Clothing Groceries Estimation Travel Costs Construction

Computation Estimation Simplified mental calculation Computation estimation is: using some computation using easy mental strategies using number sense using a variety of strategies getting close to the exact answer It is not: just a guess doing hand calculations using a calculator exact Quickly and easily get a number that is close enough to the exact answer to be useful

When do we estimate? When there is no need to have an exact answer and an estimate is good enough: for example "Do I have enough money?" When there is not enough information to get an exact answer: for example, "About how many times will my heart beat in an hour?" To check if the answer from a calculation is reasonable.

Introduce students to estimation Discuss why estimation is important Value the role of estimation Find out where students use estimation and what they know about it Use real examples Use situations where an estimate is acceptable or essential Use the language of estimation Accept a range of estimates Discuss a range of strategies Share each others' strategies Expand the students’ repertoire of estimation strategies through discussion and teaching Do examples that aren’t too hard or too easy Do a little of it often Emphasize mental strategies Sometimes limit the time students have to solve a problem to encourage estimation Link estimation with the reasonableness of exact calculations

Estimation Vocabulary Front-end: take only the first number, cut-off (truncate) the other numbers Adjusting with front-end: compensate for the values that were truncated Compensation: adjust, amend, improve, revise, modify your estimate, make it bigger, make it smaller Rounding: round to the nearest to the desired place value Clustering: numbers “cluster” around a value Compatible numbers: numbers grouped together to make computation easy

Guess vs. Estimate If a local building contractor presented you with a “guess” at how much it was going to cost you to build your new house, you'd still be in the dark on what the actual cost would be. If the builder guessed wrong, he/she would likely go broke through either accepting jobs for which the guess was far too low or lose jobs that the guess was too high. Contractors “bid” on jobs, so a close estimate is important. An actuary (a mathematician who looks at statistical information) guessing how much his company would pay in claims that year, then using that guess to set his company's insurance rates, could in the same (sinking) boat.

Use a variety of strategies There is not really any such thing as a "wrong" estimate... some estimates are less useful than others... any estimate made using the original problem is a valid estimate. The goal is not to find the one correct "estimate" but to have the skill to reason about the numbers being used, to be able to come up with a range that is suitable for using to predict the answer, and to have a quick and easy-to-do method for checking to see what a reasonable answer would be.

Questions You Should Ask "How did you get that answer?" "Why do you think it's a reasonable answer?"

Estimation Strategies

Front-End Estimation

Front-End Estimation 8857 4758 7045 + 2110 8857 4758 7045 + 2110 8000   4758 7045 + 2110 8857   4758 7045 + 2110 8000 4000 7000 + 2000 Draw the line after the first number In each addend.

Estimate to choose the most reasonable answer. Between 11,000 and 15,000 Between 16,000 and 20,000 Between 20,000 and 25,000 Between 28,000 and 33,000 8000   4000 7000 +2000 21,000

Front-end Estimation 62,899 10,236 75,000 + 37,596 62899 10236 75000   10,236 75,000 + 37,596 62899   10236 75000 + 37596 60,000 10,000 70,000 + 30,000 Draw the line after the first number In each addend.

Estimate to choose the most reasonable answer. Between 150,000 and 159,000 Between 160,000 and 169,000 Between 170,000 and 179,000 Between 180,000 and 189,000 60,000   70,000 10,000 + 30,000 170,000

Use Front-end Estimation Estimate the sum: 357,289 + 238,499 = _________ 500,000 580,000 595,000 600,000 Is the front-end estimation reasonable? What could you do to improve the front-end estimate? What is the best estimate and why? 500,000 No, it is low. Estimate with compensation. 600,000 is a closer. Use compatible numbers since 60,000 + 40,000 would be about 100,000 more.

Front-end with Compensation Front-end estimation with compensation gives a check and an easy way to get a "ballpark" answer. Previous Example: 357,289 + 238,499 = _____ "3 hundred thousand plus 2 hundred thousand is 5 hundred thousand. There is about another hundred thousand looking at the remainder of the two numbers, since 57 thousand is almost 60 thousand, and 38 thousand is almost 40 thousand. The answer is about 600,000. Compensation is a great way to develop place value number sense and a very quick way to find an estimate.

Front-End is Precursor to Rounding In real life people estimate in a variety of ways, and the accuracy needed is determined by the reason for the estimate. It is important that children learn that they must consider the context in which they are working plays a part in the strategies they choose to solve a problem. Front-end estimation can be considered a precursor to rounding, since it uses the leading digits and doesn't involve any changing of amounts. It is a great way to introduce estimation, and as students become more proficient with using just the leading digits, the skill of making adjustments should be introduced. It is just one way to find a reasonable answer.

What error is acceptable? The numbers: “59”, “54”, “55” are all “50” using front-end estimation. 1,999,999 + 1,999,999 = 2,000,000 What is a better estimate? Is the error acceptable? 4,000,000 No, 2,000,000 is too low. Learn multiple estimation strategies. Always ask the question: Is the answer reasonable? http://www.aaamath.com/grade3.html

Rounding

Round each number to the thousands place to estimate the sum. 8857   4758 7045 + 2110 8857   4758 7045 + 2110 9000 5000 7000 + 2000

Estimate to choose the most reasonable answer. Between 11,000 and 15,000 Between 16,000 and 20,000 Between 20,000 and 25,000 Between 28,000 and 33,000 9000   5000 7000 + 2000 23,000 Exact: 22,770

Round each number to the hundreds place to estimate the sum. 247 6542 489 + 92 247 6542 489 + 92 200 6500 500 + 100

Choose the most reasonable answer. A little less than 7,000 A little more than 7,000 A little less than 8,000 A little more than 8,000 200 6500 500 + 100 7300 Exact: 7,370

Round each number to the thousands to estimate the sum. 5028 6732 1285 + 835 5028 6732 1285 + 835 5000 7000 1000 + 1000 Exact: 13,880 14,000

Round each number to the hundreds to estimate the sum. 5028 6732 1285 + 835 5028 6732 1285 + 835 5000 6700 1300 + 800 Exact: 13,880 13,800

Identify the estimation strategy Front-End Rounding

Describe the estimation strategy used. Front-End Rounding 3,876 5,814 3,176 + 7,895 4,000 6,000 3,000 + 8,000 3,000 5,000 + 7,000 20,761 21,000 18,000 Which method would you use and why? Closer estimate

Describe the estimation strategy used. Front-End Rounding 9,876 8,514 6,092 + 3,895 9,000 8,000 6,000 + 3,000 10,000 9,000 6,000 + 4,000 28,377 26,000 29,000 Which estimation strategy would you use and why? Closer estimate

Estimate Front-End vs. Rounding

Estimate to choose the most reasonable answer. Between 13,000 and 14,000 Between 12,000 and 13,000 Between 11,000 and 12,000 Between 10,000 and 11,000 5000 6700 1200 + 800 Exact:13700

Estimate to choose the most reasonable answer. Between 6,500 and 7,000 Between 7,000 and 7,500 Between 7,500 and 8,000 Between 8,000 and 8,500 290 6840 481 + 94 Exact: 7705

Real World Problems

Can I buy four for $100? 23 Explain Yes 4 x $25 = $100 and $23 is less than $25 $23 rounds to $20 and 4 x $20 = $80 $23 rounds to $25 and 4 x $25 = $100 (high estimate)

$37 Can I buy three for $100? Explain No 3 x $33 is $99, so 3 x $37 would be more than $100. 3 x $40 is $120

$13.87 Can I buy five for $60? Explain No 5 x $15 = $75 which is more than $60. 5 x $14 = $70 which is more than $60. $60 divided by 5 = $12, and $13.87 is more than $12.00. The soccer ball would have to cost ______ or less for me to purchase five of them for $60. $12.00

Can I buy three for $20? CHESS Explain $4.95 Yes 3 x $5 = $15

Would you have enough money? Estimate to find your answer. If you buy 3 items that cost $4.93 each, will $15.00 be enough to buy all 3 items? Explain. If you buy 2 items for $6.29 and 1 item for $3.55 will $15.00 be enough? Explain. If you buy items for $4.32, $6.90, and $7.86, will $18.00 be enough? Explain. Yes, 3 x $5 = $15 (high estimate) No, 2 x $6 + 1 x $4 = $16 No, $4 + $7 + $8 = $19

Round Exact Estimation Front-End

Round Exact Estimation Front-End 0.1 or 10 0.01 or 100 0.001 1000 0.0001 10,000 1 digit 2 digits Exact Estimation Front-End