Transcript Estimation Strategies

Chapter 13
Using Computational Estimation
with Whole Numbers
Presented by Kelly Dau, February 14, 2012
How do you use estimation?
• Curriculum Focal Points (NCTM, 2006)
includes computational estimation with whole
numbers stating the goal is for students to
“select and apply appropriate methods.”
• NCTM Standards, Number and Operations:
Instructional programs from prekindergarten through grade 12 should enable
all students to compute fluently and make
reasonable estimates.
• Principles and Standards defines
computational fluency as “having and using
efficient and accurate methods for
computing” (NTCM, 2000, p.32).
Big Ideas
• Multidigit numbers can be built up or
taken apart in a wide variety of ways.
436 = 400 + 30 + 6 or 400 + 36
• Computational estimations use ‘nice’ or
‘friendly’ numbers so that the resulting
computations can be done mentally.
17 ~ 20 or 269 ~ 300
Content Connections
• Operations, Place Value, and WholeNumber Computation: Many of the skills
of estimation grow out of invented
strategies for computation.
• Estimation with Fractions, Decimals, and
Percents: Once you understand whole
number estimation, few new strategies are
required for estimation with other types
of numbers.
By itself, the term
estimate
refers to a number that
is a suitable approximation
adequate for the situation.
I like to look at estimation as
a roadmap to the answer!
Estimation is a higherlevel thinking skill!
• Students may find computational
estimation uncomfortable.
• Explicitly teaching strategies (as
early as grade two) will help children
develop this understanding.
Three Types of Estimation
• Measurement estimation – determining
the approximate measure (length,
weight, volume).
• Quantity estimation – approximating
the number of items in a collection.
• Computational estimation – determining
a number that is an ‘approximation’ of a
computation (this is NOT a guess!).
Computational Estimation
in the Curriculum
• Is often underemphasized!
• Teachers MUST NOT use the word
guessing when working on estimation.
• Teachers should explicitly help
students see the difference between
a guess and an estimate!
It takes a Decision!
Estimation is a decision of
whether you should solve or
estimate and, if you are going
to estimate, what strategy will
you use!
Use REAL examples of
why we estimate!
I want to serve pizza at my
birthday party. How much will I
need to order? I am having 7
guests, and I usually eat two
pieces myself (I’m a big eater)!
Language of Estimation
•
•
•
•
•
About
Close
Just about
A little more (or less) than
Between
NEVER SAY GUESS!
Accept a RANGE of estimates!
What estimate would you give for 27 x 325?
Focus on Flexible
Methods, NOT Answers!
Ask questions that provide a possible result.
$21 + $15 + $27
For three prices, the question “About how
much?” is quite different from “Is it more
or less than $60?”
Practice
Over or Under?
•
•
•
•
37 + 75
712 – 458
17 x 38
349 / 45
over/under 100
over/under 300
over/under 400
over/under 10
More excellent examples in Chapter 13!
Computational
Estimation
Strategies
Front-End Method
Estimation is made on the basis of front-end digits with
an adjustment made for what was ignored.
Addition:
429
37
+651
1000 + 100 more = 1100 (actual 1117)
Subtraction:
347
-221
100 + 20 more = 120 (actual 126)
*This works also with multiplication/division but stress the importance
of adjustments…the margin of error is much greater. Also, for
instruction, write the division problems horizontally to discourage the
thinking of ‘goes into.’
Let’s try a front-end
estimation!
$ 398 + $4250 + $2725 = ________
Rounding Methods
• Rounding simply means to substitute
a ‘nice’ or ‘friendly’ number to make
the problem easier to compute
mentally (i.e., 5, 10, 100).
• Number lines and multiplication
tables are useful tools for teaching
‘friendly’ numbers!
Rounding
• When several number are to be added,
round them to the same place value.
• For addition/subtraction problems with
only two terms, round only one.
456 + 338 + 271 = 1065
6724 – 1863 = 4861
500 + 300 + 300 = 1100
6724 – 2000 = 4724
*When multiplying/dividing, the error can be significant; a good strategy
is to round one number up and the other number down. In division, find
two ‘nice’ numbers rather than round to the nearest benchmark (4325/7
can be estimated to 4200/7).
Let’s try a problem with
rounding!
$4827 + $710 + $85 =
46 x 83 =
*Round to the nearest 10; it’s important to identify place value for
students!
Compatible Numbers
Look for two or three numbers that can be grouped to
make benchmark values (e.g., 10, 100, 500). This works
especially well in division.
A box of 36 cards is $6.95. How much is that per card?
36 x 2 = 72…or…36 x 20 = 720
$6.95 is close to $7.20, so the cards cost
a little less than $.20 each!
*Language of estimation!
Now, let’s try
Compatible Numbers!
$14.20
11.50
8.79
6.15
2.75
2.00
Two more methods!
• Clustering: Useful for a large list of addends that are
relatively close in value. All the values in this problem
68 + 52 + 81 + 5 are close to 60; 4 x 60=240 (actual
256).
• Using Tens and Hundreds: Sometimes one number can
be changed to take advantage of benchmark numbers.
456 x 5 could be viewed as 456 x 10 = 4560 when
divided by 2 is approximately 2300; 429/5 could be
seen as 429/10 ~ 43 x 2 = 86 (actual 85.8).
Review
• Estimation skills are a tool for
everyday living as well as a tool for
sense making in other areas of math.
• Do NOT use the term ‘guessing’.
• DO explicitly teach estimation skills!
Why???
YOU may be asked this…
How many basketballs would fill this pit - The Fort Knox Gold
Mine? This is an exercise in estimation - a really valuable math skill!
Useful Links
• http://www.fi.uu.nl/toepassingen/00062/sch
atten/welcome_en.html
• http://www.ixl.com/math/grade-2/estimatesums
• http://www.ixl.com/math/grade-5/estimatesums-and-differences-word-problems
• http://www.quia.com/jg/65924.html
• http://www.shodor.org/interactivate/activiti
es/ (numerous activities)