Level C Lesson 21
Volume and Mass with Word Problems
In lesson 21 the objective is, the student will use addition, subtraction, multiplication and division to solve one-step word problems involving masses and/or volumes in the same units.
The skills students should have in order to help them in this lesson include basic multiplication and division facts zero through nine, SOLVE, multiplication and division equations, measure and estimate masses and liquid volumes and building equations using an unknown value for quotient, product, difference, and sum.
We will have three essential questions that will be guiding our lesson. Number 1, how can I solve for an unknown quotient or difference in a word problem using mass or volume? Number 2, how can I solve for an unknown product or sum in a word problem using mass or volume? And number 3, how can SOLVE help me solve a word problem?
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.
During this lesson we will be completing several word problems involving mass and volume. Each pair of students will need a set of centimeter cubes to complete the concrete representation of each problem. We will also be representing each problem at the pictorial and abstract levels of representation.
Let’s take a look at the first SOLVE problem together. Mrs. Gill is watering flowers in her garden. She has twenty one liters of water. If each container holds seven liters of water, how many containers of water will Mrs. Gill have?
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. How many containers of water will Mrs. Gill 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 number of containers of water.
In Step O, we Organize the Facts. We start by identifying the facts. Mrs. Gill is watering flowers in her garden, fact. She has twenty one liters of water, fact. If each container holds seven liters of water, fact, how many containers of water will Mrs. Gill have? Now that we have identified the facts, we want to eliminate the unnecessary facts. These are the facts that will not help us to find the number of containers of water. Mrs. Gill is watering flowers in her garden. Knowing that she’s watering flowers in her garden will not help us to find the number of containers of water. So we will eliminate this fact. She has twenty one liters of water. Knowing how many liters of water she has will help us to find how many containers of water she has. So we will keep this fact. If each container holds seven liters of water, knowing how much each container holds will help us to find the number of containers. So we will keep this fact as well. Now that we have eliminated the unnecessary facts, we will list the necessary facts. Twenty one liters; each container holds seven liters of water.
In Step L we will choose an operation or operations and write in words what your plan of action will be. Before we do this let’s talk about what we know and do not know about this problem so far. What is the unknown value? It is the number of containers holding seven liters of water. If we know the total number of liters and the number of liters that each container will hold, then what operation can we use to find the number of groups? We can use division. So now let’s decide how to set up a division plan using the unit cubes to represent liters.
Now let’s go back and Line up our Plan. First we choose an operation or operations. We said that division would help us to solve this problem. Now let’s write in words what your plan of action will be, using the unit cubes to represent liters. We can place the total number of unit cubes on the workspace. Then divvy up the number of liters or unit cubes into groups to represent the amount in each container.
In Step V, we Verify Your Plan with Action. First we want to estimate your answer. We know that we have twenty one liters and each container holds seven liters. We can estimate three or four containers of water. Now it’s time to carry out your plan. We want to place twenty one unit cubes, which represent the liters, on the workspace. Let’s do that now. Now that we have placed twenty one unit cubes on our workspace to represent the total number of liters. We are going to divvy up the cubes into groups of seven unit cubes. Here’s one group of seven unit cubes; two groups of seven unit cubes; three groups of seven unit cubes. We have represented twenty one liters, in three rows of seven. Or to answer the question this would represent three containers. Now that we have created a concrete representation of this problem using the unit cubes. Let’s go on to create a pictorial representation of the problem. Let’s answer a few questions first. Do you know the total number of liters? Yes, we know that there are twenty one liters. Do you know the number of liters each container will receive? Yes, we know that each container can receive seven liters. So what operation will you use to find the unknown value? We will use division. When we completed this problem using the unit cubes we said that we would use division and that we would place the total number of unit cubes on the work space, then divvy up the number of liters or unit cubes into groups to represent to amount in each container. Now we want to line up our plan creating a picture to help us to solve the problem. For the operation we will use division. Now let’s write in words what our plan of action will be for a picture. We can draw an array to show the total liters of water. Then circle the number of liters in each group.
In Step V, we start by estimating your answer. We said that our estimate was three or four containers of water. Now let’s carry out your plan. Your plan in Step L was to draw the total number of cubes for the array. Let’s do that first in the picture box that’s provided. We can draw squares to represent each unit cube. We will draw twenty one total squares to represent twenty one liters. Next we divide the cubes into groups of seven by circling the groups. There are three groups of seven. We have drawn three rows of seven, which represents three containers. We have now solved this problem using a concrete representation with the unit cubes and a pictorial representation by drawing squares to represent each unit cube. Now let’s create an equation to help us to solve the problem. First let’s talk about what we know. What is the total number of items, or liters? There are a total of twenty one liters. What is the number of items, or liters, in each container? There are seven liters in each container. What is the unknown value in the equation? The unknown value is the number of containers, or groups, with seven liters. With your partner, determine how to write an equation with the facts that you have. Let’s complete this in Step L. We’ve already written what we would do to draw a picture to solve the problem. Now let’s write in words what we will do with an equation to solve the problem. For an equation: We want to take the total number of liters divided by the number of liters in each container, which equals c or the number of containers.
In Step V, we can Verify Your Plan with Action. We said that we would take the total number of liters, which is twenty one, divided by the number of liters in each container, which is seven and said that equaled to c, which will represent the number of containers. Our equation is twenty one divided by seven equals c. Twenty one divided by seven equals three. So c or the number of containers equals three. Let’s rewrite the equation using three instead of c and check to see if the equation is correct. Twenty one divided by seven equals three. So our equation is correct. Does the answer for the equation match the answer for the picture? Yes.
We are now ready to Examine Your Results. Does your answer make sense? Here compare your answer to the question. Yes, it does, because we are looking for the number of containers of water. Is your answer reasonable? Here compare your answer to the estimate. Yes, because it is close to our estimate of three or four containers of water. And is your answer accurate? Here you want to check your work. Yes, the answer is accurate. Now let’s write your answer in a complete sentence. There will be three containers of water.
We are going to complete another SOLVE problem together. Again starting with a concrete representation using the unit cubes, then drawing a pictorial representation and finally moving to an abstract representation by writing an equation. Let’s take a look at the SOLVE problem together. Pia and her friends have been running laps. They are hungry, and Pia has boxes of trail mix with eight grams of mix in each box. If each of the four girls each eats a box of the trail mix, how many grams of trail mix will they eat altogether?
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 many grams of trail mix will they eat altogether? 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 number of grams of trail mix the girls will eat altogether.
In Step O, we will Organize the Facts. We want to start by identifying the facts. Pia and her friends have been running laps, fact. They are hungry, and Pia has boxes of trail mix with eight grams of mix in each box, fact. If each of the four girls eat a box of trail mix, fact, how many grams of trail mix will they eat altogether? Now that we have identified the facts, we want to eliminate the unnecessary facts. These are the facts that will not help us to find the number of grams of trail mix the girls will eat altogether. Pia and her friends have been running laps. Knowing what they are doing will not help us to find the number of grams of trail mix. So we will eliminate this fact. They are hungry, and Pia has boxes of trail mix with eight grams of mix in each box. Knowing how many grams mix are in each box will help us to find, how many grams of trail mix they will eat altogether. So we will keep this fact. If each of the four girls eat a box of the trail mix, knowing how many girls are going to be eating the trail mix will help us to find the total number of grams of trail mix that they will eat. So we will keep this fact as well. Now that we have eliminated the unnecessary facts we are ready to list the necessary facts. Eight grams of trail mix in a box; four girls; each girls eats the same amount.
In Step L, we will choose an operation or operations to help us to solve the problem And we will write in words what your plan of action will be. Let’s talk about what we do not know about this problem. What is the unknown value? it is the total number of grams of trail mix the girls will eat. If we know the number of grams in each box and the number of girls that each a box of trail mix, then what operation can we use to find the total amount of trail mix that the girls ate? We can use multiplication. Now let’s decided how to set up a multiplication plan using the unit cubes to represent grams. In Step L we said we first need to choose an operation or operations. We decided that multiplication could help us to find our unknown value. Now let’s write in words what your plan of action will be. Remember that our plan should include the unit cubes representing grams. We can place the total number of unit cubes on the workspace. Make a group that will represent each girl with the number of unit cubes representing the grams eaten by each girl.
In Step V, we Verify Your Plan with Action. First we need to estimate your answer. We know that there are four girls, and that each girls eats eight grams of trail mix. We can estimate between thirty and thirty five grams will be eaten altogether. Now let’s carry out your plan from Step L. We will place eight unit cubes in each of four groups, and determine how many unit cubes are in the array. Here’s one group of eight unit cubes; two groups of eight unit cubes; three groups of eight unit cubes; and four groups of eight unit cubes. Our unit cubes represent each of the girls and the number of unit cubes in each group represent how many grams of trail mix each girl ate. There are a total of thirty two cubes. This tells us that the girls ate thirty two grams of trail mix.
Next we are going to create a pictorial representation for this problem. First let’s answer a few questions. Do you know the total number of groups, or girls, who will eat the trail mix? Yes, there are four girls. So we need to create four groups. Do you know the number of items, or unit cubes, in each group? Yes, there are eight items in each group. What is the unknown value? The unknown value is the total number of grams of trail mix.
When we completed our concrete representation using the unit cubes our plan was to place the unit cubes on the workspace. We placed the number of unit cubes to represent what each girl ate in groups to represent the number of girls and created an array.
Now let’s crate a plan for drawing a picture to help us to solve the problem. For our picture we can draw groups of items where the items are the amount of trail mix and the groups are the number of girls.
In Step V, we Verify Your Plan with Action. First we estimate your answer. We estimated our answer to be between thirty and thirty five grams. Now we need to carry out your plan for our picture. We’re going to draw the unit cubes for the array with eight items in each of four groups. Here’s one group of eight items; two groups of eight items; three groups of eight items; and four groups of eight items. There are a total of thirty two cubes that have been drawn, which tells us that the girls ate thirty two grams of trail mix. At this point we have created our concrete representation using the unit cubes and a pictorial representation by drawing squares to represent the unit cubes.
Now let’s figure out how we can use an equation to help us to solve the problem. Let’s answer a few questions first. Do you know the number of groups, or girls? Yes, there are four girls or four groups. Do you know the number if items, or grams, each girls will eat? Yes, we know that each girl will eat eight grams. So we have eight items in each group. What is the unknown value in the equation? The unknown value is the total number of items, or grams, the girls ate.
With your partner, decide how to write an equation with the facts that you have. We are going to do this in Step L. We will include this information in words underneath the description of what we would do to draw the picture to solve this problem. When creating an equation we will start with the number of girls multiplied by the number of grams in each box which equals the total number of grams the girls will eat, this will be represented by p.
In Step V, we Verify Your Plan with Action. Remember our estimate is between thirty and thirty five grams. To carry out our plan we will complete what we said in Step L. We will start with the number of girls, which is four, multiplied by the number of grams in each box, which is eight. This will equal the total number of grams the girls will eat, represented by p as this is our unknown value. Our equation is four times eight equals p. Let’s look at the picture of multiplication that we created in the last activity and recorded in the picture box. How many grams of trail mix did the girls eat? They ate thirty two grams of trail mix. So what number value represents the p? Four times eight equals thirty two. So p equals thirty two. Let’s place our answer of thirty two back into our equation in place of p, to check to see if the equation is correct. Four times eight equals thirty two. Does the answer for the equation match the answer for the picture? Yes, it does.