6th Grade - Are Your Results Reasonable?

 
     
 
     
 
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6th
Reasoning
Are Your Results Reasonable?
Use estimation to verify the reasonableness of calculated results.
Round numbers in a given problem and use to calculate an estimate. Compare an estimate to a solution to check for reasonableness of a result. Make estimations to check: whole number, fraction and decimal calculations involving addition, subtraction, multiplication and division.
 
 

Building Blocks/Prerequisites

 

Sample Problems

(1)

Solve the problem. Estimate to check.

39 + 73

(112, Estimate: 40 + 70 = 110)

(2)

Solve the problem. Estimate to check.

55.95 – 12.43

(43.52, Estimate: 56 – 12 = 44)

(3)

Solve the problem. Estimate to check.

4 ¼ x 5 ¾

(24 11/12, Estimate: 4 x 6 = 24)

(4)

Tiffany bought 5 friends movie passes. Each pass was $4.95. How much did Tiffany spend? Estimate to check.

($24.75, Estimate: 5 x 5 = 25)

(5)

Randy’s family went to the movies. They bought two adult tickets for $8.50 and three children’s tickets for $5.75. How much money did they spend in all? Estimate to check and explain why your answer is reasonable.

($34.25, Estimate: 2 X 9 = 18, 6 x 3 = 18, 18 + 18 = 32. My answer is reasonable because my estimate of 32 is fairly close to $34.25.)

Learning Tips

(1)

In the fifth grade, students learned how to round, estimate and manipulate very large numbers. In order to round numbers, children will need to know place value. You may want to review place value with your child.

(2)

In order to make estimations, your child will need to be able to round numbers. The same rules apply for rounding a whole number, decimal or fraction. To round a number, you first need to decide what place value you’d like to round to. Next, you look at the number to the right of the place value you’ve chosen. If the number to the right is 5 or greater (5,6,7,8, or 9), you will round up. This means you will make the number in the place value you’ve chosen one more. For example, if we were rounding 638 to the nearest 10, 3 is in the tens place. The number to the right of the 3 is 8. The number 8 is equal to or greater than 5, so you will round up. This means the number in the tens place, 3 will be made one more, 4. So, 638 rounded to the nearest ten is 640. Notice how the 8 has been replaced with a zero. The last step of any rounding problem is replacing all numbers to the right of the place value you’re looking at with zeros. Now, if the number to the right of the place value you have chosen is 4 or less (4, 3, 2, 1, 0), you will round down. This means you will keep the number in the place value you have chosen and replace all numbers to the right with 0s to make the number less (rounded down). For example, if we were rounding 4,358 to the nearest thousand, 4 is in the thousands place. To the right of the 4 is a 3. Three is 4 or less, so we will round down. To do this we will keep the 4 and replace every number behind it with a 0. So, 4,358 is 4,000 when rounded down. One mistake commonly made by children in rounding down a number is that they make the number in the place value they are looking at one less. Watch or this and show your child how making this mistake ruins an estimate, because the rounded number is much less than the original number. If, for instance, this mistake were made for the problem above, we would get 3,000 instead of 4,000. As we can see, 3,000 is nowhere near 4,358.


An alternative to teaching rounding is to have your child write down both choices for rounding and then choosing the one that is less for rounded down and more for rounded up. An example of how to do this is shown below.

Round the number 12,082 to the nearest hundred.

Choices: rounded down 12,000 or rounded up 12,100

Look at the number to the right of the hundreds place (8). Round up.

Answer: 12,100

This strategy works well for visual and kinesthetic learners. Have kinesthetic learners touch and read aloud each place value starting with one. Have them stop with their finger on the hundreds place. Next, have him/her write down the choices, touching and saying each one. Lastly, have your child point to the number to the write of the hundreds place. Ask if that number is greater or less than 5. Since it is greater, have your child put his/her finger on the greater estimate.

(3)

Make sure that your child understands that there are many ways to estimate for a given problem. This is true, because you can round numbers at different place values. Below are examples of three different estimations from the same original problem: 4,589 + 346 = 4,935.

Estimate 1: 5,000 + 300 = 5,300

Estimate 2: 4,600 + 300 = 4,900

Estimate 3: 4,590 + 350 = 4,940

All three estimates are correct. The first estimate has the first addend rounded to the nearest thousand and the second addend rounded to the nearest hundred. The second estimation has both addends rounded to the nearest hundred. The last estimate has both addends rounded to the nearest ten. The last estimate would be the closest to the answer to the original problem, but is more difficult to solve using mental math. In this case, estimate two would probably be the best to use, because it is easy to solve mentally and it is fairly close to the original answer. However, remind your child that there is not wrong way in making estimates, as long as each number has been rounded correctly.

(4)

The rules for rounding decimals are the same as the rules for rounding whole numbers. However, most commonly, sixth graders will round decimals to the nearest one or nearest dollar to make estimates. Many children understand this concept much better if you relate it to money. Just be sure to remind your child that any decimal with a 5 or more behind the decimal point will round up (a dollar will be added) and any number that is 4 or less behind the decimal point will stay the same. Don’t forget that we are rounding to the nearest whole number, so we don’t want any change (cents). Here’s an example.

Example: 4.58 + 3.22

5 + 3 = 8


4.58 was rounded up to 5, because the underlined number behind the decimal point is 5 or more.

3.22 was rounded down to 3 because the underlined number behind the decimal point is 4 or less.

(5)

The rules for rounding fractions to estimate are the same as decimals. Since fractions can be written as decimals and decimals as fractions, some children may want to convert fractions into decimals when deciding how to round. To do this, they will need to divide the numerator (top number) by the denominator (bottom number.) However, most children will be able to round the fractions themselves. The rules for rounding a fraction are that you will round up any fraction that is ½ (.5) or larger and you will round down and fraction less than ½. Here’s an example to show you how it works.

Example: 3/8 + 9/8

0 + 1 = 1

3/8 is less than ½, we can figure this out because ½ can be written as 4/8 and 3/8 is less than 4/8, so this fraction was rounded down to 0.

9/8 is greater than ½ or 4/8, so this fraction was rounded up to 1.

The same rules can be used to make estimations with mixed numbers. However, the whole number will become one more for fractions that are rounded up and they will stay the same for fractions that are rounded down (see below).

Example: 5 ½ x 4 ¼

6 x 4 = 24

In this problem, 5 ½ must be rounded up, so the whole number 5 becomes a 6. However, 4 ¼ has a fraction that is less than ½, so it must be rounded down to 4.

You may also to keep fractions that are at ½ and just round other fractions. This will give a more precise answer, as seen below. This is especially helpful when adding fractions or mixed numbers.

5 ½ + 4 ¼

5 ½ + 4 = 9 ½

(6)

Making estimates to find a quotient (solve a division problem), can be challenging. In order for children to be able to solve a division problem in mentally, the child will need to round numbers so that they are compatible. Compatible numbers allow for a number to be divided without having any remainders. The division problem, 400 8 is an example of dividing compatible numbers. We know this because 8 x 5 = 40. We use this information to show that 400 8 = 50.

In order to find compatible numbers, students will not be able to use typical rounding rules. Instead, they will look for numbers near the ones in the original problem that will allow them divide mentally. Here’s an example.

656 32

660 33 = 20

656 can be rounded up to 660, but as you can see 32 was not rounded, it was increased to 33 so it could be divided into 660 evenly.

Extra Help Problems

(1)

Estimate the sum by rounding to the nearest ten.

325 + 516

(330 + 510 = 840)

(2)

Estimate the difference by rounding to the nearest one.

98.15 + 64.459

(98 + 64 = 162)

(3)

Estimate the product by rounding to the nearest one.

3 1/3 x 7 5/8

(3 x 8 = 24)

(4)

Estimate the quotient by rounding each number to a compatible number.

672 69

(sample estimate 700 70 = 10)

(5)

Estimate the product by rounding to the nearest ten thousand.

112,600 x 49,500

(110,000 x 50,000 = 5,500,000,000

(6)

Solve the problem. Estimate to check.

3,456 + 6,976

(10,432; 10,000 or 10,500)

(7)

Solve the problem. Estimate to check.

8,978 – 572

(8,406; 8,300 or 8,400)

(8)

Solve the problem. Estimate to check.

5,462 x 1,321

(7, 215, 302; 7,020,000)


(9)

Solve the problem. Estimate to check.

8,081 93

(86 r 83; 90)

(10)

Solve the problem. Estimate to check.

26,567.42 + 281

(26,848.42; 27,300)

(11)

Solve the problem. Estimate to check.

102,547.01 – 2,782.67

(99,764.34; 100,000)

(12)

Solve the problem. Estimate to check.

72.56 x 3.4

(246.704; 219)

(13)

Solve the problem. Estimate to check.

89.7 9.2

(9.75; 10)

(14)

Solve the problem. Estimate to check.

7/8 + 2-1/4

(3 1/8; 3)

(15)

Solve the problem. Estimate to check.

6-2/7 – 1-1/14

(5-3/14; 5)

(16)

Solve the problem. Estimate to check.

11-1/3 x 4 1/2

(51; 55)

(17)

Solve the problem. Estimate to check.

12-1/3 4 1/6

(2-24/25; 3)

(18)

Sammy ran 8.5 miles in March, 9.8 miles in April, 12.3 miles in May. How many miles did Sammy run in those three months? Estimate to check the reasonableness of your answer.

(30.6 miles; about 30)

(19)

Anissa had $3,548.26 in her savings account. She bought a plane ticket to Africa with $2,378.19 of the money. How much money does she now have in her savings account? Estimate to check the reasonableness of your answer.

($1,170.07; about $1,200)

(20)

Indya is making a cake. She needs 2 ½ cups of sugar for the cake batter and ¾ cup of sugar for the frosting. How much sugar does she need in all? Estimate to check the reasonableness of your answer.

(3 ¼; about 3 ½ )

(21)

Chuck had a 4 1/8 inch plank. He cut a 3 ½ inch piece from it. How much of the plank was left? Estimate to check the reasonableness of your answer.

(5/8; about 0)

(22)

Nina had $563.42 in her bank account. She sent a check for half that amount to her mom. How much money did Nina keep for herself? Estimate to check the reasonableness of your answer.

($281.71; about $300 or about $280)

(23)

Steven played with his online math game for 1.5 hours on Monday, 1/2 hour on Tuesday, Wednesday, Thursday and Friday, and for 2.85 hours on Saturday and Sunday. How many hours total did Steven spend playing his online math game? Estimate to check the reasonableness of your answer.

(9.20, 9 1/5 or 9 hours and 12 minutes; about 10)


 

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