Babysitting powers

Babysitting powers

Pencil and paper
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This task is about solving maths problems involving powers.
 
 

Uncle Tipene, who likes maths, agreed to pay Ana to babysit. He said she would get paid using the formula 2n (2 to the power of n) dollars, where n is the number of hours that Ana worked. This meant that if Ana worked for 5 hours, she would get paid 32 dollars.

 

a)
 
How much would Ana get paid if she worked for (Write your answers as whole numbers.)
 
 
i)

ii)

iii)
 
1 hour? $__________

3 hours? $__________

6 hours? $__________
 
b)

 
Uncle Tipene agreed that he would still use the same formula if Ana wasn't needed. This would mean n = 0.
 
How much would Ana get if she wasn't needed? $__________
 
c)
If Ana got paid $256, show how to work out the number of hours she had babysat for.
 

Show your working.

 

 

 

Answer: __________ hours
 
d)
 
Explain how the rate Ana gets paid (2n) compares to getting $4 per hour.
 
 
 
 
 
 
Task administration: 
This task is completed with pencil and paper only.
Level:
5
Curriculum info: 
Maths, Number and Algebra, Number Strategies
Keywords: 
powers, indices, index form, money, algebraic expressions
Description of task: 
Students solve problems involving rates of pay based on a formula using powers, and compare this to a fixed rate of pay.
Learning Progression Frameworks
This resource can provide evidence of learning associated with
Multiplicative thinking, sets 8-9
within the Mathematics Learning Progressions Frameworks.
Read more about the Learning Progressions Frameworks.
Answers/responses: 

  

Y9 (03/2004)

a)

i)
ii)
iii)

2
8
64

2 marks (all 3 correct)
or
1 mark (2 correct)

difficult  (all 3 correct)

          
moderate (2 correct)

b)

  

1

1 mark

very difficult

c)

  

 8
Working that recognises the multiplicative nature of indices, e.g., for any 1 of:

  • 256 ÷ 2 ÷ 2 ÷ 2 ÷ 2 ÷ 2 ÷ 2 ÷ 2 ÷ 2 = 8
  • 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 256
  • Use answers in a) to factor up to 8 hours
  • Other working that shows the multiplicative relationship of the indices and attempts to solve the number of hours for $256.
 

1 mark
1 mark

difficult
difficult

d)

  

 For any 1 of:

  • If Anna works for more than 4 hours she will get a higher hourly rate which will continue to increase.
  • If Anna works for less than 4 hours she will get a lower hourly rate.
  • Statement about getting paid more when working longer including the breakeven point.

General statement about getting paid more using 2n formula.

2 marks
or 

1 mark

very difficult

difficult
Diagnostic and formative information: 
 

Common error

Likely calculation

Likely misconception

a) i)

4

2 × 2 (times itself)

  

a) ii)
    iii)
c)

6
12
128

2 × 3
2 × 6
256 ÷ 2

Mistakes the 2n relationships for 2 × n.

a) i) 
    ii) 
    iii) 

21
23
26

  

Does not indicate understanding of how to write powers as whole numbers.

b)

0
2

2 × 0
2 × 1

Does not apply the rule that any number to the power of zero equals one, i.e., a0 = 1.