## Thursday, 14 February 2013

### ERROR ANALYSIS - NUMBERS

By Mr Johari
courtesy of Ms Loh KY

Identify the Errors in each of the following solutions as shown by some of the students.
State the nature of errors (conceptual or careless)  and why part of the solution is wrong.

Example 1
Diagnostic Test 2 Q 4 (c)
QA
Nature of errors:  Conceptual
Error:
Line 1   (-1/2)^2 = 1/4
Line 4   (-) x (-) = +

Example 2
Diagnostic Test 2 Q 4 (c)
QB
Nature of errors:  Conceptual
Error:
Line 2   (-16)(-1/2) = +8
Line 4   (+8) x (-1/2) = -4 incidentally answer right but method wrong therefore 0 mark

Example 3
Diagnostic Test 2 Q 4 (c)
QC
Nature of errors:  Careless
Error:
Line 2   (-1/2)^2 = 0.25

1. 1) Diagnostic Test 2 Q 4 (c)
QD:
Nature of errors: Careless and conceptual (accordingly)
Error:
Line 1 11-9 equals to 2,
Line 2 -1/2 multiply by -1/2 is 1/4

2) Diagnostic Test 2 Q 4 (c)
QE:
Nature of errors: Conceptual
Error:
Line 1 -1/2 x -1/2 is 0.25, or 1/4

3) Diagnostic Test 2 Q 4 (c)
QF:
Nature of errors: Conceptual
Error:
Line 1 16(-1/2 x -1/2) does not equal to (16 x -1/2) x (16 x -1/2)

4) Diagnostic Test 2 Q 4 (d)
QA:
Nature of errors: Careless
Error:
Line 2 2/12 does not equal to 6. It should be 1/6 instead.

5) Diagnostic Test 2 Q 4 (d)
QB:
Nature of errors: Conceptual
Error:
Line 2 1/2 divide by 1/6 does not equal to -1/3. We should change the sign divide to multiplication, and therefore, changing the 1/6 to 6/1. Then, it would be easier to find the answer.

6) Diagnostic Test 2 Q 4 (d)
QC:
Nature of errors: Careless
Error:
Line 1 Firstly, 1/2 - 1 should be -1/2. Secondly, 2/3 x 1/4 should be 1/6.

7) Diagnostic Test 2 Q 4 (d)
QD:
Nature of errors: Conceptual
Error:
Line 4 The -1/2 should not be flipped when the division sign was being changed to multiplication. Only the second fraction should be flipped.

8) Diagnostic Test 2 Q 4 (d)
QE:
Nature of errors: Conceptual
Error:
Line 1 There is no need for the operation to be changed. We can solve the equation 2/3 x 1/4 just like that, without requiring the need to change the operation to division. If you change it to division, you will need to change it back to multiplication to solve it.

2. wait, nehal i thought only supposed to find one error each? nevermind.
Diagnostic Test 2 Q 4 (c)
QD
nature of error: careless
error:
line 1, 2(11-9) equals 2x2 not 2x3.
line 2 -1/2x-1/2 equals 1/4 not -1.

1. u have to do all, taufiq :)

3. 1) Diagnostic Test 2 Q 4 (c)
QD:
Nature of errors: Careless
Error:
Line 1: (11-9)=2, not 3
Line 2: -1/2 x -1/2 is +1/4

2) Diagnostic Test 2 Q 4 (c)
QE:
Nature of errors: Careless
Error:
Line 1: (11-9)=2, not 18; (-1/2)^2 = +0.25 or +1/4

3) Diagnostic Test 2 Q 4 (c)
QF:
Nature of errors: Conceptual then careless
Error:
Line 1: Always do the sums in the brackets first;
2(11-9)=2(2)
=4;
-16(-1/2)^2=-16(1/4)
=-4
Lines 2&3: (-8)^2 = +64

4) Diagnostic Test 2 Q 4 (d)
QA:
Nature of errors: Careless
Error:
Line 2: 2/12=1/6, not 6

5) Diagnostic Test 2 Q 4 (d)
QB:
Nature of errors: Conceptual
Lines 2&3: -1/2 / 1/6=-1/2 x 6
=6/2
=3

6) Diagnostic Test 2 Q 4 (d)
QC:
Nature of errors: Careless
Error:
Line 1: 1/2-1=-1/2, not -1 1/2
2/3 x 1/4=2/12, not 11/12
Lines 3&4: -18/12 / 11/12=-18/12 x 12/11
=-18/11
=-1 7/11

7) Diagnostic Test 2 Q 4 (d)
QD:
Nature of errors: Careless then Conceptual
Error:
Line 1: 2/3 x 1/4 is not 1/3 x 1/2
Line 4: (-1/2) / 1/6=-1/2 x 6
=-6/2
=-3

8) Diagnostic Test 2 Q 4 (d)
QE:
Nature of errors: Conceptual
Error:
Line 1: 2/3 x 1/4 = 2/12, not 2/3 / 4/1
Line 2: 2/3 / 4/1 = 2/12, not 0.5/3

4. Diagnostic Test 2 Q4(c)
QD
Nature of errors: Conceptual
Error
Line 1:(-(1/2)x(-(1/2))=1/4

Diagnostic Test 2 Q4(c)
QE
Nature of errors: Conceptual
Error
Line 1: (-(1/2)^2)=0.25

Diagnostic Test 2 Q4(c)
QF
Nature of errors: Conceptual
Error
Line 1: 16(-(1/2))^2=16(0.25)=4

__________________________________________________

Diagnostic Test 2 Q4(d)
QA
Nature of errors: Careless
Error
Line 2: 2/12=1/6

Diagnostic Test 2 Q4(d)
QB
Nature of errors: Careless
Error
Line 2:-(1/2)/(1/6)=-(1/2)x6=3

Diagnostic Test 2 Q4(d)
QC
Nature of errors: Careless
Error
Line 1: (1/2)-1=-(1/2)

Diagnostic Test 2 Q4(d)
QD
Nature of errors: Conceptual
Error
Line 3: (-(1/2))/(1/6)=(-(1/2))x(6/1)=3

Diagnostic Test 2 Q4(d)
QE
Nature of errors:
Error
Line 2: (2/3)/(4/1)=(2/3)x(1/4)=1/6

Pls tell me if there are errors I'll correct them :)

5. Sorry for replying as a photo as i could not use my LD yesterday so i wrote it down and uploaded it as a picture
http://postimage.org/image/z4r8u1r3l/ (overview)
http://postimage.org/image/tvw7w68oh/ (Question 1 )
http://postimage.org/image/4xu44sz3z/ (Question 2 )

6. Diagnostic Test 2 Q 4 (c)
QD
Nature of errors: Careless
Error:
Line 1 2(11-9) = 2x2
Line 2 (-1/2)(-1/2) = 1/4

Diagnostic Test 2 Q 4 (c)
QE
Nature of errors: Conceptual
Error:
Line 1 (1/2)^2 = 0.25

Diagnostic Test 2 Q 4 (c)
QF
Nature of errors: Conceptual
Error:
Line 1 16(1/2)^2 = 16 x (1/2)^2 = 16 x 1/4 = 4

Diagnostic Test 2 Q4(d)
QA
Nature of errors: Careless
Error:
Line 2: 2/12=1/6

Diagnostic Test 2 Q4(d)
QB
Nature of errors: Conceptual
Error:
Line 3 -1/2 / 1/6 = -1/2 x 6/1 = -3

Diagnostic Test 2 Q4(d)
QC
Nature of errors: Careless
Error:
Line 1 1/2 - 1 = 1/2
Line 1 2/3 x 1/4 = 2/12

Diagnostic Test 2 Q4(d)
QD
Nature of errors: Conceptual
Error:
(-1/2) / 1/6 = -1/2 x 6/1

Diagnostic Test 2 Q4(d)
QE
Nature of errors: Conceptual
Error:
Line 1 2/3 x 1/4 does not need to be changed

7. 1) Diagnostic Test 2 Q4 (c)
QD
Nature of errors: Careless
Error:
Line 1: (11-9) =2, not 3
Line 2: -1/2 x -1/2 is +1/4 as negative x negative = positive

2) Diagnostic Test 2 Q4 (c)
QE
Nature of errors: Conceptual
Error:
Line 1: -1/2 x -1/2 = +0.25 or +1/4 to negative

3) Diagnostic Test 2 Q4 (c)
QF
Nature of errors: Conceptual
Error:
Line 1: 16 x (-1/2)^2 = 4, not -8^2

4) Diagnostic Test 2 Q4 (d)
QA
Nature of errors: Careless
Error:
Line 2: the denominator 2/12 in line 1 =1/6, not 6

5) Diagnostic Test 2 Q4 (d)
QB
Nature of errors: Conceptual
Error:
Lines 2&3: -1/2/1/6 = -1/2 x 6=3, not 1/3

6) Diagnostic Test 2 Q4 (d)
QC
Nature of errors: Careless
Error:
Line 1: -1/2 - 1= 1/2, not -1 1/2; 2/3 x 1/4 = 2/12 or 1/6

7) Diagnostic Test 2 Q4 (d)
QD
Nature of errors: Conceptual
Error:
Line 4: When doing the inverted multiplication, the second fraction should be the one that is inverted, not the first. So it's (-1/2) x (6/1)

8) Diagnostic Test 2 Q4 (d)
QE
Nature of errors: Conceptual
Error:
Line 1: 2/3 x 1/4 cannot be inverted to division as it does not work that way. So it is 2/12, not 2/3 / 4/1

8. Diagnostic Test 2 Q4 (C)
QD
Nature of errors:Careless
Error:
Line 1: 11-9=2
Line 2: -1/2 x -1/2=1/4

Diagnostic Test 2 Q4 (C)
QE
Error:conceptual
Error:
Line 1: -1/2 x -1/2 = 1/4 not-1/4

Diagnostic Test 2 Q4 (C)
QF
Nature of errors: Conceptual
Error:
Line 1: 16(1/2)^2=4 not -8^2

Diagnostic Test 2 Q4 (D)
QA
Nature of errors: Careless
Line 2: 2/12=1/6

Diagnostic Test 2 Q 4 (D)
QB
Nature of errors: Conceptual
Error:
Line 2&3: 1/2 /1/6 =-3 not -1/3

Diagnostic Test 2 Q4 (D)
QC
Nature of errors: Careless
error:
Line 1: 1/2-1=-1/2 not- 1 and 1/2

Diagnostic Test 2 Q4 (D)
QD
Nature of errors: Conceptual
Error:
Line 4: when doing the multiplication the latter fraction is the one that should be inverted.

Diagnostic Test 2 Q4 (D)
QE
Nature of errors: Conceptual
Error:
Line 1: 2/3 x 1/4 cannot be changed to 2/3 / 4/1

9. Diagnostic Test Q4(c)
QD
Nature of error: Careless(line 1) and Conceptual Error(Line 2)
Error: Line 1 (Careless) 11- 9=2
Line 1 (conceptual) Negative x Negative = Positive. And 1/2 x 1/2 = 1/4

QE
Nature of error: Conceptual Error
Error: Line 1 Negative x Negative = Positive (- 1/2 x -1/2 = 1/4 or 0.25)

QF
Nature of error: Conceptual Error
Error: Line 1 16(-1/2 x -1/2) is not (16 x -1/2) x (16 x -1/2)

Diagnostic Test Q4(d)
QA
Nature of error: Careless Error
Error: Line 2 2/3 x 1/4 = 1/6 not 6

QB
Nature of error: Conceptual Error
Error: Line 3 - 1/2 divided by 1/6 = -3 not - 1/3

QC
Nature of error: Careless Error
Error: Line 1 1/2 - 1 should be -1/2. 2/3 x 1/4 should be 1/6 (He/she did addition)

QD
Nature of error: Conceptual Error
Error: Line 4 -1/2 should not be flipped. Only the second fraction should be flipped when changing division to multiplication.

QE
Nature of error: Conceptual Error
Error: Line 1 2/3 x 1/4 = 2/12, not 2/3 / 4/1
Line 2 2/3 / 4/1 = 2/12, not 0.5/3

10. Diagnostic Test Q4(c)
QD
Nature of error: Careless(line 1) and Conceptual(Line 2)
Error: Line 1 (Careless) 11- 9=2
Line 1 (conceptual) Negative x Negative = Positive. And 1/2 x 1/2 = 1/4

QE
Nature of error: Conceptual
Error: Line 1 Negative x Negative = Positive (- 1/2 x -1/2 = 1/4 or 0.25)

QF
Nature of Error: Conceptual
Error: Line 1 16(-1/2 x -1/2) is not (16 x -1/2) x (16 x -1/2)

Diagnostic Test Q4(d)
QA
Nature of error: Careless
Error: Line 2 2/3 x 1/4 = 1/6 not 6

QB
Nature of error: Conceptual
Error: Line 3 - 1/2 divided by 1/6 = -3 not - 1/3

QC
Nature of error: Careless
Error: Line 1 1/2 - 1 should be -1/2. 2/3 x 1/4 should be 1/6 (He/she did addition)

QD
Nature of error: Conceptual
Error: Line 4 -1/2 should not be flipped. Only the second fraction should be flipped when changing division to multiplication.

QE
Nature of error: Conceptual
Error: Line 1 2/3 x 1/4 = 2/12, not 2/3 / 4/1
Line 2 2/3 / 4/1 = 2/12, not 0.5/3

11. Diagnostic Test 2 Q 4 (c)
QD
nature of error: careless
error:
line 1, 2(11-9) equals 2x2 not 2x3.
line 2 -1/2x-1/2 equals 1/4 not -1.

diagnostic test 2 Q 4 (c)
QE
nature of error: conceptual
line 1 he thought that 2(11-9) equals 2x11 - 2x9.

Diagnostic test 2 Q (c)
QF
nature of error: conceptual
line 1 he thought 2(11-9) equals multiplying both numbers in the brackets by 2 then minus them. and also 16x(-1/2)^2 does not equal (16x -1/2)^2.

diagnostic test 2 Q (d)
QA
nature of error: conceptual
line 2 1/12 equals 1/6 not 6.

diagnostic test @ Q (d)
QB
nature of error: conceptual
line 3 1/2 divided by 1/6 equals 3 not 1/3.

diagnostic test Q (d)
QC
nature of error: conceptual
line 1, he just added -1 to 1/2 instead of substracting 1 from 1/2
nature of error: careless
line 1 he got 11/12 from 1/2x2/3. it is actually 1/12.
line 4 the answer is actually 18/11 not 7/12

diagnostic test Q (d)
QD
nature of error: conceptual
line 1 he he divided 2/3 by 2 and multiplied 1/4 by 2.

12. Diagnostic Test 2 Q 4 (c)
QD
Nature of errors: Careless
Error:
Line 1 11-9 = 2, not 3
-½ x -½ = ¼ , not -1.

Diagnostic Test 2 Q 4 (c)
QE
Nature of errors: Conceptual and Careless
Error:
Line 1 (-½ ) x (-½ ) = 0.25, not -0.25
2(11-9) = 4

Diagnostic Test 2 Q 4 (c)
QF
Nature of errors: Careless
Error
Line 1 2(11-9) = 4
16(-½) ^2 is not equal to (16 x -½ )^2

Diagnostic Test 2 Q 4 (d)
QA
Nature of errors: Careless
Error:
Line 2 2/12 = ⅙ , not 6

Diagnostic Test 2 Q 4 (d)
QB
Nature of errors: Conceptual
Error:
Line 3 -½ / ⅙ = -3, not ⅓

Diagnostic Test 2 Q 4 (d)
QC
Nature of errors: Conceptual
Error:
Line 1 1/2 - 1 = -½, not -1½ . 2/3 x ¼ = ⅙ , not 11/12

Diagnostic Test 2 Q 4 (d)
QD
Nature of errors: Conceptual
Error:
Line 4 - ½ should not be flipped, ⅙ should.

Diagnostic Test 2 Q 4 (d)
QE
Nature of errors: Conceptual
Error:
Line 1 To multiply a fraction, the 2nd fraction should not be flipped and the multiplication operation (x) should not be turned into a division operation (/). In this case, ¼ should not be flipped.

13. 1) Diagnostic Test 2 Q 4 (c)
QD:
Nature of errors: Careless and conceptual (accordingly)
Error:
Line 1 11-9 equals to 2,
Line 2 -1/2 multiply by -1/2 is 1/4

2) Diagnostic Test 2 Q 4 (c)
QE:
Nature of errors: Careless
Error:
Line 1: (11-9)=2, not 18; (-1/2)^2 = +0.25 or +1/4

Diagnostic Test 2 Q4(c)
QF
Nature of errors: Conceptual
Error
Line 1: 16(-(1/2))^2=16(0.25)=4

Diagnostic Test 2 Q4(d)
QA
Nature of errors: Careless
Error:
Line 2: 2/12=1/6

5) Diagnostic Test 2 Q4 (d)
QB
Nature of errors: Conceptual
Error:
Lines 2&3: -1/2/1/6 = -1/2 x 6=3, not 1/3

Diagnostic Test 2 Q4 (D)
QC
Nature of errors: Careless
error:
Line 1: 1/2-1=-1/2 not- 1 and 1/2

Diagnostic Test 2 Q4 (d)
QD
Nature of errors: Conceptual
Error:
Line 4 - ½ should not be flipped, ⅙ should.

Diagnostic Test 2 Q4 (d)
QE
Nature of error: Conceptual
Error:
Line 1 2/3 x 1/4 = 2/12, not 2/3 / 4/1
Line 2 2/3 / 4/1 = 2/12, not 0.5/3

14. Diagnostic test 2 Q4(c)
QA
Nature of error:conceptual
Line 1 (-1/2)^2=1/4 not -1/4
Nature of error:careless and conceptual
Line4 -12x(-1/4)=3 not-120

Diagnostic test 2 Q 4(c)
Nature of error
QB
Nature of errors:conceptual
Line 2:16x (-1/2)=-8 not 8

Diagnostic test 2 Q 4(c)
QC
Nature of errors:careless
Line2 (-1/2)^2=1/4 not 2.5

Diagnostic test 2 Q 4(c)
QD
Nature of error:careless
Line1 11-9=2 not 3
Nature of error:conceptual
-1/2 x -1/2=1/4 not -1

Diagnostic test 2 Q 4(c)
QE
Nature of error:careless and conceptual, according to order
Line1 2=2 not 22, 11-9=2 not18

Diagnostic test 2 Q 4(c)
QF
Nature of error: conceptual
Line1 16(-1/2)^2 =(-8)^2 not [16x-1/2]^2
Line3 (-8)^2=64 not-64

Diagnostic test 2 Q 4(d)
QA
Nature of error:conceptual
Line2 2/12 = 2/6 not 6

Diagnostic test 2 Q 4(d)
QB
Nature of error:conceptual
Line3 -(1/2)/(1/3)=-3 not -1/3

Diagnostic test 2 Q 4(d)
QC
Nature of error:careless
Line 1 (1/2 -1)/[(2/3)x(1/40]=(-1/2)/(1/6) not (-1 (1/2))/(11/12)

Diagnostic test 2 Q 4(d)
QD
Nature of error:careless
Line1 2/3=2/3 not 1/3 and 1/4=1/4 not/1/2
Nature of error:conceptual
Line 4 1/6 should be switched not 1/2

Diagnostic test 2 Q 4(d)
QE
Nature of errors:conceptual
Line 3 (0.5)/(0.5/3)=3 not-1/3

15. Diagnostic Test 2 Q 4(c)
QA
Nature of errors: Conceptual
Line 1: 2x2-16(1/4) not (-1/4)

Diagnostic Test 2 Q 4(c)
QB
Nature of errors: Conceptual
Line 3: -16(-1/2)=+8 not -8

Diagnostic Test 2 Q 4(c)
QC
Nature of errors: Careless
Line 2: -16(-1/2)^2= -16x0.25 not 2.5

Diagnostic Test 2 Q 4(c)
QD
Nature of errors: Careless
Line 1: 2(11-9)=2x2 not 2x3

Diagnostic Test 2 Q 4(c)
QE
Nature of errors: Conceptual
Line 1: (1/2)^2=0.25 not -0.25

Diagnostic Test 2 Q 4(c)
QF
Nature of errors: Conceptual
Line 1: -16(-1/2)^2=-16(0.25) not (16x-1/2^2)

Diagnostic Test 2 Q 4(d)
QA
Nature of errors: Careless
Line 2: (-1/2)/(2/12)= (-1/2)/(1/6) not (-1/2)/6

Diagnostic Test 2 Q 4(d)
QB
Nature of errors: Careless
Line 3: (-1/2)/ (1/6)= -3 not -1/3

Diagnostic Test 2 Q 4(d)
QC
Nature of errors: Careless
Line 1:(1/2)-1/(2/3 x1/4) = -1/2/(1/6) not (-1(1/2)/(11/12)

Diagnostic Test 2 Q 4(d)
QD
Nature of errors: Careless
Line 1: 2/3=2/3 not 1/3 and 1/4=1/4 not 1/2
Nature of errors: Conceptual
Line 4: 1/6 should be switched not 1/2

Diagnostic Test 2 Q 4(d)
QE
Nature of errors: Conceptual
Line 3 : (0.5)/(0.5/3)=3 not1/3

16. 1) Diagnostic Test 2 Q 4 (c)
QD:
Nature of errors: Careless
Error:
Line 1: (11-9)=2.
Line 2: -1/2 x -1/2 is 1/4

2) Diagnostic Test 2 Q 4 (c)
QE:
Nature of errors: Careless
Error:
Line 1: (11-9)=2,

3) Diagnostic Test 2 Q4 (c)
QF
Nature of errors: Conceptual
Error:
Line 1: 16 x (-1/2)^2 = 4,

4) Diagnostic Test 2 Q4 (d)
QA
Nature of errors: Careless
Error:
Line 2: the denominator 2/12 in line 1 =1/6

5) Diagnostic Test 2 Q4 (d)
QB
Nature of errors: Conceptual
Error:
Lines 2&3: -1/2/1/6 = -1/2 x 6=3

6)Diagnostic Test 2 Q4 (D)
QC
Nature of errors: Careless
error:
Line 1: 1/2-1=-1/2

7)Diagnostic Test 2 Q4 (d)
QD
Nature of errors: Conceptual
Error:
Line 4 - ½ should not be flipped, ⅙ should.

8)Diagnostic Test 2 Q4 (d)
QE
Nature of error: Conceptual
Error:
Line 1 2/3 x 1/4 = 2/12, not 2/3 / 4/1
Line 2 2/3 / 4/1 = 2/12, not 0.5/3