Rounding numbers sensibly

Rounding numbers sensibly

Pencil and paper
Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
This task is about decimal place value and rounding decimal numbers.

To give sensible answers to calculations we often need to round. Round the following numbers. 

a)
i) The number 5.16 rounded to the nearest tenth is __________
ii) The number 7.944 rounded to the nearest hundredth is __________
iii) The number 14.4626 rounded to the nearest thousandth is _____________
b)
For each rounded number in the right hand column, write down one possible number, which could be rounded to the stated value, using the rule in the middle column. 
Write your answers in the left hand column.
The first one has been done for you.

 

 
 
 
Example  
 
 
i)  
 
 
ii)  
 
 
iii)  
Possible number Rule for rounding Stated value
when rounded
6.46
 		 To the nearest tenth 6.5
 
 
 		 To the nearest tenth 7.3
 
 
 		 To the nearest hundredth 11.35
 
 
 		 To the nearest thousandth 14.756
 
Task administration: 
This task is completed with pencil and paper only.
Level:
4
Curriculum info: 
Maths, Number and Algebra, Number Knowledge
Keywords: 
rounding, decimals, place value
Description of task: 
Students round given decimal numbers to the nearest tenths, hundredths, and thousandths and identify possible original numbers from rounded numbers.
Answers/responses: 
  Y8 (05/2008)
a) i)
ii)
iii)		 5.2 [do not accept 5.20 as this includes hundredths in the answer]
7.94 [do not accept 7.940 as this includes thousandths in the answer]
14.463 [do not accept 14.4630 as this includes ten-thousandths in the answer]		 difficult
difficult
difficult
b) i)
ii)
iii)		 Accept any number at least 7.25 and less than 7.35
Accept any number at least 11.345 and less than 11.355
Accept any number at least 14.7555 and less than 14.7565		 easy
moderate
difficult
Based on a representative sample of 228 Y8 students.

NOTE:  Students’ responses to b) i)-iii) will vary within the range shown.

Diagnostic and formative information: 
This assessment resource involves knowing about rounding as well as the place value of decimals.  For example students may know the conventions of rounding, but not know the place value position of hundredths, thousandth, or even tenths.

Many students rounded a) ii) and a) iii) to the nearest tenth to get 7.9 (28% of students) and 14.5 (29%) respectively. This may indicate that these students equate rounding with tenths only. This highlights the importance of using a mix of tenths and hundredths (and even thousandths) to test students’ understanding of decimals.

  Common error Likely misconception
a) i)
   ii)
   iii)		 5.20  (35%)
7.940 (5%)
14.4630 (9%)		 Incorrect trailing zero
Puts in a trailing zero after the correctly rounded result. While this result is very nearly correct, students who answered this way had substantially lower mathematical competency than students who got the correct answer.
a) i)
   ii)
   iii)		 5.1, 5.10
7.95, 7.950, 8, 8.000
14.462, 14.4620		 Rounding in the wrong direction
Rounds down (truncates) instead of rounding up.
Rounds up instead of rounding down.
Rounds down (truncates) instead of rounding up.
a) i)
   ii)
   iii)		 5
7.9, 7.90, 7.900
14, 14.5, 14.5000, 14.46		 Rounding to the wrong order of magnitude (often to tenths)
Rounding to whole numbers.
Rounding to tenths: 7.900 (22%) was the most common of these.
Rounding to tenths or hundredths: 14.5000 (21% of students) was by far the most common of these.
b) i)
   ii)
   iii)		 7, 7.0
11, 11.0, 11.4, 11.40
15, 15.0, 14.8, 14.76		 Rounding the number given instead of finding the original number
Students rounded the stated value to ones, tenths, or hundredths
b) i)
   ii)
   iii)		 7.2, 7.4
11.34, 11.36
14.755, 14.757		 Students added or subtracted a tenth, hundredth, or thousandth
Next steps: 
Incorrect trailing zero
Students who put in an extra zero after the last significant figure just need to recognise that the answer needs only to be given to the degree of accuracy to which the rounding specifies, e.g. round to tenth requires answers such as 5.1 (or even 5.0) rather than 5.10 (or 5).

Rounding in the wrong direction
Students need to understand the protocols of rounding.

Rounding to the wrong order of magnitude
Students may need to consolidate their place value understanding of decimals, in particular tenths, hundredths, and thousandths.
Students who gave answers such as 7.900, 8.000, or 14.5000 need to pay attention to the number of decimal places to which the rounding specifies, not to the number of decimal places in the original number. These students also need to round to the specified accuracy, not just to whole numbers or tenths.

Rounding the number given instead of finding the original number
Students who rounded the stated value to ones, tenths, or hundredths or added or subtracted a tenth, hundredth, or thousandth need to see that part b) is asking for the reverse of part a).