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Introduction

Recycling

Jose has had many new ideas for improving life at the “Add It Up Ice Cream Stand.” His newest idea focuses on recycling.

In addition to ice cream, the stand also sells sodas that are packaged in aluminum cans. Because you can turn in cans for recycling and receive some money back, Jose thinks that this could be a way for the ice cream stand to generate a little more income.

He explained his idea to Mr. Harris who loved the concept. Jose put out recycling bins the first week of June. On the last day of each month, Jose took the recycled cans to the recycling center and collected money on his returns. He decided to keep track of the additional income in a small notebook.

Here is what Jose collected in June, July and August.

June $25.77

July $33.45

August $47.62

Julie asks Jose about how much he has made in recycling.

She also wants to know about how much more he made in August versus June.

Jose looks at his notebook and just by looking at the numbers can’t remember how to estimate.

The decimals are throwing him off.

You can help Jose, by the end of the lesson you will know how to estimate sums and differences of decimals in a couple of different ways.

Pay attention, we will return to this dilemma at the end of the lesson.

What You Will Learn

In this lesson, you learn the following skills.

  • Estimate sums and differences of decimals using rounding
  • Estimate sums and differences of decimal numbers using front – end estimation
  • Compare results of different estimation methods
  • Approximate solutions to real-world problems using decimal estimation

Teaching Time

I. Estimate Sums and Differences of Decimals Using Rounding

Do you remember what it means to estimate?

To estimate means to find an answer that is close to but not exact. It is a reasonable answer to a problem.

What does the word sum and the word difference mean?

If you think back, you will remember that you have already been introduced to the word sum and the word difference. A sum is the answer from an addition problem. The word difference is the answer of a subtraction problem.

How can we estimate a sum or a difference when our problem has decimals?

The easiest way to estimate a sum or a difference of decimals is to round the decimal.

If we round the decimal to the nearest whole number, we can complete the problem using mental math or at least simplify the problem so that finding an answer is easier.

Let’s look at an example.

Example

Estimate 15.7 + 4.9 = _____

In this problem, we only want to estimate our sum. Therefore, we can use our rules for rounding decimals to help us round each decimal to the nearest whole number.

15.7, the place being rounded is the 5, we look at the 7 and round up.

15.7 becomes 16

4.9, the place being rounded is the 4, we look at the 9 and round up.

4.9 becomes 5

Next, we rewrite the problem.

16 + 5 = 21

Our answer is 15.7 + 4.9 = 21.

We can also use rounding when estimating sums of larger numbers.

Example

Estimate 350.12 + 120.78 = _____

We round each to the nearest whole number to find a reasonable estimate.

350.12 becomes 350.

120.78 becomes 121.

350 + 121 = 471

Our answer is 350.12 + 120.78 = 471.

What about differences in estimations with subtraction?

We can work on these problems in the same way, by rounding.

Example

Estimate 45.78 - 22.10 = _____

45.78 rounds to 46.

22.10 rounds to 22.

46 - 22 = 24

Our answer is 45.78 - 22.10 = 24.

Can we use rounding to estimate sums and differences that involve money?

Of course!! Look at this example and see how it is done.

Example

Estimate $588.80 - $310.11 = _____

$588.80 becomes 589 we can leave off the zeros to make it simpler to estimate

$310.11 becomes 310

589 - 310 = 279

Our answer is $588.80 - $310.11 = $279.00.

Now it is time for you to try a few on your own. Estimate each sum or difference using rounding.

  1. 2.67 + 3.88 + 4.10 = _____
  2. 56.7 - 22.3 = _____
  3. $486.89 - $25.22 = _____

Take a minute to check your work with a peer.

II. Estimate Sums and Differences of Decimals Using Front – End Estimation

We can also estimate using something called front – end estimation.

Front – end estimation is a useful method of estimating when you are adding or subtracting numbers that are greater than 1000.

Here are the steps for front – end estimation.

  1. Keep the digits of the two highest place values in the number.
  2. Insert zeros for the other place values.

Now, let’s apply this to a problem.

Example

Estimate 4597 + 3865 = _____

We follow the rules for front – end estimation since each number is over 1000.

4597 becomes 4500. 4 and 5 are the digits of the two highest place values and we filled in zeros for the rest of the places.

3865 becomes 3800. 3 and 8 are the digits of two highest place values and we filled in zeros for the rest of the places.

Now we can rewrite the problem.

4500 + 3800 = 8300

Our answer for 4597 + 3865 is 8300.

What about a problem where we have one number over 1000 and one number not over 1000?

We can use front – end estimation for the number over 1000, and we can round to the highest place value for the number under 1000.

Example

Estimate 4496 - 745 = _____

4496 becomes 4400 using front – end estimation.

745 becomes 700 by rounding to the nearest hundred.

4400 - 700 = 3700

Our answer for 4496 - 745 is 3700.

Use front – end estimation on your own to estimate the following problems.

  1. 5674 + 1256 = _____
  2. 4632 - 576 = _____
  3. 8932 + 1445 = _____

Check your answers with a neighbor. Are your estimations reasonable?

Write down a few notes on front – end estimation before continuing on.

What about front–end estimation and decimals?

When using front – end estimation and decimals, we figure out how to keep the wholes separate from the parts and then combine them together.

Here are the steps to front – end estimation with decimals.

  1. Add the front digits of the numbers being added or subtracted.
  2. Round off the decimals of the numbers being added or subtracted.
  3. Combine or subtract the results.

Wow! That sounds confusing. Let’s walk through it by using an example.

Example

2.10 + 3.79 = _____

We start with the front digits of the numbers being added. That means we add 2 + 3 = 5.

Next, we round the decimal part of each number. .10 stays .10 and .79 becomes .80

.80 + .10 = .90

Now we add, since that is the operation, the two estimates together.

5 + .90 = 5.90

Our answer for 2.10 + 3.79 is 5.90.

Here is a subtraction example.

Example

16.79 - 14.12 = _____

We start by subtracting the front ends. 16 - 14 = 2

Next, we round the decimal parts. .79 becomes .80 and .12 becomes .10.

Subtract those decimals .80 - .10 = .70.

Combine for the answer = 2.70.

Our answer for 16.79 - 14.12 is 2.70.

Now it is time for you to try a few on your own. Use front – end estimation here.

  1. 54.77 + 22.09 = _____
  2. 18.22 + 19.76 = _____

Take a minute to check your work with a peer.

III. Compare the Results of Different Estimation Methods

Now that you have learned two different ways of estimating sums and difference, how can you decide which method is the better method?

Remember that a method is best if it provides the answer that is the most reasonable.

Let’s look at a few examples, use both methods of estimation and decide which method gives us the answer that makes the most sense.

Example

57.46 + 18.21 = _____

Now let’s apply what we have learned about estimation to the problem above.

We are going to use front – end estimation first and then we’ll apply estimating by rounding.

Here is our work for front – end estimation.

57 + 18 = 75 Now we have added the fronts

.46 becomes .50, .21 becomes .20 and .50 + .20 = .70

Put it altogether, 75 + .70 = 75.70

Now let’s see what our answer is if we use rounding.

57.46 rounds to 57

18.21 rounds to 18

Our answer is 57 + 18 = 75

How can we tell which one is the most accurate method of estimation?

Let’s see what the actual answer would be. Then we can figure out which method of estimation got us closer to the actual answer.

57.46 + 18.21 = 75.67

Wow! When we used front – end estimation, our answer was 75.70. That is very close to 75.67. Our other answer would have gotten us into the ball park, but wasn’t as close to the actual answer.

Sometimes, one method of estimation is better than the other. We have to look at each problem individually to figure this out. For the example that we just finished, the best choice of estimation would be front – end estimation.

What type of problem would be better for rounding?

Rounding is best when working with very large numbers. Then we can get an estimate of the answer without dealing with all of the fronts and ends of numbers using front – end estimation.

Let’s look at an example to help us understand this.

Example

$6927.11

$8100.89

Here are two cars that are for sale.

The first car has a price tag of $6927.11.

The second car has a price tag of $8100.89.

Let’s say that we wanted to figure out the difference between the prices of these two cars. If we just wanted to get an idea of how much one car was versus the other, we can estimate and the difference.

Let’s use rounding to figure out the difference between car 1 and car 2.

Car 1 $6927.11 rounds to $7000.00

Car 2 $8100.89 rounds to $8100.00

There is a difference of about $1100.00 between the two cars.

Let’s see if it was as easy with front – end estimation.

First, add the front ends. 6927 + 8100........

For this problem, because of its large numbers, it makes much more sense to round each number. Using front – end estimation would have required us to add each number and then round and add the decimal parts. It definitely would have been more challenging.

Real Life Example Completed

Recycling

You have learned all about front – end estimation and rounding to estimate sums and differences.

Now we are ready to help Jose sort through his recycling dilemma.

Let’s take another look at the problem.

Jose has had many new ideas for improving life at the “Add It Up Ice Cream Stand.” His newest idea focuses on recycling.

In addition to ice cream, the stand also sells sodas that are packaged in aluminum cans. Because you can turn in cans for recycling and receive some money back, Jose thinks that this could be a way for the ice cream stand to generate a little more income.

He explained his idea to Mr. Harris who loved the concept. Jose put out recycling bins the first week of June. On the last day of each month, Jose took the recycled cans to the recycling center and collected money on his returns. He decided to keep track of the additional income in a small notebook.

Here is what Jose collected in June, July and August.

June $25.77

July $33.45

August $47.62

Julie asks Jose about how much he has made in recycling.

She also wants to know about how much more he made in August versus June.

Jose looks at his notebook and just by looking at the numbers can’t remember how to estimate.

The decimals are throwing him off.

First, let’s go through and underline all of the important information.

The next thing that we need to do is to estimate the sum of the amounts of money that Jose collected in June, July and August.

Let’s start by rounding.

$25.77 becomes $26.00

$33.45 becomes $33.00

$47.62 becomes $48.00

Our estimated sum is $107.00.

After rounding, Jose decides to try front – end estimation to see if he can get an even more accurate estimate of the sum.

First, add the front ends, 25 + 33 + 47 = 105.

Next round the decimal parts and add them, .77 = .80, .45 = .50, .62 = .60.

.80 + .50 + .60 & = 1.90 \\105 + 1.90 & = \$106.90

Jose shows his work to Julie and the two of them are amazed! The answers for both methods of estimation were definitely very close!

Next, Jose works to figure out the difference between the amount of money collected in June versus August.

Since both sums were similar, he decides to use rounding to estimate this difference.

June = $25.77 which rounds to $26

August = $47.62 which rounds to $48

48 - 26 = $22.00

“Congratulations Jose! Your recycling campaign is definitely working! Keep up the good work,” Julie says to Jose after seeing his results.

Jose feels proud because of his accomplishment. The recycling campaign will remain at the ice cream stand.

Vocabulary

Here are the vocabulary words from this lesson.

Estimate
to find an answer that is reasonable and close to an exact answer.
Sum
the result of an addition problem
Difference
the result of a subtraction problem
Front end estimation
estimating by adding the front ends of each number in the problem, then rounding and adding the decimal parts of each number.
  • Works well with smaller numbers
Rounding
converting a number to its nearest whole number.
  • Works well with larger numbers

Time to Practice

Directions: Estimate each sum or difference by rounding.

1. 56.32 + 23.12 = _____

2. 18.76 + 11.23 = _____

3. 14.56 + 76.98 = _____

4. 11.12 + 54.62 = _____

5. 33.24 + 45.32 = _____

6. 18.97 + 15.01 = _____

7. 22.43 + 11.09 = _____

8. 4.52 + 3.21 = _____

9. 19.19 + 27.75 = _____

10. 87.12 + 88.90 = _____

11. 67.19 - 33.12 = _____

12. 88.92 - 33.10 = _____

13. 76.56 - 3.45 = _____

14. 65.72 - 11.12 = _____

15. 77.34 - 43.02 = _____

16. 88.02 - 11.10 = _____

17. 89.32 - 18.03 = _____

18. 24.67 - 10.10 = _____

19. 37.82 - 14.20 = _____

20. 55.88 - 44.22 = _____

21. 334.56 - 125.86 = _____

22. 456.11 + 112.18 = _____

Directions: Estimate using front – end estimation.

23. 34.66 + 11.12 = _____

24. 43.18 + 16.75 = _____

25. 2.34 + 1.56 = _____

26. 7.89 + 5.79 = _____

27. 8.90 + 3.21 = _____

28. 7.18 - 3.13 = _____

29. 12.65 - 7.23 = _____

30. 15.70 - 11.10 = _____

31. 25.67 - 18.40 = _____

32. 78.46 - 55.21 = _____

33. 88.12 - 34.06 = _____

34. 87.43 - 80.11 = _____

35. 94.12 - 7.08 = _____

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