Volume IV: Problem Solving
Problem Solving
Lesley Britton
Pittsburgh Allderdice High School
This document contains objectives, guidelines for lectures and discussions, and materials needed to guide high school students through an investigation into problem solving in mathematics. The modules in this lesson are designed in such a way that they can be used as a series or used singly and spread throughout the school year. The first task (Billiards) has students link mathematics to the game of Billiards and use ratios, proportions, and deductive reasoning to create a set of rules for determining the path of a ball on a given size table. The second task (Number Plays) provides two different types of puzzles where students need to use their problem solving skills to place numbers on a game board using a basic set of rules. The final task (A’s and B’s Are Number Too) provides arithmetic problems (fraction addition and multidigit multiplication) that challenge students to investigate the algorithms that they use on a daily basis. All of the tasks in this unit provide the teacher the opportunity to enrich their students learning and enhance their problem solving and critical thinking skills.
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Problem Solving, Puzzles and Patterns
Roseann Y. Casciato
Taylor Allderdice High School
Throughout the seminar Problem Solving: Where Education Interacts with Life, I was given the opportunity to examine numerous problem solving situations and learn various approaches to explain the solution/s. This seminar provided me with the tools necessary to enlighten my students with a variety of problem solving activities. Regardless whether or not a teacher teachers Advanced Placement, Gifted, Scholars or Mainstream students, all can benefit from this unit. As mathematics teachers, we all have specific curriculums to follow. I know from my own experience of trying to teach what is expected for a particular subject, it is sometimes very difficult to try to add anything else into your school year. The advantage of using this curriculum unit is that teachers can incorporate pieces of it, in any level of a classroom. I integrated warmups as well as other more involved problem solving activities in this unit.
By learning problem solving in mathematics, students should acquire ways of thinking, habits of persistence and curiosity, and confidence in unfamiliar situations that will serve them well outside the mathematics classroom. In everyday life and in the workplace, being a good problem solver can lead to great advantages.
 National Council of Teachers of Mathematis.
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Designing and Modeling
Dorothy A. Coates
Allderdice High School
Once this unit is completed, students will have explored many good problems. When I speak of good problems, they are problems that are challenging, motivating, intriguing and relevant. Good problems pique the interest of students and engage them in the mathematics. The warmup activities (The T Problem, On the Radio, and the Flying Squirrel) are chosen in order to build problemsolving skill and to revisit concepts that have been taught. Students will explore problem solving activities that involve functions and their behaviors. This team of functions include: linear, quadratic, exponential and trigonometric with real world events. During these activities, students are given the opportunity to use the TI 83 Plus calculator when working with very sophisticated mathematical functions. The use of modern technology allows multiple representations to be used in the classroom. This approach alone facilitates the transition from a traditional class format to discovery learning. In the various lessons, students create algebraic models, graph functions, and find solutions. This unit has application problems which connect to real world situations. The activities are designed to bring about motivation, active participation, and handson experiences. The National Council of Teachers of Mathematics (NCTM) has provided standards that allow connections across many disciplines; in a sense, this helps broaden students’ understanding of various concepts. As a whole, the primary focus of the various lessons is to provide students with connections that help to elicit and build their mathematical thinking. Real world problem solving events, New York and Pennsylvania State Population, The Washer Problem, the Golden Gate, Fun in the City, The Designing Engineer, and Navigation will be used to bring about connections across many content areas and help students become confident problem solvers.
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Problem Solving Through Probability in Higher Mathematics
Jason M. Dean
Schenley High School
Learning about the counting principle, permutations, combinations, and probability can prove to be very difficult for the precalculus mathematics student. Typically, students can understand and apply the necessary formula when prompted. However, students struggle when faced with an application that is more difficult in nature. This unit will emphasize a problem solving approach to probability, through small group explorations and discussions, that will help develop a deeper appreciation and understanding for this particular branch of mathematics. The many real life applications of probability will serve as motivation for students, and ensure that their level of interest in the subject remains at its highest level.
This unit will also include ten “Problem of the Month” problems, to be offered for extra credit in the elementary functions course. The problems will be challenging in nature, appropriate to this level of mathematics, and displayed on a bulletin board for an entire month of school. These problems will challenge students much in the same way that as problems done in class on a daily basis. Students will hopefully see the connection between problem solving skills and mathematics success.
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Recycling Math: Engagement and Pragmatism in Algebra Instruction
Jonathan Stephen Fantazier
Pittsburgh High School for the Creative and Performing Arts
In this paper, I present an argument for developing math activities for Algebra students from current events, news articles, literature, and activities conducted in the school environment – all in relation to the subject of recycling. As both an English and Math teacher in Pittsburgh Public Schools, I’ve developed what I hope is an authoritative lesson planning style based on active, rigorous student participation.
This paper includes two lesson primers that can be used to adapt existing concepts of the Algebra 1 curriculum to engage students in highlevel activities that give them a deeper comprehension of how these concepts relate to their lives and why the study of mathematics is vital to their overall education and competence in society.
In this writing I outline my attitude and approach to teaching Algebra 1, which I have synthesized from many sources, in the first few years of my mathematics instruction career. I believe that this paper can be useful to other teachers, even those with more tenure, not because I have some mathematical gift, but because I come to math teaching from a diverse career history that has informed my decisions and because, through PTI, I have read some incredible writers on math, whom most math teachers I know have not been exposed to.
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Math Games and Problems for Seventh Graders:
Keeping Students Interested and Involved
Jane H. Fraser
Pittsburgh Classical Academy
One of the toughest challenges facing math teachers is keeping students interested and involved throughout the year. With the prospect of double periods of math looming in the coming school year, it will be crucial to the achievement of the students and the sanity of the teachers to keep interest high and disruptions low. For many students, math seems unbearably boring and meaningless. Fortunately, there are countless problemsolving activities and math games that a teacher can incorporate into the curriculum to keep the students engaged.
The number of websites and books dedicated to math games and problemsolving is truly extraordinary; you could spend days and days exploring them and not even make a dent. I have tried to streamline that task a bit and target specific activities that are directly related to the Connected Mathematics 2 (CMP2) units that I teach to seventh graders, as well as activities that will help my students become more involved learners.
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Explanations Problem Solving with the Kindergarten Student
Stephanie L. Johnson
Pittsburgh Faison Primary
This unit is designed in a unique way. It explores problem solving in a way that will help my students develop skills now and that they may use later. The purpose of the unit describes a way to approach problems using an art, writing and math collaboration. I have made this unit interesting by using dramatics and riddles. The unit is repetitive in the way that it repeats itself by using a verbal signal to introduce each problem. It is intended for the early childhood student.
This is my Student’s first formal introduction to problem solving, so I have taken the time to build structure in all the three content areas. The structure is build through the use of routines. These routines are repeated through the unit. The seminar has provided such an enormous amount of information on solving problems. As I reflect on my participation the mind puzzling activities in the seminar left me questioning and scratching my head, wondering if I will get the right answer. This was a pleasant anxious moment for me. This unit will do the same for the kindergartner.
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Natalie Kratsas
Pittsburgh Arlington
Learning Support Teacher
I have encountered many difficulties when teaching problem solving skills to students with specific learning disabilities. One issue that reoccurs is that many students have significant reading disabilities. This prevents many students from entering the problem. Another problem is the time students are able to attend to the task. Students who are unable to attend to the task are often also easily frustrated by challenging academic work. Other students express frustration at being unable to begin a problem. These students often lack confidence in their ability to accomplish an academic task. For these reasons, the majority of my students are unwilling to attempt problems that require critical thinking and sustained, individual student work. This is a teaching unit that develops students’ problem solving skills. My goal is for learning disabled students to become independent problem solvers.
Through this unit, I hope to attack the main issues that learning disabled students encounter when learning problem solving skills. I will address their ability to approach the problem, to access prior knowledge, to read and comprehend the problem, and to break the problem into manageable parts. Students will become independent problem solvers and be able to use the strategies they learn in the future.
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Where Education Interacts with Life
Harold Michie
Pittsburgh Lincoln K8
The focus of this unit plan is on giving fifth grade students the opportunities to embrace some key problem solving strategies as they study the mathematical concept of measurement. The chance to use these strategies will be rooted in a real life experience.
Fifth grade students will explore the concept of perimeter and area and its impact on planning and building a garden. They will estimate, refine and verify specified measurements of garden design, as well as, convert linear measurements within the same system. Students will determine reasonable strategies for estimating the perimeters and areas of a figure.
The purpose of this unit is to have student become proficient at developing a plan for analyzing problems, taking note of the important information, and solving the problem using their strategy of choice. The student will evaluate if their answer is reasonable, and be able to explain their solution both verbally and in written form.
The students will be given the chance to display their work in various manners including words, numbers, symbols, pictures, charts, graphs, tables, diagrams and actual garden models.
The unit plan is made up of a series of lessons that will challenges students to use both inductive and deductive reasoning while exploring problem solving strategies such as making a table, an organized list, drawing a picture or diagram, etc. Fifth grade students will be given opportunities place ownership of the various problem solving strategies. The unit plan will conclude with a major project that will focus on the organization and design of Pittsburgh Lincoln’s annual garden display.
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Family Math
Barbara Misechok
Student Achievement Center
Developing problem solving skills is critical for the future success of students including students with special needs. Teachers often focus on teaching basic skills in the resource room instead of introducing students to problem solving techniques. The latest recommendation of the National Mathematics Advisory Panel March 2008 states that mathematics instruction needs to be a combination of conceptual understanding, computational fluency and problem solving. These capabilities are mutually supportive.
This paper focuses on data analysis and problem solving. Students will look at truth in advertising and how data is used in this field. The class will review claims that are made in advertisements and identify information that may be misleading in these advertisements. Students will develop their own survey, compile the data in a chart and graph, and design an advertisement.
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Plum, Level, & Square
Rosemary G .Schmitt
Pittsburgh South Brook
Not all students have the need or desire to go to college and therefore live with the idea that they will have no need for math. They take an interest in things that only serve them here and now. One size doesn’t fit all, and sometimes we need to tailor make our content to fit the size of our students without compromising our curriculum and standards set forth by the PA Department of Education.
Lots of students may follow in the footsteps of their fathers and look for a future in construction, pluming, electricity and soon. These are occupations that are needed and respected, yet sometimes forgotten in the world of higher education. I hope that through this problem solving unit, students are exposed to everyday problem solving involving creating floor plans of a house and calculating the amount of materials and cost associated with painting, carpeting and dry walling their house.
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