Journalling about estimation

Journalling about estimation

Pencil and paperOnline interactive
Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
This task is about describing how a method of estimation works.

Question 1Change answer

a)  Pat was away from school when his class learnt about using the front-end method of estimation for addition problems.
 
Write an explanation to Pat to show him how to use the front-end method for estimating an addition problem.
Task administration: 
This task is completed with pencil and paper or online.
  • Students need to have been introduced to the front-end method prior to being asked this question.
  • Other methods of estimation can be asked about. The scoring needs to be adjusted accordingly.
Prior knowledge needed

  • Good place value concepts.
  • It is essential that the students are comfortable with the concept of computational estimation (getting answers that are close to the real answer using mental computational strategies).
Level:
4
Curriculum info: 
Maths, Number and Algebra, Number Strategies
Key Competencies: 
Using language, symbols, and texts
Keywords: 
estimation, addition, journalling
Description of task: 
Students write down an explanation of how the front-end estimation strategy works in addition problems to give information on their understanding of the strategy.
Curriculum Links: 
Key competencies
This resource involves describing an estimation strategy for addition.  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: 
Description of front-end (typical student responses are given)
 
Complete answer. 
Mentions using only the most significant digit or the highest place-value digit.
"Its when you add the first numbers in the highest place value together with the other numbers in the highest place value. Basically you start with the front-end."
 
Near complete answer. 
Mentions using only the first (left-most) digit.
"The trick is just add the first digits 6 + 7 + 8 = 2 100 (is an estimate for 614 + 721 + 809)."
 
Partial answer. 
Use more than one of the first (left-most) digits.
"You could take the first 2 or 3 digits and then add them [for example in a vertical algorithm] with the extra zeros on the end."
Teaching and learning: 
Information of estimation method used

  • The front-end estimation strategy generally uses only the most significant (left-most) digit of the numbers being estimated. This strategy is most powerful when adding.
  • The most sophisticated explanations require mentioning the most significant digit only because of examples such as: 7 542 + 436 + 6 125 is about 7 000 + 6 000 (i.e., the 400 in the second number is ignored because the initial estimate is in the thousands only). Refer to Estimating sums of money part d) to see what your students do with numbers of different "lengths" (orders of magnitude). Students who have near complete answers may have this understanding, but have not communicated it sufficiently. You may need to conference them.
  • Final compensation occurs after the initial estimate is made. The estimate is updated to take account of about how far out the initial estimate is, usually using the second most significant digit.
Diagnostic and formative information: 
Description of compensation methods (typical student responses are given)

  • Complete response. Use next most significant digit to update the estimate.
    "If you want a more exact answer, take the next (most significant) numbers and add them together (then add this to the initial estimate)."
  • Near complete response. Use the second digit to update estimate.
    "If you want a more accurate answer, add on the next numbers 8 228, 3 928, 4 827."
  • Partial responses
    Use more than 2 digits to update estimate.
    "You could just take the first 2 or 3 digits . them add them with the extra zeros."
    Use all digits to update estimate in a left-to right order (i.e., exact computation).
    "Keep working your way down to the lowest place value digit."

(Marks not apportioned, as compensation was not asked for.)

Examples of student's working

  • "8 428 + 3 929 + 4 827, then add together like this 8 + 3 + 4 = 15, (using the extracted digits) or 8 000 + 3 000 + 4 000 = 15 000." (using the same number of digits)
  • 342 (lays it out as a vertical algorithm but only processes the left-most digit)
    + 597
    800
  • "694 382 + 365 938 + 126 131

Do (front) 6 + 3 + 1 = 10, (next in line) 9 + 6 + 2 = 17, (next after that) 4 + 5 + 6 = 15|
Then add the with the zeros 1 000 000
                                             170 000
                                            + 15 000
                                           1 185 000 " (encourage as much as possible to be done mentally)

Links with Numeracy framework
The concepts needed for front-end estimation involve early or advanced additive thinking and part-whole thinking (i.e., stages 5 or 6 of the Number Framework).
See The Number Framework: Book 1 (2004) p.10. Wellington: Ministry of Education.

For other resources involving front-end with addition see Estimating sums of money.
For further information about estimation, refer to the Computational estimation concept map.