Microscopic numbers

Microscopic numbers

Pencil and paper
Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
This task is about ordering some numbers written in standard form.

Tanu used a microscope that had a special ruler to measure the length in centimetres of some tiny animals. She wrote their sizes in standard form.

Here are her results:

5.6 × 10–2 3.4 × 10–5 1.6 × 10–4 3.1 × 10–2 6.8 × 10–4 2.1 × 10–5

 
a)		  
Sort these six numbers from smallest to largest and write them in the spaces below.
 
 
i)
 
ii)
 
iii)
 
iv)
 
v)
 
vi)
 
 
____________________ (smallest)
 
____________________
 
____________________
 
____________________
 
____________________
 
____________________ (largest)  
 
b) i)
Circle the larger of these two numbers.   2.6 × 10–4   /  2.17 × 10–4
  ii)

 

 Explain how you can tell it is larger.

 
 
 
 
 
c) i)
Circle the larger of these two numbers.   3.17 × 10–2   /   4.81 × 10–3  
  ii)

 

 Explain how you can tell it is larger.

 
 
 
 
Task administration: 
This task is completed with pencil and paper only.
Level:
5
Curriculum info: 
Maths, Number and Algebra, Number Knowledge
Keywords: 
standard form, ordering numbers, place value
Description of task: 
Students order small numbers written in standard form, and explain why they are ordered in that way. The numbers are the lengths of some microscopic animals.
Curriculum Links: 
Key competencies
This resource involves explaining why one of two numbers written in standard form is larger. This relates to the Key Competency: Using language, symbols and text.
For more information see http://nzcurriculum.tki.org.nz/Key-competencies
 
Answers/responses: 

 

Y10 (03/04)
a)  

i)  
ii)  
iii)  
iv)  
v)  
vi)  

 

 2.1 × 10–5
3.4 × 10–5
1.6 × 10–4
6.8 × 10–4
3.1 × 10–2
5.6 × 10–2
[Accept if numbers written as 0.056 etc.]
An error includes any 1 of:

  • transcription error(s) but answers correctly ordered, e.g., (0000.21);
  • numbers ordered from largest to smallest;
  • 1 pair of consecutive numbers transposed;
  • numbers ordered from 10–2 to 10–5, i.e., 3.1 × 10–2,
    5.6 × 10–2, 1.6 × 10–4, 6.8 × 10–4, 2.1 × 10–4, 3.4 × 10–4;
  • numbers ordered from 10–5 to 10–2 but incorrect within
    each pair, i.e., 3.4 × 10–5, 2.1 × 10–5, 6.8 × 10–4, etc.

2 marks  
(all 6 correct)  

or  
1 mark  
(1 error)  

difficult  

moderate  
b)   i)  
ii)		 2.6 × 10–4 and
Explanation that consists of both:

  • 2.6 is bigger than 2.17 or equivalent statements, and
  • both are raised to the same power;
    or   2.6 × 10–4 is bigger by 0.43 × 10–4 (or equivalent).

    Partially correct includes any 1 of: 

  • correct solution, but only one of above reasons given;
  • circling wrong number but giving correct explanation.

2 marks  
(fully correct)

or

 

1 mark
(partially correct)

very   difficult

 

 

moderate
c) i)
ii)		 3.17 × 10–2 and
Explanation based on any 1 of:
 

  • 10–2 is larger than 10–3.
  • 3.17 × 10–2 is in the hundredths but 4.81 × 10–3 is in the thousandths.
  • It has fewer zeros before the first significant figure.
  • Other correct statements (including how the decimal place moves, it is bigger by 0.02689 or equivalent).

    Partially correct includes any 1 of: 

  • Only compares 3.17 and 4.81;
  • –2 is bigger than –3;
  • circles wrong number but gives correct explanation.
 2 marks 
(fully correct)
or

 

 

 
1 mark
(partially correct)

 difficult 

 

 

 
difficult

 

Diagnostic and formative information: 

 

 Likely misconception
a) Orders decimal numbers and ignores powers of 10 (e.g., 1.6, 2.1, 3.1, ... , 6.8).
Orders from largest to smallest.
Transposes order within the same power of 10 (e.g., 3.4 × 10⁻5, 2.1 × 10⁻5).
b) Does not mention both numbers are to the same power of 10.
Sees 0.17 as bigger then 0.6 (because 17 > 6).
Says 2.6 has fewer digits, so it's smaller.
c)   Circles 4.81 × 10⁻3 because 4.81 > 3.17 or 0.81 > 0.17.
Sees 10⁻3 > 10⁻2.