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Book Resource—Mathematics: A Human Endeavor

Resource
Mathematics: A Human Endeavor
Jacobs, H.
New York, NY: W. H. Freeman
1994
This text was written for high school students in grades 9-12. It is appropriate for high-ability middle school students.
Problem Solving
Mathematics: A Human Endeavor is a text that was originally designed for students who are math phobic or who simply think they don’t like the subject. Its goal is to help students understand that mathematics is more than arithmetic and computation. Obtaining a broader view of mathematics is accomplished by taking students on mathematics explorations in areas they never considered before, stimulating interest and independent investigations.
Description
The lessons in this text explore mathematics found in common recreational activities such as billiards, cards, and sports. The puzzles, games, paradoxes, and cartoons presented in the text not only entertain and challenge students, but more importantly, focus their attention on the fundamental ideas of mathematics. These include deductive reasoning, inductive reasoning, number theory, geometry, probability, statistics, number sequences, topology, and functions. For example, Chapter 2 presents a series of six lessons designed to address the mathematics that relates to a variety of number sequences. In one activity, students explore geometric sequences using U.S. currency denominations and pyramid schemes. Another lesson uses poker ships to illustrate successive terms of an arithmetic sequence. Subsequent lessons address binary sequences used in circuits of computers and calculators, and sequences of squares and cubes. A final lesson in the chapter guides students in discovering the Fibonacci sequence by exploring a problem about the birth rate of rabbits.

Included at the end of each chapter are lesson extensions that further stretch students’ understanding and thinking. Answers to selected exercises are located at the back of the text.
Appeal and User Friendliness
Although written for high school students, there are a number of chapters and individual lessons that are appropriate for high-ability middle school students. Teachers need to pick and choose activities based on the individual needs and interests of their students. Chapters do not need to be presented in the order given. The friendly style and real-world connections the author uses to draw students to the mathematics captivates students’ attention as they discover important mathematics in places they would have never looked before.
Sample Problem
Rabbit Problem-Fibonacci Sequence page 99.

A pair of rabbits one month old are too young to produce more rabbits, but suppose that in their second month and every month thereafter they produce a new pair. If each new pair of rabbits does the same, and none on the rabbits die, how many pairs of rabbits will there be at the beginning of each month?

Book Resource—United We Solve

Resource
United We Solve
Erickson, T.
Oakland, CA: Eeps Media
1996
This mathematics problem-solving resource is written for students in grade 5-10. The problems presented in the book were designed to complement a high-standards middle school curriculum.
Problem Solving
United We Solve presents a collection of 116 group problem-solving activities that encourage the use of higher-level thinking and reasoning skills. These mathematical problems challenge students to work cooperatively as a group to arrive at a solution. For some tasks, students must build a structure or create a group display such as a graph. Many of the problems have multiple solutions. By working in groups, students are exposed to a greater number of problem-solving strategies, have natural opportunities to use mathematical vocabulary as they work together, and are usually more successful in solving higher-level, complex tasks.
Description
The problems in this book are arranged in four categories: proportions, open-ended, spatial, and patterns. Each category has clusters of problems that focus on common mathematical ideas, and each cluster has an average 4 problems. These problems are organized from easiest to most difficult. All clusters have a useful introductory page that outlines a list of materials needed, the mathematical content addressed in the problems, and teaching suggestions for each problem in the cluster. Every problem has either four or six clue cards that are written on a reproducible page. Students begin working on the problems by passing out clue cards to each member of the group. Members take turns reading their own clue to the group, without showing the clue to another member. Since each member’s information is essential, the group must work together to solve the problem.
Appeal and User Friendliness
Although challenging, students find working on these problems fun and engaging. Initially, teachers may want to limit the group size to two students, rather than four students. It is also strongly recommended that teachers attempt the problems before assigning them to students. The teacher notes found at the beginning of the book offer great suggestions on how to incorporate the problems into an existing curriculum, provide tips for managing and facilitating cooperative groups, and discuss problem-solving strategies that support students as they work on the group tasks. No solutions are listed in the book, however students are encouraged to visit theWeb site (www.eeps.com) to view posted samples of student solutions. If no work is posted for a particular problem, or if students have a different solution, they are encouraged to email their work so it can be posted on the site.

Web Resource—Houghton-Mifflin Brainteasers

Problem Solving
Students are presented with weekly brainteasers. Challenges include visual spatial reasoning, numeric computation, as well as higher-level evaluation.
Description
This site offers varied levels of brainteasers, allowing more advanced students to investigate more challenging questions. Each week (posted on each Wednesday), new sets of brainteasers are added and the previous stored in the archives (only for the last 3 weeks). For each archived problem, a solution is included. The current weekly questions do not supply answers until the following week. These brainteasers come from the mathematical section of the Houghton-Mifflin website. Other subject areas can be explored through the remaining website.

This site is easy to navigate and students can begin to explore problems all on their own without teacher assistance. “Helpful Hints” are provided on each question to aid students in the right direction.
Appeal and User Friendliness
Bright colors and characters make this site great to start with. Questions are varied for ability, allowing advanced students to explore higher-level problems. Answers to the brainteasers are provided a week after a problem is presented. Problems are changed every Wednesday, allowing students to begin working on a new problem throughout the weekend. Only the last 3 weeks of problems are saved in the archives. Problems past 3 weeks are not accessible or printable. This means both students and teachers must check the answers before the question is taken off the website.

Web Resource—Math Counts

Problem Solving
Students are offered many different opportunities to solve math problems, ranging in computation to applying mathematics to current events.
Description
The breadths of choices students and teachers have when entering this site makes Math Counts a necessity for teachers. While the website has multiple links within the site, the site also offers many links outside the site to other math activities. Students may find the following internal links helpful, Problem Solving, answering or submitting math problems for others to solve Meeting Place, a place to post messages for help with homework, online problems or strategies in math, and for students who want that math shirt, the Store. Teachers may find the following links more appropriate for their needs, News, where articles on research and national coverage are posted, The Program, instruction on starting a Math Counts group or simply how to get involved, and even the Meeting Place has places for teachers to chat.

Go Figure offers students a mental mountain to climb. Students must answer varied levels of questions to climb to the top. Math Counts also has Problem of the Week. Here every Monday students can find a new problem to tackle. Submitted answers are posted weekly, allowing students to reflect on their answers as well as the process to get the answers.
Appeal and User Friendliness
This site is easily navigated through clicking on desired icons. The bright colors make this site eye- catching and captivating.
Website

Web Resource—Math Forum

Gades 5-8 enrichment. This site also has problems for elementary and secondary 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 middle school problems have their own category labeled middle. Also, look for excellent problems under 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 geometry topics include area, circumference and perimeter, volume, and Pythagorean Theorem. 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
On Thursday, January 20th, the Pocono Mountain region in Pennsylvania experienced the first big snowstorm of the new millennium. Many schools closed because of the slippery conditions and cold weather. School closings are announced on local TV and radio stations, but sometimes you have to watch or listen for a long time before they announce your school. However, Carlin’s dad is the Principal, and that means Carlin gets early notification of closings. Carlin then calls his best friends, Frank and Hobart. Frank calls his friends, Julie and Kory. Hobart calls Laura and Marcellus. The phone tree continues with Julie calling Nihaar and Saloni, and Kory calling Tara and Vincent, and so on.

Since you don’t know these people, it isn’t really important to know the names. What you do need to know is that each person will call exactly two other people. We are going to assume that all the students want to get back to sleep as soon as possible, so they are not going to chat, and each call will take only one minute. Carlin finds out that there is no school and makes his first call at 6:00 a.m. Ten minutes later, at 6:10 a.m., everyone on the phone chain is back in bed. How many phone calls have been made? What time will the last calls be made if the phone chain is extended to include all 1,000 students in the school? Please be sure to answer both questions in a complete sentence.

Bonus: What number pattern is evident in this problem? Explain.