Indicate the relative advantages of exact and approximate solutions to problems and give answers to a specified degree of accuracy.

Identify the advantages of exact answers and estimates, or approximate solutions to problems. Evaluate problems to determine if an estimate or exact answer is needed.

Randy’s family went to the movies. They bought two adult tickets for $8.50 and three children’s tickets for $5.75. What is the least amount of money they need to pay for the tickets? Indicate whether you need an estimate or exact answer, then solve.

(Exact; $34.25)

(2)

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? Indicate whether you need an estimate or exact answer, then solve.

(Exact; 3 ¼)

(3)

The librarian has 329 books that she’d like to divide amongst the 3 sixth grade classrooms. How many books should she give each classroom? Indicate whether you need an estimate or exact answer, then solve.

(Estimate; She won’t be able to divide the books evenly, so she will give each class 109 or 110.)

(4)

It is 10 a.m., Amanda is driving at an average rate of 65 mph. She has to drive 100 miles to get to her destination. At about what time can she expect to arrive at her destination?

(Estimate; about 11:30 a.m.)

(5)

A bus holds 50 students. How many busses are needed to transport 116 students. ? Indicate whether you need an estimate or exact answer, then solve and explain.

(Exact or overestimate; there must be at least 3 buses. If an underestimate is made, there will be 16 students that won’t fit on the bus.)

When solving problems for math or in real life situations, sometimes an exact answer is required, and other times an estimate, or approximate answer will do. It is important for children to read word problems carefully to determine if an exact answer is needed or an approximate answer will do. Some key words that signal an estimate are; about, approximate, and close. There are two types of estimates, and overestimate and an underestimate. An overestimate is an amount that is more than the actual answer. An overestimate can be calculated by rounding the numbers from the original problem up. An underestimate is an amount that is more than the actual answer. In order to underestimate for a problem, the numbers in the original problem need to be rounded down. Children will need to use logic to determine when an overestimate is need and a when an underestimate may be used.

Sample Problem - Overestimate

Tom is having 5 guests over for lunch. The recipe he’s making calls for 15 oz. of turkey for 4 guests. How much turkey should Tom buy in order to make sure each guest has more than enough? In this problem, Tom wants more than enough. These words signal that an overestimate is needed. If we look back at the original problem, we can estimate that Tom needs about 4 oz. of turkey per guest, because 4 x 4 = 16. Now we will use this to overestimate by multiplying 4 x 5 guests = 20 oz. Tom needs at least 20 oz. of turkey per guest.

Sample Problem – Underestimate

Josephine is making party bags for the guests at her birthday party. She wants to put an equal amount of candy in each bag. After all bags have been filled, she will put any extra candy out for her guests to snack on. Josephine has 140 pieces of candy and 15 party bags to fill. About how many candies will each party bag contain? In this problem, we cannot overestimate, because there are only 140 pieces of candy. So, this number cannot be increased. An underestimate will have to be made to keep the total candies distributed under 140. We can quickly underestimate by rounding 15 party bags down to 14. When we do this, we will be able to divide into 140 evenly and mentally. Each party bag will have 10 candies.

(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)

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.

(4)

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

(5)

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.

Determine if you need an exact answer or if an estimate can be used to solve the problem. Explain.

Megan is making fruit punch for her pool party. The package says that 2 quarts serves 5 glasses. Megan is expecting 27 guests. How many quarts should she make so that she has more than enough for all of her guests to drink?

(An estimate can be used. She does not say she wants exactly enough for one serving per guest, she only wants each guest to have enough.)

(2)

Determine if you need an exact answer or if an estimate can be used to solve the problem. Explain.

William’s mom gave him $50.00 to split evenly with his 3 brothers. How much does each brother get?

(An exact answer is needed. William’s mom wants it to be split evenly, which is exactly in fourths.)

(3)

Determine if you need an exact answer or if an estimate can be used to solve the problem. Explain.

Yasmin is driving 50 mph. She needs to be at her sister’s party by noon. It is 11:00 now, she has 70 miles left to drive before she reaches her sisters house. Will she make it?

(An estimate can be used, we don’t need to know exactly what time she will be there, only an approximate time to see if she can make it.)

(4)

Determine if you need an exact answer or if an estimate can be used to solve the problem. Explain.

Amir has 4-1/2 feet of wood. If he cuts ¾ from it, how much wood will remain?

(An exact answer is needed, since the question clearly asks how much wood is remaining.)

(5)

Determine if you need an exact answer or if an estimate can be used to solve the problem. Explain.

Holly is baking cookies. She needs 3-1/4 cups of flour for one batch. She is making two batches. If she has 7 cups of flour, does she have enough for both batches?

(An estimate can be used. The problem is asking if she has enough, this can be estimated by looking at the whole numbers and fractions.)

(6)

Would an underestimate or overestimate be needed to solve the problem? Explain.

Megan is making fruit punch for her pool party. The package says that 2 quarts serves 5 glasses. Megan is expecting 27 guests. How many quarts should she make so that she has more than enough for all of her guests to drink?

(An overestimate should be used. She wants to make sure she has more than enough. The words more than enough, mean more is better; more is made with an overestimate)

(7)

Would an underestimate or overestimate be needed to solve the problem? Explain.

Yasmin is driving 50 mph. She needs to be at her sister’s party by noon. It is 11:00 now, she has 70 miles left to drive before she reaches her sisters house. Will she make it?

(It would be best to use an overestimate, so she doesn’t think she’ll be on time when she’s really late.)

(8)

Would an underestimate or overestimate be needed to solve the problem? Explain.

Mrs. Curtis has 125 pencils to distribute to her students. She would like to give each student the same amount. She has 32 students. About how many pencils will each student receive?

(Underestimate, if she only has 125 pencils, you can’t go over that amount.)

(9)

Would an underestimate or overestimate be needed to solve the problem? Explain.

Mr. Finau had $2,236 in the bank. He wrote checks for $1,296.32, $527.44, and $200.22 to pay his bills. About how much money doe he have left to spend on himself?

(It would be best to overestimate, because if he underestimates and spends too much money on himself, he will owe the bank money.)

(10)

Would an underestimate or overestimate be needed to solve the problem? Explain.

Holly is baking cookies. She needs 3-1/4 cups of flour for one batch. She is making two batches. If she has 7 cups of flour, does she have enough for both batches?

(She should overestimate so that she knows whether or not she has enough flour to make a double batch before she begins to mix the ingredients.)

(11)

Megan is making fruit punch for her pool party. The package says that 2 quarts serves 5 glasses. Megan is expecting 27 guests. How many quarts should she make so that she has more than enough for all of her guests to drink? Indicate whether you need an estimate or exact answer, then solve.

(Estimate; 30 quarts)

(12)

William’s mom gave him $50.00 to split evenly with his 3 brothers. How much does each brother get? Indicate whether you need an estimate or exact answer, then solve.

(Exact: $12.50 each)

(13)

Mr. Finau had $2,236 in the bank. He wrote checks for $1,296.32, $527.44, and $200.22 to pay his bills. About how much money doe he have left to spend on himself? Indicate whether you need an estimate or exact answer, then solve.

(Estimate; sample estimate $180.00)

(14)

Holly is baking cookies. She needs 3-1/4 cups of flour for one batch. She is making two batches. If she has 7 cups of flour, does she have enough for both batches? Indicate whether you need an estimate or exact answer, then solve.

(Estimate; 7 cups)

(15)

Amir has 4-1/2 feet of wood. If he cuts ¾ from it, how much wood will remain? Indicate whether you need an estimate or exact answer, then solve.