Level E Lesson 20
Multiply Fractions
In lesson 20 the objective is, the student will work with multiplication of fractions.
The skills students should have in order to help them in this lesson include multiplying whole numbers and understanding area of a rectangle.
We will have three essential questions that will be guiding our lesson. Number 1, why is it important to know how to build a model for multiplication of fractions? Number 2, what does a multiplication sentence for fractions mean? And number 3, what are the steps for multiplying a fraction by a fraction?
The SOLVE problem for this lesson is, Danielle is working on a design for her art project. The design is made up of rectangles and triangles. There are a total of twenty four rectangles in the design, and each rectangle has a width of one half-inch and a length of three fourths-inch. What is the area of each of the rectangles?
We will start by Studying the Problem. First we want to identify where the question is located within the problem and we will underline the question. What is the area of each of the rectangles? Now we want to put this question in the form of a statement. This problem is asking me to find the area of each rectangle.
During this lesson we will learn how to multiply fractions. We will use this knowledge to complete this SOLVE problem at the end of the lesson.
Throughout this lesson students will be working together in cooperative pairs. All students should know their role as either Partner A or Partner B before beginning this lesson.
We will begin this lesson by multiplying fractions and whole numbers using our fraction strips. Each student will need to have their fraction kit in order to participate in this part of the lesson.
Let’s start by look at the problem three times one sixth. Remember that in multiplication our first factor represents the number of groups and our second factor represents the items. The meaning of this problem is three groups of sixth items. To model this problem using our fraction strips we want to create three groups with each group having one sixth item. Here’s one group of one sixth; two groups of one sixth; three groups of one sixth. Three groups of one sixth items equals three sixths. Can we legally trade the product for fewer fraction strips in another color? Yes. Let’s simplify by using fewer fraction strips. Three sixths can be legally traded for one half. Our answer in simplest form is one half. Three times one sixth equals three sixths which when simplified equals one half.
This next problem asks us to find the product of three fifths and two. Remember that our first factor represents our groups and our second factor represents our items. The meaning of this problem is three fifths of a group of two items. When modeling a multiplication problem we want to start by representing the number of items. In this problem we have two items. So we need to represent two using our fraction strips. This would be two whole units. But since we are looking for three fifths of a group of two items, we want each item to be represented in five pieces. The fraction that will represent one whole using five pieces will be fifths. We can represent our two items using fifths. Here’s one item; and two items. Now we are ready to take three fifths of the group. We will circle three out of the five pieces that make up each item. This is our product. Three fifths of a group of two items equals six fifths. Six fifths is an improper fraction. We need to legally trade the six fifths so that our answer is not in the form of an improper fraction. We can trade five fifths for one whole. Our answer becomes one and one fifth. Three fifths of a group of two items equals six fifths, which can be legally traded for one and one fifth.
We have modeled a couple of examples of fractions, times whole numbers. Now let’s take a look at an example of multiplying of fraction by a fraction. Remember that our first factor represents our groups, and our second factor represents our items. Both of the factors in this problem are fractions. The meaning of this problem is one third of a group of three fourths items. We will start by representing our three fourths items. We want to take one third of a group as there are three pieces in three fourths, we can take one of these pieces which will represent one third of the group. One third of a group of three fourths items equals one fourth.
Next let’s look at the problem one fourth times two thirds. Our first factor represents our groups, and our second factor represents our items. The meaning of this problem is one fourth of a group of two thirds items. We will start our concrete model by representing two thirds items. We need to divide the two thirds into four parts, because we want one out of four groups. Let’s legal trade! We are going to legally trade two thirds for another fraction that will give us four pieces. Two thirds is equivalent to four sixths. Now that two thirds is divided into four sections, we can circle one fourth of our group or one out of the four sections. One fourth of a group of two thirds items equals one sixth. Can we trade for fewer pieces in one color? No, the fraction is simplified.
Now let’s take a look at the problem one half times three fifths. Instead of using our fraction strips we are going to use an area model to help us to solve the problem. The meaning of this problem is one half group of three fifths items. Let’s start by looking at our groups. We have one half of a group. What does this mean? This means that we want to shade one out of two groups on the model. Our groups are represented on the left hand side of the model. We will shade in one of the two groups. Now let’s look at the number of items in each group. There are three fifths items in each group. This means we will mark three out of the five items on our area model. Our items are represented across the top of the model. We will mark out three of the five items. Now let’s look at the intersection of the shaded areas. This is our product. One half of a group of three fifths items equals three tenths. There are three sections of our model that are shaded in both with our shading of groups and with our shading of items out of the ten total sections on our model. One half group of three fifths items equals three tenths.
Let’s do one more example together. We are going to look at the problem one third times two fourths. This problem means that we are looking for one third group of two fourths items. To represent our one third group we want to shade one out of the three groups on the model. Remember that our groups are represented on the left hand side of the model. We will shade in one of the three groups. There are two fourths items in each group. This means that we want to mark two out of the four items in our model. Remember that our items are represented across the top of the area model. We will mark out two of the four items. To find our product we want to look at the intersection of the shaded areas. This is the product of one third and two fourths. The model shows a product of two twelfths, as two of the twelve sections are shaded with our groups and items. One third times two fourths equals two twelfths. Two twelfths can be simplified to one sixth.
Now we will add to our fraction foldable helping us to organize the information we have learned in this lesson for future reference. The front of our foldable reads Fractions Foldable. When we open up the fractions foldable on pages two and three we have already included information on Addition and Subtraction with Unlike Denominators. On page four we have included information on Add and Subtract Mixed Numbers with Unlike Denominators. In this lesson we will add page five Multiply Fractions. You will include information about multiplying fractions on this page with your teacher.
We are now going to go back to the SOLVE problem from the beginning of the lesson. Danielle is working on a design for her art project. The design is made up of rectangles and triangles. There are a total of twenty four rectangles in the design and each rectangle has a width of one half-inch and a length of three fourths-inch. What is the area of each of the rectangles?
At the beginning of the lesson we Studied the Problem. We underlined the question. What is the area of each of the rectangles? And put the question in the form of a statement. This problem is asking me to find the area of each rectangle.
Now we will Organize the Facts. We will start by identifying the facts. Danielle is working on a design for her art project, fact. The design is made of up rectangles and triangles, fact. There are a total of twenty four rectangles in the design, fact, and each rectangle has a width of one half-inch, fact, and a length of three fourths-inch, fact. What is the area of each of the rectangles? Now that we have identified the facts, we want to eliminate the unnecessary facts. Danielle is working on a design for her art project. Knowing what she’s working on is not going to help us to find the area of each rectangle. So we will eliminate this fact. The design is made up of rectangles and triangles. Knowing what the design is made up of does not help us to find the area of the rectangles. So we will eliminate this fact as well. There are a total of twenty four rectangles in the design. Knowing that there are twenty four rectangles does not help us to find the area of the rectangles. So we will eliminate this fact as well. And each rectangle has a width of one half-inch. Knowing the width of each rectangle will help us to find the area of each rectangle. So we will keep this fact. We will also need to know the length of each rectangle in order to find the area. So we will keep the fact that the rectangle has a length of three fourths-inch. Now that we have eliminated the unnecessary facts, we are ready to list the necessary facts. The width is one half-inch; the length is three fourths-inch.
In Step L, we will Line Up a Plan. First we want to choose an operation or operations to help us to solve the problem. When we are looking for the area of an item, we need to use multiplication. Now we can write in words what your plan of action will be. We will multiply the width of the rectangle and the length of the rectangle.
In Step V, we Verify Your Plan with Action. First we can estimate your answer. Knowing that the width is one half-inch and the length is three fourths of an inch. We can estimate that the area is going to be less than one square inch. Let’s carry out your plan. We said that we wanted to multiply the width which is one half-inch and the length which is three fourths-inch. One half times three fourths equals three eighths inches squared. The area of each rectangle is three eighths inches squared.
Now let’s Examine Your Results. Does your answer make sense? You want to compare your answer to the question. Yes, because we are looking for the area of each rectangle. Is your answer reasonable? You want to compare your answer to the estimate. Yes, because it is close to our estimate of less than one square inch. And is your answer accurate? You want to check your work. You can use an area model or your fraction strips to check your work in a different way than you originally solved the problem. Yes. The answer is accurate. Now we are ready to write your answer in a complete sentence. Each rectangle has an area of three eights inches squared.
Now let’s go back and discuss the essential questions from this lesson.
Our first question was, why is it important to know how to build a model for multiplication of fractions? So we know what a part of a fraction or whole number looks like.
Our second question was, what does a multiplication sentence for fractions mean? It asks how many items or parts are in a group.
And our third question was, what are the steps for multiplying a fraction by a fraction? We need to decide what the problem is asking – how many groups of how many items. Then, think about the number of groups and how many items would be in each group. We draw a picture to represent the second fraction and find the number of items or fractional parts the first fraction asks for. We can also multiply the numerators and then the denominators to create a new fraction.