Estimating in sport

Estimating in sport

Pencil and paperOnline interactive
Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
This task is about using a way of estimating that you have discussed with your teacher.

Question 1Change answer

yellow-netball-ball-CL-mod.png  cricket-bat-ball-CK.png rugby-ball-CL.png
a)  Emani wrote down her team's scores from 8 games of netball.
     Here are her results:  42   45   35   37   46   33   49   34
An estimate of about how many goals were scored altogether is . 
Show how you made your estimate in this box.

Question 1Change answer

Have a group discussion with your teacher on your ways of estimating for this problem.
Make and explain estimates for each of these problems.

Question 1Change answer

b)  Jamal wrote down the scores from 7 innings of 50-over cricket.
      Here are his results: 172     216     187     192     231     207     178
The number of runs scored is about . 
Show how you made your estimate in this box.

Question 1Change answer

c)  At 6 games of rugby at The Stadium, the crowd sizes were as follows: 
     33 182     29 943     27 284     29 423     32 127     34 119
The total attendance was about . 
Show how you made your estimate in this box.
Task administration: 
This task can be completed with pencil and paper or online.
 
It is designed to be carried out with a group of students or a whole class. A suggested lesson sequence:
  1. Give the students the estimation problem in part a).
  2. Discuss students' methods. Use the language of estimation in the Computational estimation concept map for each different method.
  3. Introduce the averaging method for addition. This is where an addition problem is transformed into a multiplication problem. You may wish to write this example down as you discuss it.
    Example: Estimate 42 + 45 + 35 + 37 + 46 + 33 + 49 + 34
    Because each of the 8 numbers is close to 40, the result is about 8 × 40 = 320
  4. Students complete the sentence with the word "averaging", and then answer parts b) and c) to assess if they can apply it. Emphasise they are to use the averaging method only.
For more information about discussions see Classroom discourse (Mathematics)
Level:
4
Curriculum info: 
Maths, Number and Algebra, Number Strategies
Key Competencies: 
Using language, symbols, and texts
Keywords: 
estimation, addition, multiplication, discussion
Description of task: 
Students estimate an addition problem, share their methods, discuss the averaging method, and then do more problems using averaging.
Curriculum Links: 
This resource can help to identify students' ability to apply multiplicative strategies flexibly to whole numbers when estimating addition problems.
 
Key competencies
This resource involves recording the strategies students use to estimate in 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
Multiplicative thinking, sets 6-7
within the Mathematics Learning Progressions Frameworks.
Read more about the Learning Progressions Frameworks.
Answers/responses: 

a)

Makes a reasonable estimate close to 320 and gives a reasonable explanation of their method of estimating it.
The averaging method would be "It is 320 because there are 8 lots of about 40 so it's about 8 × 40."

1 mark
(for both correct)

b)

1,400 and evidence of using the averaging method (7 × 200).

1 mark
(for both correct)

c)

180,000 and evidence of using the averaging method (6 × 30 000).

1 mark
(for both correct)

NOTE:

  • Other strategies are possible. If they don't use the averaging method, conference them to see if they can, even if they prefer other strategies.
  • Do not accept exact answers (321, 1 383, 186 078) unless they have first made an estimate and then adjusted it using compensation.
  • Responses may be given orally to ensure that writing is not hindering the communication of ideas.
  • Students may use a combination of estimation and computation (i.e., calculations that they cannot comfortably do in their heads).
Diagnostic and formative information: 
  Common error Likely calculation Likely misconception
a)
b)
c)		 321
1 383
186 078		 42 + 45 + ... + 34
172 + 216 + ... + 178
33 182 + ... + 34 119		 Does an exact computation.
a)
b)
c)		 320
1 380 or 1 400
186 000 or 186 000		   Does an exact computation and then rounds.

Information on averaging in estimation

  • Averaging transforms an addition problem into an equivalent multiplication problem. The research literature refers to changing the mathematical structure of the problem (in this case, from addition to multiplication) as translation.
  • The averaging method is sometimes referred to as clustering, as all the numbers are clustered close to a particular one.
  • Final compensation can be done with averaging by accumulating all the differences from the target amount (e.g., how much bigger or smaller than 40, 200, or 30 000 each number is in parts a) to c) respectively).

Prior knowledge needed

  • A firm base of multiplicative facts to 10 × 10.
  • The effect of multiplying numbers by 10, 100, 1 000 (e.g., 20 × 40 = 800).
  • Good place value concepts.
  • Repeated addition is equivalent to multiplication.
For further information about estimation, refer to Computational estimation concept map.
 
Links with Numeracy Project.
Using averaging estimation with addition involves advanced additive/early multiplicative part-whole (Stage 6) knowledge and strategies. If they are still an early additive thinker, they would round each number to 40 and then add them, so they are not using averaging (see Book 1: The Number Framework (2004, p. 10 & 14). Wellington: Ministry of Education).