Chapter 16 – Developing Strategies for Fraction Computation

Big Ideas:

1. The meanings of each operation with fractions are the same the meanings for the operations with whole numbers. Operations with fractions should begin by applying these same meanings to fractional parts. For addition and subtraction, the numerator tells the number of parts and thedenomiator the unit. The parts are added or subtracted. Repeated addition and area models support development of concepts and algorithms for multiplication of fractions.Partition and measurement models lead to two different thought processes for division of fractions.

2. Estimation should be an integral part of computation development to keep students’ attention on the meanings of the operations and the expected sizes of the results.

Understanding Fraction Operations

If students enter into algebra with only memorized procedures they are at risk of struggling in algebra, which, in the long run, can limit opportunities.

A Problem-based Number-Sense Approach

Students should know they have access to a variety of ways to solve fraction computation problems. In many cases, mental or invented strategies can be applied, and a standard algorithm is not needed. The goal is to prepare students to be flexible in how they approach fraction computation. For example, students should be able to solve 1/4 + 1/2 = 1/4 + 1/4 + 1/4 = 3/4.

When developing strategies for fractions:

1. Use contextual clues. You want context that fits both the meaning of the operation and the fractions involved.

2. Explore each operation with a variety of models. Have students defend their solutions using models, including simple drawings. Connect the models to the symbolic operations.

3. Let estimation and informal methods play a big role in the development of strategies. “Should 2 1/2 x 1/4 be more or less than 2?” Estimation keeps the focus on the meanings of the numbers and the operations, it encourages reflective thinking, and helps build informal number sense with fractions.

4. Address common misconceptions regarding computational procedures. Students apply their prior knowledge – in this case whole-number computation – to new knowledge. Using whole-number knowledge can be a support to learning. For example, ask “What does 2 x 3 mean?” Follow this with, “What might 2 x 3 1/2 mean?” The concepts are the same, but the procedures are different.

Computational Estimation:

1. Benchmarks. Decide whether the fractions are closest to 0, 1/2, or 1 (or 3, 3 1/2 or 4 for mixed numbers. Example: 7/8 + 1/10. Think, “7/8 is close to 1, 1/10 is close to 0. The sum is about 1 + 0 or close to 1. Example: 5 1/3 ÷ 3/5. Think, “5 1/3 is close to 5, and 3/5 is close to 1/2. How many halves are in 5? Ten.”

2. Relative size of unit fractions. Decide how big the fraction is, based on its unit (denominator), an apply this information to adding or subtracting. Example: 7/8 + 1/10. Think, “7/8 is just 1/8 away from 1, and 1/8 is close to (but bigger than) 1/10, so the sum will be close to, but less than 1.”Example: 1/3 x 3 4/5. Think, “I need 1/3 of this value. A third of 3 is 1, and 1/3 of 4/5 is going to be just over 1/5 (since there are 4 parts), so about 1 1/5.”

Addition and Subtraction

Students are usually better at fractional computation than estimating. When students were asked the question, “Estimate the answer to 12/13 + 7/8. You will not have time to solve this using paper and pencil.” Here are your choices: 1, 2, 19, 21, Don’t know.

Here is a good warm-up to math class, or it could be a full lesson:

Activity 16.1 – Over or Under 1

Tell students that they are going to estimate a sum of difference of two fractions. They are to decide only whether the exact answer is more than 1 or less than 1. Project a sum or difference for no more than ten seconds, then hide or remove it. Ask students to write down on paper their choice of “over” or “under” one. You could also have student give you the thumbs up or down to indicate over and under.

1. 1/8 + 4/52. 9/10 + 7/83. 3/5 + 3/4 + 1/84. 3/4 – 1/35. 11/12 – 3/46. 1 1/2 – 9/10

Return to each problem and discuss how students decided on their estimates. For variations, change the target to a number other than 1. Estimate more or less than 1/2, 1 1/2, 2, or 3. Adapt this to multiplication and division problems.

Activity 16.2 – Can You Make It True?

Share equations with two missing values (in the numerators or denominators, or one of each). Explain that students cannot use digits already in the problem. Encourage students to use fraction benchmarks 0, 1/2, 1 to support their thinking.

Examples: --/6 + --/8 = 1, --/5 - --/3 = 1 9/-- + --/8 = 1

--/6 + 5/-- = 1/2 1/-- - 5/-- = 2 --/7 – 5/-- = 1

One of these is impossible. All of these equations require reasoning and thinking about the relative size of fractions, which builds a critical foundation of success in developing and understanding fraction operations.

Multiplication: We use this all the time – for example, sale items are “half off” or there is a “one-third increase in voter turn-out”. Fractions are excellent substitutes for percents. To get an estimate of 60% off of $36.60 it is useful to think of 60% as the same as 3/5 or a bit less than 2/3.

Example: 3/5 of 350. First, think 1/5 of 350 (or 1/5 of 35 is 7) – that is 70. Then think 3 x 70 = 210. First, determine the fractional part (denominator), then multiply by the number of parts you want.

Back to 60% of $36.60 – or 3/5 of $36.60. 36 is about 35, and 1/5 of 35 is 7, and so 3/5 is 3 x 7 or about $21.

Division: Estimating can help support division. Think about the problem 12 ÷ 4. This can mean how many 4’s in 12? Similarly, think 12 ÷ 1/4 which means “How many fourths in 12?” There are 48 fourths in 12. With this basic idea in mind, students should be able to estimate problems like 4 1/3 ÷ 1/2 and even 3 4/5 ÷ 2/3. Start out by having students use words to describe what these equations are asking. (How many halves are in 4 1/3) can help them think about the meaning of division and then develop an estimate. Context is important! Example: “We have 5 submarine sandwiches. A serving for one person is 2/5 of a sandwich. About how many people can get a full serving?”

Activity 16.3 – Sandwich Servings

Super Sub Sandwiches is starting a catering business. They know tat a child’s serving is 1/6 of a Super Sub and an adult serving can be either 1/3 or 1/2 of a Super Sub, depending on whether the catering customer requests small or medium. The employees must be quick at deciding the number of subs for an event based on serving size. See how you do – make decisions without computing:

6 ÷ 1/6 6 ÷ 1/3 6 ÷ 1/2

Which portion size serves the most people – child size, small, or medium? Why?

8 ÷ 1/3 5 ÷ 1/2 9 ÷ 1/6

Which serves the most people – a child’s serving of 8 sandwiches, 5 sandwiches, or 9 sandwiches? Why?

Addition and Subtraction: See if students can find a variety of ways to solve word problems.

On Saturday, Lisa walked 1 1/2 miles, and on Sunday she walked 2 1/8 miles. How far did she walk over the weekend?

Jacob ordered 4 1/4 pizzas. Before his guests arrived, he ate 7/8 of one pizza. How much was left for the party?

In measuring the wood needed for a picture frame, Elise figured that she needed two pieces that were 5 ¼ inches and two pieces that were 7 ¾ inches. What length of wood board does she need to buy to build her picture frame?

Notice these three problems 1. Use a mix of area and linear models; 2. Use a mix of whole numbers, mixed numbers, and fractions; 3. Use a variety of different contexts; 4. Include both addition and subtraction situations. The last problem involves adding four fractions, not just two. Have the students model each problem or solve in a way that makes sense to them.

Can you solve the pizza problem two ways?

Assessment: A simple problem such as 4 ¼ - 7/8 can be used as a formative assessment. You should have an Observation Checklist which includes concepts such as, “Recognizes equivalences between fourths and eighths” and “Can connect symbols to a model.”

Models: Students usually draw circle models (area models) to represent fractions. That seems to say we overuse this model to teach fractions – or maybe we always pick on pizzas. Researchers have found that circles seem to be the easiest to add and subtract fractions. It seems to be easier to develop a mental image of different sizes of different pieces of the circle.

Jack and Jill ordered two medium pizzas, one cheese and one pepperoni. Jack ate 5/6 of a pizza, and Jill ate 1/2 of a pizza. How much did they eat together?

Can you show two ways to do this problem without using the common denominator approach? 5/6 = 3/6 + 2/6, 3/6 = 1/2, so 1/2 + 1/2 = 1 with 2/6 more, so 1 2/6 pizzas. Or, if you think of 1/2 as the same as 3/6, you can combine 5/6 + 1/6 = 1 with 2/6 pizza.

You could use pattern blocks.

Al, Bill, Carrie, Danielle, Enrique, and Fabio are each given a portion of the school garden for spring planting. Here are the portions:

Al = 1/4 Bill = 1/8 Carrie = 3/16

Danielle = 1/16 Enrique = 1/4 Fabio = 1/8

They pair up to share the work. What fraction of the garden will each of the following pairs or groups have if the combine their portions of the garden? Show your work.

Bill and Danielle Al and Carrie Fabio and Enrique Carrie, Fabio, and Al

Linear Models:

Using the number line is an effective method also.

Activity 16.4 – Jumps on the Ruler

Tell students to model the given examples on the ruler. A linear context can be added (hair growing and getting it cut, growing grass in the yard), but if students have been doing many contextual tasks, it is important to see whether they can also add or subtract without a context. Use the ruler as a visual, and find the results of these three problems without applying the common denominator algorithm.

3/4 + 1/2 = 2 1/2 – 1 1/4 = 1 1/8 + 1 1/2 =

Can you think of different ways this can be solved?

Another:

Desmond runs 2 ½ miles a day. If he has just passed the 1 ¼ mile marker, how far does he still have to go?

Students may first subtract the whole numbers to get 1, then ½ - ¼ to get ¼ .

Desmond is at mile marker 2 1/2, and James is at mile marker 1 1/4. How far does James need to go to catch up to where Desmond is?

Students may use the counting up method. 1 ¼ + 1 = 2 ¼ + ¼ more = 2 ½

Do on a number line, do on a ruler.

The Algorithms:

The first thing that is necessary is that students need to be able to move back and forth with equivalent fractions. When they can do this, more than half the battle has been won. It doesn’t matter which model they are using – Area – Linear – or Set – the students can adjust the fractions to combine or subtract fractions.

Do this problem first on the board: 3/8 + 4/8 = 7/8

Then put this on the board: 6/16 + 1/2 = ? Talk about how 3/8 is the same as 6/16, and 4/8 is the same as 1/2 . Students should see the answer it 7/8. The second sum is the same as the first although the fractions look different, they are actually the same.

Like Denominators:

This is suggested by grade 4 in the CCSS, and recommended by grade 5. If the students have done a lot of working with area, linear and set models, they should be able to add and subtract fractions with like denominators. The idea that the top number counts and the bottom number tells what is counted makes adding and subtracting fractions the same as adding and subtracting whole numbers. When the denominators are the same, the key idea is that the units are the same, so they can be combined.

Unlike Denominators:

In adding or subtracting fractions with unlike denominators where only one fraction needs to be changed: have the students come up with ways to combine the fractions. ( 5/8 + 1/4) As the students explain their answers, they will come up with 1/4 and 2/8 are the same fraction. When you put it on the board, be sure to ask questions. If they don’t, you give them little clues. Can we make 5/8 into a more simple fraction? (No) Can we make 1/4 into 8ths? (Yes) And so on.

What about when both need to be changed? 2/3 + 1/4 =

Do these with models at first. This is a good time to use the circle fractions. When students are exploring this, you want to emphasize that we want to make it so we are working with equal-sized parts. That makes them easier to work with. When working with the circles, I lay the 2/3 and the 1/4 side by side on a whole circle. That leaves a small piece – then as a class we try to figure out what size of piece is not covered. That will be the common denominator! Then I go back to the original problem and ask, is there any way we can make 2/3 into twelfths? (multiply by 4/4). How about 1/4? (multiply by 3/3) – then I would put the 8/12 directly under the 2/3, and the 3/12 directly under the 1/4 to reinforce they are the same fractions.

Are common denominators required? Not always, but we can tell the class that it helps to have equal-sized parts. That makes it so the parts that we are adding and subtracting are the same size.

Halves, fourths and eighths are easy to work with because they are related.

Using circle fractions, take a whole - then lay 1/2 over the whole, then lay a 1/3 over the half. What is the size of the piece not covered by the 1/3? It is half of 1/3, or 1/6.

Fractions Greater Than One:

In many texts, they separate mixed numbers from fractions. Why? Good question. A good way to introduce this is by using the good old pizzas, or on a number line. 4 ½ + 2 ½ =

Show both on the board or doc cam. Better yet, have students go up and show it. Then go to 4 ½ + 2 1/6 =

From there go to 4 ½ + 2 1/3 =

The same goes for subtraction – and be SURE to introduce this concept on the board using models first!

3 ½ - 1 ¼ = Then go to 3 ½ - 1 5/8 =

THESE MUST BE DRAWN ON THE BOARD A NUMBER OF TIMES!!!!!

Subtract the whole numbers first: 3 – 1 which leaves 2. Now, subtract the fractions. You have ½ but need to remove 5/8. Where do you get the 5/8?

Some will just take it from the 2, leaving 1 3/8 then add on the ½ or 4/8, which gives 1 7/8.

Some will take away the 1/2 to put it at 3, and then take away the 1 5/8 to make it 1 3/8, then take that ½ or 4/8 away from that, and you have 1 7/8

Some will see they cannot take the 5/8 away from the ½, so they take a whole from the 3 and change it into 8/8 and add it to the 4/8 to make 12/8, then do the standard subtraction.

A Common Mistake:

Some students will add both the numerator and the denominator. ½ + ¼ = 3/6

Try this:

Mr. Thornapple baked a pan of brownies for the bake sale. He cut the brownies into 8 equal-sized pieces. In the morning, three brownies were sold; in the afternoon, two more were sold. What fractional part of the brownies had been sold? What fractional part is still for sale?

Draw a picture to show the problem and the solution.

For Finding Common Multiples:

Activity 16.5 – Common Multiple Flash

Make flash cards with pairs of numbers that are potential denominators. Most should be less than 12. Place students in partners and give them a deck of cards. On a student’s turn, he or she turns over a card and states a common multiple (for 6 & 8 it is 24). The student suggesting the least common multiple (LCM) gets to keep the card. Be sure to include pairs that are prime, such as 9 & 5; pairs in which one is a multiple of the other (2 & 8); and pairs that have a common divisor (8 & 12).

Common problems:

1. When given a problem like 3 ¼ - 1 3/8, students subtract the smaller fraction from the larger (3/8 – 1/4 )

2. When given a problem like 4 – 7/8, they don’t know what to do with the fact that one number is not have a fraction.

3. When given a problem like 14 ½ - 3 1/8, students focus only on the whole numbers and don’t know what to do with the fractional part.

Use models and context problems with the entire class.

Multiplication:

At first, students should be involved finding fractions of whole numbers. Such as, “If 32 is the whole, what is ½ of the whole?” Then go to “If 32 is the whole, what is ¼ of the whole?” After doing a number of these, add in a bit, such as, “If 32 is the whole, what is ¾ of the whole?” Have the class discuss how they figured out each problem.

These are fraction of the whole problems.

Activity 16.6 – Walking, Wheels, and Water

Ask students to use any manipulative or drawing to figure out the answers to the next three tasks. As you can see, they represent area, linear and set models in different contexts.

1. The walk from school to the public library takes 15 minutes. When Anna asked her mom how far they had walked, her mom said that they had gone 2/3 of the way. How many minutes have they walked? (Assume they walked at a constant speed.)<----------------------------------->

2. There are 15 cars in Michael’s matchbox car collection. Two-thirds of the cars are red. How many red cars does Michael have?

3. Wilma filled 15 glasses with 2/3 cup of water in each. How much water did Wilma use?

For 1 & 2 students might partition 15 into three equal parts, or partition a line into three equal parts and then see how many are in two parts. (2/3 of 15), which gives this: (15 ÷ 3) x 2 =

Notice that problem 3 is “15 groups of 2/3” and not “2/3 of a group of 15.” Be sure to have the students see the difference in these problems.

Problem 3 could be solved by counting up by 2/3. 2/3 x 15 = 30/3 = 10 (Wholes of fractions – 15 groups of 2/3)

Fraction of Fractions – No Subdivisions

Can you do these in your head?

You have ¾ of a pizza left. If you give 1/3 of the leftover pizza to your brother, how much of a whole pizza will your brother get? (1/3 of ¾)

Someone ate 1/10 of the loaf of bread, leaving 9/10. If you use 2/3 of what is left of the loaf to make French Toast, how much of a whole loaf will you have used? (2/3 of 9/10)

Gloria used 2 ½ tubes of blue paint to paint the sky in her picture. Each tube holds 4/5 ounce of paint. How many ounces of blue paint did Gloria use?

How would you draw a diagram for each problem?

The first problem is 1/3 of three things, the second is 2/3 of 9 things, and the last is 2 ½ of 4 things.

There are a number of ways to do the problems.

Zack had 2/3 of the lawn left to cut. After lunch, he cut ¾ of the lawn he had left. How much of the whole lawn did Zack cut after lunch? (1/2)

The zookeeper had a huge bottle of the animal’s favorite liquid treat, Zoo Cola. The monkey drank 1/5 of the bottle. The zebra drank 2/3 of what was left. How much of the bottle of Zoo Cola did the zebra drink? (8/15)

You can paper fold Zack’s problem – fold into thirds, then divide the two-thirds into fourths, and color in three of the four – which is one-half of the strip of paper.

You can go to: http://nlvm.usu.edu

Go to Grades 3-5, Numbers and Operations, and scroll way down to the fraction section.

You can use counters to show multiplication of fractions.

See page 328 in the Van de Walle text.

3/5 x 2/3 =

Use 3 counters (thirds), of which 2 are red, one white. You can’t partition the numerator (2) into 5 parts, so we need to find a multiple of three that can be partitioned into fifths – 15. 2/3 is 10 counters, 1/5 of 10 counters is 2 counters, so 3/5 is 6 counters, or 6/15.

Area Models: The area models have several advantages. It works where length can become a problem and second, it provides a visual to show that a result can be quite a bit smaller than either of the fractions used. If both the fractions are close to 1, the answer will be close to 1.

3/5 x 3/4 = See page 328 for an example.

Take out a piece of paper – this problem means 3/5 of a set of 3/4. First we must show 3/4, so draw 4 vertical lines equally spaced. Mark 3 of the 4, and that is 3/4. Now we are going to take 3/5 of the 3/4. How can we do that? Correct – divide the 3/4 up by dividing it into 5 sections (4 lines). I extend the lines all the way to the edge of the rectangle, because that shows us what fractional piece one box is of the whole. (How many boxes are there? 20 – so the kind of parts are 20ths) What is next? Correct – we fill in three of the five lines going horizontally and count how many are filled – those boxes that are shaded both ways are our product (answer) which is 9/20.

3/5 x 3/4 = the 3’s are the number of parts in product, and the 4 and 5 are the kind of parts.

Activity 16.7 – Quilting Pieces

Have the students use grid paper to sketch a drawing of a quilt that will by 8 feet by 6 feet (You could create a big one for the entire class). Explain that each group will prepare a picture that is 3 feet by 2 feet for the quilt. Ask students to tell you what fraction of the quilt a group will provide. See illustration on 328.

Then, rephrase the task. Explain that in quilt making, each group is to prepare a section of the quilt that is 1/4 of the length, and 1/2 of the width. Ask students to sketch the quilt and the portion that their group will prepare. (p. 329)

Activity 16.8 – Playground Problem

Show students the problem below. Ask students to predict which community will have the bigger playground. Record predictions. Place students in partners, and ask one to solve the problem for community A and the other to solve for community B. Once they have completed their illustrations and solutions, ask students to compare their responses and to be ready to report to the class what they decided.

Two communities, A and B, are building playgrounds in grassy lots that are 50 yards by 100 yards. In community A, they have been asked to convert 3/4 of their lot to a playground, and 2/5 of that playground should be covered with blacktop. In community B, they are building their playground on 2/5 of the lot, and 3/4 of the playground should be blacktop. In which park is the grassy playground bigger? In which lot is the blacktop bigger? Illustrate and explain.

Developing the Algorithm:

With repeated use of the area models and the linear models, the students will start to notice a pattern. Keep in mind that when I say “enough” practice, that is a lot more than is usually provided in your texts. This does not mean two or three examples – it means several weeks with different examples and representations. These exercises focus on how the denominators relate to how the grid (or line) is partitioned and how the numerator affects the solution to the problem.

Write these three problems on the board:

Before they start, ask students to estimate how big the answer will be and why.

5/6 x 1/2 3/4 x 1/5 1/3 x 9/10

In the first one it will be about ½, because 5/6 is almost one, so the answer will be about ½.

In the second, 1/5 is small, and you are taking ¾ of something small, so the answer will be even smaller.

In the third, 9/10 is almost 1, and you are taking 1/3 of that, so the answer will be about 1/3.

Then have the students draw out each problem. Then put the numbers on the board.

For each one, use a square or partition it vertically and horizontally to mode the problems. Ask, “How did you figure out what unit of the fraction (denominator) was?” or, “How did you figure out the denominator would be twelfths?”

Is that pattern true for the other examples? Is there a way you can find a pattern for how the number of parts (the numerator) is determined?

Factors Greater Than One:

Begin to use mixed numbers as they explore multiplication.

Activity 16.9 – Can You See It?

Post a partially shaded illustration like the one shown here. (At the bottom of second column on Page 329.) Ask students the following questions, and have them explain how they see it.

Can you see 3/5 of something? Can you see 5/3 of something?

Can you see 5/3 of 3/5? Can you see 2/3 of 3/5?

Many texts have the students change a mixed fraction into an “improper fraction” to multiply them. They can do that, or they can just multiply them as they are.

3 2/3 x 2 1/4 =

See the illustration on 330 of the text.

Problems Students have:

Some students will treat the denominator the same as in Adding and Subtracting. Why does the numerator stay the same when adding and subtracting and get multiplied when multiplying fractions? In adding, the process is counting parts of a whole, so the parts must be the same size. In multiplication, you are actually finding part of a part, so the part size may change. If you use a rectangle, such as in the above problem, you can see how the part size changes. The same with using circles – you can see how the part size changes.

Some students are unable to Estimate the approximate size of the answer. If students only think that multiplying makes the answer larger, they will have difficulty deciding whether or not their answer makes sense. In the problem of 1/2 x 6 1/3, they have trouble seeing the answer will be around 3, because they think multiply means bigger.

Some students will match multiplication of fractions situations with multiplication. As an example, if they see 1/3 of $24, they may think that they have to divide by 1/3 – confusing the idea that they are finding a fraction of the whole. Estimation helps in situations like this. Questions such as, “Should the result be larger or smaller than the original amount?”

Division:

Begin with building on their previous knowledge of division of whole numbers. There are two types: Partitive and Measurement.

Measurement: this is the repeated subtraction or equal groups. In these situations, equal groups are repeatedly taken away from the total. For example, “If you have 13 quarts of lemonade, how many canteens holding 3 quarts each can you fill?” This is not a sharing situation, it is an equal subtraction situation.

This is a good place to start.

You are going to a birthday party. From Mitch and Bob’s Ice Cream Company, you order 6 pints of ice cream. If you serve ¾ of a pint to each guest, how many guests can be served?

The typical solution is to draw 6 items, and divide each item into fourths, and count out how many servings of ¾ can be found. The difficulty is in seeing that this is 6 ÷ ¾. This requires some direct instruction on your part. If you change it to whole numbers, such as 24 pints, with each guest getting 3 pints, how many guests can you have? How did you figure that out?

Farmer Brown found that he had 2 ¼ gallons of liquid fertilizer concentrate. It takes ¾ of a gallon to make a tank of mixed fertilizer. How many tankfuls can he mix?

(3)

Partitive:

Here the problem asks us to partition or share the whole. We usually think of it as 24 apples shared between 4 people. It also applies to rate problems, such as, “If you walk 15 miles in 3 hours, how many miles do you walk in one hour?”

It is just a small leap to this:

Cassie has 5 1/3 yards of ribbon to make four bows for birthday gifts. How much ribbon should she use for each bow if she wants to use the same amount of ribbon for each bow? About how much ribbon will be in each bow? Can you draw a picture to show the solution?

Second situation:

This is a fraction divided by a smaller fraction:

Mark has 1 ¼ hours to finish his three chores. If he divides his time evenly, how many hours can he give to each chore? First, estimate the answer – about how long should he give to each chore?

About 25-30 minutes. How did you arrive at this?

A solution is: We start with 5/4 of an hour and have to divide it into three sections. If we change 1 hour into 4ths, that is 15 minutes is the same as ¼ of an hours. 5/4 is 75 minutes. Divide 75 into 3 equal sections and it is 25 minutes each.

See solutions on p. 332

Activity 16.10 – How Much For 1?

Pose contextual problems, like the ones here, where the focus question is, “How much for one ___?”

a. Dan paid $3.00 for a ¾ pound box of cereal. How much is that for 1 pound?b. Andrea found that if she walks quickly during her morning exercise, she can cover 2 ½ miles in

¾ of an hour. She wonders how fast she is walking in miles per hour.

First find the amount of one-fourth (partitioning) and then the value of one whole (iterating). Andrea’s walking problem is a bit harder because the 2 ½ miles, or 5 half-miles do not neatly divide into three parts. Draw pictures to help.

Most problems do not come out evenly. Sometimes we have left-overs. In the ribbons and bows problem, if she took 1 1/6 yards to make a bow, but only had 5 yards, she would have had ribbon left over. If Farmer Brown begins with 4 gallons of concentrate and makes 5 tanks of mix, he uses 15/4 or 3 ¾ gallons of the concentrate. With the ¼ gallon left over, he can make a partial tank of mix. He can make 1/3 of a tank of mix, because it takes 3 fourths to make a whole, and he has 1 fourth of a gallon (he has one of three parts he needs to make a tank).

John is building a patio. Each patio section requires 1/3 of a cubic yard of concrete. The concrete truck holds 2 ½ cubic yards of concrete. If there is not enough for a full section at the end, John can put in a divider and make a partial section. How many patio sections can John make with the concrete in the truck?

See a solution on 333 in the upper right hand corner. In this problem, you can see that you get three patio sections from each cubic yard, so that makes 6 sections from the two yards. That means there is still ½ yard left in the truck. One third more for one more section leaves 1/6 of a yard in the truck. The question asks how many sections can the cement make – 1/6 is half of 1/3, so John can make 7 full sections, and a half.

Developing the Common-Denominator Algorithm:

This method relies on the measurement or repeated subtraction concept of division. In the problem of 5/3 ÷ 1/2 we can change both fractions into 6ths, so the problem would look like this: 10/6 ÷ 3/6, which is 10 ÷ 3, or 3 1/3.

One way would be to say, “To divide fractions, first find a common denominator, and then divide the numerators.”

Another way is the common way most of you learned, and that is the Invert and Multiply Method.

First, present a series of problems to see if the students can see a pattern.

3 ÷ 1/2 = How many servings of 1/2 are in 3 containers?

5 ÷ 1/4 = How many servings of 1/4 are in 5 containers?

8 ÷ 1/5 = How many servings of 1/5 are in 8 containers?

3 3/4 ÷ 1/8 = How many servings of 1/8 are in 3 ¾ containers?

Students will begin to notice that they are multiplying by the denominator of the second fraction. In the third problem, “You get 5 servings for every whole container, so 5 x 8 = 40.”

Then move to:

5 ÷ ¾ =

8 ÷ 2/5 =

3 ¾ ÷ 3/8 =

Have the students solve these and compare the answers to the first set. Notice that if there are 40 one fifths in 8, then when you group the fifths in pairs (2/5) there will be half as many, or 20. Another way of saying this, “If you double the size of the serving, you serve half as many people.” Similarly, if the fraction is 3/4, after finding how many fourths, you will group them in threes, which means you will get 1/3 as many servings. You can see this means to divide by three.

You have 1 ½ oranges, which is 3/5 of an adult serving. How many oranges (and parts of oranges) make up an adult serving?

You might be thinking that you first need to find what a fifth would be – which would be one-third of the oranges you have – or ½ an orange (notice you are dividing by the numerator). Then, to get the whole serving, you multiply ½ by 5 (the denominator) to get 2 ½ oranges in 1 adult serving.

The denominator leads you to find out how many fourths, fifths, or eighths you have, and the numerator tells you the size of the serving, so you group according to how many are in the serving. The process is multiply by the denominator and divide by the numerator.

At some point, someone reasoned that just flipping the fraction would be more straightforward – multiplying by the top and dividing by the bottom – and that is why we have learned to “invert and multiply.”

Common Misconceptions:

Thinking the answer should be smaller. Based on their whole number division experiences, students think that when dividing by a fraction, the answer should be smaller. This is true if the divisor is greater than one (5/3) but it is not true if the fraction is less than one. A good way to help here is to have the students estimating answers.

Connecting the illustration to the answer. Students may understand 1 ½ ÷ ¼ means “How many fourths are in 1 ½ .” So they may start out to count how many fourths and get 6, but in recording their answer, they may put 6/4.

Writing remainders. Knowing what the unit is (the divisor) is critical and must be understood in giving the remainder. In 3 3/8 ÷ ¼ , students are likely to count 4 fourths for each whole (12 fourths) and one more for 2/8, but then not know what to do with the 1/8 left over. It is important they understand the measurement concept of division. Ask, “How much of the next piece do you have?” Context can help, in particular, servings. If the problem is about pizza servings, there would be 13 servings and ½ of the next serving.

Websites for Developing Strategies for Fraction Computation

Diffy

http://nlvm.usu.edu/en/nav/frames_asid_326_g_3_t_1.html

The goal in a Diffy puzzle is to find differences between the numbers on the corners of the square, working to a desired difference in the center. When working with fractions, the difference of two fractions is a fraction that can be written in many different ways, and students must recognize equivalent forms.

Fractions – Adding

http://nlvm.usu.edu/en/nav/frames_asid_106_g_2_t_1.html

Two fractions and an area model for each are given. The user must find a common denominator to rename and add the fractions.

Fraction Bars

http://nlvm.usu.edu/en/nav/frames_asid_203_g_2_t_1.html

Much like Cuisenaire rods, this applet places bars over a number line on which the step size can be adjusted, providing a flexible model that can be used for all four operations.