Estimating with addition III

Estimating with addition III

Pencil and paper
Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
This task is about estimating in addition problems.
Estimation is an approximate answer you can easily work out in your head, NOT the calculation of an exact amount. 
The following questions are about estimating, NOT calculating.
 
a)
Here is an equation: 57 + 24
i) The sum of the equation is about: (Circle one)
(A) 30   
(B) 60   
(C) 80   
(D) 100   
(E) None of these. It is about _______

ii) In the box below, show how you estimated this:

 
 
 
 
 
b) Here is another equation: 259 + 348
i) The sum is about _______  (Do NOT give an exact answer)

ii) In the box below, show how you estimated this:

 
 
 
 
 
c) Here is a third equation: 124 + 438 + 267
i) The sum is about _______  (Do NOT give an exact answer)

ii) In the box below, show how you estimated this:

 
 
 
 
 
Task administration: 
This task is completed with pencil and paper only.
Level:
3
Curriculum info: 
Maths, Number and Algebra, Number Strategies
Key Competencies: 
Using language, symbols, and texts
Keywords: 
estimation, addition
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 strategies flexibly to whole numbers.
Key competencies
This resource involves recording strategies  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 6-7
within the Mathematics Learning Progressions Frameworks.
Read more about the Learning Progressions Frameworks.
Answers/responses: 
    Y7 (03/2005)
a)

 

 C (or E and an appropriate estimate) and an acceptable estimation strategy. For example:

  • 80 (by rounding)
  • 70 (by front-end estimation)
  • 80 (using front-end and final compensation)
 difficult
b)

 

 An acceptable estimation strategy and an appropriate estimate. For example:

  • 590, 600, or 610 (by rounding)
  • 500 (by front-end estimation)
  • 590, 600, or 610 (using front-end and final compensation)
 difficult
c)

 

 An acceptable estimation strategy and an appropriate estimate. For example:

  • 800 or 830 (by rounding)
  • 700 (by front-end estimation)
  •  800, 810, 820 or 830 (using front-end and final compensation)
 difficult
Results based on a sample of 220 students. 
For examples of student responses, see Working with students.
Teaching and learning: 
Students at this level who are learning to calculate numbers appear to have a reluctance to estimate. If they can work it out then they cannot see the 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) 50 + 20 or 5 + 2
b) 200 + 300
c) 100 + 400 + 200		 70 or 80
500
700		 3%
2%
2%
Front end and final compensation a) 70 + (7 + 3)
b) 500 + (50 + 40)
c) 700 + (20 + 30 + 60)		 80
590
810		 10%
17%
16%
Rounding a) 60 + 20 or 6 + 2
b) 300 + 300 or 260 + 350
c) 100 + 400 + 300		 80
600 or 610
800		 25%
19%
11%
Grouping nice numbers a) 55 + 25
b) 250 + 350 or 260 + 340
c) 100 + (440 + 260) or 120 + (440 + 160)		 80
600
800 or 820		 1%
3%
3%

NOTE: These percentages include students who attempted that strategy, but did not necessarily come up with an appropriate estimate.

Strategy name Likely calculation Typical estimate(s) % using strategy
Exact computation only: using the vertical algorithm (approx. 80%) and other methods. a) 57 + 24
b) 259 + 348
c) 124 + 438 + 267		 81
607
829		 6%
10%
11%
Exact computation only: using place value partitioning.
(see Next steps 1)		 a) (50 + 20) + (7 + 4)
b) (200 + 300) + (50 + 40) + (9 + 8)
c) (100 + 400 + 200) + (20 + 30 + 60) + (4 + 8 + 7)		 81
607
829		 2%
3%
4%
Exact computation and then rounding.
(see Next steps 2)		 a) 57 + 24
b) 259 + 348
c) 124 + 438 + 267		 80 or 100
600 or 610
800 or 820 or 830		 20%
9%
8%
Exact computation: using place value partitioning then rounding.
(see Next steps 1 & 2)		 a) (50 + 20) + (7 + 4)
b) (200 + 300) + (50 + 40) + (9 + 8)
c) (100 + 400 + 200) + (20 + 30 + 60) + (4 + 8 + 7)		 80 or 100
600 or 610
800 or 820 or 830		 13%
7%
5%
Total percentage of students - computing exact answer (including exact answer only and rounded answer). 
 		 a)
b)
c)		 41%
29%
28%
Rounds but still cannot do calculation mentally.
(see Next steps 3)		 a) 55 + 25
b) 260 + 350
c) 120 + 440 + 270		 80
810
830		 1%
8%
15%

Results are based on a sample of 220 students.

Next steps: 
  1. Exact computation based on place-value partitioning is closely linked to front-end estimation with final compensation. The difference is that the student continues to compensate or calculate the next place-value digit until they reach an exact answer. This is effectively a left-to-right addition algorithm. It is important that students recognise that an estimation does not need an exact answer. They could work with front-end resources, begin with a single calculation of the largest place value, then perform a single compensation to get a reasonable estimate, and stop. Students who have performed an exact place-value partition for question a), but estimate in b) and c) show good estimation understanding.
  2. Discuss with students how 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 different ways students in the class estimated problems. Students may equate rounding with 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 estimation is about getting an approximate answer that they can easily work out in their head. Students may think that any rounding implies estimation.
  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 more information on estimation, click on 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 (See The Number Framework: Book 1 (2004) pp.11 & 13. Wellington: Ministry of Education).