Estimation Strategies

Strand: Number
Outcome: 2

Step 1: Identify Outcomes to Address

Guiding Questions

What do I want my students to learn?
What can my students currently understand and do?
What do I want my students to understand and be able to do, based on the Big Ideas and specific outcomes in the program of studies?

See Sequence of Outcomes from the Program of Studies

Strand: Number

Grade 4

Grade 5

Grade 6

Specific Outcomes
3.	Demonstrate an understanding of addition of numbers with answers to 10 000 and their corresponding subtractions (limited to 3- and 4-digit numerals) by: using personal strategies for adding and subtracting estimating sums and differences solving problems involving addition and subtraction.
6.	Demonstrate an understanding of multiplication (2- or 3‑digit by 1-digit) to solve problems by: using personal strategies for multiplication with and without concrete materials using arrays to represent multiplication connecting concrete representations to symbolic representations estimating products applying the distributive property.
7.	Demonstrate an understanding of division (1-digit divisor and up to 2‑digit dividend) to solve problems by: using personal strategies for dividing with and without concrete materials estimating quotients relating division to multiplication.

Specific Outcomes
2.	Use estimation strategies, including: front-end rounding compensation compatible numbers in problem-solving contexts.
5.	Demonstrate, with and without concrete materials, an understanding of multiplication (2-digit by 2-digit) to solve problems.
6.	Demonstrate, with and without concrete materials, an understanding of division (3-digit by 1-digit), and interpret remainders to solve problems.
11.	Demonstrate an understanding of addition and subtraction of decimals (limited to thousandths).

Specific Outcomes
8.	Demonstrate an understanding of multiplication and division of decimals (1-digit whole number multipliers and 1-digit natural number divisors).

Big Ideas

Number Sense
Number sense plays a major role the conceptual understanding of estimation strategies. Number sense is "an intuitive feeling for numbers and their various uses and interpretations; an appreciation for various levels of accuracy when figuring; the ability to detect arithmetical errors; and a common-sense approach to using numbers" (Reys 1992, p. 3).

Curriculum and Evaluation Standards identifies various aspects of number sense:

understanding the meanings of numbers,
having an awareness of multiple relationships among numbers,
recognizing the relative magnitude of numbers,
knowing the relative effect of operating on numbers and
possessing referents for measures of common objects and situations in the environment.

Estimation
Estimation is a mental "process of producing an answer that is sufficiently close to allow decisions to be made" (Reys 1986, p. 22). The types of estimation include quantity, computation and measurement. The focus for estimation described in the Grade 5 Number strand, specific outcome 2, is on computational estimation.

The first four aspects of number sense listed above provide the focus for computational estimation. Principles and Standards for School Mathematics states that estimating first and then calculating "provides a tool for judging the reasonableness of calculator, mental and
paper-and-pencil computations" (NCTM 2000, p. 155). Expanding upon this further it states:

"Students should be encouraged to explain their thinking frequently as they estimate. As with exact computation, sharing estimation strategies allows students access to others' thinking and provides many opportunities for rich class discussions"
(NCTM 2000, p. 156).

As students estimate first and then calculate, they refine their estimation strategies. When estimating, students focus on "the meaning of the numbers and the operations"
(Van de Walle and Lovin 2006, p. 125).

Front-end Strategy
The front-end strategy is a method of estimating computations by keeping the first digit in each of the numbers and changing all the other digits to zeros. This strategy can be used to estimate sums, differences, products and quotients. Note that the front-end strategy always gives an underestimate for sums, products and quotients (2- or 3-digit divided by 1-digit).

Example:
You buy a hamburger for $4.79 and a drink for $1.26. Will a $5 bill cover the cost?

Solution:
Total the front-end (dollar) amounts.
$4 + $1 = $5
Since the front-end strategy always gives an underestimate for a sum, $5 will not cover the cost.

Compensation Strategy
The compensation strategy is a method of adjusting a computational estimate to make it closer to the calculated answer. This strategy is used with the front-end and compatible numbers strategies to provide better estimates.

Example:
You buy a hamburger for $4.79 and a drink for $1.26. Will a $5 bill cover the cost?

Solution:
Total the front-end (dollar) amounts.
$4 + $1 = $5
Using the compensation strategy, group the cent amounts to form dollars:
$0.26 and $0.79 together make a little more than $1.

Final estimate including front-end and compensation:
$5 + $1 = $6
Answer to the problem: A $5 bill will not cover the cost because the cost is a little more than $6.

Compatible Numbers Strategy
The compatible numbers strategy is a method of estimating computations by using "friendly" or "nice" numbers that can be easily calculated mentally. Compatible numbers can be used in estimating sums, differences, products and quotients.

Example:
You are to divide $435 evenly among 7 people. About how much money should each person receive?

Solution:
Using compatible numbers, change 435 to a number closest to 435 and evenly divisible by 7;
i.e., 420.
420 ÷ 7 = 60
Each person will receive about $60.

Using compensation, think "420 is less than 435 so the estimated answer is a little less than the calculated quotient."

Final estimate using compatible numbers and compensation:
Each person will receive a little more than $60, about $62.

Note: When using compatible numbers to estimate quotients with a single digit divisor, the 3‑digit dividend is changed to the nearest multiple of the divisor using multiples of 10, keeping the divisor unchanged.