Level E Lesson 19 Add and Subtract Mixed Numbers Unlike Denominators

Level E Lesson 19
Add and Subtract Mixed Numbers – Unlike Denominators

In lesson 19 the objective is, the student will add and subtract mixed numbers with and without models.

The skills students should have in order to help them in this lesson include adding and subtracting fractions.

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

The SOLVE problem for this lesson is, Aria’s father is laying tile in their new kitchen. The length of one side of the room is seventeen feet. Each of the tiles has a length of one and three tenths feet. What is the length of two tiles put together?

We will begin 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 length of two tiles put together? Now that we have identified the question, we want to put this question in the form of a statement. This problem is asking me to find the total length of two tiles placed together.

During this lesson we will learn how to add mixed numbers. 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 adding mixed numbers using our fraction strips. Students should work with their partner to model this first example together.

We will start by representing the first addend one and one sixth using our fraction strips. We will use one whole unit and one, one sixth unit strip to represent one and one sixth. We also need to model our second addend of one half. We will place one of our one half fraction strips below one and one sixth. Before we can add our mixed number and our fraction together, we need to find a common denominator. The common denominator will be six. Since the common denominator is six we can legally trade each of the fraction strips for sixths. We will start by legally trading the whole unit for six sixths. Six sixths will be added to the one sixth that is already a part of this mixed number, giving a total of seven sixths for the first addend. We can rewrite the problem as seven sixths plus one half. The mixed number one and one sixth changed to the improper fraction seven sixths. This fraction is called an improper fraction because the numerator is greater than or equal to the denominator. Now that we have legally traded our first addend for sixths and represented this as an improper fraction, we are ready to legally trade our second addend one half for sixths. One half is equivalent to three sixths. We can rewrite our problem as seven sixths plus three sixths. Seven sixths plus three sixths equals ten sixths. We have represented with our fraction strips a total of ten sixths. One and one sixth plus one half equals ten sixths. The fractions ten sixths, is improper because the numerator is greater than or equal to the denominator. We need to change ten sixths to a mixed number by legally trading six sixths for one whole. We have one and four sixths. Ten sixths written as a mixed number is one and four sixths. Can we legally trade the sum for fewer fraction strips in another color? Yes, let’s simplify the fraction. Four sixths is equivalent to, two thirds. One and four sixths can be simplified to, one and two thirds.

Let’s model another example using our fraction strips. In this example we want to find the sum of one and one fourth plus two thirds. We will start by showing one and one fourth using our fraction strips. One whole unit and one fourth unit will be used to represent one and one fourth. Now let’s add two thirds to our model. In order to add one and one fourth plus two thirds, we need to find a common denominator first. We need to identify the common denominator for fourths and thirds. The common denominator is twelfths. Since the common denominator is twelve we can legally trade each of the fraction strips for twelfths. We can legally trade one whole unit for twelve twelfths and the one fourth unit for three twelfths. Twelve twelfths and three twelfths gives us a total of fifteen twelfths for the first addend. The mixed number one and one fourth changed first to the mixed number one and three twelfths and then to the improper fraction fifteen twelfths. Let’s rewrite our problem to reflect this legal trade. One and one fourth is equivalent to fifteen twelfths. So our problem now reads fifteen twelfths plus two thirds. Now we need to trade two thirds for a fraction in twelfths. Two thirds is equivalent to eight twelfths. We can rewrite our problem representing two thirds as twelfths. Our problem now reads fifteen twelfths plus eight twelfths; fifteen twelfths plus eight twelfths gives us a sum of twenty three twelfths. As twenty three twelfths is an improper fraction we need to change twenty three twelfths to a mixed number by legally trading twelve twelfths for one whole which leaves eleven twelfths. This means that as a mixed number the improper fraction twenty three twelfths is written as one and eleven twelfths. Can we legally trade the sum for fewer fraction strips in another color? No, the sum is in simplest form. One and one fourth plus two thirds equals one and eleven twelfths.

We want to find the sum of one and one half plus one third. Let’s start by representing this problem using fraction strips. We can represent our first addend one and one half with one blue one whole unit and one brown one half unit. We have just represented our first addend one and one half. Our second addend is one third. We can represent one third with our fraction strips, with one green one third fraction strip. Now let’s take our fraction strips and create a pictorial model of the addends for this problem. Our first addend is one and one half. We want to shade in one and one half on the first two fraction bars. We will use blue to shade in the whole unit. And on the second fraction bar we want to create two sections, shading in one section brown to represent one half. On the first two fraction bars we have represented one and one half. We will represent our second addend one third on our third fraction bar. We need to divide this bar into three sections and shade one of the three sections in green to represent one third. Our answer in the pictorial model will be represented on the last two fraction bars. In order to find our sum we need to first look at our mixed number and our fraction to see if they have a common denominator. No, currently the mixed number and the fraction do not have a common denominator. So what is the common denominator of halves and thirds? The common denominator is six. We need to legally trade each of the fraction strips for sixths. Let’s start with the mixed number one and one half. We will divide both of these fraction strips into sixths. We have shaded in nine sixths to represent one and one half as our first addend. Now we need to legally trade our second addend for sixths. We need to divide the third fraction bar into sixths. And we can see that we have shaded in two of the six sections. One third is equivalent to two sixths. So we can rewrite our problem one and one half plus one third as nine sixths plus two sixths. We can count the shaded sixths to determine the sum. There are a total of eleven sixths that are shaded. Our sum is eleven sixths. Eleven sixths is an improper fraction. We need to change our improper fraction to a mixed number. Six sixths can be legally traded for one whole and we have five sixths remaining. Eleven sixths is equal to one and five sixths.

Let’s take a look at another example together. Again let’s start by creating a concrete representation of this problem using our fraction strips. Our first addend is one and two fifths. We can represent this with our fraction strips by using one whole unit and two of the light green one fifth units. We can represent our second addend one half by using one of our brown one half units. Now let’s represent what we did with our fraction strips pictorially. Our first addend one and two fifths can be drawn on our fraction strips by shading in the first fraction strip in blue to represent one whole unit. And the second fraction strip divided into five sections, shading two of these sections in light green to represent two fifths. Our first two fraction strips represent one and two fifths. Our third fraction strip will represent our second addend one half. We need to divide this fraction strip into two sections and shade one of these sections in brown to represent one half. The last two fraction strips will be used to represent our sum. Looking at our mixed number and our fraction, we do not have a common denominator. We need to find out what the common denominator is between fifths and halves. What is the common denominator? The common denominator between fifths and halves is tenths. We need to legally trade the fraction strips to get the common denominator for the mixed number and the fraction. We will start by looking at our first addend one and two fifths. We need to divide each of these fraction strips into tenths. We have a total of fourteen tenths for our first addend. Now let’s look at our second addend. We need to divide the third fraction strip into tenths to represent our second addend. There are five tenths shaded on our second addend. One half is equivalent to five tenths. Let’s rewrite our problem now that we have made our legal trade for tenths. One and two fifths plus one half is equivalent to fourteen tenths plus five tenths. Fourteen tenths plus five tenths equals nineteen tenths. We have a total of nineteen tenths shaded between our first addend and our second addend. Nineteen tenths is an improper fraction. We need to change nineteen tenths to a mixed number. Ten tenths is equivalent to one whole and there are nine tenths left over. So nineteen tenths can be written as a mixed number as one and nine tenths. One and two fifths plus one half equals one and nine tenths.

For this next example we are not going to create our concrete representation using the fraction strips. Instead we will start at the pictorial level by drawing each of our addends using the fraction strip pictures. We will represent our first addend one and two twelfths as one whole unit shaded in blue and on a second fraction strip shade in two of twelve sections to represent two twelfths. Our second addend is one and one fourth. We will shade in one whole fraction strip in blue to represent one whole and on a second fraction strip we will divide this into four sections shading one section to represent one fourth. We have now represented one and two twelfths plus one and one fourth. In order to find the sum we need to have a common denominator. What is the common denominator between twelfths and fourths? The common denominator is twelfths. We need to legally trade our mixed numbers for twelfths. Let’s look at the first addend. One whole will be legally traded for twelve twelfths. And we already have two twelfths represented as part of our mixed number. One and two twelfths is equivalent to fourteen twelfths. We can legally trade one and one fourth for twelfths. One whole is equivalent to twelve twelfths. And one fourth is equivalent to three twelfths. One and one fourth is equivalent to fifteen twelfths. Let’s rewrite our problem now that we have a common denominator between our mixed numbers. One and two twelfths plus one and one fourth is equivalent to fourteen twelfths plus fifteen twelfths. When we add up all of our twelfths we have a total of twenty nine twelfths, fourteen twelfths plus fifteen twelfths equals twenty nine twelfths. We want to change this improper fraction to a mixed number. Twelve twelfths is equivalent to one whole unit, twenty four twelfths is equivalent to, two whole units. We can create two whole units with five twelfths left over. Twenty nine twelfths is equivalent to, two and five twelfths. One and two twelfths plus one and one fourth is equal to, two and five twelfths.

Now let’s apply what we have done in this lesson so far to an example without using models. This problem is asking us to find the sum of two and one eighth and one and one fourth. We can see that these two mixed numbers do not have a common denominator. We need to identify the common denominator. We will do this by listing the multiples of both denominators until you find a common one. The multiples of eight are, eight, sixteen, twenty four. And the multiples of four are, four, eight, twelve, sixteen. The least common multiple of four and eight is eight. We need to change the addends to improper fractions with the common denominator. Two and one eighth is equivalent to seventeen eighths. What happened mathematically? The mixed number was changed to an improper fraction. Two wholes are equivalent to sixteen eighths with one eighth remaining. So when we add sixteen eighths and one eighth together we get seventeen eighths. Now let’s take a look at one and one fourth. We can change one and one fourth to an improper fraction five fourths. And then legally trade so that we have a denominator of eight. Five fourths is equivalent to ten eighths. What happened mathematically? The mixed number was changed to an improper fraction. One whole is equivalent to four fourths. Four fourths and one fourth give us five fourths. The four in the denominator was multiplied by two and the five in the numerator was also multiplied by two. So the improper fraction five fourths became ten eighths. We can now write our mixed numbers as improper fractions for the problem. Two and one eighth is equivalent to seventeen eighths and one and one fourth is equivalent to ten eighths. Seventeen eighths plus ten eighths equals twenty seven eighths. Twenty seven eighths is an improper fraction. Twenty seven eighths can be rewritten as the mixed number three and three eighths. Can we legally trade the sum for fewer fraction strips in another color? No, the sum is in simplest form.