Monday, 4 February 2013

INDICES & SURDS - The Mistake Fragment part 1

BY MR JOHARI

Context
Below shown the solution posted by the secondary 3 students during the 4th February 2013 Maths Quiz C.
There are errors in the solution posted.

Focus
Error Analysis and Peer Evaluation
Individual Effort

You are required to post the following as a comment:

(i) Nature of error - conceptual or carelessness (be very specific in your description)
example - careless error due to arithmetic (addition)
example - conceptual error due to error in (a + b)^2 = a^2 + b^2.

(2) Correction to the error
Ensure that your solution is concise and correct.

1. 1) The division sign changed to a multiplication sign, this is a conceptual error as when u convert cuberoot to ^1/3 it does not change the sign

2a) Conceptual error 32^2 ≠ 2^5+x, instead it should be 2^5x

2c) Careless error 3^2≠2, 3^2=9

3a) Careless error 64≠2^5, 64=2^6

3c) Careless error 16≠2^5, 16=2^4

1. Good observation - sometimes there is a fine line between carelessness n conceptual errors.

2. 1. Careless & Conceptual. He changed the divide sign to multiplication. He also made another mistake. If he continues, he should have add 1/3 to 3 instead of multiplying it, as m^n + m^b = m^n+b

2a. Conceptual. 32^x should be 2^5x. The 5 and x should be multiplied together instead of added.

2c. Careless. 3/9 is not 1/2 it is 1/3

3a. Careless. 64 is not 2^5, it is 2^6.

3c. Careless. 1/16 should be 2^-4 not 2^-5.

3. 1) It is a conceptual error, as when he changed ÷3√m it became x3^1/3, it should retain its divide sign as it is only changing its form not the equation.
m^x=m^3÷m^1/3xm^5
m^x=m^7 2/3
x=7 2/3

2a)concept error 32^x ≠ 2^5+x, 32^x =(2^5)^x
32^x =(2^5)^x
32^x=(2^x)^5
(3)^5= 243

2c) careless error 3^2= 9 ≠ 6 so instead of 1/2 you should get 1/3
3/4^x=3/(2^x)^2=3/3^2=3/9=1/3

3a)careless error 64 = 2^6 not 2^5
64 = 2^6
2^5x=2^6
5x=6
x=1.2

3c) Careless error 16 = 2^4 16 ≠ 2^5
2^5x=1/2^4
2^5x=2^-4
5x=-4
x=-0.8

4. 1) Conceptual error . Even if we change cube root to an index of 1/3 , the sign does not change and remains as divide
2) Conceptual error . 32 = 2^5 . But the x is the index of 32 so , the x is multiplied together with the 5 to become 2^5x
2c) Careless error . 3/3^2 is not half , instead it if 3/9 which is 1/3
3) Careless error . 2^5 is 32 , not 64 . So it is actually 5x =6
3c) Careless error . 16 is 2^4 or in this case 1/16 is 2^-4 .

5. 1) Conceptual error . Even if we change cube root to an index of 1/3 , the sign does not change and remains as divide
2) Conceptual error . 32 = 2^5 . But the x is the index of 32 so , the x is multiplied together with the 5 to become 2^5x
2c) Careless error . 3/3^2 is not half , instead it if 3/9 which is 1/3
3) Careless error . 2^5 is 32 , not 64 . So it is actually 5x =6
3c) Careless error . 16 is 2^4 or in this case 1/16 is 2^-4 .

6. Q1) Careless. "m^3 / m^1/3" is changed to "m^3 x m^1/3" thus getting the wrong solution. Else, the solution would be m^3 / m^1/3 x m^5 = m^7/1/3 where x = 7 and 1/3 .

Q2a) Conceptual error. 32^x = (2^5)^x which is 2^5x instead of 2^5+x . Solution would be (2^x)^5 = 3^5 = 243.

Q2c) Careless error. 3/3^2 is not 1/2 , instead its 3/3^2 = 3/9 = 1/3

Q3a) Careless error. 64 is not 2^5 , it should be 2^6, thus 5x = 6 and x = 1.2.

Q3c) Careless error. 16 is not 2^5 , it should be 2^4, thus 5x = -4 which x = -0.8.

7. 1) Conceptual error. He mistook the a^m x a^n = a^m+n law.
m^3 x m^1/3 x m^5 = m^3+1/3+5 = m^7/2/3

2a) Careless error. He converted 32^x as 2^5+x while the proper conversion should be 2^5x
2^5x = (2^x)^5 = 3^5 = 243

2c) Careless error. 3^2 does not equal 6
3/3^2 = 3/9 = 1/3

3a) Careless error. 64 does not equal to 2^5. Instead it should be 2^6

2^5x = 2^6
5x = 6
x= 1/1/5

3c) Careless error. 1/16 does not equal to 2^-5. instead it should be 2^-4
2^5x = 2^-4
5x = -4
x = -4/5

8. 1) Careless error: when converting 3√m to m^1/3, he changed the sign from ÷ to x.
On the second line, m^3 x m^1/3, he multiplied the index instead of adding. Conceptual error

Correction:

m^x = m^3 ÷ 3√m x m^5
m^x = m^3 ÷ m^1/3 x m^5
x = 3 - 1/3 + 5
x = 7 2/3

2a) Conceptual error. 32^x ≠ 2^5+x. Instead, 32^x = 2^5x.

Correction:

32^x = 2^5x = (2^x) ^ 5 = 3^5 = 243

2c) Careless error 3^2 ≠ 6. Instead, 3^2 = 9

Correction:

3/4^x = 3/(2^2)^x = 3/9 = 1/3

3a) Careless error. 64 ≠ 2^5. Instead, 64 = 2^6

Correction:

32^x = 64
2^5x = 2^6
5x = 6
x = 1.2

3c) Careless error. 16 ≠ 2^5. Instead, 16 = 2^4

Correction:

32^x = 1/16
2^5x = 2^-4
5x = -4
x = -0.8

9. This comment has been removed by the author.

10. 1) Conceptual error because the division sign was change to multiplication sign.
It should be:
m^x = m^3 ÷ ∛m x m^5
m^x = m^3 ÷ m^1/3 x m^5
x = m - 1/3 + 5
x = 23/3

2a) Conceptual error because 32^x ≠ 2^5+x.
It should be:
32^x = (2^5)^x = (2^x)^5 = 3^5 = 243

2c) Careless error due to division.
It should be:
3/4^x = 3/(2^x)^2 = 3/3^2 = 1/3

3a) Careless error because 64 ≠ 2^5
It should be:
32^x = 64
2^5x = 2^6
5x = 6
x = 6/5

3c) Careless error because 1/16 ≠ 2^-5
It should be:
32^x = 1/16
2^5x = 2^-4
5x = -4
x = -4/5

11. 1) Conceptual, division sign does not change to multiplication
Conceptual, when multiplying , it is not 3x1/3, but 3+1/3

2) Conceptual, 32^x = 2^5^x, not 2^5+x

2c) Careless, 3^2 != 6

3) Careless, 64 != 2^5

3c) Careless, 1/16 != 2^-5

12. 1. Conceptual error. Wrong concept of the laws of indices.

m^x = m^3 / 3√m x m^5
m^x = m^3 / m^1/3 x m^5
m^x = m^3-(1/3)+5
x=7 2/3

2a. Conceptual error. 32 does not equal to 2^5+x but 2^5x.

32^x = 2^5x
= (2^x)^5
= 243

2c. Careless error due to the error in 3/3^2 = 1/2

3 / 4^x = 3 / (2^x)^2
= 3/3^2
= 3/9
= 1/3

3a. Careless error due to the error in 64 = 2^5

32^x = 64
2^5x = 2^6
5x = 6
x = 1 1/5

3c. Careless error due to the error in 1/16 = 2^-5

32^x = 1/16
2^5x = 2^-4
5x = -4
x = -4/5

13. 1. Conceptual error. m^3 / √m is not equals to m^3 * m ^1/3. The sign does not change from * to / when changing the cube root to a power. Also, m^3 * m^1/3 = m^(3+1/3). The powers need to be added when multiplying.
2a. Conceptual error. 32^x = (2^5)^x = 2^5x. Not 32^x = (2^5)^x = 2^(5+x). The powers need to be multiplied and not added in this case.
2c. Careless error. 3/3^2 = 3/9 = 1/3. Not 3/3^2 = 3/6 = 1/2
3a. Careless error. 64 = 2^6 not 2^5
3c. Careless error. 1/16 = 2^-4 not 2^-5

14. 1)
carelessness
m^x=m^3 / m^1/3 *m^5
m^x=m^(3-1/3+5)
m^x=m^23/3
x=23/3=7+2/3
2a)
conceptual error
32^x=2^5^x=(2^x)^5=3^5=243
2c)
carelessness
3/3^3=3/9=1/3
3a)
conceptual error
32^x=(2^5)^x
(2^5)^x=64=2^6
x=6/5
3c)
carelessness
2^5x=2^-4
x=-4/5

15. q1) conceptual error: sign should remain as multiplication sign not change into divide, law of indices states that m^3 x m^1/3 should be m^4/3 and not m.
q2a) conceptual error: 32^2 ≠ 2^5+x, instead it should be 2^5x
q2c) careless error 3/3^2 is not 1/2, expand before simplifying, instead answer should be 3/9 = 1/3
q3) Careless error: 64 ≠2^5 instead is equal to 2^6
3c) Careless error: 1/16 ≠ 2^-5, instead is equal to 2^-4. Thus 5x=-4, therefore value of x=-0.8

16. Part 2:

1)
m^x = m^7 2/3
x=7 2/3

2a)
32^x = (2^x)^5
= (3)^5
=243

2c)
3/4^x = 3/(2^x)^2
=3/3^2
=1/3

3a)
32^x = 64
2^5x = 2^6
x = 1.2

3c)
32^x = 1/16
2^5x = 2^-4]
x = -0.8

17. 1.Conceptual error. m^1 x m^2 = m^1+2.
It should not be multiplied, which in this case shows m^3 x m^1/3, which should be m^3+1/3 and not m^3 x 1/3.
m^x = m^ 8 1/3
therefore, x = 8 1/3

2a) Conceptual error. 32^2 ≠ 2^5+2, but instead (2^5)^x.
(2^x)^5
= (3)^5
= 243

2c) Careless Error. 3^2 = 9, not 6.
3/ 3^2 = 3/ 9
= 1/3

3a) Careless error. 64 = 2^6, not 2^5
2^5x = 2^6
5x = 6
x = 1.2

3c) Careless error, 1/16 = 2^-4, not 2^-5
2^5x = 2^-4
5x = -4
x = -4/5

18. 1. CE m^3 / √m is not equals to m^3 * m ^1/3. The sign does not change from * to / when changing the cube root to a power. Also, m^3 * m^1/3 = m^(3+1/3).

2a. CE32^x = (2^5)^x = 2^5x. Not 32^x = (2^5)^x = 2^(5+x).

2c. CM 3/3^2 = 3/9 = 1/3. Not 3/3^2 = 3/6 = 1/2

3a. CM64 = 2^6 not 2^5

3c. CM 1/16 = 2^-4 not 2^-5

19. 1. Conceptual error. When he changed the cube root, the divide sign should not change to multiply sign. Both the indices of m should not be multiplied to each other, it should be added instead.
2a. Conceptual. Since 32 is 2^5 and x is the index of 32 in the question, x should be the index of (2^5) and so x will be multiplied into the index of 2.
2c. Careless. 3^2 is not 3*2. 3^2 = 9.
3a. Careless. 64 ≠ 2^5. 64 = 2^6
3c. Careless. 16 ≠ 2^5. 16 = 2^4

20. 1) Conceptual ERROR.He accidentally changed the division sign into a multiplication sign.
2a) Conceptual ERROR 32^x doesn't equal to 2^5+x . It should be 2^5x since (x^m)^n =x^mn
2c) Conceptual ERROR. 3^2 is 9 not 6 since x^m doesn't equal to xm
3a) Careless ERROR 2^5= 32 NOT 64. 2^5=64
3c) Careless ERROR. 16 = 2^4 not 2^5

21. 1. CE
m^x=m^3÷m^(1/3)*m^5
=m^(3-1/3+5)
=m^(7+2/3)
therefore, x=7 2/3

2. CE
32^x=(2^5)^x=(2^x)^5
=3^5=243

3. CL
3/(3^2)=1/3

4. CL
32^x=64
2^5x=2^6
x=1.2

5. CL
32^x=1/16
2^5x=2^-4
x=-0.8