## Monday, August 1, 2011

### Level Test (review) part 2

Objectives:
to clarify, consolidate and analyse common conceptual and careless errors committed by students during level test.

Individually, please identify one possible error in each of the error analysis solutions shown below. eg. Conceptual error due to misquoting of (a+b)^2 law. It should be a^2+2ab+b^2 NOT a^+b^2. You may correct part of OR all the errors found in each solution.
In total, you should complete 5 work in all.

A

B

C

D

E

1. A: He did not change the (–) of the (-n) to (+)
He also didn’t simplify [5m+n-4m-n] even though it is the wrong answer.

B: He did not change the (–) of the (-n) to (+)
He simplified the equation, however, he did an extra step of cancelling out (m) and the (m) in [m+n] which was not needed as there is a (+) sign in the denominator, making the cancelling wrong.

C: In 7(x-3), he did not multiply the 7 and the 3 as well.
In –(20-7x), he did not change the (–) sign of the (-7x) to (+), thus making his answer wrong.

D: He should solve (2a+1)^2 first instead of multiplying it by 3.

E: His +(-3y) became -9x^2...So instead of getting the answer as 9xy-3y, he got the answer as 9xy-9x^2

2. (a) 1. 5(m+n) was expanded into 5m+n which is incorrect. It should be 5m+5n
2. -1 x -n should be equal to +n, not -n
3. The fraction can be simplified further.
(b) 1. 5(m+n) was expanded into 5m+n which is incorrect. It should be 5m+5n
2. -1 x -n should be equal to +n, not -n
3. The numerator m and the denominator m cannot be canceled because of the + sign in the denominator.
(c) 1. 7(x-3) should be 7x-21, not 7x-3.
2. -1 x -7x should be + 7x, not - 7x
(d) 1. (2a+1)^2 was multiplied by 3 before it was squared. It should have been squared first, then multiplied by 3
2. -(6a+3)^2 should be - 6a^2-36a-9, not -36a^2 +9
3. -1 x +9 should be - 9, not +9
(e) 1. y(6x-9y) should be 6xy-9y^2, not 6xy-9xy.
2. (-3y)^2 should be +9y^2, not -9y^2

3. For:
(a) It should be 5m + 5n - 4m + n instead of 5m + n - 4m - n.

(b) It isn't correct to cancel out the two 'm's. We should just leave the answer as m/m+n.

(c)It should be 7x + 21 - 20 + 7x instead of 7x + 3 - 20 + 7x. When we open up the brackets, 7(x + 3) should become 7x + 21 instead of 7x + 3 as we are multiplying everything inside of the bracket with 7. And when we open the brackets for -(20 - 7x), it should become - 20 + 7x instead of - 20 - 7x as '-' x '-' is equals to '+'.

(d)They should square the numbers in the brackets first before multiplying it by 3.

(e)Calculation error for y(6x - 9y), it should be 6xy - 9y^2 . And for +(-3y), the person calculated wrongly i guess, it shouldn't be 9x^2...it should be -3y.

4. a) There should be a bracket over 'm+n' and it should be'-4m+n', not '-4m-n'. Hence the working of this problem should be '[5(m+n)-4m+n]/m+n'.

b) Like in 'a', there should be a bracket over 'm+n' and it should be'-4m+n', not '-4m-n'. Also, it is not possible to simplify the answer to 1/n as there is a presence of a plus sign which hence does not allow the student to cancel out the 'm'.

c) Part of the working should be '7x-21-20+7x' and not '7x-3-20-7x'. In this case, the student forgot to multiply 7 and 3 (to get the 21). Also, when negative is multiplied by negative, we would get positive. Therefore it should be '+7x' and not '-7x'.

d) (2a+1)^2 should be done first in stead of multiplying it by 3.

e) The student mixed up the symbols. Instead of '-9xy', it should be '-9y^2'. For (-3y)^2, it should be '+9y^2' and not '-9x^2'.

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6. a) It should be 5m+5n-4m+n Instead of 5m+n-4m+n. For the 5 whole, he should do in a separate working if not he would get confused. 5 whole is 5 X m+n and he should get 5m+5n. It should be +n instead of -n because negative X negative is positive. He should have put a bracket so that he would remember to multiply accordingly.
b) After solving an algebra equation, he should not cancel it with the denominator as it would lead to another answer.
c) The answer should be 7x-21-20+7x= 14x-41. He did not do this multiplication correctly.
d) He should square the brackets first before multiplying with the number outside the bracket as the square only belongs to the number inside the bracket. If you multiply first, it means you are squaring the number outside the bracket too.
e) His first step was wrong. It should be 12xy+6xy-9y^2-3y instead of 12xy+6xy-9xy-9x^2. He did his multiplication wrongly!! ><

7. a)for the numerator of the fraction it should be 5(m+n)-4m+n
b)the numerator for the first fraction should be 5(m+n)-4m+n and the m in m/n+m cannot be canceled out
c)7(x-3)-(20-7x) is 7x-21-20+7x not 7x-3-7x
d)the second line is supposed to be 3+12a^2-3(4a^2+1) as the brackets should be squared first.
e)the second line should be 12xy+6xy-9y^2-3y as 9yXy does not equal 9xy and -3y does not give 9x^2

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9. A)
5=(5m+5n)/m+n, not (5m+n)/m+n

(5m+5n-4m+n)/m+n, not (5m+5n-4m-n)/m+n

B)
You cannot cancel the denominator with a numerator,
so his solution should be m/m+n

C)
7(x-3)= 7x-21
-(20-7x)=-20+7x

Therefore his solution should be 14x-41

D)
He should have squared (2a+1) first,not multiply it. He also squared it incorrectly.
3+12a^2-3(2a+1)^2=3+12a^2-3(4a^2+4a+1)
=3+12a^2-12a^2-12a-3
=-12a
Therefore his solution should be -12a

E)
+(-3y) is not -9x^2, it is -3y.He also multiplied incorrectly, which gave him 9xy, where it should have been 9y^2

12xy+y(6x-9y)+(-3y)=12xy+6xy-9y^2-3y
=18xy-9y^2-3y
Therefore his answer should have been 18xy-9y^2-3y

10. A)He/her forgot to multiply the fraction by 5. In order to complete the sum, he/her should have made them common by multiplying 5 by m+n

B)He/her canceled the m and n. He/her should have left it in the fraction.

C) He/her didn't multiply the 3 by 7. He/her should have multiplied everything in the bracket. Thus his/her answer should have been -41

D)He/she didn't expand the brackets properly, leading to a miscalculation. So the answer should have been -12a.

E)He/her multiplied the bracket wrongly. Instead of 9xy, it should have been y^2 since y x y = y^2. Also the bracket (-3y)^2 had a wrong sign. Since it was negative and was squared, it should have been positve 9y^2 not 9x^2. Thus the answer would be 12xy+6xy=18xy

11. A)In the second step the student forgot to multiply n by 5, he only multiply m by 5.Also since the n is negative and 5 will be multiplied with a negative sign the n in the second step would be positive. Also he forgot to continue the equation to the final step.

B)In the second step the student forgot to multiply n by 5.Also since the n is negative and 5 will be multiplied with a negative sign the n in the second step would be positive.The third step should be m+6n/m+n

C)The student only multiplied 7 with x since the its 7(x-3),7 must be multiplied with both x and 3.Thus the equation should be (7x-21)-(20-7x).Also the student forgot that (-)x(-)=(+).So the equation would be 7x-20-21+7x=14x-41

D)The student should square the equation before multiplying.So the equation would be 3+12A^2-3(4A^2+4A+1)=3+12A^2-(12A^2-12A-3)=-12A

E)The student multiplied y(6x-9Y) wrongly as it should be 6xy-9Y^2.Also since its(-3Y)^2is would be +9Y^2,not 9X^2Thus the whole equation would be 12XY+(6XY-9Y^2)+9Y^2=12XY+6XY-9Y^2+9^2=12XY+6XY=18XY

12. Valery~ (conceptual errors)8/03/2011 08:30:00 AM

A: Conceptual error of the negative and positive signs.
He did not change the (–) of the (-n) to (+)
Simplification not done.
He also didn’t simplify [5m+n-4m-n] even though it is the wrong answer.

B: Conceptual error of the negative and positive signs.
He did not change the (–) of the (-n) to (+)
Conceptual error of cancelling out one part of the denominator when it is not needed to in algebraic context.
He simplified the equation, however, he did an extra step of cancelling out (m) and the (m) in [m+n] which was not needed as there is a (+) sign in the denominator, making the cancelling wrong.

C: In 7(x-3), he did not multiply the 7 and the 3 as well.
Conceptual error of the negative and positive signs.
In –(20-7x), he did not change the (–) sign of the (-7x) to (+), thus making his answer wrong.

D: Conceptual error due to the misquoting of (2a+1)^2.
He should solve (2a+1)^2 first instead of multiplying it by 3.

E: Conceptual error of mixing up the variables.
His +(-3y) became -9x^2...So instead of getting the answer as 9xy-3y, he got the answer as 9xy-9x^2

13. A) 5m+n-4m+n/m+n = 5m-4m+2n/m+2n = m+2n/m+2n
Mistake: (-)x(-) does not equal to (-)

B) 5m+n-4m+n/m+n = 5m-4m+2n/m+2n = m+2n/m+2n
Mistake: (-)x(-) does not equal to (-), you cannot simplify the equation any further

C) 7(x-3) - (20-7x) = 7x-21-20+7x = 14x-41
Mistake: Times both numbers that are in the bracket, (-)x(-) does not equal to (-)

(D) 12xy+y(6x-9y)+(-3y)^2 = 12xy+6xy-9y^2+9y^2 = 18xy-9y^2+9y^2 = 18xy
Mistake: Question has been copied wrongly, y has changed to x, 9+6 does not equal to 9, -9xy-9x^2 does not equal to -9x^2

14. a) Instead of 5m+n-4m-n/m+n, it should be 5(m+n)-4m+n/m+n
b)Instead of 5m+n-4m-n/m+n = m/m+n, it should be 5(m+n)-4m+n/m+n = m+6n/m+n.
c)It should be 7x + 21 - 20 + 7x instead of 7x + 3 - 20 + 7x. When we open up the brackets, 7(x + 3) should become 7x + 21 instead of 7x + 3 as we are multiplying everything inside of the bracket with 7. And when we open the brackets for -(20 - 7x), it should become - 20 + 7x instead of - 20 - 7x as - x - = +
d) (2a+1)^2 should be solved first before multiplying by 3.
e) The student made a careless mistake by changing -9x^2, it should be 9y^2.

15. A: In this answer, the signs on the fraction 4m-n/m+n are not corrected properly to -4m+n/m+n when making the bigger fraction of 5m+5n-4m+n/m+n. The correct equation should be 5- (4m-n/m+n) = 5m+n-4m+n/m+n. Also, in this answer, the person missed out a bracket for the m+n after the number 5, so he missed out expanding the 5 (m+n).

B: This answer includes the same mistake made in the previous question, which leads to another wrong answer. The fraction should be 5m+5n-4m+n/m+n instead of 5m+n-4m-n/m+n, and calculating the correct fraction would give m+6n/m+n.

C: The person who calculated this did not expand the numbers properly. When done correctly, the working should give 7x-21-20+7x, and calculating that should give 14x-41.

D: In this working the person did not do the square first before expanding, thus causing him to do the rest of the question wrongly and get the wrong answer.

E: The person who did this question added a -9x^2 instead of -3y and he also did not expand y (6x-9y) properly, and so he got the wrong answer.

16. a) 5 - (4m-n)/m+n
= 5(m+n)-(4m-n)/ m+n
= 5m+5n-(4m-n)/m+n
= 5m+5n-4m+n/m+n
=6n+m/m+n
The student has been careless and missed out the 5 for the n and he has to introduce brackets for the 4m-n so that it would be correct.

b) In this qn also the student missed out the 5 for the n and he could not cancel out the m and m because the sign beside is not multiplication and it's a addition.
5 - (4m-n)/m+n
= 5(m+n)-(4m-n)/ m+n
= 5m+5n-(4m-n)/m+n
= 5m+5n-4m+n/m+n
=6n+m/m+n

c)The student has been careless an did not multiply the 3 and while removing the brackets he did not change the sign of -7x to + 7x ( because (-) x (-) = (+). The correct way should be:
7(x-3)-(20-7x)
= 7x-21-(20-7x)
= 7x-21-20+7x
= 14x -41

d) The student has to expand the equation before even he multiplies it with 3. And this is a conceptual error. He did not expand the equation properly. The correct answer should be:
3+12a^2 - 3(2a+1)^2
= 3+12a^2 - 3(4a^2+4a+1)
= 3+12a^2 - (12a^2+12a+3)
= 3+12a^2 - 12a^2-12a-3
= -12a

e) 9y x y is 9y^2 so this is a careless error and (-3y)^2 is 9y^2 and not 9x^2
so this is the correct answer:
12xy+y(6x-9y)+(-3y)^2
= 12xy+ 6xy-9y^2 + 9y^2
= 18xy

17. A) The mistake in the answering of A) was that 5-(4m-n)/(m+n) is supposed to be (5m+5n-4m+n)/m+n and not (5m+n-4m-n)/m+n

B) The mistake in answer B) was that 5-(4m-n)/(m+n) is supposed to be (5m+5n-4m+n)/m+n and not (5m+n-4m-n)/m+n, and when simplified, is supposed to be m+6n/m+n

C) The conceptual error in C) was that the answerer had forgotten to multiply the -3 by 7. Thus the answer should be 7(x-3)-(20-7x) = (7x-2)-(20-7x)=7x+7x+21+20=14x+41

D)The conceptual error in the answer was that the square should be separated first and not multiplied by 3. Therefore the answer shoud be 3+12a^2-3(2a+1)^2=3+12a^2-(6a+3)(2a+1)=3+12a^2-(12a^2+6a+6a+3)=3=!2a^2-(12a^2+12a+3)=-12a

E)The carelesss mistake in the answer was that the person had squared the (-3y) wrongly. Therefore the answer should be 12xy+y(6x-9y)+(-3y)^2=12xy+6xy-9y^2+(-3y)(-3y)=12xy+6xy-9y^2+9y^2+18xy

18. A) There are careless mistakes in the working. 5xn=5n, not n. Next, +n is copied incorrectly as -n.

B) Related to A), the final answer is wrong. m+n cannot be divided by m. The fraction cannot be simplified.

C) There is a careless mistake in this working. In the working, it is stated: 7(x-3) -(20-7x) = 7x-3-20-7x. When 7 x is multiplied by 3, it is 21, not 3. The negative sign behind (20-7x) had been ignored too. When (-) is multiplied by (-), the answer is a (+). Therefore the answer is 20+7x. The final answer is 7x-21-20+7x= 14x-41

D) There is conceptual error in the working. In the working, it is stated:
-3(2a+1)^2 = -(6a+3)^2. This is wrong, as expansion must always be done first, before multiplying. The next answer is also wrong as expansion was done incorrectly. Hence, the correct answer is -3(4a^2+4a+1) = -12a^2+12a+3. Then, we add the rest of the working into the answer.
3+12a^2-12a^2-12a-3 = 6

E) There is a careless mistake in this working. The front part of the working is fine, but -9^2 is not the correct answer for +(-3y). The correct answer should be(-3y)

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20. A) When you add a number with a fraction, the denominator and numerator must be cross multiplied. Eg 2-(3m/n)=(2n-3m)/n
Thus the numerator for A should be 5m+5n-4m+n.

B) The same mistake is made but another mistake is that he should not cancel out the two 'm' and that is an extra step.

C) When you multiply the negative sign into the bracket, you multiply it with both digits and its sign. Thus, the answer should be 7x-3-20+7x=14x-23

D) The numbers in the brackets should be squared first then be multiplied by the number outside the bracket. Thus, the answer is 3+12a^2-3(4a^2+4a+1)=3+12a^2-12a^2-12a-3=-12a

E) For the second step, the sign used is wrong. The person also copied the wrong factor. (-3y)^2=9y^2 not -9x^2

21. a) It supposed to be 5(m+n) over (m+n) instead 5m+n over (m+n) as only 5(m+n) over (m+n) is 5
b) You are not supposed cancel out the ms
c) The person forgot to multiply 3 by 7 and only multiplied it by x
d) You are supposed to square (2a+1) first then you multiply it by 3 so the answer is (12a^2+3)
e) y multiplied by y is not xy but it is y^2 instead

22. A:

5 - 4m - n/m+n
= 5m+5n - 4m+n/m+n

B:

5 - 4m - n/m+n
= 5m+5n - 4m+n/m+n
= m+ 6n/m+n
[Divided and simplified the wrong way, should not have cancelled out the m in the end.]

C:

7(x - 3) - (20 - 7x)
= (7x - 21) - (20 - 7x)
= (14x -21) - 20
= 14x - 41
[7(x-3) is equivalent to (7x - 21), not (7x - 3). There should be a bracket around (20 - 7x).]

D:
3 + 12a^2 - 3(2a + 1)^2
3 + 12a^2 - 3(4a^2 + 4a + 1)
3 + 12a^2 - 12a^2 - 12a - 3
= -12a
[3(2a+1)^2 = 3(4a^2 + 4a + 1) = 12a^2 - 12a - 3]

E:

12xy + y(6x - 9y) + (-3y)^2
= 12xy + 6xy - 9y^2 + 9y^2
= 12xy + 6xy
= 18xy
[y(6x + 9y) = 6xy - 9y and (-3y)^2 = 9y^2]

23. A. Conceptual error. m+n should belong to 5, and thus it should have brackets.
5 = 5(m+n) ÷ (m+n) = 5m+5n ÷ m+n
5 ≠ 5m+n ÷ m+n

B. The top part of this is exactly the same as A:
5 = 5(m+n) ÷ (m+n) = 5m+5n ÷ m+n
5 ≠ 5m+n ÷ m+n

There is a second problem in this one though. If the answer is m÷(m+n), both m's cannot cancel out each other as there is still addition needed to be done. This should be the correct answer, and there isn't a need to cancel out m anymore.

C. This is a problem due to the brackets. 7 belongs to x AND 3.
7(x-3) = 7x - 7(3) = 7x - 21, ≠ 7x - 3

There is another problem. - and - = +

D. The person should square (2a+1) first before multiplying it by 3.

E. When a negative number is squared, it becomes positive
(-3y)^2 = 9y^2