Level E Lesson 22
Model Fractions as Division
In lesson 22 the objective is, the student will work with modeling and interpreting fractions as division of the numerator by the denominator and solve word problems with division that have answers in the form of mixed numbers or fractions.
The skills students should have in order to help them in this lesson include division of whole numbers, SOLVE, and division equations.
We will have three essential questions that will be guiding our lesson. Number 1, what does a fraction represent? Number 2, what does a division sentence mean? And number 3, what are the steps for dividing when representing division as a fraction?
The SOLVE problem for this lesson is, Brandon’s scout troop is going on a camping trip. The fifteen campers will be hiking and camping for three days. Each camper is responsible to bring a food item. Brandon volunteered to bring trail mix. His mom went to the store and purchased twelve pounds of trail mix. If she divides the trail mix into fifteen equal bags, how much trail mix will each camper have?
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 trail mix will each camper have? Now that we have identified 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 trail mix each camper will have.
During this lesson we will learn how to interpret a fraction as division of whole 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 by working with the fraction one half. Since we are working with fractions and fractions always relate back to one whole unit, let’s place the one whole unit fraction strip on our workspace. Every fraction can be interpreted as a division problem with the fraction bar as the division symbol. We are going to show one whole unit divided into two groups. We can show this by placing two one half units underneath our one whole unit. The value of each group is one half. The fraction one half can be written as a division problem as one divided by two. As we said the fraction bar represents the division symbol.
Let’s take a look at the fraction one third. Again we will place our one whole unit fraction strip on our workspace. We want to model one whole unit divided into three groups, to represent the fraction one third. We can do this by placing the one third units underneath the one whole unit. The value of each of the groups is one third. We can write this fraction as a division problem, one divided by three as our fraction bar represents division.
Now let’s take a look at the division problem four divided by three. This problem means how many groups of three are in four? We want to create a model to show how many groups of three are in four. We can do this by representing four items and circling groups of three items. The fraction represented by the division problem is four thirds. We can write this division problem as the fraction four over three. How many groups of three were circled in the model? We had one group of three circled in our model. Were there any shapes left after circling the group of three? Yes. There was one shape left over. We can represent this one shape that was left over as one third. We do this because we have one out of a group of three that was left over. Our answer to the problem, how many groups of three are in four, is that there, is one and one third groups of three in four.
Let’s take a look at the problem seven divided by five. This problem is asking, how many groups of five are in seven? We want to draw a model of what this problem looks like. We have a total of seven items. We want to circle groups of five items. We can circle one group of five items. With two items left over. The fraction represented by the division problem is seven over five. We had circled one group of five and there were two of the shapes left over after circling the group of five. We can represent our left over shapes as two fifths, because we have two out of a group of five left over. To answer the question, how many groups of five are in seven, we have one and two fifths groups of five in seven.
Next let’s look at the problem twelve divided by eight. The meaning of this problem is twelve divided by eight. The fraction represented by the division problem is twelve over eight. We need to complete the division to determine the quotient. When we divide eight into twelve, eight goes into twelve one time, with four left over. We write this as one and four eighths. This fraction can be simplified. One and four eighths simplifies to one and one half. The quotient of twelve divided by eight is one and one half. Between what two whole numbers is our quotient? The quotient one and one half falls between one and two.
Now let’s look at the problem nine divided by five. The meaning of this problem is nine divided by five. We can represent this division problem as the fraction nine over five. When we complete the division, five goes into nine one time, with four left over. We write this as one and four fifths. Can this fraction be simplified? No. The quotient of nine divided by five is one and four fifths. Between what two whole numbers does our quotient fall? One and four fifths is between one and two.
We are now going to go back to the SOLVE problem from the beginning of the lesson. Brandon’s scout troop is going on a camping trip. The fifteen campers will be hiking and camping for three days. Each camper is responsible to bring a food item. Brandon volunteered to bring trail mix. His mom went to the store and purchased twelve pounds of trail mix. If she divides the trail mix into fifteen equal bags, how much trail mix will each camper have?
At the beginning of the lesson we Studied the Problem. We underlined the question, and put this question in our own words in the form of a statement. This problem is asking me to find the amount of trail mix each camper will have.
Now we are ready to Organize the Facts. First we want to identify the facts. Brandon’s scout troop is going on a camping trip, fact. The fifteen campers, fact, will be hiking and camping for three days, fact. Each camper is responsible to bring a food item, fact. Brandon volunteered to bring trail mix, fact. His mom went to the store and purchased twelve pounds of trail mix, fact. If she divides the trail mix into fifteen equal bags, fact, how much trail mix will each camper have? Now that we have identified the facts, we are ready to eliminate the unnecessary facts. Or those facts that will not help us to find the amount of trail mix each camper will have. Brandon’s scout troop is going on a camping trip. Knowing where they are going is not going to help us to find the amount of trail mix each camper will have. So we will eliminate this fact. The fifteen campers, we need to know how many campers are going on the trip in order to figure out how much trail mix each camper will have. So we will keep this fact. Will be hiking and camping for three days. Knowing how long the camping trip is, will not help us to find the amount of trail mix each camper has. So we will eliminate this fact. Each camper is responsible to bring a food item. Knowing that everyone is bringing something is not going to help us to find the amount of trail mix each camper will have. So we can eliminate this fact. Brandon volunteered to bring trail mix. Knowing that Brandon is bringing trail mix is not going to help us to find the amount of trail mix each camper has. So we will eliminate this fact as well. His mom went to the store and purchased twelve pounds of trail mix. This is important to knowing how much trail mix each camper will have. So we will keep this fact, if she divides the trail mix into fifteen equal bags. Knowing that the bags are equal is going to help us to find the amount in each bag. So we will keep this fact as well. Now that we have eliminated the unnecessary facts, we are ready to list the necessary facts. Fifteen campers; Twelve pounds of trail mix; evenly divided.
In Step L, we will Line Up a Plan. First we are going to choose an operation or operations to help us to solve the problem. Remember that the problem is asking us to find the amount of trail mix each camper will have. Step O, gives us information about the amount of trail mix each camper will receive. We know the total number of campers and the total amount of trail mix. The unknown value is the amount of trail mix each camper will receive. The operation that we can use to determine the amount of trail mix per camper is division. Now let’s write in words what your plan of action will be. We are going to divide the total quantity of trail mix by the number of campers.
In Step V, we Verify Your Plan with Action. First we estimate about how much trail mix each camper will receive. Since there are fifteen campers and twelve pounds of trail mix we can estimate that each camper will receive less than one pound. Now we can carry out your plan. We said that we wanted to divide the total quantity of trail mix by the number of campers. We will divide twelve by fifteen as we have twelve pounds of trail mix and fifteen campers. Twelve divided by fifteen equals p. P is our unknown value, which represents the amount of trail mix each camper will receive. Twelve divided by fifteen can be represented as twelve over fifteen, which when simplified equals four fifths. Each camper will receive four fifths of a pound of trail mix.
Now let’s Examine Your Results. Does your answer make sense? Here compare your answer to the question. Yes, because we are looking for the amount of trail mix each camper will receive. Is your answer reasonable? Here compare your answer to the estimate. Yes, because it is close to our estimate of less than one pound. Is your answer accurate? Here check your work. Yes. The answer is accurate. Now we are ready to write your answer in a complete sentence. Each camper will have four fifths of a pound of trail mix.
We are going to look at another SOLVE problem together in this lesson. Julie is having a party for her birthday. She has invited seven friends from school. Julie is planning on playing games and watching a movie. Julie’s mom is ordering three pizzas. If the eight girls share the pizza equally, how much pizza will each girl have?
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. How much pizza will each girl have? Now that we have identified the question, we want to put the question in our own words in the form of a statement. This problem is asking me to find the amount of pizza each girl will have.
In Step O, we will Organize the Facts. We will start by identifying the facts. Julie is having a party for her birthday, fact. She has invited seven friends from school, fact. Julie is planning on playing games and watching a movie, fact. Julie’s mom is ordering three pizzas, fact. If the eight girls, fact, share the pizza equally, fact, how much pizza will each girl have? Now that we have identified the facts, we can eliminate the unnecessary facts. Those are the facts that will not help us to find the amount of pizza each girl will have. Julie is having a party for her birthday. This fact is not necessary to finding out how much pizza each girl will have. So we eliminate this fact. She has invited seven friends from school. Knowing who’s been invited is not going to help us to answer how much pizza each girl at the party will have. So we will eliminate this fact as well. Julie is planning on playing games and watching a movie. Knowing what they are going to do at the party will not help us to find the amount of pizza each girl will have. So we eliminate this fact as well. Julie’s mom is ordering three pizzas. We need to know the number of pizzas being ordered, to find out the amount of pizza each girl will have. So we keep this fact. If the eight girls, knowing that there are eight girls who will share the pizza is going to be important to finding out the amount of pizza each girl will have. So we will keep this fact. Share the pizza equally. It’s important to know that they are sharing the pizza equally, so that we know that they are sharing the pizza equally, so that we know that each girl gets the same amount. So we will keep this fact as well. Now that we have eliminated the unnecessary facts, we can list the necessary facts. Eight girls; three pizzas; share pizzas evenly.
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. The problem is asking us to find the amount of pizza each girl will have. In Step O, we organized the facts about the amount of pizza each girl will have. We know the total number of pizzas and the total number of girls. The unknown value in this problem is the amount of pizza each girl will have. We can use the operation of division to help us to determine the amount of pizza per girl. Now we can write in words what your plan of action will be. For this problem we are going to be using a pictorial model to help us to solve. We can draw a picture of each pizza and divide the number of pizzas by the number of girls.
In Step V, we Verify Your Plan with Action. First we want to estimate your answer. We know that there are eight girls and three pizzas. So each girl should receive about half of a pizza. Now let’s carry out your plan. We said that we were going to draw a picture to help us to solve this problem. We will start by drawing a picture of each pizza. There were three pizzas. So we will draw three circles to represent the three pizzas. Then we said that we wanted to divide the number of pizzas by the number of girls. There are eight girls, so we will divide each pizza into eight slices. We have three pizzas divided by eight girls. Each girl will receive three eighths of a pizza.
Now let’s Examine Your Results. Does your answer make sense? Here you want to compare your answer to the question. Yes, because we are looking for the amount of pizza each girl will have. Is your answer reasonable? Here we want to compare your answer to the estimate. Yes, because it is close to our estimate of about half of a pizza. And is your answer accurate? Here you want to check your work. Yes. The answer is accurate. Now we will write your answer in a complete sentence. Each girl will have three eighths of a pizza.
Now let’s go back and discuss the essential questions from this lesson.
Our first question was, what does a fraction represent? It represents division of the numerator by the denominator.
Our second question was, what does a division sentence mean? How many groups of blank are in blank items?
And our third question was, what are the steps for dividing when representing division as a fraction? Divide the numerator by the denominator and write the quotient as a fraction or mixed number.