NCTM Annual Conference St. Louis, MO April 2006 “Investigative Problems to Enhance Students’ Learning and Motivation” Dr. Jeremy Winters Dr. Dovie Kimmins

The Locker Problem A school in Costaville, New Hampshire, has a peculiar orientation ceremony. All entering students are required to either open or close all the lockers in the school according to an old tradition. The school has 500 lockers for 500 students all of which are closed. The student who enters the school first opens all of the lockers. The next student closes every even locker. The next entering student changes the state of every third locker by either opening or closing it. (Changing the state of a locker means opening a closed locker or closing an open locker.) The process continues with subsequent students changing the state of a locker by opening and closing it according to their entering position. When all the students have entered the building which of the lockers stay open?

Why Use Investigative Problems? “Algorithms alone can make understanding of mathematics more difficult.” - Wisconsin Center for Educational Research “In order to acquire mathematics expertise in a durable and useful form, students need to construct mathematical knowledge and create their own meaning of the mathematics they encounter.” - Siegel and Borasia (1992) “Worthwhile tasks should be intriguing with a level of challenge that invites speculation and hard work. Such tasks often can be approached in more than one way.” - National Council of Teachers of Mathematics

Aim High Aim Low

Objective: Explore patterns related to place value Replace each question mark with one of the digits 2,5,6,8,9 to produce the LARGEST possible SUM. Do not repeat digits. ??? +??

Objective: Explore patterns related to place value Largest possible sum is Is there more than one way to achieve this sum? How many ways in all?

Objective: Explore patterns related to place value Replace each question mark with one of the digits 2,4,6,8,9 to produce the LARGEST possible PRODUCT. Do not repeat digits. ???? x?

Objective: Explore patterns related to place value Largest possible product is 77,778. How do we know this is the maximum product? 8642 x9

Objective: Explore patterns related to place value Replace each question mark with one of the digits 2,4,6,8,9 to produce the LARGEST possible PRODUCT. Do not repeat digits. ???? x??

Objective: Explore patterns related to place value The largest product is 79,488. What is the pattern for placing the digits? 864 x92

Designing Place Mats The Family and Consumer Sciences Department has contracted this classroom to find all possible place mats with the following criteria for their upcoming Parent Night. Place mats are constructed using black and white circular chips. The black chips are used for the sides of the place mat and the white chips for the interior. The place mats must be constructed so that the number of black chips used and the number of white chips used are the same. The Family and Consumer Science Department tried to complete the task prior to contracting this classroom to do the job. The following was the department’s failed attempt at constructing a mat. Once you found all possible mats, state your method for finding the possible mats.

Fold to the Max

When the top left corner of a piece of paper is folded down to touch the bottom edge, a triangle is formed in the lower-left hand corner, as shown in the diagram. At what point along the bottom edge should the top left corner be placed in order to maximize the area of the triangle in the lower- left corner? Adapted from #64 on page 102 of 101 Project Ideas for the Geometer’s Sketchpad, 2001 by Key Curriculum Press.

The Gateway Arch Problem On a trip to St. Louis, you visit the Gateway Arch. Since you have plenty of time on your hands, you decide to estimate its altitude. The distance across the base of the arch is 162 meters. You set up a Cartesian coordinate system with one end of the arch at the origin. The other end of the arch is at x = 162 meters. To find a third point on the arch, you measure a value of y = 4.55 meters when x = 1 meter. You assume that the arch is a parabola. 1.Find the particular equation of the underside of the arch. 2.What is the x-coordinate of the vertex? Predict the height of the arch using this coordinate. 3. An airplane with a wingspan of 40 meters tries to fly through the arch at an altitude of 170 meters. Will the plane make it? Justify your answer.

Statue of Andrew Jackson In the statue of Andrew Jackson, his nose measures 3 feet and 4 inches. How long is the arm of Andrew Jackson’s Statue (shoulder to fingertip)?

You may obtain the problems from this presentation at Click on Research and then follow the appropriate links. If you have problems that could be added to these, please them to