Estimating food numbers

Estimating food numbers

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

The method we discussed today is called the interval strategyfront-end estimationrounding upaveraging method
For each of these problems:
  • Make a first estimate of what the total score is at least.
  • Make a second estimate of what the total score is no more than.
  • Use the method you discussed today.

Question 1Change answer

Mr Cooke needed to order food for a school picnic.  
 
a)  There were 678 students, and each student needed 4 sandwiches.
 
     I estimate that he will need at least  sandwiches altogether.
     This is because:
 
     I estimate that he will need no more than  sandwiches altogether
     This is because:

Question 1Change answer

b)  There were 23 classes. Each class got 46 bananas.
     I estimate that he will need at least  bananas altogether.
     This is because:
 
     I estimate that he will need no more than  bananas altogether.
     This is because:
Task administration: 
This assessment activity is designed to be carried out with a group of students or a whole class. A suggested lesson sequence:

  1. Introduce the interval strategy in multiplication.
  • Get a first estimate that is smaller than the actual answer by using just the first (most significant) digit in each number (Front-end estimation).
  • Get another estimate that is bigger than the actual answer by rounding-up each number to the next biggest 10, 100, 1 000, etc.

Example. Estimate 138 × 3.
A first estimate is at least 100 × 3 = 300 (front-end).
A second estimate is that it is no more than 200 × 3 = 600 (rounding-up).

  1. Get the students to complete the sentence at the top of their worksheet with "interval". Emphasise they are to use this method when they do parts a) and b).
Level:
4
Curriculum info: 
Maths, Number and Algebra, Number Strategies
Key Competencies: 
Using language, symbols, and texts
Keywords: 
estimation, multiplication
Description of task: 
Students discuss making an interval estimate in multiplication problems (i.e., getting a lower and an upper limit for the actual answer using the front-end and rounding-up estimation methods). They then use this method on two problems.
Curriculum Links: 
This resource can help to identify students' ability to apply multiplicative strategies flexibly to whole numbers when estimating multiplication problems.
Key competencies This resource involves recording the strategies students use to estimate in multiplication problems. This relates to the Key Competency: Using language, symbols and text.
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) i) 2 400 or 2 600
ii) Any 1 of:

  • 600 × 4 = 2 400 or
  • 650 × 4 so it is about 2 600 or
  • other acceptable explanations for an underestimate.

iii) 2 800
iv) 700 × 4 or other acceptable explanations for an overestimate.

 1 mark (for both i) & ii) correct)
1 mark (for both iii) & iv) correct)
b) i) 800 or 900
ii) Any 1 of:

  • 20 × 40 or
  • 20 × 45 or
  • other acceptable explanations for an underestimate.

iii) 1 500
iv) 30 × 50 or other acceptable explanation for an overestimate.

 1 mark (for both i) & ii) correct)
1 mark (for both iii) & iv) correct)
  Total 4 marks

 

Diagnostic and formative information: 
  Common error Likely calculation Likely misconception
a) i)
b) i)		 2 712
1 058		 678 × 4
23 × 46		 Performs exact calculation rather than estimation.

Information on interval estimation

  • Front-end always underestimates the true results because both the numbers being multiplied are smaller than the numbers in the original product. Front-end is where just the most significant digit of a number is used in the estimation.
  • Rounding-up always overestimates the true results because both the numbers being multiplied are bigger than the numbers in the original product. Rounding-up takes each number to the next biggest 10, 100, 1 000, etc.
  • The interval method recognises that the answer lies somewhere between these two extremes.

Prior knowledge required

  • 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.
For more information on estimation, click on Computational estimation concept map.
 
Links with Numeracy Project
Using interval estimation with multiplication involves advanced additive/early multiplicative part-whole (Stage 6) knowledge and strategies (see The Number Framework, Book 1 (2004, p. 10 & 14). Wellington: Ministry of Education).
 
Other ARB resources
For similar resources about estimation with multiplication refer to: