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Introduction

Building a House

Travis is hoping to work with his Uncle Larry for the summer. Uncle Larry is a contractor who works on building houses. Travis has always loved working with his hands and construction seems to be a perfect fit for him. He also loves seeing a house start from nothing and be built.

Travis’ Uncle Larry is a bit concerned because Travis is a little young to be working on a construction site, but Travis is sure that he is up to the task. To test things out first, Uncle Larry has asked Travis to come and work with him during school vacation week. He is finishing a house and there are some jobs that Travis can help him with. Travis is thrilled. He can hardly wait for the first day, and after what feels like forever, it has finally arrived.

Travis and Uncle Larry arrive at the site. They are going to be working on finishing a part of a wall. When they arrive, there are bunch of boards and tools waiting for them.

Here is the dilemma.

Two wall studs have already been nailed into the floor. Travis and Uncle Larry need to add in the brace that goes between the two studs. The space between the wall studs measures 43 \frac{5}{8}”. The board that they wish to use is 4 \frac{1}{2}’ long.

“Travis, this is your first task,” Uncle Larry says. “While I go and check on some other work, I need you to do a few estimations. First, figure out if the board we have will fit. Then, figure out how much of the board we need to cut off to fit between these two wall studs. Do you have any questions?”

“Nope,” says Travis getting out a piece of paper and a pencil.

Travis knows how to figure this out, do you? Well, if you don’t, you will by the end of the lesson. This lesson is all about estimating with fractions and whole numbers. Pay close attention, we come back to solve Travis' problem later!

What You Will Learn

In this lesson, you will learn the following skills:

  • Round fractions to the nearest half.
  • Round mixed numbers to the nearest whole number.
  • Estimate sums and differences of fractions and mixed numbers.
  • Estimate sums and differences involving real-world fractional amounts.

Teaching Time

I. Round Fractions to the Nearest Half

We use fractions in everyday life all the time. Remember that when we talk about a fraction, we are talking about a part of a whole. Often times, we need to use an exact fraction, but sometimes, we can use an estimate. If you think back to our earlier work on estimation, you will remember that an estimate is an approximate value that makes sense or is reasonable given the problem.

Example

What fraction does this picture represent?

If we wanted to be exact about this fraction, we could say that there are \frac{12}{20} shaded boxes.

However, it makes much more sense to say that about half of the boxes are shaded.

We call this rounding to the nearest half.

How do we round to the nearest half?

To round a fraction to the nearest half, we need to think in terms of halves. We often think in terms of wholes, so this is definitely a change in our thinking. There are three main values to round to when we round a fraction to the nearest half.

The first is zero. We can think of 0 as \frac{0}{2}, or zero halves. The second value is \frac{1}{2}, or one half. The third value is 1, which can be thought of as \frac{2}{2}, or two halves. When rounding to the nearest half, we round the fraction to whichever half the fraction is closest to on the number line 0, \frac{1}{2}, or 1. If a fraction is equally close to two different halves, we round the fraction up.

Let’s look at an example.

Example

\frac{5}{6}

To figure out which value five-sixths is closest to, we must first think in terms of sixths. Since the denominator is six, that means that the whole is divided into six parts. The fraction \frac{0}{6} would be the value of zero, \frac{3}{6} would be the value of \frac{1}{2}, and \frac{6}{6} is the same as 1. The fraction \frac{5}{6} is closest to \frac{6}{6}, so rounding to the nearest half would be rounding to 1.

Our answer is 1.

Try a few of these on your own. Round each fraction to the nearest half.

  1. \frac{1}{5}
  2. \frac{3}{8}
  3. \frac{7}{9}

Take a few minutes and check your work with a neighbor.

II. Round Mixed Numbers to the Nearest Whole Number

We can also estimate by rounding mixed numbers. Remember that a mixed number is a number that has a whole number and a fraction. A mixed number refers to a number that is between one whole number and another.

How do we round mixed numbers to the nearest whole?

To do this, we need to look at both the whole number part of the mixed number and the fraction part of the mixed number. The whole will tell us which two numbers the fraction part is between.

Example

5 \frac{1}{6} is found between the whole numbers 5 and 6.

Our answer is 5. 5 \frac{1}{6} is closer to 5.

In the example we just looked at, one-sixth is a very small fraction. If the fraction part of the mixed number had been one-half or greater, then we would have said that five and one-sixth was closer to six.

We can think in this way whenever we are rounding mixed numbers.

Practice by rounding these mixed numbers.

  1. 7 \frac{6}{9}
  2. 4 \frac{1}{4}
  3. 6 \frac{5}{10}

Check your work with a friend. Discuss any answers that do not match.

Take a few minutes to take notes on rounding fractions and mixed numbers to the nearest half and whole.

III. Estimate Sums and Differences of Fractions and Mixed Numbers

Now that you know how to round fractions to the nearest half and mixed numbers to the nearest whole, we can apply this information to estimating sums and differences?

How do we estimate a sum?

Remember back that a sum is the answer in addition problem. You have already learned how to estimate a whole number sum and a decimal sum. Now we are going to learn how to estimate fraction sums.

To estimate a sum of two fractions, you must use what you have learned about rounding to the nearest half. That is the first thing that you do when estimating sums of fractions.

Example

\frac{3}{8} + \frac{1}{7} = \underline{\;\;\;\;\;\;\;\;\;}

To estimate this sum, we must first round each fraction to the nearest half. Let’s start with three-eighths.

Is \frac{3}{8} closest to zero, one – half or one whole?

We know that  \frac{4}{8} = \frac{1}{2}, so we can say that \frac{3}{8} is closest to one-half.

Is \frac{1}{7} closest to zero, one-half or one whole?

We can say that \frac{1}{7} is closest to zero because it such a small part of a whole.

Now we rewrite the problem.

\frac{1}{2} + 0 = \frac{1}{2}

Our answer is \frac{1}{2}.

How do we estimate a sum for two mixed numbers?

When working with mixed numbers, we round to the nearest whole number, not the nearest half. We round each mixed number and then add to find our estimate.

Example

3 \frac{4}{5} + 2 \frac{1}{9} = \underline{\;\;\;\;\;\;\;\;\;}

First, let’s round 3 \frac{4}{5} to the nearest whole. We know that we are either going to round down to 3 or up to 4 because this mixed number is between those two whole numbers. Four-fifths is almost one whole. We can see this because if we had five-fifths, we would have another whole. This mixed number is closest to 4. We need to round up to 4.

Next, let’s round 2 \frac{1}{9} to the nearest whole. We know that we are either going to round down to 2 or up to 3 because this mixed number is between those two whole numbers. One-ninth is a very small fraction. Think about it, we would need eight-ninths more to make one whole. Therefore, we round down to 2.

Now we can rewrite the problem.

4 + 2 = 6

Our estimate is 6.

Practice some of these on your own. Estimate each sum by rounding.

  1. \frac{4}{9} + \frac{7}{8} = \underline{\;\;\;\;\;\;\;\;\;}
  2. 3 \frac{1}{5} + 7 \frac{6}{8} = \underline{\;\;\;\;\;\;\;\;\;}

Take a minute to check your work with a peer.

What about estimating the differences of fractions?

Remember that when we talk about a difference, that we are talking about subtraction. We can approach estimating the differences of fractions and mixed numbers in the same way that we approached sums. We need to round each fraction or mixed number and then subtract to find the estimate.

Example

\frac{4}{5} - \frac{3}{7} = \underline{\;\;\;\;\;\;\;\;\;}

\frac{4}{5} rounds up to 1.

\frac{3}{7} rounds to \frac{1}{2}.

Next, we rewrite the problem.

 1 - \frac{1}{2} = \frac{1}{2}

Our estimate is \frac{1}{2}.

Now let’s look at one with mixed numbers.

Example

6 \frac{3}{4} - 2 \frac{1}{8} = \underline{\;\;\;\;\;\;\;\;\;}

6 \frac{3}{4} rounds to 7.

2 \frac{1}{8} rounds to 2.

Now we rewrite the problem.

7 - 2 = 5

Our estimate is 5.

Now it is time for you to practice. Estimate the following differences.

  1. \frac{6}{7} - \frac{1}{11} = \underline{\;\;\;\;\;\;\;\;\;}
  2. 5 \frac{1}{3} - 2 \frac{3}{4} = \underline{\;\;\;\;\;\;\;\;\;}

Check your answers with a partner. Be sure that you both have the same answers.

Real Life Example Completed

Building a House

Now that you have finished this lesson, you are ready for some estimating with Travis and Uncle Larry. Here is the problem once again.

Travis is hoping to work with his Uncle Larry for the summer. Uncle Larry is a contractor who works on building houses. Travis has always loved working with his hands and construction seems to be a perfect fit for him. He also loves seeing a house start from nothing and be built.

Travis’ Uncle Larry is a bit concerned because Travis is a little young to be working on a construction site, but Travis is sure that he is up to the task. To test things out first, Uncle Larry has asked Travis to come and work with him during school vacation week. He is finishing a house and there are some jobs that Travis can help him with. Travis is thrilled. He can hardly wait for the first day, and after what feels like forever, it has finally arrived.

Travis and Uncle Larry arrive at the site. They are going to be working on finishing a part of a wall. When they arrive, there are bunch of boards and tools waiting for them.

Here is the dilemma.

Two wall studs have already been nailed into the floor. Travis and Uncle Larry need to add in the brace that goes between the two studs. The space between the wall studs measures 43 \frac{5}{8}”. The board that they wish to use is 4 \frac{1}{2}’ long.

“Travis, this is your first task,” Uncle Larry says. “While I go and check on some other work I need you to do a few estimations. First, figure out if the board we have will fit. Then, figure out how much of the board we need to cut off to fit between these two wall studs. Do you have any questions?”

“Nope,” says Travis getting out a piece of paper and a pencil.

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

The first thing to notice is that the space is being measured in inches, and the boards are being measured in feet. Let’s change the feet to inches first.

4 \frac{1}{2}’ = 48” + 6” = 54” is the board length.

The space measures 43 \frac{5}{8}”.

The first thing that Uncle Larry wanted Travis to figure out was if the board would be long enough to fit the space. 54” is greater than 43 \frac{5}{8}”, so it will work, but the board will need to be cut.

To figure out how much board to cut, we need to find a difference. We can estimate the difference by rounding.

54” is already a whole number.

43 \frac{5}{8} is closest to 44. We round it up to 44”.

54 - 44 = 10”.

Travis and Uncle Larry will need to cut approximately 10” from the board to have it fit into the space. Fractions and mixed numbers are used all the time in real life dilemmas like Travis’. Contractors use fractions all of the time!

Vocabulary

Here are the vocabulary words that are found in this lesson.

Fraction
a part of a whole written with a fraction bar, a numerator and a denominator.
Estimate
to find an approximate answer that is reasonable and makes sense given the problem.
Mixed number
a number made up of a whole number and a fraction.
Sum
the answer to an addition problem.
Difference
the answer to a subtraction problem.

Technology Integration

This video shows you how to estimate with fractions.

Estimating with fractions

Time to Practice

Directions: Estimate the following sums.

1. \frac{1}{5} + \frac{4}{5} = \underline{\;\;\;\;\;\;\;\;\;}

2. \frac{8}{9} + \frac{4}{6} = \underline{\;\;\;\;\;\;\;\;\;}

3. \frac{2}{9} + \frac{4}{5} = \underline{\;\;\;\;\;\;\;\;\;}

4. \frac{3}{6} + \frac{2}{3} = \underline{\;\;\;\;\;\;\;\;\;}

5. \frac{5}{6} + \frac{2}{3} = \underline{\;\;\;\;\;\;\;\;\;}

6. \frac{1}{12} + \frac{9}{11} = \underline{\;\;\;\;\;\;\;\;\;}

7. \frac{6}{12} + \frac{10}{11} = \underline{\;\;\;\;\;\;\;\;\;}

8. 1 \frac{1}{10} + 2 \frac{1}{2} = \underline{\;\;\;\;\;\;\;\;\;}

9. 4 \frac{2}{3} + 5 \frac{4}{5} = \underline{\;\;\;\;\;\;\;\;\;}

10. 7 \frac{1}{9} + 8 \frac{1}{5} = \underline{\;\;\;\;\;\;\;\;\;}

11. 14 \frac{5}{9} + 8 \frac{4}{5} = \underline{\;\;\;\;\;\;\;\;\;}

12. 4 \frac{2}{3} + 7 \frac{1}{7} = \underline{\;\;\;\;\;\;\;\;\;}

13. 18 \frac{1}{13} + 7 \frac{2}{10} = \underline{\;\;\;\;\;\;\;\;\;}

14. 11 \frac{12}{13} + 4 \frac{1}{10} = \underline{\;\;\;\;\;\;\;\;\;}

15. 22 \frac{5}{7} + 11 \frac{1}{5} = \underline{\;\;\;\;\;\;\;\;\;}

Directions: Estimate each difference.

16. \frac{4}{5} - \frac{1}{4} = \underline{\;\;\;\;\;\;\;\;\;}

17. \frac{4}{5} - \frac{3}{4} = \underline{\;\;\;\;\;\;\;\;\;}

18. \frac{9}{10} - \frac{3}{6} = \underline{\;\;\;\;\;\;\;\;\;}

19. \frac{11}{12} - \frac{1}{7} = \underline{\;\;\;\;\;\;\;\;\;}

20. \frac{10}{13} - \frac{1}{10} = \underline{\;\;\;\;\;\;\;\;\;}

21. \frac{6}{13} - \frac{5}{11} = \underline{\;\;\;\;\;\;\;\;\;}

22. 2 \frac{1}{13} - 1 \frac{11}{12} = \underline{\;\;\;\;\;\;\;\;\;}

23. 9 \frac{5}{6} - 4 \frac{11}{13} = \underline{\;\;\;\;\;\;\;\;\;}

24. 23 \frac{1}{6} - 14 \frac{2}{9} = \underline{\;\;\;\;\;\;\;\;\;}

25. 33 \frac{5}{7} - 10 \frac{1}{11} = \underline{\;\;\;\;\;\;\;\;\;}

26. 42 \frac{1}{15} - 10 \frac{10}{11} = \underline{\;\;\;\;\;\;\;\;\;}

27. 19 \frac{1}{4} - 6 \frac{1}{11} = \underline{\;\;\;\;\;\;\;\;\;}

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