Estimating addition II

Estimating addition II

Pencil and paper
Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
Estimation is an approximately correct answer you can easily work out in your head, NOT the calculation of an exact amount.
 
Estimate the totals of the following sets of numbers. Do not calculate the exact total.
a)

 

 364 + 238 + 174

Show how to get the approximate total.

 
 
 
 
 
 

 The total is about __________
 
b)

 
 
2 387 + 3 628

Show how to get the approximate total.

 
 
 
 
 
 

 The total is about __________
 
c)

 
 
438 + 531 + 247 + 571 + 361

Show how to get the approximate total.

 
 
 
 
 
 

 The total is about __________
Task administration: 
This task is completed with pencil and paper only.
Level:
4
Curriculum info: 
Maths, Number and Algebra, Number Strategies
Key Competencies: 
Using language, symbols, and texts
Keywords: 
estimation, addition
Description of task: 
Students to show how they estimate in some addition problems.
Curriculum Links: 
This resource can help to identify students' ability to apply additive  strategies flexibly to whole numbers.
 
Key competencies
This resource involves recording the strategies students use to estimate 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 7-8
within the Mathematics Learning Progressions Frameworks.
Read more about the Learning Progressions Frameworks.
Answers/responses: 
    Y8 (11/2004)
a) An acceptable estimation strategy with an appropriate estimate. For example:

  • 760 or 770 (by grouping nice numbers)
  • 800 (by rounding)
  • 760 or 770 (using front-end and final compensation)
 difficult
b) An acceptable estimation strategy with an appropriate estimate. For example:

  • 6 000 (by grouping nice numbers)
  • 6 000 (by rounding)
  • 5 900 or 6 000 (using front-end and final compensation)
 difficult
c) An acceptable estimation strategy with an appropriate estimate. For example:

  • 2 150 (by grouping nice numbers)
  • 6 100 (by rounding)
  • 5 140 or 6 000 (using front-end and final compensation)
 very difficult

Results based on a sample of 228 students at a representative sample of schools.

NOTE:
These questions are difficult because students often attempt to use an estimation strategy, but either could not appropriately apply it, or could not produce a reasonable estimate with it.

For further information about estimation, refer to Computational estimation concept map.
Teaching and learning: 
Students at this level who are learning to calculate numbers appear to have a reluctance to estimate. If they can work out the answer then they cannot see a reason for estimating. 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.
 
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) 300 + 200 + 100 or 3 + 2 + 1
b)2 000 + 3 000 or 2 + 3
c) 400 + 500 + 200 + 500 + 300 or 4 + 5 + 2 + 5 + 3		 600
5 000
1 900		 1%
1%
2%
Front end and final compensation a) 600 + 60 + 30 + 70 or 600 + 60 + 40 + 70
b) 5 000 + 300 + 600 or 5 000 + 400 + 600
c) 1 900 + 40 + 30 + 40 + 70 + 60		 760 or 770
5 900 or 6 000
2 140		 7%
6%
5%
Rounding a) 400 + 200 + 200
b) 2 000 + 4 000
c) 400 + 500 + 200 + 600 + 400		 800
6 000
2 100		 6%
5%
9%
Grouping nice numbers a) (230 + 170) + 360 or (360 + 240) + 170
b) (2 400 + 3 600)
c) (440 + 360) + (530 + 570) + 250		 760 or 770
6 000
2 150		 21%
16%
14%

NOTE:

  • The percentages include students who attempted that strategy, but did not necessarily come up with an appropriate estimate.
  • Results are based on 228 students from a representative sample of schools.
  • Other estimation adjustment strategies are possible for example rounding with final compensation, interval estimation, or rounding one number.
 
 
 Strategy name Likely calculation Typical estimate(s) % using strategy
Exact computation a) 364 + 238 + 174
b) 2 387 + 3 628
c) 438 + 531 + 247 + 571 + 361		 776
6 015
2 148		 20%
23%
26%
Exact computation only: using place value partitioning.
(see Next steps 1)		 a) 300 + 200 + 100 = 600
600 + (60 + 30 + 70) = 760
760 + (4 + 8 + 4)
b) Analogous to above
c) Analogous to above		 776
6 015
2 148		 10%
11%
8%
Exact computation and then rounding.
(see Next steps 2)		 a) 364 + 238 + 174
b) 2 387 + 3 628
c) 438 + 531 + 247 + 571 + 361		 770, 775, etc.
6 000, 6 100.
2000, 2100, etc.		 7%
4%
4%
Rounds but still cannot do calculation mentally
(see Next steps 3)		 a) 365 + 240 + 170
b) 2 380 + 3 620
c) 440 + 530 + 250 + 570 + 360		 770 etc.
6 000
2 150		 22%
29%
28%

Results based on a sample of 228 students at a representative sample of schools.

Next steps: 
  1. Exact computation based on place-value partitioning is closely linked to front-end estimation with final compensation. The student continues to compensate until they reach an exact answer. This is effectively a left-to-right addition algorithm. Work with front-end resources, and emphasise that estimation does not need an exact answer.
  2. Discuss that estimation aims to eliminate exact computation. Expose students to other estimation strategies other than rounding (refer to Computational estimation information). This could be through a discussion of the ways different students in the class estimated with these problems. Students may equate rounding and estimation.
  3. Question the student if they can easily do the mental computation after the rounding that they have specified. If they cannot, they need to be exposed to the idea that numbers should be rounded to ones that they can easily add mentally. Students may think that any rounding implies estimation.

 

Links with Numeracy framework
Estimating these numbers with an appropriate estimation strategy involves advanced additive part-whole (Stage 5) knowledge and strategies (See The Number Framework: Book 1 (2004) pp.11 & 13. Wellington: Ministry of Education).
 
For further information about estimation, click on Computational estimation concept map.