Generation powers

Generation powers

Pencil and paper
Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
This task is about writing whole numbers in index form and solving maths problems involving powers.
  
       Great grand-father Fraser

a) Mr Fraser had 5 sons.

  • i) Each of his sons also had 5 sons (Mr Fraser's grandsons).
  • ii) Each of those sons had 5 sons (Mr Fraser's great-grandsons). 
 
b) Complete the table below to show how many grandsons and great-grandsons Mr Fraser had:

  • i) as a whole number; and
  • ii) in index form.

Generation

Number
(in each generation)

Number
(in index form)

Mr Fraser's sons

5

51

a) Mr Fraser's grandsons

i)

ii)

b) Mr Fraser's great-grandsons

i)

ii)
 
c)
 
If this trend continued, show how many great-great-grandsons Mr Fraser would have. 
(Write your answer as a whole number.)
 
Show your working.

 

 

 

 


Number of great-great-grandsons: __________
Task administration: 
This task is completed with pencil and paper only.

 

Level:
4
Curriculum info: 
Maths, Number and Algebra, Number Strategies
Keywords: 
powers, index form, indices
Description of task: 
Students complete a table showing a family of sons increasing by a power, and calculate the next power showing their working.
Answers/responses: 

  

Y9 (03/2004)

a)
 

i)
ii) 

25, and
52

difficult

b)
 

i)
ii) 

125, and
53

very difficult

c)

 

 625 [Conditional marking: accept answer in b) × 5]
For any 1 of:

  • 54 = 625
  • 5 × 5 × 5 ×5 = 625
  • 125 × 5 = 625
  • other acceptable working.

moderate
moderate
Teaching and learning: 
A significant number of students correctly identified the

  • whole numbers for a) i) 25 (56%) and b) i) 125 (38%); and
  • the index form for a) ii) 52 (50%) and b) ii) 53(46%).

However, when the whole number and the index form were both required, it became:

  • for 52 and 25 (24%), and
  • for 53 and 125 (18%).

Students are required to answer both the index form and the whole number form to get a mark – otherwise it is possible that they simply continue a pattern without understanding the nature of indices.
NOTE: These results were based on a trial sample of 155 Year 9 students in March 2004.

Diagnostic and formative information: 
  Common error Likely calculation Likely misconception
a) i)
b) i)		 5
5		   Does not show an understanding of writing numbers from a scenario in index form, (continues a pattern without understanding).
a) ii)
b) ii)		 252
1253		 (5 × 5)2 or (52)2
(5 × 5 × 5)3 or (53)3		 Multiplies by the factor and increments the index number (performs operation twice).
a) ii)
b) ii)		 55
255		   Incorrect understanding of powers. Believes the index of a number 5 is equivalent to multiplying the number by five.
a) ii) 251 52 = 5 × 5 Partial understanding of solving powers, but not how to write as index form.