Math Resources

(1) Problems with many approaches but 1 solution

(2) Problems with many solutions

(3) Problems that lead to many new problems.

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 Standards-Based 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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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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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 3-4-5, 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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