# Mathematics   >>   Problem Solving   >>   Secondary

Web Resource—IBM’s “Ponder This”

Problem Solving
Here students must read word problems and solve challenging math questions. Questions involve both analyzing given data as well as the application of mathematical equations.
Description
IBM monthly supplies problems and “You are cordially invited to match wits with some of the best minds in IBM Research.” Each previous month’s challenge is posted with its solution. All problems are challenging and involve high level thinking skills. Students who answer the questions correctly may have their name posted on the website, if they so choose. Some problems go unsolved for quite a while due to the level of challenge, while others are solved by many students.

Students may choose to explore the entire IBM site. The site presents current research as well as technology advancements. Much of the website is informative and research based.

And as Don Coppersmith says “So give your mind a break from its routine—you never know what other problems you may solve in the process!”
Appeal and User Friendliness
A great part of this website is that it allows students to submit answers to the questions presented. After the problem has been up for a few weeks the solution is provided and IBM provides a list of students who have correctly answered the question. IBM also allows students to submit problems they feel are challenging, a great collaboration between students and educators.
Sample Problem
Ponder This Challenge:Find four points in the plane such that the distance between each pair of points is an odd integer, or show that this is impossible. (The points need not have integer coordinates.)

We are interested in your original solutions, not those that you may have seen elsewhere.

Web Resource—Math Forum

Grades 9-12 enrichment. This site also has problems for elementary and middle school students.
Problem Solving
There are a wide variety of problems across math subject areas at this site. They are creative and authentic, focus on student interest, and foster higher order thinking.
Description
This site allows you to pick an age group, content area (based on NCTM standards) and type of story problem and then offers many problems under each one. The site also offers links for teachers such as a rubric for coding the problem difficulty, a chance to read what other teachers say about how they use the problems in their class, and a link to look at student solutions. The students have an opportunity to send solutions to the site and mentors from across the country will respond to the solutions and communicate with the students working on the problems. If the problem is solved correctly, the students will be recognized on the site.

The secondary problems of the week are categorized into the topic areas of geometry, algebra, discrete math, and trigonometry & calculus. Under each topic heading, problems are further broken down into specific areas. For example, under Calculus the categories include derivatives, integration, series, proof, area, perimeter, and volume. Story topics are categorized by age, animals, food, travel/geography, holiday, sports, and money. Although these topics seem elementary-school focused, you will find very challenging problems for your high school students. Each problem is also rated on a difficulty level of 1-5 to help meet the appropriate challenge level of your students.
Appeal and User Friendliness
This site is very well organized, easy to use, and contains excellent problems set in real world contexts. The links described above are rarely found in other sites. The availability of mentors enhances student independence and individualization.
Sample Problem
This problem was found under Trig/Calculus/calculus/derivatives with a difficulty level of 4.

The Y. O. Ming Construction Company is slated to build a new multilevel building in downtown New York City. All of the steel girders they are using to construct the building are 27 feet long. One of the corridors the girders are to be carried down has a 90-degree turn at the end. The width of the hall before the turn is 8 feet, and the width of the hall after the turn is to be determined. The owner of the building has instructed the lead architect of the construction company to make all the halls as close to 8 feet wide as possible, and no less. However, the architect, knowing trigonometry, figures out that the construction company’s girders won’t fit around the corner if the hall after the turn is 8 feet wide. Wishing to keep his job and the company’s contract, the architect sets out to find the minimum width he can make the second part of the corridor to please the owner, while still making it possible for the Y. O. Ming Construction Company to carry the girders down the corridor. The only problem is, the architect was more interested in designing houses than paying attention in calculus class, so now he is turning to you to figure out how wide the hall should be.

Assume the width of the girders is negligible. Show all work so the architect can understand your solution to the problem.

Bonus: What is the longest the girders can be if the second part of the corridor is exactly 8 feet wide?