Drawing a diagram.
In the attempt to understand and solve a problem, sometimes words alone are not
enough. Information not apparent in the verbal format may become obvious or clearer
when presented visually. A standard procedure for solving some problems is to draw a
picture or diagram to display important relationships. Drawing a diagram organizes the
information spatially helping to interpret the problem. This organization can be the key
to problem solving.
Many problems lend themselves to visual representation. Any situation that
evokes a mental image or picture can be clarified through the use of a diagram.
Obviously, this includes any problem that involves a geometric shape. The problem
visualization can be aided by the drawing of that shape.
Example 1: Pool Deck
Grace has just put in a new 8-foot wide concrete deck completely around her 12
by 20 foot rectangular pool. The deck is slick when wet so she decided to paint it with
non-skid paint for safety. What is the area, in square feet, she is painting?
The solution for this problem is shown on the following USV worksheet.
UNDERSTAND
U - 1: READ the problem.
U - 2: Make two LISTS.
KNOWN
NOT KNOWN
12 ft. x 20 ft. rectangular pool
length of deck
8 ft. wide deck
width of deck
deck is to be painted
area of deck
area of a rectangle = length x width
area of pool*
deck is around pool - pool is inside deck
area of deck and pool*
U - 3: STRATEGY: Name the problem solving strategy you intend to use. Explain in
complete short sentences how you intend to use and implement the strategy.
I will use drawing a diagram to solve this problem. I can draw the pool and then
the deck around it so that I can label all the known distances and verify the shape of the
deck. By combining the distances, I will be able to determine the length and width of the
deck. I will then use the appropriate area formula for the deck.
Now that the strategy and plan of attack has been decided, the solving process can begin.
Notice that the diagram has been fully labeled and statements clarify any assumptions,
interpretations, or determinations.
SOLVES - 1: SOLVE: Show your strategy in use. Label any information including
columns, drawings, etc. Include any explanation that justifies your actions or thoughts.
The diagram shows that the deck makes another rectangle. Using arrows to show
the direction of each measurement, I can see that the length of the deck includes the
length of the pool plus 8 ft. on the left and 8 ft. on the right. Through addition, the length
of the deck is 36 ft. I can also see that the width of the deck includes the width of the
pool plus 8 ft. above and 8 ft. below. Through addition, the width of the deck is 28 ft.
Since the shape is a rectangle, I multiply these to get the area of the large rectangle made
by the boundaries of the deck. 36 ft. x 28 ft. = 1008 sq. ft.
S - 2: REVISIONS to plan: What information has newly appeared? What assumptions
had to be altered? How did this affect your plan?
I found the area of the deck boundaries but it had the pool inside it. The pool area
should not be painted so I must remove the area of the pool. Therefore, I must also find
the area of the pool and subtract it from the deck rectangle I have. My solution will then
be only the area of the deck. I added the area of the pool and the area of the deck and
pool to my list of unknowns [see asterisks in the list] when I realized these were also
needed to solve the problem.
Area of the pool = 20 ft. x 12 ft. = 240 sq. ft.
Area of the deck = Area of the boundary - area of the pool
1008 sq. ft.
-
240 sq. ft.
= 768 sq. ft.
Mathematically, the problem has been solved. Use the final section of the
worksheet to state the answer clearly in a complete sentence. The sentence should
include a portion of the question to indicate the information that was sought. Reread the
problem to assure that the answer found answers the question asked. Then, check and
verify the numerical answer and the label attached to that answer.
VERIFY
V - 1: ANSWER(S): Write your answer in short complete sentences including all
appropriate labels.
Grace will be painting 768 sq. ft. of deck with non-skid paint.
V - 2: APPLY: Show that your answer satisfies the problem. Verify that
everything is consistent and makes sense.
I needed to find the area in square feet. The boundary of the deck was found to be
36 ft. by 28 ft. The pool had the dimensions 20 ft. by 12 ft. The area of the pool had to
be subtracted from the area enclosed by the deck since the pool would not be painted.
[Continued on next page]
Area is feet x feet = feet2 or square feet
so area deck boundary = 36 ft. x 28 ft. = 1008 sq. ft.
area of the pool = 20 ft. x 12 ft. = 240 sq. ft.
area of the deck alone = 768 sq. ft.
The area of the deck alone is to be painted so 768 sq. ft. will be painted. The mathematics
checks and the label concur with the desired result.
Also, diagrams can enable a problem solver to orchestrate a method of counting
when trying to solve problems that ask “How much?”
Example 2: Tournament
There are six teams in a basketball tournament. Each team needs to play each of
the other teams one time. If you are in charge of scheduling the games, how many games
must you schedule? Team A playing team B is the same as team B playing team A.
To diagram this problem, use a letter to represent each team. Draw lines to show
the connection or a game played between the teams. By counting the lines, the number
games can be determined. This is demonstrated in two forms below.
Although either diagram provides the same conclusion, the first may appear to be less
confusing making it easier to count the 15 connections.
Lastly, a diagram can enhance information and understanding beyond that which is
gained from the words.
Example 3: Bookworm
A hungry bookworm begins to eat through the front cover of Book One and does
not stop until he is through the back cover of Book Two of a set of Harry Potter books. If
each volume contains one-and-one half inches of pages wrapped with a one eighth inch
bound cover, how much did the bookworm eat?
The first impression is that the bookworm eats through the eighth inch front
cover, the one-and-one half inches of pages, and the eighth inch back cover for 1 3/4
inches per volume making his total path 3 1/2 inches. However, if a diagram is drawn of
a book on a shelf, the front cover is to the right and the back is to the left as shown below.
Furthermore, if the two books are together in order on the shelf, the front cover of Book
One is adjacent to the back cover of Book Two. In reality, the bookworm has therefore
only eaten 1/4 inch of the books. This demonstrates the value of a diagram to reason
through a problem.
Problems
1.
Michael is painting one wall the 15.5-foot length of his living room. The room
has a ceiling height of 9.5 feet. If there is an 8 ft. by 5 ft. window in the exact middle of
that wall, what is the actual square footage that he is painting?
2.
You are the first to arrive at a party (besides your host). After you, seven more
people arrived at the party. If everyone shook hands once to greet each other, how many
handshakes took place?
3.
There are ten high schools in your district. As the District Athletic Coordinator,
you must schedule the football and soccer games between each school. If each football
team plays each other and each soccer team plays each other, how many games must you
schedule?
4.
A carpenter has five boards. He needs to cut each board into four pieces. If he
takes 6 seconds to make one cut, how long would it take him to cut all five boards
assuming his assistant is moving all the pieces for him?
5.
The Rusty Stein Restaurant has ten rectangular tables in their special event room.
Each table seats four people along each side and one person on each end. These tables
are placed end-to-end to form one long rectangular table. How many people can be
seated in this arrangement?
6.
The new Burleton office building has 37 floors, each 12 feet high. The elevators
travel at a constant rate of speed for a smooth ride. If it takes 12 seconds for an elevator
to go from the first floor to the fourth floor, how long will it take to go from the fourth
floor to the middle floor?
7.
You have been called to organize the local dog show for the Great Dane Society.
You were lucky enough to secure the location at Wiltshire Hall; however, they have just
redone the floors and do not want dogs scratching the floor. The Wiltshire agreed with
your committee’s solution that everyone (dogs and people) would wear foot covers. To
make the foot covers will require one square yard of material for either a person or a dog.
The seamstress is on the phone. The 41 square yards of material has arrived, she
needs to know how many people, and how many dogs are attending so that she makes the
correct number of each type of foot covers. You do not know the number but you
remember your secretary saying that half the people are bringing two dogs, two people
are bringing three dogs, three are not bringing any dogs, and the rest are bringing one
dog. Since she needs to start immediately, what are you going to tell the seamstress?