Level D Lesson 17
Subtract Fractions – Like Denominators
In lesson 17 the objective is, the student will work with subtracting fractions with like denominators.
The skills students should have in order to help them in this lesson include basic subtraction facts and knowledge of fractions.
We will have three essential questions that will be guiding our lesson. Number 1, how does building with concrete materials help our understanding of fractions? Number 2, how does the use of pictures help our understanding of fractions? And number 3, how can we subtract fractions with like denominators?
The SOLVE problem for this lesson is, Miguel and his brother, Roberto, are pouring water into different containers to water their grandmother’s flowers. Miguel has filled nine-tenths of his container, and Roberto has filled three-tenths of his container. What is the difference between the amounts in the containers?
We will begin by Studying the Problem. First we want to identify the question within the problem and we will underline the question. What is the difference between the amounts in the containers? Now we want to take this question and put it in our own words in the form of a statement. This problem is asking me to find the difference between the amount of water in each container.
During this lesson we will learn how to subtract fractions with like 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.
In problem 1, we will use our fraction kits to find the difference of four-fifths take away one-fifth. We will put our whole unit at the top of our workspace, as all of our fraction pieces relate back to one whole. We will start by representing the first fraction which is our minuend. We will place four-fifths underneath our one whole. Notice that all strips are light green, as all strips are fifths. Next we will subtract or take away the second fraction which is our subtrahend. We will subtract one-fifth, so we will take away one-fifth from the four-fifths that we have represented. We are left with three-fifths. Four-fifths take away one-fifth equals three-fifths. Let’s talk about what we just did at the concrete level using our fraction strips. Our difference is three light green or three-fifths. Four-fifths take away one-fifth equals three-fifths. What happened to the denominators when the fractions were subtracted? The denominators stayed the same. What happened to the numerators when the fractions were subtracted? The second numerator was subtracted from the first to find the difference. Can we legally trade the difference for fewer fraction strips in another color? No. This tells us that the difference is in simplest form.
In this next example we will use our fraction strips to find the difference of five-eighths take away three-eighths. We will start by representing the first fraction, which is our minuend. We will place five-eighths under our one whole unit. Notice that all strips are red as they all represent eighths. Next we want to subtract or take away the second fraction. This is our subtrahend. We will take away three-eighths. We are left with two-eighths. Five-eighths take away three-eighths equals two-eighths. Let’s talk about what we just did with our fraction strips. Our answer or difference is two red or written as a fraction two-eighths. Five-eighths take away three-eighths equals two-eighths. What happened to the denominators when the fractions were subtracted? The denominators stayed the same. What happened to the numerators when the fractions were subtracted? The second numerator was subtracted from the first to find the difference. Can we legal trade the difference for fewer fraction strips in another color? Yes. Legally trading for the fewer fraction strips in another color is simplifying. We can legally trade two reds for one yellow. Two-eighths is equivalent to one-fourth. So two-eights in simplest form is one-fourth.
In this next example we want to build five-eighths take away one-eighth using our fraction kits. We will start by representing the first fraction, which is our minuend. We will place five-eighths below our one whole unit. Now we want to subtract or take away the second fraction our subtrahend. We will take away one-eighth. Five-eighths take away one-eighth is equal to four-eighths. Can we legally trade the difference for fewer fraction strips in another color? Yes. Legally trading for the fewer fraction strips in another color is simplifying. We can legally trade four red for one brown. Four-eighths is equal to one-half. So we can simplify our answer as one-half.
Now let’s take what we did using our fraction strips and create a pictorial model. Again, we are working with five-eighths take away one-eighth. We are given fraction bars that have already been broken up for us in order for us to model pictorially what is occurring in this problem. Our first fraction bar will represent the minuend. We will shade five-eighths of our fraction bar. We will use red to represent eighths. We will take away our subtrahend by crossing out one-eighth of our five-eighths that have been represented. What we are left with is the difference. We will represent the difference on the second fraction bar that has been provided for us. Five-eighths take away one-eighth leaves four-eighths. Can this fraction be traded for fewer strips in another color? Yes. We can legally trade four-eighths for one-half. We will use the third fraction strip to represent our difference simplified. We will separate this fraction strip into two equal pieces, so that we can shade in one of these pieces to represent one-half. Five-eighths take away one-eighth equals four-eighths, which when simplified is equal to one-half.
Let’s take a look at another problem together. This time we have three-fourths take away two-fourths. On our first fraction bar it has been broken up into four equally sized pieces. We will represent the minuend on this fraction bar. We will do this by shading in three of the four sections. This represents three-fourths. To subtract our subtrahend of two-fourths we will cross out two of the three shaded pieces. We have crossed out two-fourths. On our second fraction bar we will represent the difference. Three-fourths take away two-fourths leaves us with one shaded piece. So we will shade in one piece for the difference, which represents one-fourth. Can this fraction be traded for fewer strips in another color? No, it is simplified.
Now let’s look at the problem five-sixths take away four-sixths. We are not given our fraction strips at the pictorial level so we need to draw these in. We will have two fraction strips. The first will represent our minuend subtract our subtrahend. And the second will represent the difference. On the first fraction strip we will represent five-sixths. We need to break up our fraction strip into six pieces and we will shade five of those six pieces to represent the minuend. Now we want to take away four-sixths, so we will cross out four of the five shaded pieces to represent four-sixths. Our difference is one-sixth. We will use our second fraction bar to represent the difference. We need to break this fraction strip into six equal pieces and we will shade one piece to show the difference or what is left when we subtract four-sixths from five-sixths. Five-sixths take away four-sixths equals one-sixth. Can this fraction be traded for fewer strips in another color? No, it is simplified.
Let’s draw one more example together. This time we have four-fifths take away two-fifths. We need to draw our fraction strips. The first one representing our minuend, subtract our subtrahend, and the second to represent the difference. On the first fraction bar we will break it up into five pieces as we need to represent four-fifths. We will represent four-fifths by shading in four of the five sections that make up one whole. Now we want to subtract two-fifths from four-fifths. So we will cross out two of these four sections. What we are left with is our difference. Our difference is in fifths, so our second fraction bar will be split up into fifths as well. We will shade two of these fifths to represent the difference. Four-fifths take away two-fifths equals two-fifths. Can this fraction be traded for fewer strips in another color? No, it is simplified.
Now we are going to take what we’ve learned from the concrete representations using our fraction strips and the pictorial representations by shading in fraction strips on our paper and apply it to an example without the models. This problem is asking us, what is five-twelfths take away four-twelfths. What do we do with the numerators? Let’s think back to what we did with the numerators when we completed an example using our fraction strips or at the pictorial level. We subtracted the second numerator from the first numerator. So we will subtract. What do we do with the denominators? In the examples that we did using our fraction kits and at the pictorial level the denominators stayed the same. So we leave them the same. Let’s rewrite the number sentence horizontally. Five-twelfths take away four-twelfths. If we subtract the numerators we will end up with one as our numerator in our answer. And if we leave the denominators alone we will end up with twelve as our denominator. Five-twelfths take away four-twelfths equals one-twelfth. In the example five-twelfths take away four-twelfths equals one-twelfth. Can this fraction be traded for fewer strips in another color? No, it is simplified.
Let’s take a look at one more example together. This problem is asking us, what is seven-tenths take away two-tenths. What do we do with the numerators? We subtract. What do we do with the denominators? We leave them the same. Let’s rewrite the number sentence. Seven-tenths take away two-tenths is equal to what? We need to subtract the numerators seven take away 2 equals five, and leave the denominators the same. So our difference is five-tenths. Seven-tenths take away two-tenths equals five-tenths. Can this fraction be traded for fewer strips in another color? Yes, let’s simplify. Five-tenths simplifies to one-half. So we can represent our difference in simplest form as one-half.
Now we will add to a foldable helping us to organize the information we have learned in this lesson for future reference. We will be working with our fractions foldable. So far we have completed the front and back of our foldable as well as the inside left hand page. We will add to the next page, Subtraction Like Denominators, this will be page 3 of our foldable. You will work with your teacher to include the steps for subtraction with like denominators on this page.
We are now going to go back to the SOLVE problem from the beginning of the lesson. Miguel and his brother, Roberto, are pouring water into different containers to water their grandmother’s flowers. Miguel has filled nine-tenths of his container, and Roberto has filled three-tenths of his container. What is the difference between the amounts in the containers?
At the beginning of the lesson we Studied the Problem. First we underlined the question, what is the difference between the amounts in the containers? And then we put that question into our own words in the form of a statement. This problem is asking me to find the difference between the amount of water in each container.
In Step O, we will Organize the Facts. First we want to identify the facts. Miguel and his brother Roberto, are pouring water into different containers to water their grandmother’s flowers, fact. Miguel has filled nine-tenths of his container, fact, and Roberto has filled three-tenths of his container, fact. What is the difference between then amounts in the containers? Now we want to eliminate the unnecessary facts. Miguel and his brother, Roberto, are pouring water into different containers to water their grandmother’s flowers. This fact will not help us to find the difference between the amounts in the containers, so we will eliminate this fact. Miguel has filled nine-tenths of his container. This is necessary to find the difference between the amounts in the containers, so we will keep this fact. And Roberto has filled three-tenths of his container. This is also necessary to finding the difference between the amounts in the containers. So we will keep this fact as well. Now we want to list the necessary facts. Miguel has filled nine-tenths of his container; Roberto has filled three-tenths of his container.
Next we will Line Up a Plan. First, we want to choose an operation or operations to help us to solve the problem. We know that we are looking for the difference between the amounts in the containers. So we will use subtraction to find the difference. Now we will write in words what your plan of action will be. We want to subtract the amount of water in Roberto’s container from the amount of water in Miguel’s container. Simplify the fraction if necessary.
In Step V, we Verify Your Plan with Action. First, we will estimate your answer. We can estimate a difference of about one-half. Now we will carry out your plan. We said that we want to subtract the amount of water in Roberto’s container from the amount of water in Miguel’s container. Miguel’s container is nine-tenths full, so we will subtract Roberto’s container, which is three-tenths full from nine-tenths. We want to subtract the second numerator from the first numerator, nine minus three is six. And leave the denominator alone. So our answer is six-tenths. We can simplify this fraction, six-tenths in fewer pieces of one color is equivalent to three-fifths.
In Step E, we will Examine Your Results. Does your answer make sense? Here compare your answer to the question. Yes, the answer makes sense, because we are looking for the difference between the amounts in the containers. Is your answer reasonable? Here compare your answer to the estimate. Yes, our answer is reasonable, because it is close to our estimate of a difference of about one-half. Is your answer accurate? Here go back and check your work. We completed our work by following the steps for a subtraction problem with like denominators. You can use your fraction strips or use a picture to go back and check your work. Yes, our answer is accurate. Finally, we write your answer in a complete sentence. Miguel has filled three-fifths more of his container than Roberto.
Now let’s go back and discuss the essential questions from this lesson.
Our first question was, how does building with concrete materials help our understanding of fractions? Using concrete materials helps us see and touch the fractions.
Number 2, how does the use of pictures help our understanding of fractions? Using pictures helps us see the fractions.
And number 3, how can we subtract fractions with like denominators? Represent the first fraction, take away the second, then simplify.