Estimating with addition

Estimating with addition

Pencil and paper
Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
This task is about estimating the sum of whole numbers.
An estimate is sometimes called a "good guess" that is easy to do in your head.
The following questions are about estimating, NOT about working out the exact answer.		  
a)

 
i)  Estimate the sum of: 86 + 47 + 65  __________
 
ii) Show or explain how you estimated the sum in the box below.
 
 
 
 
 
 
 
 
b)

 
  
i)  Estimate the sum of: 254 + 463 + 129  __________
 
ii) Show or explain how you estimated the sum in the box below.
 
 
 
 
 
 
 
 
c)

 
 
i)  Estimate the sum of: 386 + 76 + 9 + 457  __________
 
ii) Show or explain how you estimated the sum in the box below.
 
 
 
 
 
 
 
Task administration: 
This task is completed with pencil and paper only.
Levels:
3, 4
Curriculum info: 
Maths, Number and Algebra, Number Strategies
Key Competencies: 
Using language, symbols, and texts
Keywords: 
estimation, addition, key competencies
Description of task: 
This task requires students to estimate some addition problems using any appropriate method.
Curriculum Links: 
This resource can help to identify students' ability to apply additive and simple multiplicative ideas flexibly to combine or partition whole numbers to make sensible estimates of addition problems.
Key competencies
This resource involves explaining how they performed estimates of several addition problems. This relates to the Key Competency: Using language, symbols and text.
Learning Progression Frameworks
This resource can provide evidence of learning associated with
Additive thinking, sets 6-7
Additive thinking, sets 7-8
within the Mathematics Learning Progressions Frameworks.
Read more about the Learning Progressions Frameworks.
Answers/responses: 
    Y7 (03/2005)
a) An acceptable estimation strategy and an appropriate estimate. For example:

  • 210 (by rounding)
  • 180 (by front-end estimation)
  • 180-210 (using front-end and final compensation)
 difficult
(for both strategy and estimate)
b) An acceptable estimation strategy and an appropriate estimate. For example:

  • 900 (by rounding)
  • 700 (by front-end estimation)
  • 850 (using front-end and final compensation)
 difficult
(for both strategy and estimate)
c) An acceptable estimation strategy and an appropriate estimate. For example:

  • 900, 940  (by rounding)
  • 700 (by front-end estimation)
  • 800, 810, 820 or 830 (using front-end and final compensation)
 difficult
(for both strategy and estimate)

Results based on a representative sample of 174 students.

Teaching and learning: 
Prior knowledge needed

  • A firm base of addition and subtraction facts.
  • Good place value concepts.
  • Ability to mentally add numbers in tens, hundreds, etc, using standard place value concepts.
Diagnostic and formative information: 
Strategy name   Likely calculation Typical estimate(s) % using strategy

Front-end

 

 a)
b)
c)		 80 + 40 + 60
200 + 400 + 100
300 + 400		 180
700
700		 3%
2%
2%
Front end and final compensation a)
b)
c)		 180 + (6 + 7 + 5)
700 + (50 + 60 + 20) or 700 + (50 + 60 + 30)
700 + (80 + 70 + 50) or 700 + (90 + 80 + 60)		 198 or 200
830 or 840
900 or 930		 13%
14%
11%

Results are based on a representative sample of 174 students.

NOTE: The percentages include students who attempted that strategy, but did not necessarily come up with an appropriate estimate. Less than half of those using estimation strategies gave tidy numbers as their answers.

Strategy name   Likely calculation Typical answer(s) % using strategy
Place-value partitioning*
Exact computation only:
(see Next steps 2)		 a)
b)
c)		 (80 + 40 + 60) + (6 + 7 + 5)
(200 + 400 + 100) + (50 + 60 + 20) + (4 + 3 + 9)
(300 + 400) + (80 + 70 + 50 )
+ (6 + 6 + 9 + 7)		 198
846
928		 29%
31%
27%
Exact computation:
using the vertical algorithm.
(see Next steps 1)		 a)
b)
c)		 86 + 47 + 65
254 + 463 + 129
386 + 76 + 9 + 457
(laid out as vertical algorithm)		 198
846
928		 7%
8%
6%
Exact computation using other methods.
(see Next steps 2)		 a)
b)
c)		 86 + 47 + 65
254 + 463 + 129
386 + 76 + 9 + 457		 198
846
928		 20%
16%
18%

Results are based on a sample of 220 students.

Next steps: 
  1. Exact computation. To help students understand the need for estimation encourage discussion about situations when estimation should be used, e.g., for a quick total at the supermarket. 
  2. Place-value partitioning. This is where students do a separate computation for the number of hundreds, the number of tens, and the number of ones, then combine these three to get an exact answer. Students who are doing an exact computation based on place-value partitioning are applying the principles of front-end and final compensation especially when they compute the most significant digits first. However, they then continue to compensate until they reach an exact answer (this is effectively a left-to-right addition algorithm). They could work with front-end with resource, Estimating sums of money, and then on final compensation with resource Estimating team scores. It is important that these students recognise that an estimate does not need an exact answer (see step 1. above).
  3. Students may equate estimation with the rounding method only. Expose students to other estimation strategies other than rounding (refer to Computational estimation concept map). This could be done through a discussion of the different ways students in the class estimated in these problems.
  4. Numbers for an estimation question should be chosen to match students ability to calculate (i.e., numbers they can not easily and quickly calculate – more numbers, larger numbers, or less time to work them out).
For further information about estimation, refer to Computational estimation concept map.
 
Links with Numeracy framework.
Estimating numbers with an appropriate estimation strategy involves advanced additive part-whole (Stage 5) knowledge and strategies (Book 1: The Number Framework, pages 11-13).