reasoning in quadratic equations with one unknown - Cerme 7

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were correct since 0∙0=0. Namely, they thought that the two x’s stood for differentnumbers.StatementsQuestion 6A student hands in thefollowing work for thefollowing problem.Solve ;x 2 -14x+24=3(x-12)∙(x-2)=3(x-12)∙(x-2)=3∙1x-12=3 x-2=1x=15 x=3Ç.K= {3, 5}Is the student correct?Explain your answer withits reasons?“The answeris Wrong”Because,firstly, 3 mustcarry the leftside of theequation andequalize the0. Then, theotheroperationsmust be done.In this way,the equationx 2 -14x+21=0Students’ types of responses with their reasoningI. II. III. IV.“Theanswer isWrong”.Becausewhen wesubstitute 3and 15 forx, theequation isnotprovided.“The answer isRight”Since the result isequal to 3, weequate 3 rather than0 while factoring it.Therefore, theresult is true.Students againsolve as:“x 2 -14x+24=3”(x-12)∙(x-2)=3(x-12)∙ (x-2)=3∙1x-12=3 x-2=1x=15 x=3(3,1)“The answer isWrong”Since theequations areseparated as(3,1) there is noerror when(x-12)=3however, thereis error when(x-2)=1.It must be(x-2)=3 then,x=5. Therefore,the solution willbe {5, 15}rather than{3, 15}.Table 2: Common examples of students’ types of responses with their reasoning forquestion 6.In statements I and II (see Table 2), students were aware of the error in the solution ofthe given question. However, to explain the reasons for the mistake, they presentedprocedural explanations like the responses in statements I, II, III for question 5 (seeTable 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 0while factoring it”, it can be said that they wrongly tried to transfer the null factor lawto this context. That is, they equated the factors of equation x 2 -14x+24 with theinteger factors of 3. In statement IV, students correctly claimed “the answer of thequestion wrong”; however, their explanations were fully erroneous. Similar tostatement 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 factorsof 3. In both statements III and IV, students did not check whether the roots theyfound were appropriate or not.6
StatementsQuestion 7The solution of thequadratic equation“2 x 2 =3x”is given in the following;According to you, is thissolution correct or not?Explain your answer withits reasons?Solution:I. step 2 x 2 =3xII. step 2∙x∙x=3∙xIII. step 2∙x =3IV. step x= 3/2Ç.K = {3/2}Students’ types of responses with their reasoningI. II. III. IV.“The answer isWrong”Because 3x mustbe carried the leftside of theequation andequalized the 0.Then,2 x 2 =3x2 x 2 - 3x=0x∙(2x-3)=0x=0, x=3/2.“Theanswer isRight”Thesolution isright;however, itmust beadded 0 tothe solutionset.“The answerRight”Because whenwe substitutethe value forx, the equationis satisfied.“The answer isRight”2 x 2 =3x and x 2 isopened.2∙x∙x=3∙xYes the x issimplified.2x=3 so x=3/2.Table 3: Common examples of students’ types of responses with their reasoning forquestion 7.In statement I (see Table 3), students stated that the answer was correct. Theyexplained an appropriate procedure required for solving the equation. Since theymemorized the rule without its reasons, they could only exhibit how the proceduremust be worked. In statement II, on the other hand, students were aware that the rootsof the equation were 0 and 3/2. However, they did not recognize that when wassimultaneously canceled from both sides, the root 0 disappeared. Furthermore, instatement III, the explanation for solution was just based on the substitution method.In statement IV, students incorrectly stated that the answer was right. Like instatement II, students were not aware of the missing root 0 when canceling an in theequationDISCUSSIONThe results indicate that most of the students used the factorization technique to solvequadratic equations. This result supports Bosse and Nandakumar (2005), whoclaimed that a large percentage of the students preferred to apply the factorizationtechniques to find the solutions of quadratic equations. Also, in parallel with theresults of Bossé and Nandakumar (2005) and Kotsopoulos (2007), the result of thisstudy revealed that factoring the quadratic equations was challenging when they werepresented to students in non-standard forms and structures. After looking at theexamples of students’ solutions (see Figures 1, 2, and 3), it can be said that thestudents 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 ifwhat they were doing was mathematically correct. These results give some cluesabout students’ instrumental understanding of solving quadratic equations with oneunknown. However, to make an exact judgment about students’ relational or7
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Page 3 and 4: understanding quadratic equations.
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