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Difficulty Level: At Grade Created by: CK-12

## Introduction

The Blueberries

Teri and Ren are both in the seventh grade. They are baking pies and muffins for the bake sale. Teri has decided to make a blueberry pie and Ren has decided to make blueberry muffins. While Teri works on making her pie crusts. Ren offers to go to the grocery store with his Mom to get the blueberries that they will need.

Teri tells Ren that she needs \begin{align*}5 \frac{1}{2}\end{align*} cups of blueberries for the pie. Plus she will need an additional \begin{align*}\frac{1}{4}\end{align*} of a cup for decorating the top of the pie. Ren know that he will need \begin{align*}1 \frac{1}{3}\end{align*} cups of blueberries for his pie.

If blueberries come in pints and there are two cups in one pint, how many pints of blueberries will Ren need to buy at the store?

Ren takes out a piece of paper and a pencil. You can figure this out too. In this lesson you will learn all that you need to know to help Ren figure out the blueberry dilemma.

What You Will Learn

In this lesson, you will learn the following skills.

• Add fractions and mixed numbers.
• Estimate sums of fractions and mixed numbers.
• Identify and apply the commutative and associative properties of addition in fraction operations, using numerical and variable expressions.
• Model and solve real-world problems using simple equations involving sums of fractions or mixed numbers.

Teaching Time

I. Add Fractions and Mixed Numbers

Adding fractions and mixed numbers is as easy as adding whole numbers. The only trick is to make sure that the fractions we are adding have the same denominator.

Imagine adding \begin{align*}\frac{1}{2}\end{align*} cup of flour to \begin{align*}\frac{1}{3}\end{align*} cup of flour. We know that the new mixture of flour is more than \begin{align*}\frac{1}{2}\end{align*} cup of flour and more than \begin{align*}\frac{1}{3}\end{align*} cup of flour. We also know that the new mixture is less than 1 cup of flour and greater than \begin{align*}\frac{2}{3}\end{align*} cup of flour.

We have to divide the whole into a new number of parts, that is, find a common denominator in order to get a fraction which accurately describes the new amount of flour.

When we use the common denominator of 6 and add the fractions \begin{align*}\frac{3}{6}\end{align*} (equivalent fraction of \begin{align*}\frac{1}{2}\end{align*}) to \begin{align*}\frac{2}{6}\end{align*} (equivalent fraction of \begin{align*}\frac{1}{3}\end{align*}), we simply add the numerators and keep the denominator the same. If we add \begin{align*}\frac{1}{2}\end{align*} cup of flour to \begin{align*}\frac{1}{3}\end{align*} cup of flour, we get \begin{align*}\frac{5}{6}\end{align*} of a cup of flour.

How do we do this when we add mixed numbers and fractions?

Mixed numbers and fractions can be a little tricky because you are dealing parts and wholes. You can find the sum of them though by keeping in mind that you add parts with parts and wholes with wholes. Here is an example.

Example

\begin{align*}\frac{3}{4}+ 2 \frac{1}{3}\end{align*}

Here we are going to add a fraction and a mixed number together. You can see that the fractions have different denominators. This is the first thing that we need to change. Both fractions must have the same denominator before we can add them.

To do this, we find the common denominator of 3 and 4. That number is 12. Now we rename the fractions in terms of twelfths, and create equivalent fractions with denominators of 12.

\begin{align*}\frac{3}{4} &= \frac{9}{12}\\ \frac{1}{3} &= \frac{4}{12}\end{align*}

Now we can add them. When the denominators are the same, we have to add the numerators only.

\begin{align*}\frac{9}{12}+\frac{4}{12}=\frac{13}{12}\end{align*}

We can change \begin{align*}\frac{13}{12}\end{align*} into the mixed number \begin{align*}1 \frac{1}{12}\end{align*}.

Now we had a 2 from the original mixed number. We add this to our sum.

The answer is \begin{align*}3 \frac{1}{12}\end{align*}.

Here our answer is in simplest form so we leave it alone. If you can simplify an answer you must do so or the answer is incorrect.

Here are the steps.

3. Add the sum of the fractions to the whole numbers

Write these steps down in your notebook before continuing with the lesson.

3D. Lesson Exercises

1. \begin{align*}9 \frac{1}{2} + 22 \frac{1}{4}\end{align*}
2. \begin{align*}2 \frac{1}{3} + 8 \frac{2}{3}\end{align*}
3. \begin{align*}5 \frac{1}{3} + \frac{1}{7}\end{align*}

Take a few minutes to check your answers with a partner. Be sure that your work is accurate before continuing.

II. Estimate Sums of Fractions and Mixed Numbers

Estimation is a method for finding an approximate solution to a problem. For example, the sum of 22 and 51 is exactly 73. We can estimate the sum by rounding to the tens place and adding \begin{align*}20 + 50\end{align*}. Then we can say the sum of 22 and 51 is “about 70.”

In the last lesson, we used three benchmarks (0, \begin{align*}\frac{1}{2}\end{align*} and 1) to get a sense of the approximate value of different fractions. We can use this same technique to estimate sums of two or more fractions and mixed numbers.

First, we approximate the value of each fraction or mixed number using the benchmarks 0, \begin{align*}\frac{1}{2}\end{align*} and 1.

Next, we find the sum of the approximate values.

When you are approximating the value of mixed numbers, first figure out the approximate value of the fraction and then add it to the whole number. For example, the approximate value of \begin{align*}2 \frac{3}{4}\end{align*} is 3 because the approximate value of \begin{align*}\frac{3}{4}\end{align*} is \begin{align*}1 (2 +1 = 3)\end{align*}.

Even when you are asked to find an exact answer, estimation is a useful way to get an idea of a reasonable solution to a problem. Once you have finished solving for an exact answer to a problem, you can check your answer against the estimate. Refer back to the following steps if necessary.

Estimating sums of fractions and mixed numbers:

1. Approximate the value of each of the fractions or mixed numbers by using the benchmarks 0, \begin{align*}\frac{1}{2}\end{align*} and 1
2. Add these approximate values to get an estimated sum

Write these steps in your notebook and then continue with the lesson.

Example

Estimate the following sum, \begin{align*}\frac{5}{9} + \frac{1}{77}\end{align*}

First, we approximate the value of the individual fractions. \begin{align*}\frac{5}{9}\end{align*} is approximately \begin{align*}\frac{1}{2}\end{align*} and \begin{align*}\frac{1}{77}\end{align*} is approximately 0. Now, we rewrite the problem substituting the approximate values: \begin{align*}\frac{1}{2} + 0\end{align*} is about \begin{align*}\frac{1}{2}\end{align*}.

Example

Estimate the following sum, \begin{align*}3 \frac{6}{7} + 1 \frac{4}{9}\end{align*}

First, we approximate the value of each of the mixed numbers. The approximate value of \begin{align*}3 \frac{6}{7}\end{align*} is 4 because the approximate value of the fraction \begin{align*}\frac{6}{7}\end{align*} is \begin{align*}1 (3 + 1 = 4)\end{align*}. The approximate value of \begin{align*}1 \frac{4}{9}\end{align*} is \begin{align*}1 \frac{1}{2}\end{align*} because the approximate value of \begin{align*}\frac{4}{9}\end{align*} is \begin{align*}\frac{1}{2}\end{align*}. We rewrite the problem with the approximate values of the mixed numbers and it looks like this: \begin{align*}4 + 1 \frac{1}{2}\end{align*}. We estimate that the sum of \begin{align*}3 \frac{6}{7}\end{align*} and \begin{align*}1 \frac{4}{9}\end{align*} is about \begin{align*}5 \frac{1}{2}\end{align*}.

3E. Lesson Exercises

Estimate the following sums.

1. \begin{align*}\frac{6}{7}+\frac{1}{2}\end{align*}
2. \begin{align*}\frac{29}{30}+7 \frac{8}{10}\end{align*}
3. \begin{align*}1 \frac{1}{2}+ 3 \frac{5}{6}\end{align*}

III. Identify and Apply the Commutative and Associative Properties of Addition in Fraction Operations, using Numerical and Variable Expressions

Now that we know the basics of adding fractions, we can use two mathematical properties of addition to help us solve more complicated problems.

The Commutative Property of Addition states that the order of the addends does not change the sum. Let’s test the property using simple whole numbers.

\begin{align*}& 4 + 5 + 9 = 18 && 5 + 4 + 9 = 18 && 9 + 5 + 4 = 18\\ & 4 + 9 + 5 = 18 && 5 + 9 + 4 = 18 && 9 + 4 + 5 = 18\end{align*}

As you can see, we can add the three addends (4, 5, and 9) in any orders. The Commutative Property of Addition works also works for four, five, even 100 addends.

It works for fraction addends, too. This means that the order that you add fractions in does not change the sum of the fractions.

Parentheses are grouping symbols used in math to let us know which operations to complete first. The order of operations tells us that operations in parentheses must be completed before any other operation. The Associative Property of Addition states that the way in which addends are grouped does not change the sum. Once again, let’s test the property using simple whole numbers.

\begin{align*}& (4 + 5) + 9 = 18 && (5 + 4) + 9 = 18 && (9 + 5) + 4 = 18\end{align*}

Clearly, the different way the addends are grouped has no effect on the sum. The Associative Property of Addition also works for fraction addends.

How do we use these properties when adding fractions?

These two properties are extremely useful when adding fractions. If you are adding three fractions and two of the fractions have like denominators, you can add those two fractions together and then find a common denominator with the third. This can be a big time saver.

When you are working with variable expressions or with expressions which contain an algebraic unknown (like \begin{align*}x\end{align*}) you can use the commutative and associative properties of addition to simplify the expression. Let’s check out some examples.

Example

Simplify the following variable expressions using the associative and commutative properties of addition.

\begin{align*}3 \frac{2}{3} + x + \frac{1}{3}\end{align*}

To simplify means to make smaller. We are going to simplify this expression. We use the Commutative Property of Addition to do this.

That’s a great question. If you think about it, because it doesn’t matter which order you add fractions in, you can add the mixed number and the fraction and ignore the \begin{align*}x\end{align*}. This will help you to simplify the expression. Let’s add the two numbers together.

\begin{align*}3 \frac{2}{3}+\frac{1}{3}=4\end{align*}

One-third plus two-thirds is three-thirds which is the same as one. We add one to the whole number three and get four.

Our simplified expression is \begin{align*}4 + x\end{align*}.

Example

\begin{align*}\frac{3}{10} + \left(\frac{1}{4} + x\right)\end{align*}

To simplify this expression, we are going to use the Associative Property of Addition. The hint is that there are parentheses in this expression.

That’s a good question. We use that property because we can change the grouping. You know from the order of operations that operations inside the parentheses are done first. Well, you can’t complete this operation in the grouping symbol because you don’t know what \begin{align*}x\end{align*} is. But if you change the grouping, you can simplify the expression.

\begin{align*}\left(\frac{3}{10} + \frac{1}{4}\right) + x\end{align*}

Now we add three-tenths and one-fourth.

\begin{align*}\frac{3}{10} &= \frac{6}{20}\\ \frac{1}{4} &= \frac{5}{20}\\ \frac{6}{20}+\frac{5}{20} &= \frac{11}{20}\end{align*}

Our answer is \begin{align*}\frac{11}{20}+x\end{align*}.

3F. Lesson Exercises

Simplify each expression using the Commutative and Associative Properties of Addition. Be sure your answers are in simplest form.

1. \begin{align*}\frac{2}{3}+ y+ \frac{1}{5}\end{align*}
2. \begin{align*}\frac{1}{2}+ \left(\frac{1}{2}+ x\right)\end{align*}
3. \begin{align*}x+ \frac{4}{9}+\frac{2}{9}\end{align*}

IV. Model and Solve Real-World Problems Using Simple Equations Involving Sums of Fractions or Mixed Numbers

Fractions describe parts of a whole. Because of their accuracy, they are useful in many real-world situations, but especially situations involving measurement. When you confront a situation that requires your math skills, it is helpful to think the problem out before you jump in and try to find an answer. Ask yourself: "What is the problem? What are we trying to find out? What information do we have that will help us solve this problem? What mathematical tools can you use to get the answer?" Let’s look at some real-world situations involving addition of fractions and learn more about problem solving.

Example

Donte is making a costume with blue, red and black fabric. He has \begin{align*}6 \frac{1}{2}\end{align*} yards of blue fabric, \begin{align*}3 \frac{2}{3}\end{align*} yards of red fabric and \begin{align*}5 \frac{4}{5}\end{align*} yards of black fabric, how many yards of fabric does Donte have altogether?

Let’s look at the problem carefully and define the values that we know and the value or values that we want to know.

We know that Donte has 3 types of fabric (blue, red and black) and we know also the lengths of each type of fabric. We want to find out how much fabric Donte has altogether. If we represent this problem in an equation, it would look like this:

Length of blue fabric + length of red fabric + length of black fabric = total length of fabric

Since we know the lengths of the individual colors of fabric, we can rewrite the expression like this:

\begin{align*}6 \frac{1}{2} + 3 \frac{2}{3} + 5 \frac{4}{5} =\end{align*} total length of fabric.

If we add the mixed numbers together, we will learn what we want to find out. First, we will add the first two mixed numbers. We use the common denominator of 6 for the fractions and we find the sum of the two mixed numbers:

\begin{align*}6 \frac{3}{6} + 3 \frac{4}{6} = 9 \frac{7}{6}\end{align*}

Notice that seven-sixths is improper meaning that it is larger than one whole. We can convert this improper fraction to a mixed number.

\begin{align*}9 \frac{7}{6}=10 \frac{1}{6}\end{align*}

Now we can this new mixed number, \begin{align*}10 \frac{1}{6}\end{align*} to the length of the black fabric, \begin{align*}5 \frac{4}{5}\end{align*} yards.

We use the common denominator of 30 for the fractions and we find the sum of these mixed numbers. \begin{align*}10 \frac{5}{30} + 5 \frac{24}{30} = 15 \frac{29}{30}\end{align*}

We can use the exact sum or we can say that Donte has just about 16 yards of fabric.

Remember the dilemma with the blueberries? You now have all the skills that you need to help solve that problem. Let’s go back to the introductory problem and work it through.

## Real Life Example Completed

The Blueberries

Reread this problem once again. Then underline any important information before solving it.

Teri and Ren are both in the seventh grade. They are baking pies and muffins for the bake sale. Teri has decided to make a blueberry pie and Ren has decided to make blueberry muffins. While Teri works on making her pie crusts. Ren offers to go to the grocery store with his Mom to get the blueberries that they will need.

Teri tells Ren that she needs \begin{align*}5 \frac{1}{2}\end{align*} cups of blueberries for the pie. Plus she will need an additional \begin{align*}\frac{1}{4}\end{align*} of a cup for decorating the top of the pie. Ren know that he will need \begin{align*}1 \frac{1}{3}\end{align*} cups of blueberries for his pie.

If blueberries come in pints and there are two cups in one pint, how many pints of blueberries will Ren need to buy at the store?

Ren takes out a piece of paper and a pencil.

First, we will need to find the sum of the blueberries. We will figure out how many cups of blueberries both Teri and Ren will need for their recipes.

\begin{align*}5 \frac{1}{2}+\frac{1}{4}+1 \frac{1}{3}\end{align*}

Next, we need to rename each fraction with a common denominator. We can use 12 as our common denominator.

\begin{align*}5 \frac{6}{12}+\frac{3}{12}+1 \frac{4}{12}\end{align*}

If we add the fraction parts, we get a sum of \begin{align*}\frac{13}{12}\end{align*}. This improper fraction changes to \begin{align*}1 \frac{1}{12}\end{align*}.

Next we add this to our whole numbers.

The sum of the blueberries is \begin{align*}7 \frac{1}{12}\end{align*}.

Now we have to figure out how many pints Ren will need to purchase. There are two cups in a pint. To have enough blueberries, Ren will need to purchase 4 pints of blueberries. There will be some left over, but having some left over is better than not having enough!!

## Vocabulary

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

Fraction
a part of a whole
Mixed Number
a whole number and a fraction
states that the order in which you add values does not change the sum of the values.
the way that you group numbers does not change the sum of the numbers being added.

## Technology Integration

1. http://www.mathplayground.com/howto_fractions_diffden.html – This video t

Other Videos:

http://www.mathplayground.com/howto_fractions_diffden.html – This video teaches you how to add fractions with uncommon denominators.

## Time to Practice

Directions: Add the following fractions and mixed numbers.

1. \begin{align*}\frac{3}{7} + \frac{1}{14}\end{align*}

2. \begin{align*}\frac{3}{4} + \frac{1}{3}\end{align*}

3. \begin{align*}\frac{2}{5}+ \frac{3}{10}\end{align*}

4. \begin{align*}\frac{1}{9} + \frac{1}{6}\end{align*}

5. \begin{align*}2 \frac{3}{5} + \frac{17}{20}\end{align*}

6. \begin{align*}8 \frac{5}{12} + 2 \frac{1}{6}\end{align*}

7. \begin{align*}1 \frac{2}{3} + 1 \frac{3}{4}\end{align*}

8. \begin{align*}2 \frac{1}{5} + 4 \frac{14}{15}\end{align*}

Directions: Estimate the sums.

9. \begin{align*}\frac{1}{29} + \frac{4}{5}\end{align*}

10. \begin{align*}\frac{9}{11} + \frac{4}{10}\end{align*}

11. \begin{align*}\frac{2}{5} + \frac{12}{13}\end{align*}

12. \begin{align*}\frac{2}{71} + \frac{1}{29}\end{align*}

Directions: Estimate the sums.

13. \begin{align*}3 \frac{6}{7}+ 2 \frac{10}{11}\end{align*}

14. \begin{align*}8 \frac{1}{12} + 6 \frac{3}{7}\end{align*}

15. \begin{align*}2 \frac{9}{10} + 3 \frac{1}{17}\end{align*}

16. \begin{align*}1 \frac{2}{12} + \frac{44}{46}\end{align*}

17. \begin{align*}\frac{1}{6}+\frac{2}{6}+\frac{3}{7}\end{align*}

18. \begin{align*}\frac{1}{4}+ 3 \frac{5}{8} + 4 \frac{3}{4}\end{align*}

19. \begin{align*}\frac{2}{9} + \left(\frac{1}{3} + \frac{5}{9}\right)\end{align*}

20. \begin{align*}\left(2 \frac{7}{8} + \frac{2}{3}\right) + 1 \frac{1}{8}\end{align*}

Directions: Simplify the expressions using the associative and commutative properties of addition.

21. \begin{align*}x + 3 \frac{2}{3} + 5 \frac{1}{6}\end{align*}

22. \begin{align*}\frac{1}{4} + x + \frac{5}{8}\end{align*}

23. \begin{align*}\left(\frac{1}{9} + x\right) + \frac{2}{9}\end{align*}

24. \begin{align*}2 \frac{1}{14} + \left(x + 3 \frac{5}{7}\right)\end{align*}

25. One-third of the CDs in Joseph’s CD collection are classical music CDs. Two-sevenths of the CDs are hip-hop CDs. What fraction of Joseph’s collection is classical and hip-hop.

26. Naira is making pinecone stew. First, she mixes \begin{align*}3 \frac{1}{5}\end{align*} cups of chopped pinecones with \begin{align*}1 \frac{1}{2}\end{align*} cups of mud. For the snail sauce on top, she uses another \begin{align*}1 \frac{3}{8}\end{align*} cups of chopped pinecones. How much chopped pinecone does Naira use for her recipe?

27. Jennifer is trying to determine if the cheerleading squad has enough ribbon for the pep rally on Friday. Kurt contributes \begin{align*}9 \frac{1}{6}\end{align*} feet of gold ribbon. Estelle contributes \begin{align*}\frac{3}{4}\end{align*} foot of red ribbon. Aaron brings in \begin{align*}5 \frac{2}{7}\end{align*} feet of gold ribbon at the last minute. How much ribbon does the cheerleading squad have?

28. Marcus and Anita are trying to complete a project. They finished \begin{align*}\frac{1}{4}\end{align*} of the project on Saturday. They got another four-sevenths of the project done on Tuesday. If the project needs to be finished by Thursday, write an algebraic expression that shows how much they have done on Saturday, how much they have done on Tuesday and how much they need to do on Wednesday.

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