Using Computational Estimation with Whole Numbers 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 pre-kindergarten 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 Whole-Number 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 higher-level 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? Use the first idea that comes into your head and write down the result. Then try a different approach. Is one superior to the other? Are all answers reasonable?

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 over/under 100 17 x 38 over/under 400 349 / 45 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/schatten/welcome_en.html http://www.ixl.com/math/grade-2/estimate-sums http://www.ixl.com/math/grade-5/estimate-sums-and-differences-word-problems http://www.quia.com/jg/65924.html http://www.shodor.org/interactivate/activities/ (numerous activities)