6.1: Fraction Estimation
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
“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 realworld 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
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
Let’s look at an example.
Example
To figure out which value fivesixths 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
Our answer is 1.
Try a few of these on your own. Round each fraction to the nearest half.

15 
38 
79
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
Our answer is 5.
In the example we just looked at, onesixth is a very small fraction. If the fraction part of the mixed number had been onehalf or greater, then we would have said that five and onesixth was closer to six.
We can think in this way whenever we are rounding mixed numbers.
Practice by rounding these mixed numbers.

769 
414 
6510
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
To estimate this sum, we must first round each fraction to the nearest half. Let’s start with threeeighths.
Is
We know that
Is
We can say that
Now we rewrite the problem.
Our answer is
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
First, let’s round
Next, let’s round
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.

49+78=−−−−− 
315+768=−−−−−
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
Next, we rewrite the problem.
Our estimate is
Now let’s look at one with mixed numbers.
Example
Now we rewrite the problem.
7  2 = 5
Our estimate is 5.
Now it is time for you to practice. Estimate the following differences.

67−111=−−−−− 
513−234=−−−−−
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
“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.
The space measures
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 \begin{align*}43 \frac{5}{8}''\end{align*}, 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.
\begin{align*}43 \frac{5}{8}\end{align*} 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.
Time to Practice
Directions: Estimate the following sums.
1. \begin{align*}\frac{1}{5} + \frac{4}{5} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
2. \begin{align*}\frac{8}{9} + \frac{4}{6} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
3. \begin{align*}\frac{2}{9} + \frac{4}{5} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
4. \begin{align*}\frac{3}{6} + \frac{2}{3} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
5. \begin{align*}\frac{5}{6} + \frac{2}{3} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
6. \begin{align*}\frac{1}{12} + \frac{9}{11} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
7. \begin{align*}\frac{6}{12} + \frac{10}{11} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
8. \begin{align*}1 \frac{1}{10} + 2 \frac{1}{2} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
9. \begin{align*}4 \frac{2}{3} + 5 \frac{4}{5} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
10. \begin{align*}7 \frac{1}{9} + 8 \frac{1}{5} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
11. \begin{align*}14 \frac{5}{9} + 8 \frac{4}{5} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
12. \begin{align*}4 \frac{2}{3} + 7 \frac{1}{7} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
13. \begin{align*}18 \frac{1}{13} + 7 \frac{2}{10} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
14. \begin{align*}11 \frac{12}{13} + 4 \frac{1}{10} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
15. \begin{align*}22 \frac{5}{7} + 11 \frac{1}{5} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
Directions: Estimate each difference.
16. \begin{align*}\frac{4}{5}  \frac{1}{4} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
17. \begin{align*}\frac{4}{5}  \frac{3}{4} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
18. \begin{align*}\frac{9}{10}  \frac{3}{6} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
19. \begin{align*}\frac{11}{12}  \frac{1}{7} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
20. \begin{align*}\frac{10}{13}  \frac{1}{10} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
21. \begin{align*}\frac{6}{13}  \frac{5}{11} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
22. \begin{align*}2 \frac{1}{13}  1 \frac{11}{12} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
23. \begin{align*}9 \frac{5}{6}  4 \frac{11}{13} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
24. \begin{align*}23 \frac{1}{6}  14 \frac{2}{9} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
25. \begin{align*}33 \frac{5}{7}  10 \frac{1}{11} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
26. \begin{align*}42 \frac{1}{15}  10 \frac{10}{11} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
27. \begin{align*}19 \frac{1}{4}  6 \frac{1}{11} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
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