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Math Resources
Problem Archive
Using rich and open mathematical problems is a hallmark of a Japanese
math research lesson. While lesson study does not have to revolve around
open problems, we have found that these kinds of problems have enabled
our participants to engage in deeper conversations about student learning
of mathematics.
Based on a presentation by Jerry Becker at NCTM Boston, we have organized
the "open" math problems in this section into 3 general categories.
If not stated otherwise, the problem came from Becker's materials.
(1) Problems with many approaches
but 1 solution
1a. How many dots?
1b. Fractions, decimals and percents using an Area
Model
1c. Ships in the Fog
1d. Chances Are
1a. How many dots?
Problem Statement:
In how many ways can you determine the number of marbles on this
page?
Handout:
How many dots? (pdf)
Comments:
When we used this problem at the March, 2003 workshop, our participants
not only came up with many approaches to count the 25 marbles,
but they also came up with more generalizable approaches to counting
similar sets of marbles based on the size of the diagonal.
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1b. Fractions, decimals and percents
Problem Statement:
Shade 6 squares in a 4 by 10 rectangular grid. Explain how you
would determine what percent, what decimal, and what fractional
part are shaded (without using an algorithm).
Handout:
Linking Fractions, Decimals and Percents
(pdf)
Comments: This problem was taken from Implementing
StandardsBased Mathematics Instruction: A Casebook for Professional
Development by Mary Kay Stein, Margaret Schwan Smith, Marjorie
A. Henningsen and Edward A. Silver. (New York: Teachers College
Press, 2000) We used this problem at our May 21, 2003 workshop
along with the case.
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1c. Ships in the Fog
Problem Statement:
Two ships are sailing in a fog and are being monitored by tracking
equipment. As they come onto the observer’s screen, the
ship Andy Daria (AD) is at a point 900 mm from the bottom left
screen along the lower edge. The other one, the Helsinki (H) is
located at a point 100 mm above the lower left corner along the
left edge. One minute later the positions have changed. The AD
has moved to a location on the screen that is 3mm “west”
and 2mm “north” of its previous location. The H has
moved 4 mm “east” and 1 mm “north”. Assume
that they will continue to move at a constant speed on their respective
linear courses. Will the two ships collide if they maintain their
speeds and directions? If so, when? If not, how close do they
actually come to each other?
Handout:
Ships in the Fog (pdf)
Comments:
This problem was taken from Windows on Teaching Math: Cases
of Middle and Secondary Classrooms. Katherine Merseth, Ed.
(New York: Teachers College Press, 2003) We used this problem
at our May 21, 2003 workshop along with the case.
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1d. Chances Are
Problem Statement:
A town has two hospitals. On the average, there are 45 babies
delivered each day in the larger hospital. The smaller hospital
has about 15 births each day. Fifty percent of all babies born
in the town are boys. In one year each hospital recorded those
days in which the number of boys born was 60% or more of the total
deliveries for that day in that hospital. Do you think that it’s
more likely that the larger hospital recorded more such days than
the smaller hospital or that the two recorded roughly the same
number of such days?
Handout:
Chances Are (pdf)
Comments:
This problem was taken from Windows on Teaching Math: Cases
of Middle and Secondary Classrooms. Katherine Merseth, Ed.
(New York: Teachers College Press, 2003) We used this problem
at our May 21, 2003 workshop along with the case.
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(2) Problems with many solutions
2a. How "scattered" are the marbles?
Problem Statement:
Three students each threw five marbles, which came to rest as
shown. The winner is the student with the smallest scattering.
The degree of scattering seems to decrease in the order A,B,C.
Devise as many methods as you can to determine the degree of scattering.
Handout:
How scattered are the marbles?
(pdf)
Comments:
We used this problem at the March, 2003 workshop.
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(3) Problems that lead to many new
problems.
3a. "Beep Me Home, Mamma"
3b. Origami Folding
3c. Trapezoid Proof
3a. "Beep Me Home, Mamma"
Problem Statement:
Your little brother is allowed to travel around your neighborhood,
but he must be home in 1 hour when mom “beeps” him.
On his bike, he travels at 10mph along the road, but only 6mph
through the grass. Where can he be in the neighborhood, if he
only has 1 hour to get home?
Handout:
Beep Me Home,
Mamma (pdf)
Comments:
The Lesson Study Working Group at the 2002 Park City Mathematics
Institute (PCMI) developed this problem to launch a problem solving
lesson for high school geometry students. They adapted the problem
from "Don't Fence Me In" in Balanced Assessment,
Advanced High School Assessment, Package 2, Dale Seymour Publications,
2000. More about their work can be found at http://www.mathforum.com/pcmi/hstp/sum2002/wg/lesson/
We used this problem at the Introductory workshop in August, 2002
and February, 2003.
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3b. "Origami Folding"
Problem Statement:
If you fold one vertex (A) of a square to any point on the opposite
side (DC), three triangles are formed. What can you say about
the relationship among those three triangles.
Handout:
Origami Folding Problem (pdf)
Comments:
The Origami Fold Problem is from a lesson report distributed at
the International Congress of Mathematics Educators (ICME) in
August 3, 2000. It is taken from a research lesson at Oizumi Junior
High School (attached to Tokyo Gakugei University) and was taught
by Mr. Shinichirou Matsumoto.
We used this problem at the December, 2002 workshop.
When we used this problem during the workshop with our lesson
study participants, they explored some of the following questions
and relationships:
 Is there a way to use calculus to find the place where the sum
of the areas is maximum?
When middle triangles are 345, do the areas have ratio 1 :
9 : 16?
 Can the Pythagorean theorem be used to get side lengths expressed
using one variable?
 Is the area of the trapezoid constant?
 Is the height of the trapezoid constant?
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3c. "Trapezoid Proof"
Problem Statement:
Given trapezoid ADCD with segment AD parallel to segment
BC and M is the midpoint of BC. Prove that segment PQ is parallel
to segment BC.Then, changing the conditions of this problem,
pose new problems.
Handout:
Trapezoid Proof (pdf)
Comments:
This geometry problem comes from a a booklet entitled "School
Mathematics in Japan" distributed at the International Congress
of Mathematics Educators (ICME) on August 4, 2000. It is taken
from a research lesson at Setagaya Junior High School (attached
to Tokyo Gakugei University) and was taught to an 8th grade class
by Mr. Kohji Yamazaki.
We used this problem at the March, 2003 workshop.
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