Level E Lesson 27
Volume of Complex Figures: Word Problems
In lesson 27 the objective is, the student will be able to solve problems involving the
volume of two non-overlapping right rectangular prisms by adding the volumes of the
non-overlapping parts.
The skills students should have in order to help them in this lesson include multiplying
whole numbers and finding the volume of a right rectangular prism.
We will have three essential questions that will be guiding our lesson. Number 1, what
are the two volume formulas for rectangular prisms? Number 2, how are the two
formulas related? And number 3, describe how to find the volume of a rectangular prism
made of two non-overlapping rectangular prisms.
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.
To participate in the first activity of this lesson each pair of students will need a set of
centimeter cubes.
Let’s take a look at the chart. It describes the length, width, and height for Prism A and
Prism B. Partner A will build Prism A and Partner B will build Prism B. Go ahead and
do this now. Prism A should be built with a length of three units, a width of two units
and a height of four units. Prism B should be built with a length of two units, a width of
three units, and a height of three units. What is the volume of Prism A? We used a total
of twenty four cubes to create Prism A. The volume of Prism A is twenty four cubic
units. What is the volume of Prism B? We used a total of eighteen cubes to create Prism
B. The volume of Prism B is eighteen cubic units. Let’s record our answers in the chart.
We can also find volume by multiplying the length, width and height together. Let’s
check our answers by doing so in the chart. For Prism A, three times two equals six
times four equals twenty four. Prism A has a volume of twenty four units cubed. For
Prism B, two times three equals six times three equals eighteen. The volume of Prism B
is eighteen units cubed. How did you find the volume of each prism? We counted the
unit cubes or multiplied length by width by height.
Now, slide the two prisms together. This is called a complex figure because there is not
one consistent length, width or height to the figure, but the figure can be split into two
non-overlapping prisms. Did you change the number of cubes? No. What is the volume
of the two prisms together? We know the volume of Prism A is twenty four units cubed
and volume of Prism B is eighteen units cubed. Since we did not change the number of
cubes, twenty four plus eighteen is forty two. The volume of the two prisms together is
forty two cubic units. Let’s record our answer in the chart. The volume of Prism A and
B together is forty two units cubed. How did you find the volume of the combined
figure? We counted the number of cubes or added together the two volumes from the
chart. The formula for finding the volume of the rectangular prism is length times width
times height. Can we multiply all of the dimensions of both prisms together to get the
volume? Let’s see. If we take all of the dimensions and multiply them together we get
four hundred thirty two. The volume of the two prisms together was forty two units. So
no, we can not multiply all of the dimensions of both prisms together, because we do not
get forty two cubic units. Can we add the lengths, add the widths, and add the heights,
then multiply to get the volume? Let’s see. We can add the length two and three
together to get five. Add the width, two and three together to get five. And add the
heights four and three to get seven. Then multiply five times five times seven, which
equals one hundred seventy five. So can we add the lengths, add the widths, and add the
heights, then multiply to get the volume? No, we do not get forty two cubic units. If you
did not have cubes to count, how could you find the volume of this complex figure? We
would need to find the volume of each prism and add them together.
Now let’s take a look at a SOLVE problem together. A birthday cake is made of two
rectangular layers. The bottom layer is twenty four inches by eight inches by three
inches. The second layer is twelve inches by four inches by one inch. What is the total
volume of the cake?
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. What is the total volume
of the cake? Now that we have identified the question, let’s put this question in our own
words in the form of a statement. This problem is asking me to find the volume of the
entire cake.
In Step O, we Organize the Facts. First we identify the facts. A birthday cake is made of
two rectangular layers, fact. The bottom layer is twenty four inches by eight inches by
three inches, fact. The second layer is twelve inches by four inches by one inch, fact.
What is the total volume of the cake? Now that we have identified the facts, we can
eliminate the unnecessary facts. These are the facts that will not help us to find the
volume of the entire cake. The facts that we have are that the birthday cake was made up
of two rectangular layers. We are given the dimensions of the bottom layer and the
dimensions of the second layer. Are any of these facts unnecessary? No, we need all of
these facts to solve the problem. So we will keep all three facts. Now let’s list the
necessary facts. Two rectangular prism layers; first layer is twenty four by eight by three
inches; and the second layer is twelve by four by one inch.
In Step L, we will Line Up a Plan. First we need to choose an operation or operations to
help us to solve the problem. We know that we want to find the volume of the entire
cake. When we find the volume of a rectangular prism we need to use multiplication.
Since the cake has two layers we will need to add the volumes together. So we will also
need to use addition. Our operations are multiplication and addition. Now let’s write in
words what your plan of action will be. We want to find the volume of each layer using
the volume formula and then add the two volumes together.
To complete Step V, let’s place the measurements on the picture. Identify the location of
the twenty four inch measure. The length of the bottom layer is twenty four inches. Now
identify the location of the eight inch measure. The width of the bottom layer is eight
inches. And identify the location of the three inch measure. The height of the bottom
layer is three inches. Now let’s look at the second layer. Identify the location of the
twelve inch measure. The length of the second layer is twelve inches. Next, identify the
location of the four inch measure. The width of the second layer is four inches. And last
identify the location of the one inch measure. The height of the second layer is one inch.
Now that we have placed the measurements on the picture let’s complete Step V, Verify
Your Plan with Action. Let’s estimate your answer. We are looking for the volume of
the entire cake. The volume of the entire cake will be about six hundred forty inches
cubed. We can say that this is a good estimate by rounding the measurements of the
bottom layer and the second layer. Finding the volume of each of those and then adding
them together. Now lets carry out your plan. To find the volume of the entire cake we
need to take the volume of prism one and add it to the volume of prism two. The volume
of prism one can be found by taking the length times the width times the height. And the
volume of prism two can be found by taking the length times the width times the height.
Once each of the volumes is found we will add these values together to find the volume
of the entire cake. Let’s substitute in the value of the length, width and height for each
prism into our formula. The volume equals the quantity twenty four times eight times
three plus the quantity twelve times four times one. For order of operations we know that
we need to multiply before we can add. We will multiply twenty four times eight times
three which gives us five hundred seventy six. And we will multiply twelve times four
times one which gives us forty eight. The volume of the entire cake will be found by
adding five hundred seventy six plus forty eight. Five hundred seventy six plus forty
eight equals six hundred twenty four. The volume of the entire cake is six hundred
twenty four inches cubed.
In Step E, we Examine Your Results. Does your answer make sense? Here compare
your answer to the question. Yes, because I found the volume of the entire cake. Is your
answer reasonable? Here compare your answer to the estimate. Yes, because it is close
to my estimate of about six hundred forty inches cubed. And is your answer accurate?
Here check your work. Yes, the answer is accurate. We are now ready to write your
answer in a complete sentence. The volume of the cake is six hundred twenty four inches
cubed.
Now let’s take a look at another SOLVE problem together. A game box is made of two
different rectangular prisms. The base of the bottom prism has an area of eighty inches
squared and a height of two inches. The top prism has an area of thirty two inches
squared and a height of one inch. What is the volume of the game box?
First we Study to problem. We underline the question, what is the volume of the game
box? And put this question in our own words in the form of a statement. This problem is
asking me to find the game box’s volume.
In Step O, we will Organize the Facts. First we will identify the facts. A game box is
made of two different rectangular prisms, fact. The base of the bottom prism has an area
of eighty inches squared, fact, and a height of two inches, fact. The top prism has an area
of thirty two inches squared, fact, and a height of one inch, fact. What is the volume of
the game box? Now that we have identified the facts, we need to eliminate the
unnecessary facts. We know that we are looking for the game box’s volume. Know that
the game box is made up of two different rectangular prisms and knowing the dimensions
of those rectangular prisms, are all important facts to figuring out the volume of the entire
game box. So we will keep all of the facts in the problem. Since we are not eliminating
any unnecessary facts, we are ready to list the necessary facts. Two rectangular prisms;
first prism has a base area of eighty inches squared and a height of two inches; and the
second prism has a base area of thirty two inches squared, and a height of one inch.
In Step L, we will Line Up a Plan. We will start by choosing an operation or operations
to help us to solve the problem. What formula will we need to use? We will use the
formula, volume equals base area times height. Why aren’t we using the formula volume
equals length times width times height? We are not using this formula because we
already know the area of the base of each prism. So what operations will we need? To
find the volume of each rectangular prism we will need to use multiplication. Then we
want to add the two volumes together to find the total volume of the game box. Our
operations will be multiplication and addition. Now let’s write in words what your plan
of action will be. We will find the volume of each prism using the volume formula and
then add the two volumes together.
In Step V, we Verify Your Plan with Action. We will start by estimating your answer.
We can estimate that the volume of the game box will be about one hundred ninety
inches cubed. We could arrive at this estimate by rounding the measurements of each
prism. Multiplying them together to find the volume and then adding the volumes
together. Now let’s carry out your plan. The volume of the entire game box will equal
the volume of prism one plus the volume of prism two. To find the volume of each prism
we will multiply the base area times the height. Now let’s substitute in the values of the
base area and the height for each prism. To find the volume of the game box we will
multiply the quantity eighty times two plus the quantity thirty two times one. Eighty
times two equals one hundred sixty and thirty two times one equals thirty two. We need
to add one hundred sixty plus thirty two, to find the volume of the game box. One
hundred sixty plus thirty two equals one hundred ninety two. The volume of the game
box is one hundred ninety two inches cubed.
In Step E, we Examine Your Results. Does your answer make sense? Here compare
your answer to the question. Yes, because I found the volume of both prisms. Is your
answer reasonable? Here compare your answer to the estimate. Yes, because it is close
to my estimate of about one hundred ninety inches cubed. And is your answer accurate?
Here check your work. Yes, the answer is accurate. We are now ready to write your
answer in a complete sentence. The volume of the game box is one hundred ninety two
inches cubed.
Now let’s go back and discuss the essential questions from this lesson.
Our first question was, what are the two volume formulas for rectangular prisms?
Volume equals length times width times height, or volume equals base area times height.
Our second question was, how are the two formulas related? The base area is the same as
length times width.
And our third question was, describe how to find the volume of a rectangular prism made
of two non-overlapping rectangular prisms. We find the volume of each prism separately
and then add then volumes together.