In Lesson 18 the Objective Is, the Student Will Subtract Fractions with Unlike Denominators

Level E Lesson 18
Subtract Fractions – Unlike Denominators

In lesson 18 the objective is, the student will subtract fractions with unlike denominators.

The skills students should have in order to help them in this lesson include equivalent fractions and subtracting fractions with like denominators.

We will have three essential questions that will be guiding our lesson. Number 1, how does it help our understanding of subtracting fractions to build with concrete materials? Number 2, how does it help our understanding of subtracting fractions to build with pictorial models? And number 3, how can we subtract fractions with unlike denominators?

The SOLVE problem for this lesson is, Mr. Martinez is building a short walkway to the playground area. There is a muddy area from the door to the gate of the playground. He needs to cut a length of wood that is three fourths of a meter, and the length of wood he has is seven eighths of a meter. How much wood will he need to cut from the longer piece of wood to have the length he needs?

We will begin by Studying the Problem. First we need to identify where the question is located within the problem and we will underline the question. How much wood will he need to cut from the longer piece of wood to have the length he needs? Now that we have underlined the question, we want to put this question in our own words in the form of a statement. This problem is asking me to find the amount of wood that needs to be cut from the longer piece of wood.

During this lesson we will learn how to subtract fractions with unlike denominators. 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 using our fraction kits to model three eighths take away one fourth. When modeling subtraction we start by representing the first fraction. We also want to be sure that we represent one whole unit on our workspace. As all of our fraction pieces relate back to one whole. We will place the fraction three eighths underneath the one whole unit. Three eighths is the minuend in this example. What is the color of this fraction? Our minuend, three eighths, is red. Before we can subtract one fourth from three eighths we need to determine if the denominators in the subtraction problem are the same. No, the denominators are different. Before the fractions can be subtracted, we must have a common denominator. Let’s legally trade the fraction strips so that it is possible to subtract. Looking at the two fractions that we are working with in this example, we can legally trade one fourth for two eighths so that we can subtract. One fourth is equivalent to, two eighths. We can now take away one fourth or two eighths from the three eighths that we started with. We identify what is left over to find out the difference which is one eighth. Three eighths take away one fourth equals one eighth.

Now let’s take a look at what we did in this example. We started with three eighths and we took away one fourth. In order to do this we needed to have our fraction strips the same color. So we legally traded one fourth for two eighths. Our problem became three eighths take away two eighths which gave us a difference of one eighth. What happened to the denominators when the fractions were subtracted? The denominators were not alike, so one had to be changed to make a fraction that was equivalent with a common denominator. What happened to the numerators when the fractions were subtracted? The numerator of one fraction changed when the denominator was changed. Once the two fractions had a common denominator, the numerators were subtracted to find the difference. Can we legally trade the difference for fewer fraction strips in another color? No, the difference is in simplest form.

Let’s take a look at another example together. This time we will use our fraction kits to find one half take away one third. We want to place the fraction one half underneath the whole unit. One half represents the minuend. What color fraction strip is the minuend? The minuend is brown. We need to look at the denominators of our minuend and subtrahend and see if they are the same before we can subtract. The denominators are different. Before the fractions can be subtracted, we must have a common denominator. Let’s legally trade the fraction strips so that it is possible to subtract. We are going to need to legally trade one half for three sixths so that we can subtract. One half is equivalent to three sixths. Then we subtract two sixths from the three sixths that we just legally traded for. The reason that we are subtracting two sixths is because one third is equivalent to, two sixths. We can now take away one third by taking away two sixths. We identify what is left over to find out the difference which is one sixth. One half take away one third equals one sixth.

Let’s take a look at what we just did with our fraction strips. Our problem is one half take away one third. We needed our fraction strips to be the same color in order to subtract. So we legally traded one half for three sixths. And can now take away two sixths because it is equivalent to one third from the original problem. Three sixths take away two sixths equals one sixth. What happened to the denominators when the fractions were subtracted? The denominators were not alike, and they both had to be changed to make fractions that have common denominators. What happened to the numerators when the fractions were subtracted? The numerators both changed when the denominators were changed. Once the two fractions had common denominators, the numerators were subtracted to find the difference. Can we legally trade the difference for fewer fraction strips in another color? No, the difference is in simplest form.

We are now going to use our fraction kits to build the next example six ninths take away one third. We will start by representing six ninths underneath our one whole unit. We will do this by placing six of the purple one ninth fraction strips on our workspace. We want to take away one third. One third is equivalent to three ninths. So we will need to take away three ninths from the six ninths that we have. Six ninths take away one third equals three ninths. Can we simplify this fraction using fewer pieces? Can we legally trade the difference for fewer fraction strips in another color? Yes, three ninths is equivalent to one third. Six ninths take away one third equals three ninths which when simplified equals one third.

We will now move to the pictorial representation of the fraction strips with the same example that we just completed. There are three fraction strips that are provided for us. The first strip represents the first number in the problem, which is the minuend. The second strip represents the difference. And the third strip represents the difference in simplest form if it is possible to legally trade our fraction strips for fewer fraction strips in another color. As the first fraction strip represents the minuend, we will start by representing six ninths on the first strip. There are nine sections on the first fraction strip. We will shade in six of these sections in purple to represent six ninths. We need to subtract one third from six ninths. We need to legally trade one of the fractions for another to get a common denominator. One third is equivalent to three ninths. So we can take away three ninths. The difference is three ninths, which we represent on the second fraction strip. Let’s rewrite the problem to show the steps that we have completed. Six ninths take away one third is equal to six ninths take away three ninths. Remember that we needed to change the subtrahend which is the second fraction in our problem, so that we had a common denominator to subtract. One third was equivalent to three ninths. So we rewrite one third as three ninths in the problem. Six ninths take away three ninths equals three ninths. Now can we legally trade the difference for fewer fraction strips in another color? Yes, let’s simplify. Three ninths is equivalent to one third. Our difference in simplest form is one third.

Let’s take a look back at what happened when we did our legal trade. One third was equivalent to three ninths. What happened mathematically? The three in the denominator was multiplied by three and the one in the numerator was also multiplied by three. One times three equaled three in the numerator and three times three equaled nine in the denominator.

Let’s take a look at another example together. Again we will use our fraction kits to build the example one half take away one fifth. We will start by representing the minuend which is one half, underneath our one whole unit. We want to subtract one fifth from one half. In order to do this our fraction strips need to be the same color. We can legally trade one half for five tenths. And we can legally trade on fifth for two tenths. Five tenths take away two tenths leaves us with three tenths, so one half take away one fifth equals three tenths.

We will now move to the pictorial representation of the fraction strips using the same example. Our first fraction strip will represent the minuend in our example. And the second fraction strip will represent the difference. Our third fraction strip represents the difference in simplest form. If we are able to legally trade for fewer pieces of one color after we find the difference. We will begin the pictorial model by representing the minuend on the first fraction strip. The first fraction strip is divided into two sections. We will shade one of these two sections in brown to represent the minuend one half. Since one half and one fifth have unlike denominators we need to list the multiples of both denominators until you find a common one. The multiples of two are, two, four, six, eight, ten and twelve. And the multiples of five are, five, ten, fifteen, the least common multiple is ten. Our new denominator for both fractions needs to be ten, in order for us to complete this subtraction problem. Let’s start by looking at one half. One half is equivalent to five tenths. What happened mathematically in this legal trade? The two in the denominator was multiplied by five and the one in the numerator was also multiplied by five. One times five equals five in the numerator and two times five equals ten for the denominator. Let’s look at our second fraction one fifth. One fifth is equivalent to two tenths. What happened mathematically in this legal trade? The five in the denominator was multiplied by two and the one in the numerator was also multiplied by two. One times two equals two in the numerator and five times two equals ten in the denominator. Now that we have found the equivalent fractions of one half and one fifth, using tenths. Let’s rewrite the problem as five tenths take away two tenths. On our first fraction strip we are going to need to divide the fraction strip into tenths. One half was equivalent to five tenths. Since we are going to take away two tenths we will cross our two of the tenths on the first fraction strip to represent taking away two tenths. How many tenths are left? There are three tenths left over. We will represent three tenths as the difference on the second fraction strip. We need to divide this fraction strip into tenths, and shade three of these tenths in order to represent our difference. Five tenths take away two tenths equals three tenths. Can we legally trade the difference for fewer fraction strips in another color? No, the difference is in simplest form.

In the next example we will start with the pictorial model to solve the problem one half take away two sixths. Remember that in a subtraction problem we start by representing the minuend. The minuend in this example is one half. So we will draw one fraction strip divided into two sections and shade one of these two sections to represent one half. Since one half and two sixths have unlike denominators. We need to find a common denominator. We will list the multiples of both denominators until you find a common one. The multiples of two are, two, four, six. And the multiples of six are, six, twelve. We do have a common multiple. This is our least common multiple, which is six. Six will be our common denominator for both fractions. We represented one half as a pictorial model by shading in one out of the two sections of a fraction strip that represents one half. We found that our common denominator will need to be six. So underneath one half we will draw another fraction strip, this time divided into six sections. We can see that one half is equivalent to three sixths. We will shade in three of the six sections to represent three sixths. One half is equivalent to three sixths. What happened mathematically in this legal trade? The two in the denominator was multiplied by three and the one in the numerator was also multiplied by three. One times three equals three in the numerator and two times three equals six in the denominator.

Going back to our pictorial model let’s rewrite our problem. We legally traded one half for three sixths. So our problem can be rewritten as three sixths take away two sixths. Our subtrahend two sixths means that we will cross out two of the six sections on our fraction strip. We want to make sure to cross out sections that are already shaded from the minuend. Three sixths take away two sixths gives us a difference of one sixth. We will represent this difference on the last fraction strip. We need to divide this fraction strip into sixths and shade one sixth. Our difference is one sixth. Can we legally trade the difference for fewer fraction strips in another color? No, the difference is in simplest form.

Next let’s take a look at the example two sixths take away one fourth. Pictorially we will represent the minuend using a fraction strip divided into six sections. We will represent two sixths by shading in the first two sections of this fraction strip in orange to represent sixths. We can see that we have unlike denominators in this problem as we did in the last example. So we need to list the multiples of both denominators until you find a common one. The multiples of four are, four, eight, twelve and sixteen. The multiples of six are, six, twelve, eighteen. The least common multiple is twelve. This means that we will use twelve as our common denominator. Let’s start by looking at two sixths and how we can legally trade two sixths for twelfths. We can draw a pictorial representation underneath our two sixths representing twelfths. We can see that two sixths is equivalent to four twelfths. So we can legally trade two sixths for four twelfths. Looking at this legal trade what happened mathematically? The six in the denominator was multiplied by two and the two in the numerator was also multiplied by two. Two times two equals four in the numerator and six times two equals twelve in the denominator. Let’s look at the subtrahend one fourth and see how we can legally trade fourths for twelfths. We can start by drawing a pictorial representation of one fourth using a fraction strip divided into four sections and shading one of the four sections in yellow to represent one fourth. Underneath this we will draw another fraction strip. This time divided into twelfths. We can see that one fourth is equivalent to three twelfths. We can legally trade one fourth for three twelfths. Let’s take a look at what happened mathematically in this legal trade? The four in the denominator was multiplied by three and the one in the numerator was also multiplied by three. One times three equals three in the numerator and four times three equals twelve in the denominator.