Estimating sweets

Estimating sweets

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.
sweets-candy-chocolates-CK.png
Some children want a quick estimate of the number of bags they need to put some sweets into for the school fair.
 
Make an estimate of about how many bags they each need, and explain how you estimated this.

Question 1Change answer

a)  Zac has 225 pieces of chocolate, and he decides to put 6 pieces in each bag. 
The number of bags he needs is about .
This is because ...

Question 1Change answer

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

Question 1Change answer

b)  Troy has 198 pieces of toffee, and he decides to put 7 pieces in each bag. 
The number of bags he needs is about .
This is because ...

Question 1Change answer

c)  Lex has 1,433 jaffas, and she decides to put 30 in each bag. 
The number of bags she needs is about .
This is because ... 

Question 1Change answer

d)  Wendy has 3,271 jubes, and she decides to put 67 in each bag. 
The number of bags she needs is about .
This is because ...
Task administration: 
This task can be completed with pencil and paper or online.
 
This assessment activity is designed to be carried out with a group of students or a whole class. A suggested lesson sequence:
  1. Give the students the problem of estimating 225 ÷ 6 to identify whether they already know how to use nice numbers and factors.
  2. Discuss students' methods. Use the language of estimation in the Computational estimation concept map for each different method.
  3. Introduce the nice numbers and factors method in division, i.e., change at least one of the numbers so that one number is a factor of the other. This should be based on basic multiplication facts. You may wish to write this example down as you discuss it.
    Example: Estimate 225 ÷ 6
    Change 225 to 240
         because 24 ÷ 6 = 4, so 240 ÷ 6 = 40 or
         because 6 × 40 = 240
    Discuss why this is closer to the exact answer than changing 225 to 180 because there is  a smaller change to go to 240. Also, 30 is an underestimate of the number of bags so he  will not have enough.
  4. Students complete the sentence with the word "nice numbers and factors", and then answer parts b) to d) to assess if they can apply it. Emphasise they are to use the nice numbers and factors method only.
For more information about discussions click on  Classroom discourse (Mathematics)
 
 
Level:
4
Curriculum info: 
Maths, Number and Algebra, Number Strategies
Key Competencies: 
Using language, symbols, and texts
Keywords: 
estimation, division, discourse
Description of task: 
Students estimate a division problem, share their methods, discuss the nice numbers and factors method, and then do more problems using nice numbers and factors.
Curriculum Links: 

This resource can help to identify students' ability to apply multiplicative strategies flexibly to whole numbers when estimating quotitive division problems.

 
Key competencies
This resource involves recording the strategies students use to estimate in quotitive division problems. This relates to the Key Competency: Using language, symbols and text.
For more information see http://nzcurriculum.tki.org.nz/Key-competencies
 
Learning Progression Frameworks
This resource can provide evidence of learning associated with
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 40 and gives a reasonable explanation of their method of estimating it.  The nice numbers method would be:
40 because 240 ÷ 6 = 40 or
24 ÷ 6 = 4, so it is 40

b)

30 because 210 ÷ 7 = 30 (or because 7 × 30 = 210)
or equivalent estimates and explanations.

c)

50 because 1 500 ÷ 30 = 50 (or because 30 × 50 = 1 500)
or equivalent estimates and explanations.

d)

Any 1 of:

  • 50 because 3 500 ÷ 70 = 50 (or because 70 × 50 = 3 500);
  • 50 because 3 000 ÷ 60 = 50 (or because 70 × 50 = 3 500);
  • equivalent estimates and explanations.

NOTE:

  • Other strategies are possible. If they don't use the nice numbers and factors methods, conference them to see if they can, even if they prefer other strategies.
  • Students may use a combination of estimation and computation (i.e., computations that they cannot easily do in their heads and may need to write down, e.g., 230 ÷ 6).
Diagnostic and formative information: 
  Common error Likely calculation Likely misconception
a)
b)
c)
d)		 37.5
28.28
47.76
48.82		 225 ÷ 6
198 ÷ 7
1 433 ÷ 30
3 271 ÷ 67		 Performs exact calculation rather than estimation.
a)
b)
c)
d)		 30
20
40
40		 180 ÷ 6
140 ÷ 7
1 200 ÷ 30
2 800 ÷ 70		 Rounds down to a nice number and therefore underestimates the number of bags. Indicates they have the idea of nice numbers, but they need to choose a factor that gives an answer closer to the exact one.
a)
b)
c)
d)		 38.33
28.57
46.67
47.14		 230 ÷ 6
200 ÷ 7
1 400 ÷ 30
3 300 ÷ 70		 Rounds to the nearest 10, 100, etc., then does an exact calculation.
a)
b)
c)
d)		 37 or 38
28 or 29
47 or 48
48 or 49		 225 ÷ 6 (or 38 × 6 = 228)
198 ÷ 7 (or 29 × 7 = 203)
1 433 ÷ 30 (or 48 × 30 = 1 440)
3 271 ÷ 67 (or 49 × 67 = 3 283)		 Does an exact calculation but either rounds up or down to get an answer (or works out an exact multiple of one number rather than using basic facts ones).

Information on using nice numbers and factors in estimation

  • Nice numbers is often known as compatible numbers. It involves making the numbers compatible with respect to the operation being undertaken. In division, this means making one of the two numbers a factor of the other. The relationship should be based on basic multiplication facts.
  • Changing the value of at least some numbers is referred to in the literature as reformulation.
  • Nice numbers division gives a reasonably accurate estimate only if the amount of rounding of one number is proportionally about the same as the amount of rounding of the other number. Both numbers need to be rounded in the same direction, i.e., both rounded up, or both rounded down. Recognising that one number being rounded roughly balances the other one being rounded in the same direction is referred to as intermediate compensation.
  • Final compensation (i.e., updating an initial estimate to a more accurate one) can be done with nice numbers division.

Prior knowledge needed

  • A firm base of multiplicative basic facts.
  • Good place value concepts.
  • Understanding that estimation is a quick mental way of getting an answer close to the exact one.
  • Ability to evaluate the order of magnitude of answers in division.
Links with Numeracy Project
Using nice numbers and factors as an estimation method involves at least advanced multiplicative part-whole (Stage 7) knowledge and strategies (see The Number Framework, Book 1 (2004, p. 11 & 15). Wellington: Ministry of Education).

For similar ARB resources about nice numbers and factors as an estimation method refer to Estimating scores and crowds.
For further information about estimation, refer to Computational estimation concept map.