More informationSecond Grade Mars 2009 Overview of Exam. Task Descriptions. Algebra, Patterns, and
Second Grade Mars 2009 Overview of Exam Task Descriptions Core Idea Task Algebra, Patterns, and Same Number, Same Shape Functions The task asks students to think about variables and unknowns in number
More information3.2. Solving quadratic equations. Introduction. Prerequisites. Learning Outcomes. Learning Style
Solving quadratic equations 3.2 Introduction A quadratic equation is one which can be written in the form ax 2 + bx + c = 0 where a, b and c are numbers and x is the unknown whose value(s) we wish to find.
More informationDRAFT. New York State Testing Program Grade 8 Common Core Mathematics Test. Released Questions with Annotations
DRAFT New York State Testing Program Grade 8 Common Core Mathematics Test Released Questions with Annotations August 2013 THE STATE EDUCATION DEPARTMENT / THE UNIVERSITY OF THE STATE OF NEW YORK / ALBANY,
More informationTennessee Department of Education
Tennessee Department of Education Task: Pool Patio Problem Algebra I A hotel is remodeling their grounds and plans to improve the area around a 20 foot by 40 foot rectangular pool. The owner wants to use
More informationPolynomials and Factoring. Unit Lesson Plan
Polynomials and Factoring Unit Lesson Plan By: David Harris University of North Carolina Chapel Hill Math 410 Dr. Thomas, M D. 2 Abstract This paper will discuss, and give, lesson plans for all the topics
More informationMath Review. for the Quantitative Reasoning Measure of the GRE revised General Test
Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important
More informationOperations and Algebraic Thinking Represent and solve problems involving addition and subtraction. Add and subtract within 20. MP.
Performance Assessment Task Incredible Equations Grade 2 The task challenges a student to demonstrate understanding of concepts involved in addition and subtraction. A student must be able to understand
More informationAlgebra I. In this technological age, mathematics is more important than ever. When students
In this technological age, mathematics is more important than ever. When students leave school, they are more and more likely to use mathematics in their work and everyday lives operating computer equipment,
More informationContents. How You May Use This Resource Guide
Contents How You May Use This Resource Guide ii 7 Factoring and Algebraic Fractions 1 Worksheet 7.1: The Box Problem......................... 4 Worksheet 7.2: Multiplying and Factoring Polynomials with
More informationMathematics Objective 6.) To recognize the limitations of mathematical and statistical models.
Spring 20 Question Category: 1 Exemplary Educational Objectives Mathematics THECB Mathematics Objective 1.) To apply arithmetic, algebraic, geometric, higherorder thinking, and statistical methods to
More informationParallelogram. This problem gives you the chance to: use measurement to find the area and perimeter of shapes
Parallelogram This problem gives you the chance to: use measurement to find the area and perimeter of shapes 1. This parallelogram is drawn accurately. Make any measurements you need, in centimeters, and
More informationOperations and Algebraic Thinking Represent and solve problems involving multiplication and division.
Performance Assessment Task The Answer is 36 Grade 3 The task challenges a student to use knowledge of operations and their inverses to complete number sentences that equal a given quantity. A student
More informationTotal Student Count: 3170. Grade 8 2005 pg. 2
Grade 8 2005 pg. 1 Total Student Count: 3170 Grade 8 2005 pg. 2 8 th grade Task 1 Pen Pal Student Task Core Idea 3 Algebra and Functions Core Idea 2 Mathematical Reasoning Convert cake baking temperatures
More informationWentzville School District Algebra 1: Unit 8 Stage 1 Desired Results
Wentzville School District Algebra 1: Unit 8 Stage 1 Desired Results Unit Title: Quadratic Expressions & Equations Course: Algebra I Unit 8  Quadratic Expressions & Equations Brief Summary of Unit: At
More informationAlgebra 2: Q1 & Q2 Review
Name: Class: Date: ID: A Algebra 2: Q1 & Q2 Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Which is the graph of y = 2(x 2) 2 4? a. c. b. d. Short
More informationCAN X=3 BE THE SOLUTION OF AN INEQUALITY? A STUDY OF ITALIAN AND ISRAELI STUDENTS
CAN X=3 BE THE SOLUTION OF AN INEQUALITY? A STUDY OF ITALIAN AND ISRAELI STUDENTS Pessia Tsamir Tel Aviv University Luciana Bazzini University of Torino This paper describes some findings of a study regarding
More informationALGEBRA I (Common Core)
The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA I (Common Core) Wednesday, June 17, 2015 1:15 to 4:15 p.m. MODEL RESPONSE SET Table of Contents Question 25..................
More information1. I have 4 sides. My opposite sides are equal. I have 4 right angles. Which shape am I?
Which Shape? This problem gives you the chance to: identify and describe shapes use clues to solve riddles Use shapes A, B, or C to solve the riddles. A B C 1. I have 4 sides. My opposite sides are equal.
More informationSouth Carolina College and CareerReady (SCCCR) Algebra 1
South Carolina College and CareerReady (SCCCR) Algebra 1 South Carolina College and CareerReady Mathematical Process Standards The South Carolina College and CareerReady (SCCCR) Mathematical Process
More informationTeaching Approaches Using Graphing Calculator in the Classroom for the HearingImpaired Student
Teaching Approaches Using Graphing Calculator in the Classroom for the HearingImpaired Student Akira MORIMOTO Tsukuba College of Technology for the Deaf, JAPAN Yoshinori NAKAMURA Miyagi School for the
More information5. Factoring by the QF method
5. Factoring by the QF method 5.0 Preliminaries 5.1 The QF view of factorability 5.2 Illustration of the QF view of factorability 5.3 The QF approach to factorization 5.4 Alternative factorization by the
More information6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives
6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise
More informationFactoring Polynomials
Factoring Polynomials 412014 The opposite of multiplying polynomials is factoring. Why would you want to factor a polynomial? Let p(x) be a polynomial. p(c) = 0 is equivalent to x c dividing p(x). Recall
More informationGRADE 8 MATH: TALK AND TEXT PLANS
GRADE 8 MATH: TALK AND TEXT PLANS UNIT OVERVIEW This packet contains a curriculumembedded Common Core standards aligned task and instructional supports. The task is embedded in a three week unit on systems
More informationCOLLEGE ALGEBRA. Paul Dawkins
COLLEGE ALGEBRA Paul Dawkins Table of Contents Preface... iii Outline... iv Preliminaries... Introduction... Integer Exponents... Rational Exponents... 9 Real Exponents...5 Radicals...6 Polynomials...5
More informationSOLVING QUADRATIC EQUATIONS BY THE DIAGONAL SUM METHOD
SOLVING QUADRATIC EQUATIONS BY THE DIAGONAL SUM METHOD A quadratic equation in one variable has as standard form: ax^2 + bx + c = 0. Solving it means finding the values of x that make the equation true.
More informationDYNAMICS AS A PROCESS, HELPING UNDERGRADUATES UNDERSTAND DESIGN AND ANALYSIS OF DYNAMIC SYSTEMS
Session 2666 DYNAMICS AS A PROCESS, HELPING UNDERGRADUATES UNDERSTAND DESIGN AND ANALYSIS OF DYNAMIC SYSTEMS Louis J. Everett, Mechanical Engineering Texas A&M University College Station, Texas 77843 LEverett@Tamu.Edu
More informationIOWA EndofCourse Assessment Programs. Released Items ALGEBRA I. Copyright 2010 by The University of Iowa.
IOWA EndofCourse Assessment Programs Released Items Copyright 2010 by The University of Iowa. ALGEBRA I 1 Sally works as a car salesperson and earns a monthly salary of $2,000. She also earns $500 for
More informationALGEBRA 2/ TRIGONOMETRY
The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA 2/ TRIGONOMETRY Friday, June 14, 2013 1:15 4:15 p.m. SAMPLE RESPONSE SET Table of Contents Practice Papers Question 28.......................
More informationLecture 5 : Solving Equations, Completing the Square, Quadratic Formula
Lecture 5 : Solving Equations, Completing the Square, Quadratic Formula An equation is a mathematical statement that two mathematical expressions are equal For example the statement 1 + 2 = 3 is read as
More informationUnit 12: Introduction to Factoring. Learning Objectives 12.2
Unit 1 Table of Contents Unit 1: Introduction to Factoring Learning Objectives 1. Instructor Notes The Mathematics of Factoring Teaching Tips: Challenges and Approaches Additional Resources Instructor
More informationLAKE ELSINORE UNIFIED SCHOOL DISTRICT
LAKE ELSINORE UNIFIED SCHOOL DISTRICT Title: PLATO Algebra 1Semester 2 Grade Level: 1012 Department: Mathematics Credit: 5 Prerequisite: Letter grade of F and/or N/C in Algebra 1, Semester 2 Course Description:
More informationActually, if you have a graphing calculator this technique can be used to find solutions to any equation, not just quadratics. All you need to do is
QUADRATIC EQUATIONS Definition ax 2 + bx + c = 0 a, b, c are constants (generally integers) Roots Synonyms: Solutions or Zeros Can have 0, 1, or 2 real roots Consider the graph of quadratic equations.
More informationHIBBING COMMUNITY COLLEGE COURSE OUTLINE
HIBBING COMMUNITY COLLEGE COURSE OUTLINE COURSE NUMBER & TITLE:  Beginning Algebra CREDITS: 4 (Lec 4 / Lab 0) PREREQUISITES: MATH 0920: Fundamental Mathematics with a grade of C or better, Placement Exam,
More informationTom wants to find two real numbers, a and b, that have a sum of 10 and have a product of 10. He makes this table.
Sum and Product This problem gives you the chance to: use arithmetic and algebra to represent and analyze a mathematical situation solve a quadratic equation by trial and improvement Tom wants to find
More informationALGEBRA I (Common Core)
The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA I (Common Core) Wednesday, August 13, 2014 8:30 a.m. MODEL RESPONSE SET Table of Contents Question 25...................
More information3.1.1 Improve ACT/SAT scores of high school students; 3.1.2 Increase the percentage of high school students going to college;
SECTION 1. GENERAL TITLE 135 PROCEDURAL RULE West Virginia Council for Community and Technical College SERIES 24 PREPARATION OF STUDENTS FOR COLLEGE 1.1 Scope  This rule sets forth minimum levels of knowledge,
More informationA COMPARISON OF LOW PERFORMING STUDENTS ACHIEVEMENTS IN FACTORING CUBIC POLYNOMIALS USING THREE DIFFERENT STRATEGIES
International Conference on Educational Technologies 2013 A COMPARISON OF LOW PERFORMING STUDENTS ACHIEVEMENTS IN FACTORING CUBIC POLYNOMIALS USING THREE DIFFERENT STRATEGIES 1 Ugorji I. Ogbonnaya, 2 David
More informationIndiana State Core Curriculum Standards updated 2009 Algebra I
Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and
More information3.1. Solving linear equations. Introduction. Prerequisites. Learning Outcomes. Learning Style
Solving linear equations 3.1 Introduction Many problems in engineering reduce to the solution of an equation or a set of equations. An equation is a type of mathematical expression which contains one or
More informationHow Students Interpret Literal Symbols in Algebra: A Conceptual Change Approach
How Students Interpret Literal Symbols in Algebra: A Conceptual Change Approach Konstantinos P. Christou (kochrist@phs.uoa.gr) Graduate Program in Basic and Applied Cognitive Science. Department of Philosophy
More informationMATHEMATICAL CREATIVITY THROUGH TEACHERS PERCEPTIONS
MATHEMATICAL CREATIVITY THROUGH TEACHERS PERCEPTIONS Maria Kattou, Katerina Kontoyianni, & Constantinos Christou University of Cyprus This study examines elementary school teachers conceptions of creativity,
More informationSpringfield Technical Community College School of Mathematics, Sciences & Engineering Transfer
Springfield Technical Community College School of Mathematics, Sciences & Engineering Transfer Department: Mathematics Course Title: Algebra 2 Course Number: MAT097 Semester: Fall 2015 Credits: 3 NonGraduation
More informationFactoring Trinomials: The ac Method
6.7 Factoring Trinomials: The ac Method 6.7 OBJECTIVES 1. Use the ac test to determine whether a trinomial is factorable over the integers 2. Use the results of the ac test to factor a trinomial 3. For
More informationNorwalk La Mirada Unified School District. Algebra Scope and Sequence of Instruction
1 Algebra Scope and Sequence of Instruction Instructional Suggestions: Instructional strategies at this level should include connections back to prior learning activities from K7. Students must demonstrate
More informationhttp://www.aleks.com Access Code: RVAE4EGKVN Financial Aid Code: 6A9DBDEE3B74F5157304
MATH 1340.04 College Algebra Location: MAGC 2.202 Meeting day(s): TR 7:45a 9:00a, Instructor Information Name: Virgil Pierce Email: piercevu@utpa.edu Phone: 665.3535 Teaching Assistant Name: Indalecio
More informationSOLVING POLYNOMIAL EQUATIONS
C SOLVING POLYNOMIAL EQUATIONS We will assume in this appendix that you know how to divide polynomials using long division and synthetic division. If you need to review those techniques, refer to an algebra
More informationMultidigit Multiplication: Teaching methods and student mistakes
Cynthia Ashton Math 165.01 Professor Beck Term Paper Due: Friday 14 December 2012 Multidigit Multiplication: Teaching methods and student mistakes Abstract: Description of the paper. Why is it important/interesting,
More informationis identically equal to x 2 +3x +2
Partial fractions.6 Introduction It is often helpful to break down a complicated algebraic fraction into a sum of simpler fractions. 4x+7 For example it can be shown that has the same value as + for any
More informationLecture 7 : Inequalities 2.5
3 Lecture 7 : Inequalities.5 Sometimes a problem may require us to find all numbers which satisfy an inequality. An inequality is written like an equation, except the equals sign is replaced by one of
More informationNew York State Testing Program Grade 3 Common Core Mathematics Test. Released Questions with Annotations
New York State Testing Program Grade 3 Common Core Mathematics Test Released Questions with Annotations August 2013 THE STATE EDUCATION DEPARTMENT / THE UNIVERSITY OF THE STATE OF NEW YORK / ALBANY, NY
More information