How do we use rounding to estimate differences?
Sometimes subtraction is like addition. There are times when we do not need to compute the exact number. We can estimate a close answer.
To estimate a difference, we subtract numbers that are close to the numbers in the problem but that are easier to work with.
One way we estimate is by rounding. In this strategy, we round each number to the nearest ten, hundred, thousand, or greater place value.
Let’s look at an example of how this strategy is
used to estimate the difference between 68 and 41.
68
41 First we round 68 to the nearest ten.
The number 68 is between 60 and 70. It is closer to 70 than 60. So 68 rounds up to 70. This is the nearest ten.
Then we round 41 to the nearest ten.
The number 41 is between 40 and 50. It is closer to 40 than 50.
So 41 rounds down to 40. This is the nearest ten.
Finally, we subtract the rounded numbers: 68 − 41 rounds to 70 – 40.
68
41 S rounds to S 70
40 30 The extended fact is 70 − 40 = 30.
The estimated difference is 30.
0 10 20 30 40 50 60 6870 80 90 100
Think about where 68 is on the number line. Is it closer to 60 or 70?
0 10 20 30 4041 50 60 6870 80 90 100
Think about where 41 is on the number line. Is it closer to 40 or 50?
Problem Solving:
Finding Distances on a Map
6
Estimating Differences
There is a simple way to decide whether to round a number up or down.
• When rounding to the nearest ten, if the number is halfway or more to the next ten, round up. If the number is less than halfway to the next ten, round down.
• When rounding to the nearest hundred, if the number is halfway or more to the next hundred, round up. If the number is less than halfway to the next hundred, round down.
Let’s use the rounding strategy to estimate a difference by rounding to the nearest hundred.
Example 1
Estimate the difference between 472 and 213.
Round each number to the nearest hundred.
472 – 213 Round 472 to the nearest hundred.
The number 472 is between 400 and 500. It is closer to 500 than 400.
So 472 rounds up to 500. This is the nearest hundred.
Round 213 to the nearest hundred.
The number 213 is between 200 and 300. It is closer to 200 than 300.
So 213 rounds down to 200. This is the nearest hundred.
Finally, subtract the rounded numbers.
The extended fact is 500 − 200 = 300.
The estimated difference is 300.
0 100 200 300 400 500 600
472
0 100 200 300 400 500 600
213
472
213 S rounds to S 500
200 300
How do we determine which place value to round to?
When estimating differences, we follow these steps to round each number:
• Look at the lesser number.
• Round that number to its greatest place value.
• Round the other number to the same place value.
Let’s look at an example of how this works.
Example 1
Estimate the difference between 12,100 and 8,860.
12,100 – 8,860
The number 8,860 is the lesser number. Its greatest place value is the thousands. So round both numbers to the nearest thousand.
12,100
– 8,860 S rounds to S 12,000 – 9,000 3,000 The estimated difference is 3,000.
The number 12,100 is between 12,000 and 13,000. It is less than halfway to 12,000, so it rounds down.
5,000 6,000 7,000 8,000 9,000 10,000 11,000 12,000 13,000
8,860 12,100
Remember to round to the greatest place value of the lowest number.
Reinforce Understanding
Use the mBook Study Guide to review lesson concepts.
Apply Skills
Turn to Interactive Text, page 65.
How do we use estimation to solve problems about distance?
We often use estimation when we determine driving distances. Most of the time, we do not need to know the exact distance from one place to another, so we use estimation.
The most common data displays for driving distances are maps. The map below shows the driving distances between three U.S. cities.
Example 1
Solve the word problem using estimation.
Problem:
The distance from Seattle to Phoenix is 1,508 miles. Rounded to the nearest hundred, the estimated distance is 1,500 miles. The distance from Seattle to San Francisco is 810 miles. Rounded to the nearest hundred, the estimated distance is 800 miles. About how much farther is the distance from Seattle to Phoenix than the distance from Seattle to San Francisco?
We can create an extended fact and subtract to find the estimated distance: 1,500 − 800 = 700.
SAN FRANCISCO SAN FRANCISCO SEATTLE
PHOENIX 810 m
iles 1,5
08 m iles
The distance from Seattle to Phoenix is about 700 miles more than the distance from Seattle to San Francisco.
Problem Solving: Finding Distances on a Map
Problem-Solving Activity
Turn to Interactive Text, page 66.
Reinforce Understanding
Use the mBook Study Guide to review lesson concepts.
Estimation is a good skill to use when reading maps because we usually do not need to know the exact distance.
Activity 1
Find the difference using expanded form. Then write the answer in standard form.
1. 53
27
2. 429
352
3. 697
368
4. 78
59 Activity 2
Find the difference using traditional subtraction.
1. 375
128
247
2. 291
78
213
3. 515
124
391
Activity 3
Estimate the difference. Then use a calculator to compute the exact answer and compare.
1. 98
57
2. 593
257
3. 582
319 Activity 4 • Distributed Practice
Add.
1. 26
+ 25
51
2. 317
+ 146
463
3. 4. 1,120
+ 287
1,407
Model 276
157 S
200 70 6
− 100 50 7 S
200 70 6
− 100 50 7 S
200 60 + 10 6
− 100 50 7
S
200 60 10 + 6
− 100 50 7 S
200 60 16
− 100 50 7
100 10 9 S 100 + 10 + 9 = 119
Model 91
37 S
91 rounds down to 90 37 rounds up to 40 S
90
40 50
The exact answer (using a calculator) is 91 − 37 = 54.
Since 50 is close to 54, the answer is reasonable.
803 + 902
