PROBLEMS, PROBLEMS:
SOLUTIONS, SOLUTIONS!
Grades 5-8
This lesson has been designed to provide students with an understanding
of strategies used for effectively solving problems in various areas. Video,
hands-on activities and interaction among students show the need for forming
an hypothesis and to recognize correlation between an educated guess and
a solution. Video and simulated situations provide opportunities for students
to internalize recognition of the real problem and how to incorporate problem
solving strategies to solve it successfully.
MATH WORKS
Problem Solving: Identifying the Problem
#102
Problem Solving: Simplifying the Problem
#107
GEOGRAPHY SKILLS
Problem Solving #112
Students will be able to:
- identify concrete and abstract patterns;
- discuss correlation between possible and actual outcomes;
- demonstrate logical reasoning in solving problems;
- incorporate probability reasoning and predictions to determine intended
outcomes;
- list ways and methods to solve problems;
- explain importance of recognizing the real problem;
- separate unnecessary parts of a problem from those parts needed to
reach a solution;
- demonstrate critical problem solving strategies.
(per class)
- 1 chalkboard, chalk, eraser
- 1 chart tablet
- 1 black marking pen
- 1 overhead projector
(per group of 4)
- 1 sheet notebook paper
- 1 pencil
- 2 grocery items or their containers/wrappers
- 1 calculator
- 1 Activity Sheet
- 1 sealable plastic bag containing 15 Red beans, 15 Lima beans and
15 Peas
- 1 paper plate
(per student)
- 2 sheets notebook paper
- 1 pencil
Say, "Today we are going to focus on how to solve a problem."
Engage students in discussion pointing out: often a problem can be solved
in more than one way; and the method of solution may not be obvious or readily
identifiable when someone says they don't get it or they don't understand.
Divide the class into groups of four students each. Distribute a sheet of
notebook paper and a pencil to each group; instruct them to select a recorder
from within the group. Say, "Brainstorm within your group and decide
on strategies that can be employed to figure out a problem you don't understand."
Emphasize there are several things a good mathematician can do to solve
a difficult problem. Say, "As your group agrees on a strategy that
can be used to solve a problem, it should be written down by the recorder."
Instruct students to begin the task; allow a reasonable period of time for
the task to be accomplished.
Allow groups to interact as they share strategies decided upon. List strategies
on chalkboard as they are suggested by individual groups, then direct discussion
toward combining strategies which are similar but stated differently. Guide
discussion to a logical sequencing of problem solving strategies as you
ultimately conclude and record the sequenced strategies on a chart tablet
as: Understand the problem. (What is being asked? What are the facts? Which
infor-mation is important to solving the problem? Which is not?) Plan a
solution. (What options do I have for solving the problem? Which can help?
(e.g.) Draw a picture; construct a model; guess and check; work backwards;
make a graph or table; or use a computation skill of addition, subtraction,
multiplication or division.) Solve the problem. Check your work. After completion
of the strategies list, distribute a sheet of notebook paper and a pencil
to each student. Say, "Use the chart tablet list and record the strategies
for your own personal use and referral as you decide how to solve various
problems today."
Ask, "Why do you suppose understanding the problem is sequenced
first on the list?" Allow students to tell it is necessary to know
what a problem is before you can solve it. Say, "You are going to see
a video about two girls attempting to figure out how much money is needed
to buy their grandmother a jewelry box. The problem may appear simple to
you, however, to a younger person it would not be easy." To give students
a specific responsibility while viewing say, "As you watch the video,
refer to your list of problem solving strategies and decide if and how the
first strategy is employed by the girls, then be prepared to defend what
you decide."
MATH WORKS
Problem Solving: Identifying the Problem #102
BEGIN tape immediately following opening credits. PAUSE tape
on visual of narrator; audio is, "...some step to take when you're
sure the clues are there but you just can't find them." Allow students
to tell and defend their belief about whether the girls properly implemented
the first strategy. If the following wasn't included in students' rationale,
ask: "What was the mistake in computation made by the girls? What gave
the mistake away?" (needed to raise more money than the jewelry box
cost) Emphasize the need to always use common sense when problem solving.
Write the term on chalkboard as you encourage students to share their perception
of common sense and as you reinforce the need to apply logic. Say, "Refer
to the list of strategies and decide which ones weren't employed by the
girls." Lead discussion to conclusion that the girls didn't understand
the problem (strategy #1) and they didn't check their work (strategy #4).
Say, "Had the girls understood the problem and had they checked their
work, what should be done would have been clear to them."
Ask, "Do you think either girl recognizes their mistake?" Allow
for response. To give students a specific responsibility while viewing say,
"Watch the next video and be prepared to tell what information is needed
to solve the problem and which information isn't needed." RESUME
tape. PAUSE tape on visual of narrator and girls; audio is, "Well,
it looks like Jenny and Arlene are going to be able to buy their granny
that beautiful jewelry box." Allow students to tell which information
they believe is needed and which they believe isn't. To give students a
specific responsibility while viewing say, "Watch as I freeze frame
the video on a list of things we know about the jewelry box, then decide
if your belief about needed information is correct." RESUME
tape. FREEZE frame on graphic list of things known about the problem.
Allow students to identify correct information as $40.00 and $19.20 difference.
Say, "Always disregard any information that isn't needed to solve a
problem so it won't interfere or be confused with information you do need."
To give students a specific responsibility while viewing say, "In the
next video, see if you can identify which is needed information and which
is not needed as the narrator states the problem to be solved." RESUME
tape. PAUSE tape on graphic list of what is known and what needs
to be known about the problem; audio is, "What would you do with this
information?" Allow students to internalize separating needed from
un-needed information as you ask, "If a farm is 120 km from where they
started and the occupants of an automobile have already traveled 70 km,
how many more km must they travel to reach the farm?" Allow students
to discuss and answer 50 km. Ask, "How did removing unnecessary information
make the problem easier to solve?" Allow students to share opinions.
To give students a specific responsibility while viewing say, "Watch
the next video to test the accuracy of your opinion." RESUME
tape. PAUSE tape on visual of the information blinking on the screen;
audio is, "...and bingo, that solves the problem." Allow time
for students to tell whether their opinions were valid.
Say, "At times, a problem can appear confusing because you aren't familiar
with its setting. If so and the setting can be changed to one you're more
familiar with, it should be easier for you to understand." To give
students a specific responsibility while viewing say, "Watch the next
video which shows a problem and be prepared to tell how substituting a familiar
situation for one that was unfamiliar helped." RESUME tape;
PAUSE tape with audio, "...that the situation you create is
parallel with the problem." Allow students to describe how bicycles
were substituted for chickens, thus, eliminating an unfamiliar farm setting
for city children and providing a setting they could relate to with the
bicycles.
Identify a city very different from the area or city your students have
grown up in and are familiar with. (e.g.) New York vs Nashville. Have students
contrast how the two would differ in: recreational opportunities, housing,
means of trans-portation, etc. Encourage them to consider why a youngster
from Nashville might have difficulty relating to a problem about subway
schedules but not the youngster from NYC; and why the youngster from NYC
might have trouble relating to rabbits or squirrels playing on your lawn
but not someone from Nashville. Use the discussion to reinforce the desirability
of substitute situations to help solve problems.
Say, "We have discussed two important elements to help you understand
problems, what are they?" List on chalkboard as students tell: eliminate
unnecessary information and change an unfamiliar setting to a familiar one.
Say, "A third key element to help you understand a problem is to look
for a relationship between two things." To give students a specific
responsibility while viewing say, "As you watch the next video, see
if you can figure out which is the better buy as two children look for a
contrast in price between different packages of hot dogs." RESUME
tape; PAUSE tape with audio of girl, "...so two of the six packs
of hot dogs would be a better buy than the package of ten." Allow for
student response. Ask, "Who solved the problem but in a different way?"
Discuss various methods that might have been used for solving the problem.
Add number three to the list of elements to help you understand (solve)
a problem on the chalkboard: find the relationship between two things.
Ask, "In what ways do many supermarkets try to help with problem solving
for shoppers?" (calculators on carts to help you find better buys;
items are marked with price and unit price; store brand products priced
lower than a similar product with a national brand name.) Ask, "How
many keys to help with better understanding a problem do you remember?"
Allow students to respond. To give students a specific responsibility while
viewing say, "As you watch the next video, check to see if you remembered
all the keys to understanding a problem." RESUME tape. STOP
tape after narrator has restated all methods. Engage students in discussion
as they tell: 1. What do I know? 2. What do I need to know? 3. Change the
setting. 4. Find the re-lationship by restating the problem in your own
words.
Distribute to each group of four students a calculator and a set of two
competing grocery items which are priced differently and are of different
sizes. Suggested items are:
- 2 unlike multi-packs of bath soap
- 2 unlike sized boxes of cereal
- 6 cans of soft drink and 1 two liter bottle
- 2 unlike dish washing detergents
- 1 bottle and 1 box laundry detergent
- 1 half gallon and 1 gallon milk
- 2 varieties of cookies
- 2 rolls unlike paper towels
- 2 unlike toothpastes, etc.
Explain the items are to be used for solving a problem. (Distribute additional
notebook paper as needed) Say, "Use the items distributed to your group
and determine which is the better buy." Review the problem solving
strategies previously presented, then instruct students to begin the task.
Upon completion, allow each group to present and defend its findings as
other groups use calculators to check the presenting group for accuracy.
Say, "Simplifying a problem will at times allow you to understand it
more easily. In the next video two students use this strategy to determine
how much candy they will have to sell in order to purchase a trampoline."
To give students a specific responsibility while viewing say, "Watch
and be prepared to tell how they simplified their problem."
MATH WORKS
Problem Solving: Simplifying the problem #107
BEGIN tape with visual of scene inside a gym; audio is, "Very
nice." PAUSE tape with visual of scene inside an office; audio
is, "... and that's how it's done." Allow students to tell how
the video problem was simplified. Ask, "Did you find it easier to decide
what to do with smaller numbers? How was it made easier? (easier to see
relationships between smaller numbers) What are examples of situations when
this might be a desirable strategy to employ?" Reinforce that an understanding
of any problem can very well be the most critical of all steps in problem
solving.
Say, "Just as mathematicians, scientists must also determine the problem
before they can begin to solve it. Breaking a problem down to make it more
specific will often create a clearer picture of exactly what the problem
is." Explain; in the next video there is a very frustrated student
attempting to prepare a science report. To give students a specific responsibility
while viewing say, "Watch to learn why he is having difficulty developing
the report and decide what he could do to make the task more manageable
before the teacher shows him." RESUME tape on visual of the
science topic poster; audio is, "Any problem is easier to solve if
you think about a way to simplify it." STOP tape on visual of
narrator; audio is, "It's a perfect way to simplify any problem."
Ask, "Why was the boy having problems developing his report?"
(His topic was too broad.) Ask, "When accepting a topic for research
and development, why is it wise to be very specific when choosing your topic?"
Accept responses. Emphasize the need to select a specific part of a topic
as it permits a more in depth research strategy which will always result
in a more comprehensive and thorough report. Encourage students to interact
as they identify specific topics that could be selected as opposed to the
following, (e.g.): Salt Water Animals; Western Ski Resorts; Balancing the
National Budget; Kings and Queens of England; and The Life and Times of
Galileo.
Reinforce a need to be specific as you use Galileo's theory of gravity as
a specific topic that might be developed, segueing into the definition and
a discussion on theory. Encourage students to share their knowledge of theory,
leading to conclusion that a theory is a "best guess" based on
a study of all available information about something or how something occurred.
Allow students to have fun as they respond to "What is your theory
about which came first, the chicken or the egg?" After students have
had a moment of being frivolous, bring the responses to closure. Write the
name Alfred Wagner on the chalkboard. Ask, "What famous theory is Alfred
Wagner credited with developing?" Accept all responses. To give students
a specific responsibility while viewing say, "Watch the next video
and find out what Wagner's theory was?"
GEOGRAPHY SKILLS
Problem Solving #112
BEGIN tape with visual of the world map; audio is, "Let's compare
shapes. Back in 1913..." STOP tape at end of video before closing
credits. Ask, "What was Alfred Wagner's theory?" (Continental
Drift) Ask, "Would you have been able to solve the problem Wagner theorized?"
Allow students to respond. Ask, "What in-formation and clues were available
for Wagner to study and base his hypothesis and theory on?" (Land formations
on different continents that matched at the shorelines.) "Do patterns
sometimes provide clues in other areas of problem solving?" Encourage
students to interact and share their ideas. "What did Wagner name the
super-continent?" (Pangea) Ask, "In what ways did patterning help
Wagner solve the problem and develop his theory?" Guide discussion
to realization that understanding the problem and his forming an hypothesis
about known patterns between the continents were critically important to
the development of his theory.
Distribute to each group of four students: 1 copy of the Activity
Sheet; 1 sealable plastic bag containing 15 red beans, 15 lima beans and
15 peas; and 1 paper plate. Give all students in each group a piece of notebook
paper and a pencil. Use an overhead projector to show the beans and a clean
sheet on the chart tablet to record results and strategies which are used
by students to solve the problems.
Say, "You are now going to solve problems in a hands-on process using
the materials that were distributed to your group. Remember to follow the
sequence of strategies discussed today in solving the problems." Explain
a person familiar to the group has written a Bean Cook Book, however, the
eight recipes on the Activity Sheet were returned by the publisher because
some of the ingredients and amounts were not included. Say, "It is
now your responsibility to solve the problems so the cookbook can be published.
Each recipe must contain all three types of beans."
Instruct students to begin with problem number one and use their bag of
beans to solve the problem in a hands-on manner. Say, "Then, show your
calculations on notebook paper." Allow time for groups to complete
the task. Select a volunteer to explain how her/his group solved the problem,
then have them use the overhead projector to show all beans that were used.
Next, select another volunteer from the group to demonstrate their cal-culations
on the chalkboard. If other groups incorporated strategies that differed
from the presenting group, have them share the strategy they used. (Ans:
2 Lima beans; 4 Red beans; 4 Peas) Record the answer on the chart tablet.
Say, "Some problems can have more than a single solution; others will
have just one." Instruct students to continue and solve problem number
two. Repeat the previous process as another group is selected to present.
(Ans: 4 Red beans; 2 Peas; 4 Lima beans) Record results on the chart tablet.
Instruct groups to proceed with problem three. Use the established pattern
for student presentation as other groups validate correctness of the presenting
group's work. (Ans: 2 Red beans; 4 Lima beans; 2 Peas)
Direct groups to continue with problem number four. Again, use the previous
process of presenting as another group explains and visually demonstrates
their work using the overhead projector. (Ans: 5 Red beans; 5 Lima beans;
8 Peas) After presentation, validation and up-dating of the chart are completed
for this problem, engage students in discussion related to examples of when
concrete methods and abstract methods of problem solving have thus far been
used.
Instruct students to continue with problem number five. Follow the procedure
as a different group is appointed presenter. (Ans: 6 Red beans; 3 Lima beans;
3 Peas)
Have students continue with problem number six. After groups have completed
the task, point out that possible solutions can be endless as the recipe
stated "at least 12 beans." Any answer is correct that contains
no less than a total of 12 beans and follows the other two criteria.
Direct students to compute problem number seven. (Ans: 1 Pea; 3 Red beans;
4 Lima beans)
Have students solve problem number eight. Upon completion, inform students
there are several solutions to this problem. Each that does not exceed a
total of 20 beans is correct.
Conduct a brief review of strategies to be used in problem solving. Have
students consider the desirability for creating a mental picture to be used
in visualizing important information and determining possible strategies
they might use to solve a problem.
Plan a field trip and visit the Meteorology Department of a
local television station. Prepare students to look for methods of problem
solving used to forecast weather. As a forecast is prepared, have students
look for activities which reveal problem solving procedures. They should
be able to identify and describe how each is used. Invite a law enforcement
officer to visit the classroom and discuss how problem solving is an on-going
responsibility in her/his profession. Ask them to include how knowledge
of patterns is an integral part of identifying suspects. Use the information
to create a list of situations in which students solve "cases"
where patterns link a suspect(s) to a crime or to one another.
Creative Writing
Have an interested group of students write a short drama about preventing
poachers from destroying animals in a wildlife preserve in Kenya.
Performing Arts
Use the creative writing drama and plan a performance. Appoint students
with special interests or skills to compose or select appropriate music,
design a set and costumes, serve as prop and stage hands and serve as performers.
Encourage them to utilize problem solving strategies from inception and
planning throughout the actual time of performance.
Science/Research
Select an inventor, a doctor who discovered a cure for a disease, a great
artist or performer, etc., and research their life to learn how they became
successful. Apply how they succeeded to strategies for solving problems,
then share findings by preparing a report to be presented to the class.
Master Teachers: James Parsons and Sharon Braden
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