Completing Exercises vs Solving Problems
What is Problem Solving?
"A good problem is something you don't know how to
solve. That's what makes it a good puzzle and a good
opportunity. A good problem does not just sit there in
isolation, but serves as a springboard to other
interesting questions."
- Paul Lockhart, A Mathematicians Lament
I.
Who’s Responsible for What?
II.
Unit Assessment
Objectives of this Session
0Solve Problems
0List Problem Solving Skills
0View Resources
0View Sample Lesson Plans
Let’s Solve a Problem
Crossing the River ...
0 Level 1 – with a wolf, goat, cat,
and cabbage
0 Level 2 – with Missionaries
and Cannibals
0 Level 3 – with Dogs
In groups of ___, you have __ minutes
to solve the problem.
III.
Level 1: Wolf, Goat,
Cat and Cabbage
0 Sailor Cat needs to bring a wolf, a goat, and a cabbage
across the river. The boat is tiny and can only carry
one passenger at a time.
If he leaves the wolf and the goat alone together, the
wolf will eat the goat.
If he leaves the goat and the cabbage alone together,
the goat will eat the cabbage.
How can he bring all three safely across the river?
0 Online version
0 Flipchart
Level 2: Missionaries
and Cannibals
0 Three missionaries and three cannibals must cross a
river using a boat which can carry at most two people,
under the constraint that, for both banks, if there are
missionaries present on the bank, they cannot be
outnumbered by cannibals (if they were, the cannibals
would eat the missionaries.) The boat cannot cross
the river by itself with no people on board. How can
they all cross the river alive?
0 Online version
0 Flipchart
Level 3: Crossing the River with Dogs
0 The situation: The five members of a family and their five dogs
(each family member owned one of the dogs) were hiking when
they encountered a river to cross. They rented a boat that could
hold three living things: people or dogs. Unfortunately, the dogs
were temperamental. Each was comfortable only with its owner
and could not be near another person, not even momentarily,
unless its owner was present. The dogs could be with other dogs,
however, without their owners. The crossing would have been
impossible except that Lisa’s dog had attended a first-rate
obedience school and knew how to operate the boat. No other
dogs were that well educated.
0 The task: Explain in detail how do the five family members and
their dogs cross the river?
How did you solve the problem?
0 What skills were required?
0 What previous knowledge was required?
0 Is there more than one solution? Is one solution better
than another?
text
text
text
The Locker Problem
Rubrics Drive Instruction
IV.
My
V.
st
1
Quiz
Problem Solving Skills
0 Draw a Diagram
0 Systematic Lists
0 Eliminate Possibilities
0 Matrix Logic
0 Look for a Pattern
0 Guess and Check
0 Sub-problems
0 Unit Analysis
Problem Solving Skills
0 Solve an Easier Related Problem
0 Physical Representations
0 Work Backwards
0 Venn Diagrams
0 Algebra
0 Finite Differences
0 Other Ways to Organize Information
0 Other Ways to Change Focus
0 Other Forms of Spatial Organization
A text by Ted Herr and Ken Johnson
0 Problem Solving
Strategies: Crossing the
River With Dogs and
Other Mathematical
Adventures
(ISBN: 1559530685 / 1-55953-068-5 )
0 Available at AbeBooks.com
for as little as $1 (used).
0 GAHS Math text samples
VI.
Reading and Writing Across
the Curriculum
VII.
Is this worth
teaching?
TIPC Chart
Integrating Mathematics
What are the rules?
0 Level 1: Coordinates of a Point
0 Level 2: Equation of a Line
0 Level 3: Equation of a Quadratic
0 Free web based “software”.
0 No download necessary.
0 Geogebra.org (
VIII.
)
Context of each sample:
Why are we learning this?
0 Level 1: Coordinates of a Point
0 Flat Map verse GPS coordinate system
0 Level 2: Equation of a Line
0 Business Models
0 Level 3: Equation of a Quadratic
0 Trajectory motion of a ...
0 [basketball, model rocket, arrow, etc.]
Sample Lesson Plans
0 Option #1 – Traditional + Technology
0 Option #2 – Problem Based Learning
0 Option #3 – PBL + Lecture
0 Option #4 – Vodcast Enhanced
IX.
My County’s Pacing Guide
My Pacing Guide 2010-2011
Lesson 1-1: Point, Line, Plane
0 Standard: none
0 Objective:
0 The student will be able to make and describe fundamental
sketches. The student will also be able to understand basic
concepts such as intersect, horizontal, vertical, collinear and
coplanar.
0 Essential Questions:
0 How does a system of rules help us create sketches from
verbal descriptions and vice versa?
0 What is the vocabulary of Geometry in terms of nouns, verbs,
and adjectives?
PBL Lesson Plan 1-1
0 Setting: small group (2-5)
0 Activity Directions:
1. One student (team leader) will have the picture to describe. All
other group members will have one sheet of blank paper.
2. The team leader must describe the picture verbally while the rest of
the group, while working together, attempts to draw the picture.
The team leader may only use words. This team will have no more
than two minutes to create the sketch.
3. Each team has a different picture and exchange pictures (rotate to
another team) until each group has tried every picture.
4. Team sketches will be shared with the whole class (show and tell)
5. Each team will be assigned one picture and must describe it in a
tweet.
0 Picture Sources: Not so well know paintings by famous artists such
as Monet, Van Gogh, etc..
[the whole plan]
Lesson 1-2: Segments and Rays
0 Standard: none
0 Objective:
0 The student will be able to define segment, ray, opposite rays,
congruent, midpoint, bisect, length and postulate. The "Ruler
Postulate" and "Segment Addition Postulate" will be
introduced.
0 Essential Question:
0 What is the difference between equality and congruence?
PBL Lesson Plan 1-2
0 Setting: small group (2-5)
0 Activity Directions:
1.
2.
3.
4.
5.
6.
7.
8.
Each student in the group has a piece of paper, metric ruler, and pencil.
Each group is assigned a length (in millimeters). This length is to be kept
secret from the other groups in the classroom.
Each member of the group is to draw a segment exactly the assigned length.
Groups exchange/rotate papers to another group.
Each group measures the length of the segment and records the length on
the paper. Each member of the group shares their finds on each paper and
comes to a consensus of the mystery length.
[optional] Exchange papers again to a new group.
Return papers to the original group.
Discussion Question: Name two objects that are [exactly] equal?
0 Leading Questions from the Teacher:
0 What is “margin of error”?
0 If two “pencils” appear to be equal in length, could we not “zoom in” on the
ruler and find that they are ever so slightly different?
[the whole plan]
What are the Good Problems?
video
blog post
The 4 Ds
Define:
Design:
Write a list of questions but do not answer them yet.
“What do you need to do or know in order to solve this
problem?”
Create a working sketch and equation that will solve the
problem.
Do:
Gather information needed to substitute into the equation
and solve.
Debrief:
Turn in your work with the grade you believe you have
earned. Discuss with the teacher why you deserve this
grade.
Always with a Rubric
Thank You.