How I estimate: Multiplication

How I estimate: Multiplication

Pencil and paper
Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
This task is about different ways of estimating in multiplication.

There were 14 friends at a birthday party. They were each given a bag, and each bag had about 26 jellybeans in them.

 
a)
Estimate about how many jellybeans there were altogether and show how you did this in the box below.
 
 
 
 
 
 
 

My estimate is __________ jellybeans.
 
b)
  
Show another way of estimating about how many jellybeans there were.
 
 

 
 
 
 
 

My estimate is __________ jellybeans.
 
c)		  
Show a third way of estimating about how many jellybeans there were.

 

 
 
 
 
 

My estimate is __________ jellybeans.
Task administration: 
This task is completed with pencil and paper only.

The task is best done after students have had teaching on alternative strategies for estimation. 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. Emphasise that this is a multiplication problem (14 × 26).
  2. Ask the student(s) to make an estimate of the number of jellybeans and show how they did it in the box in a).
  3. Ask if they can think of up to two more different ways of making an estimate for this.
  4. Have a discussion sharing the different methods used. Prompt students if they think that each estimate is too small, about right, or too large (see Next steps 4. and 5. at the end of this resource).
Level:
4
Curriculum info: 
Maths, Number and Algebra, Number Strategies
Key Competencies: 
Using language, symbols, and texts
Keywords: 
estimation, multiplication
Description of task: 
Students show how they estimate in some multiplication problems and come up with sensible estimates.
Curriculum Links: 
This resource can help to identify students' ability to apply multiplicative strategies flexibly to whole numbers.
 
Key competencies
This resource involves recording multiple strategies they could use to estimate a multiplication problem. This relates to the Key Competency: Using language, symbols and text and Thinking (seeing multiple strategies).
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)
b)
c)

 Acceptable estimation strategies with an appropriate estimate.
For example:

  • 300 (10 × 30 – rounding)
  • 200 (10 × 20 – front-end)
  • 350 (7 × 50 – doubling and halving)
 

Results based on a sample of nine Year 8 students in one-to-one interviews.
 
NOTE:
  • Rounding each number to the nearest 10 (i.e., 10 × 30) was by far the preferred option.
  • A group of nine Year 8 students were given this problem both before and after they had completed a unit emphasising the use of a wide range of estimation strategies. Before the unit most students could come up with just one estimation strategy with three giving two strategies. After the unit, all but one student could come up with at least two strategies, with two suggesting four different ways of estimating. A wider range of strategies were used overall.
     
Diagnostic and formative information: 

Strategy name

Likely calculation

Typical estimates

Front-end

10 × 20

200

Front-end and final compensation

10 × 20 + 4 × 20 + 6 × 10

340

Rounding

10 × 30 or
20 × 20

300
400

Rounding and intermediate compensation

10 × 36 or
10 × 34

360
340

Rounding up

20 × 30

600

Nice numbers

15 × 25 (only accept if it can easily be done mentally using facts about the 25 times table)

375

Rounding one number

10 × 26

260

Rounding one number and final compensation

10 × 26 + 4 × 25

360

Doubling and halving

7 × 52 7 × 50

350

NOTE:

  • Accept if students attempt a strategy, but do not necessarily come up with an appropriate estimate.
  • Results are based on a sample of nine Year 8 students in one-to-one interviews. 

Strategy name

Likely calculation

Typical answer(s)

Exact computation

14 × 26

364

Exact computation and then rounds
(see Next steps 2)

14 × 26

360

Rounds but still cannot do calculation mentally (see Next steps 3)

15 × 25 (but does vertical algorithm or other exact method)

375

Prior knowledge needed

  • A firm base of multiplicative facts.
  • Good place value concepts.
  • Ability to mentally add numbers in tens, hundreds, etc, using standard place value concepts.
Next steps: 
  1. Expose students to a wide range of estimation strategies (refer to Computational estimation information). This could be through a discussion of the ways different students in the class estimated with these problems. Students may equate rounding and estimation.
  2. Discuss that estimation aims to eliminate exact computation.
  3. Question the student if they can easily do the mental computation after the rounding that they have specified. If they cannot, they need to be exposed to the idea that numbers should be rounded to ones that they can easily add mentally. Students may think that any rounding implies estimation.
  4. Discuss the concept of intermediate compensation. i.e. Reduce one number by a similar proportion as you increase the other one by.
  5. Students who see 10 × 30 = 300 as "About right (because I rounded down by 4 and up by 4)" can be challenged with the question "What about 20 × 20 = 400? (Is it about right?)". They should be perplexed that 300 and 400 are both "about right". One response was "The answer must be between them" (Interval estimation). A more sophisticated answer, based on proportional reasoning, was "Taking 14 down to 10 is taking off 4/14 so I need to add about 4/14 to 26, which is about 8, so its about 10 × 34 = 340".
For more information about computational estimation see the Computational estimation concept map.
 
Links with Numeracy Project

Estimating these numbers with an appropriate estimation strategy involves advanced additive (early multiplicative) part-whole (Stage 6) knowledge and strategies (See The Number Framework: Book 1 (2004) pp.10 & 14. Wellington: Ministry of Education).