# Mathematics   >>   Critical Thinking   >>   Middle

Book Resource—Looking for Pythagoras: The Pythagorean Theorem

Resource
Looking for Pythagoras: The Pythagorean Theorem
Lappan, G., Fey, J., Fitzgerald, W., Friel, S., and Phillips, E.
Glenview, IL: Prentice Hall
2002
This text is written for all students in grade 8. It is appropriate for high ability students in grades 6 and 7.
Critical Thinking
This sequential unit of instruction teaches about two important ideas: the Pythagorean Theorem and irrational numbers. Students are challenged to use reasoning and critical thinking skills as they discover the Pythagorean relationship and investigate real-world application problems. In solving the problems, students make connections among the concepts of area, distance, slope and rational numbers.
Description
Looking for Pythagoras is a unit from the NSF-funded Connected Mathematics Project middle school curriculum series. It features six investigations that engage students in explorations concerning the relationship among the side lengths of right triangles and introduces students to the meaning of irrational numbers. Students begin the unit by discovering the distance between dots on a grid, and explore finding the length of line segments by thinking of the segments as the sides of squares. Using grid paper, students identify the area of the squares, and then determine the length of the sides. Students discover that some squares have side lengths that are irrational numbers. Students then build on this understanding as they investigate the patterns among the areas of three squares that can be drawn on the sides of a right triangle. This leads students to discover the Pythagorean Theorem. The explorations encourage students to use the Pythagorean relationship and irrational numbers to investigate a variety of real-world applications. Mathematical reflections offered at the end of each investigation can be used as journal entries for assessment purposes.
Appeal and User Friendliness
The teacher notes are excellent throughout the unit and provide background information about the mathematics behind and beyond the activities. They also list guiding questions for teachers to use to stimulate high-level thinking and mathematical discussion. Each investigation is followed by 3 different types of exercises, providing for built-in differentiation.

Book Resource—Variables and Patterns: Introducing Algebra

Resource
Variables and Patterns: Introducing Algebra
Lappan, G., Fey, J., Fitzgerald, W., Friel, S., and Phillips, E.
Glenview, IL: Prentice Hall
2002
This text is written for all students in grade 7. It is appropriate for high ability students in grade 5.
Critical Thinking
This sequential unit of instruction teaches students important algebraic ideas about variables and patterns of change. The investigations challenge students to use reasoning and critical thinking skills as they observe, describe, and record the way in which one variable changes in relation to another. In analyzing data sets from a variety of situations and searching for patterns, students eventually discover how to write the patterns as a rule (equation) using symbolic representations. This unit serves as an introductory unit from which to study further key algebraic concepts.
Description
Variables and Patterns is a unit from the NSF-funded Connected Mathematics Project middle school curriculum series. It is the first unit in the program’s algebra strand. It features five investigations that engage students in exploring a variety of situations in which change occurs. The context for the problems is for students to organize and run a bicycle tour company. Students begin the unit by exploring three ways of representing a changing situation. First, students describe the change narratively, then create data tables to document the changes in two variables. Lastly, students make graphs depicting the changes and discover how to express patterns of change as symbolic rules. You may be able to increase the pace in these initial investigations depending on the ability level of your students. Students are encouraged to compare the various methods of recording change and identify the strengths and weaknesses of each. This is important to focus on. The last investigation in the unit exposes students to graphing calculators. Students are encouraged to use the calculators to compare graphs of various equations and to create and use tables to find values of x when given the value of y. Mathematical reflections offered at the end of each investigation can be used as journal entries for assessment purposes.
Appeal and User Friendliness
The context of organizing and running a bike tour provides wonderful opportunities for students to discover and analyze patterns of change in a variety of situations. In one investigation, students analyze data tables and graphs that involve distance, rate and time for bike tour participants to return home from an event. Students also have opportunities to analyze the profit of running a bike tour depending on the number of participants. These real-world connections help students see the importance and usefulness of recognizing relationships and patterns of change. The teacher notes are excellent throughout the unit and provide background information about the mathematics behind and beyond the activities. They also list guiding questions for teachers to use to stimulate high-level thinking and mathematical discussion. Each investigation is followed by 3 different types of exercises, providing for built-in differentiation.

Web Resource—Fractals

Critical Thinking
Critical thinking is highlighted on this extensive site on fractals. Excellent questions focusing on looking for patterns and developing formulas highlight critical thinking. Students are also encouraged to develop creative thinking skills by designing their own fractals.
Description
Most of the information on the internet regarding fractals is either just a photo display of fractals or very sophisticated mathematics. This site is definitely designed for student exploration and provides an extensive study of fractals including the history, biographies of mathematicians, the mathematics behind fractals, the creation of famous fractals including the Koch Snowflake, The Jurassic Park Fractal, and the Sierpinski Triangle and the study of their properties. It is an excellent site for talented students to explore on their own. An extra bonus is the coding of the activities to mathematics standards and benchmarks including those of NCTM, TIMSS, NAEP, and International Baccalaureate.
Appeal and User Friendliness
This site is organized efficiently by categories displayed as a sidebar menu, provides easy-to-access direct links to other information, and has very attractive fractal displays. Clear step-by-step instructions for creating fractals along with good diagrams are included. It is highly motivational for students. Teachers will find the print versions of lesson plans with suggested assessments useful.
Sample Problem
Do you see a pattern? Use the pattern to predict the fraction of the Sierpinski Triangle you would NOT shade in the Step Four Triangle. Confirm your prediction and explain.
Website
www.mcrel.org/lesson-plans/plus/math/fractals.asp (website no longer active)

Web Resource—Math Forum: Geometry Forum Project of the Month

Critical Thinking
Students are asked to evaluate given situations, come up with multiple solutions and well as construct answers to solve problems.
Description
Within the Math Forum website, monthly geometry projects/challenges are presented. Questions are submitted by different people, including professors, tutors, and faculty. Student may choose to submit their answers via email or web form. Answers are posted in the middle of the following month. Winners as well as Honorable Mentions are posted. Questions may be submitted individually or in groups. Winners may receive certificates and Geometry Forum t-shirts or key chains. The cooperating teacher (if used, not necessary) may also receive a certificate.
Appeal and User Friendliness
This website is easily navigated. Below the title of each project is a short description of the activity. With so many choices it will be easy to find a problem that works for any situation.
Sample Problem
Questions range from straightforward computations (The Euler Line) to analytic questions and observations (Building a Mobile Sept. 1998).

Web Resource—NCTM Illuminations

These I-math investigations are geared towards Middle school.
Critical Thinking
These I-Math investigations are interactive and require students to compare, evaluate and predict what they feel the outcome will be on a certain problem. Students are asked to reflect before and after performing problems. Questions like “Will it hold for figures with more than five sides?” and “What do you notice about the list of surface areas for a fixed number of cubes? Do you think that this is always true for isometric drawings? Give a justification for your answer.” encourage analysis and higher level thinking.
Description
This site offers I-Math interactive activities for students. Pre-K through Grade 12 correlated to the NCTM Principles and Standards for School Mathematics. Each example consists of an interactive math applet with suggestions for student activities and a discussion of the mathematics and reflective questions for teachers. The math applet is also offered as a stand alone for the student to work on independently. The activities are written for all students at the specific grade bands, however there are extensions in many of the activities for greater challenge. It would also be appropriate to use the next high level grade band for activities that will meet the needs of your brightest students. You will need to be selective since not all investigations are appropriate for talented students.

The Middle School I-Math applets work with geometric figures, graphs, and the number concepts of factors and products. Some applets are in game format, such as the Fraction Track game that explores equivalence. Other applets move toward more abstract thinking. For example, one investigation explores how the mean and median are related. Can you change the mean and keep the median the same? Which is a better measure of central tendency for a specific data set? Applets range from single-day investigations to multi-day investigations, allowing both students and teachers further exploration. Some investigations require a formal written report including a summary, any pictures or charts needed and an explanation of your results.
Appeal and User Friendliness
This site is very interactive with applets ranging from an interactive geoboard to a fraction track game. It is easy for both teachers and students to use. Students can do most activities independently although some games require pairs of students. Good reflective questions for teachers aid in assessing student understanding. The student applets also provide questions for students including analysis and evaluation questions.

Web Resource—SetGame

Critical Thinking
This site stretches the limits of critical thinking.

Each Set puzzle challenges students to figure out the card combinations to 6 sets. To solve the puzzle, students need to evaluate all cards and then group them by given criteria. Xactika, another game offered, pairs the participant against the computer in a card against card battle based on card value or card tricks.
Description
This website is not set up in the conventional question/answer form. Rather it offers daily challenges that emphasize perception as well as discriminatory ability. The site offers 3 interactive games, with Set and Xactika being math relevant. The goal of each daily Set puzzle is the successful identification of 6 sets (3 cards in each set), determined by the card attributes of symbol, color, shading and number of shapes. The site provides cues if a submitted set is incorrect and why it does not match the characteristics of a set. The second game, Xactika, involves strategic playing of cards in a player’s hands. Both the opponent and the player’s card hand can be seen. The object is to win every battle. Playing a trick or playing a higher card than the opponent determines a winning hand. If the player at any time loses a hand the game is over. The game can be restarted and played again until the battle is won.

The site is fully interactive for all ages and the navigation within the site can be easily explored. Setgame.com also offers multiple games with some challenging perception, others numbers and comparisons.
Appeal and User Friendliness
The bright colors and easy navigation make this site available for all ages. A sidebar menu displays each game available as well as the Daily Puzzles. Within each game menu, directions, history and many other links are offered. The section for Teachers provides articles that connect math concepts to the games involved. While the site only offers one daily game without varied levels, these games cover all ages.
Website