Finding the value of n
Show how to find the value of n in the following equations. [5n means 5 × n.]
a) 
n – 4 = 13


b)

25 = n – 6


c)

12 = 3n


d)

8n = 4


e)

4n + 2 = 22


f)

15 = 7 + 2n

Y8 (05/2008)  
a) 
Working showing: n = 13 + 4 or other correct methods of solution. 17 
moderate
easy 
b) 
Working showing 25 + 6 = n or other correct methods of solution. 31 
moderate
easy 
c) 
Working showing 12 ÷ 3 = n or other correct methods of solution. 4 
difficult
easy 
d) 
Working showing n = 4 ÷ 8 or other correct methods of solution. ½, 0.5 
very difficult
difficult 
e) 
Working showing 4n = 22 – 2; n = 20 ÷ 4 or other correct methods of solution. 5 
difficult
easy 
f) 
Working showing: 15 – 7 = 2n; 8 ÷ 2 = n or other correct methods of solution. 4 
difficult
moderate 
Based on a representative sample of 227 students.
NOTE:
 Do not accept expressions that just show that the answer they gave is correct (e.g. 17 – 4 = 13), as these do not show how they arrived at their answer.
 Do accept trial and improvement methods. We found these to be very uncommon.
 One step problems involving addition or subtraction.
 One step problems involving multiplication or division.
 Two step problems involving all four arithmetic operators.
Equations are also written in different forms to support:
 the idea that they are not restricted to using the form a + b = c; or
 that the "as yet unknown" (in these cases n), is always on the lefthand side of equations.
Equations are about balance of what is on each side of the equals sign (see balance, in Algebraic Thinking: Equality).
Common error  Likely calculation  Likely misconception  
a) b) e) f) 
9 19 6 11 
13 – 4 25 – 6 (22 + 2) ÷ 4 (15 + 7) ÷ 2 
Inappropriate inverse operator (+ or –) Student uses an incorrect inverse operator to solve the equation (subtraction instead of addition). 
c) d) 
36 32 
12 × 3 4 × 8 
Inappropriate inverse operator (× or ÷) Student uses an incorrect inverse operator to solve the equation (multiplication instead of division). 
e) f) 
16 6 
22 – 2 – 4 15 – 7 – 2 
Incorrect inverse operator Student uses an incorrect inverse operator to solve the equation (subtraction instead of division) 
c) d) e) f) 
12 4 20 8 
22 – 2 15 – 7 22 – 2 15 – 8 
Ignores the coefficient of the unknown (n) The student does not take account of the effect of the number of n's in the equation. 
e)  2  8 ÷ 4 = 2 
Fails to manipulate equation to the form n = … Does not recognise that division by 4 leads to the equation 2n = 1. 
In questions e) and f), students' answers may reflect a combination of the errors above.
Students need to explore:
 the concept of the additive inverse (a – a = 0 for any value of a);
 the concept of additive identity (a + 0 = a, for any value of a). and
 that adding or subtracting the same number to each side of an equation leave it "still in balance".
Click on the link or use the keyword additive identity or on the link Algebraic Thinking Concept Map: additive identity for resources relating to these properties.
Alternatively solving this sort of equation can be explored using the model of reverse addition shown in Reversing Addition and When Subtraction Becomes Addition, Book 8: Teaching Number Sense and Algebraic Thinking (PDF).
Inappropriate inverse operator (× or ÷)
Students need to explore:
 the concept of the multiplicative inverse (a ÷ a = 1 for any value of a);
 the concept of multiplicative identity (a × 1 = a, for any value of a). and
 that multiplying or dividing each side of an equation by the same number leaves it "still in balance".
Alternatively solving this sort of equation can be explored using a model similar to that shown in the two activities above from Book 8: Teaching Number Sense and Algebraic Thinking (PDF) but with several lots of n being drawn. This model starts to break down with fractional lots of n, or with negative lots of n.
Incorrect inverse operator
The student needs to explore both of the above concepts, seeing the link between addition and subtraction, and between multiplication and division.
Ignores the coefficient of the unknown (n)
The student needs to see that an equation such as 12 = 3n is equivalent to 3 × ? = 12.
They may need to explore the same issues as for Inappropriate inverse operator (× or ÷) (see above)
Fails to manipulate equation to the form n = …
The student needs to see that finding the value of n is the same as completing the equation n = ???
The student may need to explore the first two issues of inappropriate inverse operators
(+, –, ×, or ÷ ) above.
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