Transcript Problem Solving

DEEP THOUGHTS
Instead of having answers on math
tests, can’t we just have opinions, and
if my opinion is different than that of
everyone else, hey, can’t we all just get
along.
Jack Handy
THE HORSE PROBLEM
A man named Joe buys a horse named Ed for
$60 from a woman named Flo.
Joe then sells the horse for $70, buys it back
again for $80 and sells it again for $90.
How much does Joe make or lose in the horse
trading business?
POLYA’S Problem-Solving Process:
Getting To Know The Problem:
This involves making sense of the context of the problem, what
information is given (needed and extraneous), and what it is that is being
asked?
Devising a Plan to Solve the Problem:
Students should be encouraged to share possible plans and to discuss
alternative approaches.
ESTIMATION should be encouraged at this point.
Implementing a Solution Plan:
(See strategies to follow)
Look Back and Beyond:
Does the result correlate with the estimation? Were all conditions met and
accounted for?
Encourage students to extend the problem through “What if…” questions.
Problem:
A family has three children, Adam, Beatrice and
Caroline. Each child has been given a chore. Adam is
to vacuum the carpets in the house every third day.
Beatrice must take out the garbage every fourth day
and Caroline must mow the lawn every sixth day. In
August, on which day(s) will all the children be doing
their chores together?
Problem:
I have an unlimited supply of pennies, nickels and
dimes in my pocket. If I take three coins out of my
pocket at a time, how much could I have in my hand?
Problem:
Roll the dice 20 times. Subtract the small number from
the large number. If they are the same, count as zero.
Plot your results on a line graph.
Problem:
You have been given a piece of land for a garden. You must fence this
land in totally to keep rabbits out. Assuming you need 24 square metres
for your garden, what is the least amount of fence you could use?
Problem:
You are working in a bicycle repair shop and are in charge of making
tricycles and bicycles out of extra parts. You have 18 frames (some for
tricycles, some for bicycles) and 46 wheels. Assuming that the wheels
on bicycles and tricycles are the same size, how many of each type of
cycle can you make (ensure you use all the parts you have).
Problem:
In the bus loading area there are four school busses lined up, front to
back, waiting for students to get on. Each school bus is three metres
long and there is a two metre space between each one. How long is the
loading area?
Problem:
A woman appeared on “Who Wants to Be a Millionaire” and won a large
sum of money. When she arrived back home she met her three best
friends, one at a time. To each friend she met she gave half of the money
she had then. After all three meetings she had $8 000.00 remaining. How
much did she win on the show?
PROBLEM SOLVING STRATEGIES
(Any one or more of these can, and should, be used for a variety of problems)
 Dramatize (act it out) or create a model of the
situation
 Draw a picture
 Construct a table or chart
 Find a pattern
 Solve a simpler problem
 Guess and check
 Work backwards
 Consider all possibilities
 LISTEN TO THEM
TALK!!!!!!!
 LET STUDENTS
DEVELOP THEIR
OWN STRATEGIES!
CHARACTERISTICS OF A GOOD
(RICH) LESSON:
(see: Flewelling (2002). Realizing a Vision of Tomorrow’s Classroom, Rich Tasks)
 Curriculum relevance
 Student relevance
 Authentic content and structure
 Flexible- for different levels
 Problem solving and question posing
 Inquiry/exploration/investigation/experimentation
 Communication
 Reflect on learning
 Creative
Go to:
http://math.unipa.it/~grim/AFlewelling70-72
PROBLEM OF THE
DAY/ WEEK/ MONTH/ YEAR?
Assuming you have access to pennies,
nickels, dimes and quarters, how many
combinations of change can you make for a
dollar?
Come ready to talk about your method of
solving this problem and your thinking while
you did so!
NATIONAL COUNCIL OF TEACHERS OF
MATHEMATICS
(NCTM)
Ontario Association of Mathematics Educators
(OAME)
 The primary professional organization for teacher
of mathematics of grades K-12
 1989- released Curriculum and Evaluation
Standards
 1991- released Professional Standards for
Teaching Mathematics
CURRICULUM STRANDS
 Number Sense and Numeration
 Geometry and Spatial Sense
 Measurement
 Patterning
 Data Management and Probability
THE ONTARIO CURRICULUM
GRADES 1-8
(read p 1-9 of the mathematics curriculum)
FIVE STRANDS:
Number Sense and Numeration
Measurement
Geometry and Spatial Sense
Patterning and Algebra
Data Management and Probability
Number Sense and
Numeration
Counting, numeral representation, more
and/or less than, equal to, part-whole
relationships, base ten…
Measurement
Linear measure, perimeter, area, volume,
mass, time, money, comparing sizes of
objects, non-standard and standard units
of measure…
Geometry and Spatial Sense
Simple and complex shapes (two and
three dimensional), transformational
geometry (flips, slides, turns), attributes
of shapes (vertices, sides, faces),
graphing coordinates…
Patterning and Algebra
Simple repeating patterns, growing
patterns, shape designs, sets of
numbers, patterns in art, graphs, data
collection, equations, relationships,
variables…
Data Management and
Probability
Describing and organizing graphs,
statistics, trends, estimations, rations,
fractions, collecting, presenting and
comparing data…
MATHEMATICS EXPECTATIONS
EXPECTATION:
What a child should be able to demonstrate
ACTIONS:
This is what a child should
do to demonstrate their learning,
e.g. model, compare, build,
connect
OBJECTS:
This is the content that
will be demonstrated, e.g.
equivalent fractions, place
value
FIVE PROCESS STANDARDS
1. Problem Solving
2. Reasoning and Proof (conceptual vs.
procedural)
3. Communication (oral, written, drawn,
kinesthetic…
4. Connections (within and outside
mathematics
5. Representation (symbols, diagrams,
graphs, charts, pictures)
SHIFTS IN CLASSROOM
ENVIRONMENT
☺ Classrooms as communities and not just collection
of individuals
☺ Toward logic and mathematical evidence as
verification (away from teacher as authority)
☺ Toward mathematical reasoning (concepts) and
away from memorization
☺ Toward conjecturing, inventing, problem solving
and creating and away from mechanics of getting
the right answer
☺ Toward connecting mathematics to other disciplines