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Book Resource—Covering and Surrounding: Two-Dimensional Measurement

Resource
Covering and Surrounding: Two-Dimensional Measurement
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 6. It is appropriate for high ability students in grades 3-5.
Critical Thinking
This sequential unit of instruction features two important kinds of measures-perimeter and area. Students engage in problem-solving explorations and use critical thinking to discover strategies, procedures, and formulas for finding area and perimeter. Analyzing bumper-car floor plans for an amusement park company and designing mountain storm shelters for stranded hikers are some of the explorations students engage in as they develop an understanding of these measurement concepts.
Description
Covering and Surrounding is a unit from the NSF-funded Connected Mathematics Project middle school curriculum series. This unit addresses perimeter and area in seven investigations. Initially students discover strategies for finding the area and perimeter of rectangular shapes, and also work on irregular figures such as footprints. As students continue through the unit, they apply their discoveries to develop new strategies and procedures for finding the perimeter and area of other two-dimensional figures such as triangles, parallelograms, and circles. Students are challenged to discover relationships between perimeter and area. Two investigations encourage students to explore what happens when one measure (area or perimeter) is held constant. These activities help students confront the misconceptions that figures with the same area always have the same perimeter and vice versa. Mathematical reflections found at the end of each investigation can be used as journal entries for assessment purposes. A unit project, Plan a Park, offers students a wonderful opportunity to use and apply newly learned concepts from the unit in a more extensive and personal manner.
Appeal and User Friendliness
The real-world contexts presented in the investigations appeal to students’ interests. The activities make excellent use of materials such as square tiles, string, and transparent grids in exploring the concepts of area and perimeter. The teacher notes are very detailed 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—Cranium Crackers

Resource
Cranium Crackers
Pacific Grove, CA: Critical Thinking Books and Software
1997
Book 1 is written for students in grades 3-4.
Book 2 is written for student in grades 5-6. High ability students may be introduced to the books at an earlier grade level.
Critical Thinking
Cranium Crackers is a supplemental curriculum resource that gives students experience in building important critical thinking skills that are not only necessary for success in mathematics, but for success in real-life problem solving as well.
Description
Each book presents a wide range of mathematical and verbal problems that are fun and thought provoking. Students are able to approach solving the problems in a variety of ways. Grid problems, magic squares, Diophantine problems, clock arithmetic, Cross NumberTM Puzzles, and Math Mind Benders® are some of the activities featured in this series. The problems in each book range from relatively easy to challenging, and skill levels increase as the series progresses. Examples and practice problems are given when new critical thinking concepts are introduced. Each page takes approximately 15-20 minutes for students to complete. Teacher notes and discussion questions can be found in the answer section located at the back of each book.
Appeal and User Friendliness
The rich assortment of problems encourages students to develop diverse thinking skills in an interesting fashion. Clear directions are given for each activity and the examples presented are easy for students to follow. The detailed comments and suggestions found in the answer section give teachers helpful insights and strategies to better guide and stimulate students as they work through the activities. The comments also provide excellent rationale and background information concerning the mathematical skills and concepts that relate to the problems.
Sample Problem

DIRECTIONS
in the clues, “A” means across, and “D” means down. For example. “4-D” would refer to clue number 4 DOWN.

Each square takes a single digit from 0 through 9.

No answer begins with 0.

PROBLEM

1. Emil’s age is double Dorothy’s and 3 years more than Francine’s. Dorothy’s great-aunt Martha is a crop duster in Australia.

ACROSS

1. Age of Emil’s cousin Luke
1. Years Luke has been a swim champion
2. Francine’s age
3. 1-D x 2
DOWN

1. Emil’s age
2. Dorothy’s age
1. Age of great-aunt Martha with digits reversed
2. Age of Woofer, Emil’s dog

1

2

3

4

5

Reproduced by permission from page 160

Book Resource—Filling and Wrapping: Three Dimensional Measurement

Resource
Filling and Wrapping: Three Dimensional Measurement
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 grades 4-5
Critical Thinking
his unit of instruction teaches important concepts related to surface area and volume of three-dimensional objects. The hands-on investigations engage students in rich mathematical explorations that promote the use of critical thinking strategies. The activities included high-level extensions that will challenge students with math talent in grades 4-5.
Description
Filling and Wrapping is a sequential unit of instruction from the NSF-funded Connected Mathematics Project middle school curriculum series. Filling and Wrapping addresses measurement concepts related to surface area and volume by engaging students in seven, hands-on investigations. In some activities, Building Boxes and Designing Juice Containers, students explore the surface areas and volumes of rectangular prisms and cylinders in great depth. Students informally investigate other solids, such as cones, spheres, and irregular shapes to develop volume relationships. In addition, students look at how changes in the scale of rectangular prisms affect surface area and volume. Mathematical reflections found at the end of each investigation can be used as journal entries for assessment purposes. Also included at the end of the unit is an optional student project. This project gives students an opportunity to apply what they have learned about volume and surface are in an extensive, real-world application problem. Students are asked to participate in a package design contest for a sporting goods company. In addition to submitting package designs, students must include a convincing written explanation outlining how the designs meet the company’s requirements.
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. Although this unit was written for middle school students, the hands-on explorations are within the reach of high-ability elementary students. This unit assumes that students are comfortable with finding the area of a variety of two-dimensional figures. Covering and Surrounding, a grade-6 unit from the same series, would be a good prerequisite resource for students who need experience with area before attempting this unit.

Book Resource—Groundworks: Algebra Puzzles and Problems

Resource
Groundworks: Algebra Puzzles and Problems
Greenes, C. and Fendell, C.
Chicago, IL: Creative Publications
1998
This algebra resource consists of a series of books written for grades 4-7 and appropriate for talented elementary students.
Critical Thinking
Algebra Puzzles and Problems is a supplemental mathematics series designed to give students opportunities to explore the big ideas of algebra before a more formal and sophisticated study of the subject. The problems and puzzles provide interesting experiences for students to make connections between arithmetic reasoning and algebra. These activities are essential in helping prepare students for success in future algebra studies.
Description
Each book in this series contains reproducible problems and puzzles that are organized around the six big ideas of algebra: representation, proportional reasoning, balance, variable, function, and inductive reasoning. Sets of problems are presented for each big idea, and each set consists of six problems. The problem sets begin with an introductory whole-class lesson, followed by problems that have similar reasoning structures. One of the big ideas of algebra, balance, deals with the concept of equality among variable expressions. “Balance Beam” and “Balance Challenge” are the problem sets in the Grade 4 book that relate to the idea of balance. In “Balance Beam,” students must determine where to place a one-ounce mass to make the given beam balance, and then write an equation to represent the balanced beam. In “Balance Challenge,” students must determine which missing block will balance a third scale and explain how they arrived at their solution. These experiences support the development of equality, and encourage students to discover ways to change inequalities in order to achieve equality. This work is necessary in preparing students for later work with solving equations and inequalities in algebra.
Appeal and User Friendliness
The unique variety of problems presented in this series captures students’ interest as they work on developing important algebraic reasoning skills. The goals and objectives for each set of problems are clearly outlined in the teacher notes, and are located an adjacent pages to problems A and D in each of the problem sets. The teacher notes also list some guiding questions to help stimulate children’s thinking as they solve problems. Solutions are included in each book, as well as a class management chart to assist teachers in tracking students as they progress through the problem sets. The big ideas and sets of problems do not have to be completed in the order presented. Teachers may want to use their own judgment and assign problems based on the mathematics content, their alignment with the curriculum, students’ interests, or students’ needs. A Certificate of Excellence blackline master is included for successful completion of all the problems related to one of the big ideas in algebra.

Book Resource—How Likely Is It?: Probability

Resource
How Likely Is It?: Probability
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 6. It is appropriate for high ability students in grades 3-5.
Critical Thinking
This sequential unit of instruction is designed to help develop students’ abilities to understand and reason about probability. The investigations provide opportunities for students to explore experimental and theoretical probability and discover the relationship between them. In solving the problems presented in the activities, students also make important connections between probability and rational numbers, geometry, statistics, science and business.
Description
How Likely Is It? is a unit from the NSF-funded Connected Mathematics Project middle school curriculum series. It features seven investigations that engage students in developing an understanding of probability. Being able to determine probabilities helps students understand past events and empowers them to make informed decisions about future events. The first three investigations focus on experimental probability and offer students many opportunities to collect data by conducting experiments using objects such as coins, marshmallows, and spinners. Applying critical thinking and reasoning skills, students analyze the data to determine experimental probabilities. The next two investigations engage students in playing games that involve spinners, rolling dice, and drawing colored cubes from a bag. While working on these explorations, students are formally introduced to theoretical probability and are challenged to analyze and compare both the experimental and theoretical probabilities. The last two investigations present opportunities for students to apply their new knowledge about probability to interesting situations such as winning contests, or predicting the occurrence of genetic traits such as “tongue curling.” The questions that accompany each investigation encourage students to go beyond just playing a game or conducting an experiment, and help them focus on the rich mathematics behind the activities. Mathematical reflections offered at the end of each investigation can be used as journal entries for assessment purposes.
Appeal and User Friendliness
The games and activities presented in this unit are very motivational and the contexts that frame the problems appeal to students’ interests. The experiments used to collect data are easy to set up and very doable in classrooms. 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—Prime Time: Factors and Multiple

Resource
Prime Time: Factors and Multiple
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 6. It is appropriate for high ability students in grades 3-5.
Critical Thinking
This sequential unit of instruction teaches important concepts in number theory including high-level extensions that will challenge students with math talent in grades 3-5. The first two investigations, the Product Game and the Factor Game, are excellent critical thinking activities presented as motivational games. Using the questions and discussion in the teacher notes, teachers can encourage students to go beyond the playing of each game and explore rich mathematical thinking strategies that are embedded in each activity.
Description
Prime Time is the first unit in the NSF-funded Connected Mathematics middle school curriculum series. Prime Time addresses the basics of number theory in five investigations: factors, multiples, prime and composite numbers, even and odd numbers, square numbers, greatest common factors, and least common multiples. It includes extensions on abundant numbers, deficient numbers, and perfect numbers. Mathematical reflections at the end of each investigation can be used as journal entries for assessment purposes. Throughout the unit, students are engaged in an ongoing project in which they select a favorite number and then apply the newly learned concepts in each investigation to their number. This allows students to take a personal interest in the material and gives them an opportunity to communicate about the mathematics they are learning.
Appeal and User Friendliness
The teacher notes are excellent throughout the unit and provide solid background in 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—Roads to Reasoning: Developing Thinking Skills Through Problem Solving

Resource
Roads to Reasoning: Developing Thinking Skills Through Problem Solving
Krulik, S. and Rudnick, J.
Chicago, IL: Wright Group/McGraw Hill
2001
This series consists of curriculum resource books, written for grades 1-8. Teachers may elect to use more advanced grade level books with high-ability students.
Critical Thinking
Roads to Reasoning is a supplemental curriculum resource series designed to develop reasoning and problem-solving skills, while at the same time reinforce math computation skills that are appropriate for the specified grade level.
Description
This series consists of a variety of problem-solving activities that encourage students to become more flexible in their thinking and reasoning. Problems in each book are organized into six sections; each focuses on a particular facet of the problem-solving process. For example, in the section “What Number Makes Sense?”, problems are listed from which the numerical data has been removed. Students must choose from a list of numbers to fill in the blanks so the problems and solutions make sense. In the section “What’s Wrong?”, students are given problems with solutions that contain errors. Students must identify the errors and then find the correct solution. In the section “What’s the Question if You Know the Answer?”, students are given situations with data as well as several possible answers. Students must then determine an appropriate question that would match the given solutions. Other sections ask students to identify missing information from a given problem, and to generate a list of questions that can be answered based on given mathematical situations and sets of numerical data. Solutions to the problems are listed at the end of each section. Assessment tips and a general 3-point scoring rubric are also provided.
Appeal and User Friendliness
The teacher notes found at the beginning of each section are helpful in explaining the reasoning approach that is being developed and offer great discussion ideas for during and after the completion of the problems. Also listed in the notes are the math computation skills and topics that are reinforced in each of the problems. The problems appeal to students’ interests and often require students to explain their thinking. Suggestions an how to use the problems as class openers and closers are also provided.

Web Resource—Hands-On Equations (Resource Kit)

Critical Thinking
Students use analytical thinking skills as they physically manipulate pieces on a balance scale to solve algebraic equations.
Description
Hands-On Equations is a visual and kinesthetic teaching system for introducing algebraic concepts to students in grades 3 to 8. This system enables young children, as early as 3rd grade, to gain access to algebraic concepts normally presented in the 7th, 8th, or even 9th grades. This system consists of 26 lessons that span three increasing levels of difficulty. At each level, the students use manipulative pieces including a balance scale to physically set up and solve the algebraic equations.
Appeal and User Friendliness
While worksheets are provided, teachers need to make sure that they are encouraging students to think critically. High-level questions should encourage oral and written discussion on algebraic reasoning. The manipulation of objects or symbols to solve equations is easy for talented students. Understanding the rationale for doing this should be the goal of instruction. If this is stressed, this can be a valuable resource to challenge and interest elementary students in algebraic thinking.
Sample Problem
The following is a sample problem from the website.

Web Resource—NCTM Illuminations

Critical Thinking
The I-Math investigations at this National Council of Teachers of Mathematics site are interactive and encourage high level thinking, especially the extensions. Many explorations are open-ended analyses, for example students are asked to find how many different triangles they can make on a geoboard. They are also asked to justify how they know they are different.
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, suggests for student activities, with 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 higher level grade band for activities that will meet the needs of bright students. You will need to be selective since not all investigations are appropriate for talented students.
The activities are categorized into different topics according to the NCTM Standards. I-Math Investigations at the Pre-K-2 level include investigating the concept of a triangle, creating and analyzing patterns, learning about number relationships, learning geometry and measurement concepts, developing geometry understanding, and developing estimation strategies.
Grades 3-5 investigations cover exploring geometric solids and their properties, communicating about mathematics using games, collecting, representing, and interpreting data using spreadsheets and graphing software, and exploring properties of rectangles and parallelograms using dynamic software.
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