Who is estimating? Multiplication

Who is estimating? Multiplication

Pencil and paper
Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources

Your teacher will show you responses from five people who were asked to estimate this multiplication:

67 × 34

Some of them were estimating and some were not. For each flash card:

  1. circle either the tick or the cross in the box to show whether that person is / is not estimating;
  2. explain your answer.
 

Estimating or Not? ( or )    (Circle one)

Explanation why they are estimating or not.
a)  

Sam
       

  
b)  

Aroha
       

  
c) 
Gina
       

  
d)
Peter
       

  
e)
Josef
       

  
 

 

Task administration: 
This task is completed with pencil and paper, and other equipment.
 
Equipment: Six flash cards (one with the problem on it, and five each with one person's response on it).

Each students needs a copy of just the first page of the printed task.
Only one set of the six flash cards that follow are needed. These could be laminated for future use.
 
This task can be administered to individual students orally, with the teacher recording responses or with small groups of students or a whole class, with students recording their responses on their own worksheets. A suggested lesson sequence:
  1. Show the students the flash card with the original problem. Emphasise that the people have been asked to estimate the answer to the problem.
  2. Show one flash card at a time. For each one ask your students:
      •  Is this person estimating?
      •  Explain why they are estimating or not.
  3. You may lead into a discussion about estimation, why you estimate, and what estimation is. 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, multiplication, concept cartoons, discourse, key competencies
Description of task: 
In this task students decide which people are estimating and which are not, and explain their reasoning.
Curriculum Links: 
This resource can help to identify students' ability to recognise estimation strategies 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)
ii)

and
Sam has done an exact calculation using a calculator.

1 mark
(for both correct)

b)

 i)
ii)

and
Aroha has used the front-end (first digit) of each number and multiplied them.

1 mark
(for both correct)

c)

 i)
ii)

and
Gina has guessed (and given a specific number).

1 mark
(for both correct)

d)

 i)
ii)

and
Peter has done a calculation that is still too hard to comfortably do in his head.

1 mark
(for both correct)
e) i)
ii)

and
Josef has rounded the numbers and then multiplied them.

 1 mark
(for both correct)

NOTE: Responses may be given orally to ensure that writing is not hindering the communication of ideas. See "Working with Students" for examples of student responses.

Diagnostic and formative information: 
Name Student response Likely misconception
Peter "Yes he is, because he rounded them." Equates rounding with estimation (even when the product cannot be calculated easily).
Aroha "Kind of. She's rounding, but she's not rounding the right way." The estimate or method is not accurate enough. Does not see front-end estimation as legitimate.

 

Name Student response Conception about estimation
Aroha "Yes she is, she rounded it. It made it kind of even easier than the other ways, its really quick." Estimation is quick.
Josef "Yes, he's just rounding it."
"Yes ... because he's making 67 an easier number by making it 70 and he's making 34 an easier number by making it 30."		 Estimation uses rounding.
Aroha "Yeah, she's estimating. She's gone 60 by 30, and that's really simple." Estimation is simple.
Peter

Sam

 "I don't think he's estimating because he's working it out exactly."
"He's not estimating the way I see it. He's just calculated it."
"No, because he might as well get what the (exact) answer is."
"No, he's using a calculator."		 Estimation is neither an exact (right) answer nor using a calculator.
Aroha "Yeah, that's estimating because ... she's rounded it down to an easier way of doing the sum (sic)." Estimation can be front-end (i.e., rounding down or truncation). This can look at the left-most digits i.e., face value (8, 4, and 3) or the total value (800, 400, and 300).
Josef "Yeah, that's an estimation, 'cause he's doing the rounding, and it's probably quite close to what the answer actually is, cause he's rounded it up 3 and down 4 so that's only 1 off." Estimation can use compensation to get a more precise estimate. This example is of pre-compensation (i.e., done before computation).

Quotes from one-to-one interviews of nine Year 8 students after a unit on estimation.

For further information about estimation, refer to Computational Estimation Information.