Department of Curriculum and Instruction
Lillie Sipp
Director of Curriculum Chief Academic Officer
Sheryl Klein Supervisor Mathematics
MATHEMATICS CURRICULUM GUIDE GRADE 8
Table of Contents
Administration ………………………………………………………………………………………………………………………………………………………. .. 3 Board of Education …………………………………………………………………………………………………….………………………………………….........4 Mission Statement ……………………………………………………………………………………………………………………………………………………...5 Acknowledgements …………………………………………………………………………………………………………………………………………….……... 6 Beliefs About Curriculum ……………………………………………………………………………………………………………………………… ……………. 7 Philosophy and Goals …………………………………………………………………………………………………………………………………........................ .8 Instructional Practices ……………………………………………………………………………………………………………………………………………….. .10 Units of Study …………………………………………………………………………………………………………………………………………………………12 Modifications/Differentiating Instruction ……………………………………………………………………………………………………………… …………….25 About the Connected Mathematics Program ……….…………………………………………………………………………………………………………………26 Suggested Timeline………………………………………………………………………………………………………………………………………………….....28 Technology in the Connected Mathematics Program …………………………………………………………………………………………………………………29 Assessment in the Connected Mathematics Program
…..……………………………………………………………………………………………………………31
Preparation for Standardized Assessment …………………………………………………………………………………………………………………………......32 Appendix …..………..…………………………………………………………………………………………………………………………………………………37 A. B. C. D. E.
NJCCS and Cumulative Progress Indicators …………………………………………………………………………………………………….…………….. 38 Assessment Based on Standards ……………………………………………………………………………………………………………………………… 49 Accountable Talk ....…………………………………………………………………………………………………………………………………………… 52 Questioning …………………………………………………………………………………………………………………………………………………...... 53 Professional Resources …………………………………………………………………………………………………………………………………………. 59
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2005-2006 ADMINISTRATION Superintendent
Dr. Paula Howard
Chief of Staff
Mrs. Angelina Chiaravalloti
Business Manager
Mr. Victor Demming
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PLAINFIELD BOARD OF EDUCATION 2005-2006
Mr. Martin P. Cox, President Mr. Agurs Linward Cathcart, Jr., Vice President Ms. Patricia Barksdale Reverend Tracey Brown Mrs. Wilma G. Campbell Mr. David Graves Mrs. Lisa C. Logan-Leach Ms. Bridget Rivers Ms. Vickey Sheppard
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ACKNOWLEDGEMENTS A project of this magnitude required the efforts and talents of many people over an extended period of time. The staff and community in Plainfield would like to acknowledge the contributions of the members of the Mathematics Committee in the development of the K - 8 Mathematics Curriculum.
K-8 Mathematics Curriculum Members Joyce Alston Evergreen Joanne Barrett Woodland Joyce Corriero Emerson Kristina Jerome Jefferson Kimberley Morris Jefferson Cheri Phillips Washington Denise Thir Barlow Jean A. Williams Emerson
Valerie Atkins Clinton Sandra Burton Cedarbrook Dana Gaines Maxson Luanne Lohman Jefferson Kathleen O’Connor Cook Steven Stibich Evergreen Helen Wiley Hubbard Eleanor Wilson Stillman
Staff members from the Department of Curriculum and Instruction who contributed to the preparation of this document are Sheryl Klein, Supervisor of Mathematics, Kathryn Ballin, High School Mathematics Content Supervisor, Lillie Sipp, Chief Academic Officer/Director of Curriculum and Instruction, and Ann Nettingham, Department of Mathematics Secretary.
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BELIEFS ABOUT CURRICULUM
Board policies establish the expectation that the district will provide a written, taught, and tested curriculum. Curriculum and its delivery are the major work of the schools. Curriculum development is a participatory process, involving at appropriate junctures in the process, teams that include various stakeholders, including teachers, administrators, students, parents and the community. A high quality rigorous curriculum is a critical component in providing all children with access to an equal education. Every child learns when the curriculum is delivered in a manner that actively engages the student. A high decree of commitment to the individual needs of students is reflected in the design and delivery of the curriculum. The ongoing process of curriculum development, instructional delivery, and assessment is a natural part of implementing a well-articulated curriculum. A well-articulated curriculum is vertically and horizontally aligned. The curriculum development process benefits the district in the following ways: • It enhances communication and articulation between and among teachers and administrators • It deepens teachers and administrators understanding of the knowledge and skills needed for students to be proficient in the various curriculum areas • It promotes a cohesive program of learning for all students
Technology is an integral tool for learning and should be infused throughout the curriculum. Curriculum must align with district assessments, state assessments, and standards.
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PHILOSOPHY AND GOALS OF THE MATHEMATICS PROGRAM The Core Curriculum Content Standards for Mathematics, released in May 1996, and revised in July 2002, by the New Jersey State Department of Education, has focused attention on a new set of goals and expectations for school mathematics. This visionary document provides a broad framework for what the mathematics curriculum in grades K – 12 should include in terms of content priority and emphasis. Mathematics is central in the development of knowledge, understanding, appreciation, values and beliefs that are required by individuals as they embark upon a lifetime of learning. The purpose of the K – 12 Mathematics Curriculum is to support implementation of the New Jersey Core Curriculum Content Standards in the Plainfield Schools. It specifies a set of high level outcomes for the math undertaking and performance of ALL students. The themes of problem solving, reasoning, communication, and connections have been woven throughout. The use of technological tools as a means of guiding instruction is an integral part of the curriculum. The curriculum also contains a strong focus on integrating the Cross-Content workplace Readiness Standards into mathematics instruction. Teachers can no longer focus on repetitive and low level cognitive activities if they are to prepare all students to reach the new standards. Instruction must, instead, focus on problem solving, critical thinking, and active learning. Students develop their innate problem-solving power through a program of rich and challenging experiences. Faced with a variety of individual and cooperative challenges, students begin to cultivate their organizational, analytic, decision-making, and communication skills. Sustainable change must occur in the classrooms of teachers and in the school district if we are to realize the vision set forth in the standards. It will not be easy to reach this challenge, but working collectively we can make the vision of the Core Curriculum Content Standards a reality in the Plainfield Public Schools. The need to understand mathematics in order to succeed in all walks of life is without precedent. Students need to develop mathematical competence and confidence when formulating and solving problems and proficiency in applying mathematical reasoning. More than ever, students need to communicate mathematically and be able to link mathematics to other curricular areas and situations in their daily lives. Further, the effects of technological innovations continue to permeate mathematics; students need to incorporate technology when learning and applying mathematics. The students in the Plainfield Schools will need increasingly advanced levels of knowledge and skills if they are to be successful throughout their lives. They will need to be life-long learners to be able to compete in a global, information-based economy. Cognizant of these mathematical needs, the K-12 mathematics curriculum was developed to educate and challenge all students to become mathematically literate. Mathematically literate students demonstrate an ability to analyze and use quantitative information, and to show an appreciation for the beauty and value of mathematics. Further, mathematically literate students have the ability to explore, make connections, draw conclusions, communicate, reason logically, and use effectively a variety of mathematical methods to solve problems.
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Educational goals for all students, based upon the National Council of Teachers of Mathematics (NCTM) Standards and the New Jersey Core Curriculum Standards for Mathematics, must reflect the importance of mathematical literacy. Toward this end, our goals are to have students in the Plainfield Public Schools: • • • • •
Comprehend mathematical concepts, operations, and relations – know what mathematical symbols, diagrams, and procedures mean. Carry out mathematical procedures, such as adding, subtracting, multiplying and dividing numbers flexibly, accurately, efficiently, and appropriately. Formulate problems mathematically and devise strategies for solving them using concepts and procedures appropriately. Use logic to explain and justify a solution to a problem or to extend from something known to something not yet known. See mathematics as sensible, useful, and doable if you work at it and be willing to do the work.
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INSTRUCTIONAL PRACTICES How we teach is as important as what we teach. The instructional practices described here are to be used by every teacher of mathematics from kindergarten through twelfth grade. 1.
Create a climate that promotes thinking and problem solving. The room should be arranged to encourage the use of cooperative groups, thereby maximizing the interaction among students.
2.
Make math active, hands-on. Help students see that mathematics is not just arithmetic, not just performing calculations, and not just seeing how many problems on the worksheet can be accomplished during the class period. Students need to know that "to do" mathematics is to explore, justify, represent, solve, compare, construct, discuss, use, investigate, describe, develop, and predict.
3.
Verification, justification and interpretation. Require students to verify the correctness and reasonableness of results and interpret them in the context of the problem being solved.
4.
Provide students with opportunities for exploration and confirmation. Students should be allowed to explore and discover concepts. They should be able to describe and confirm their individual hypothesis, thereby making them conscious of their own thinking processes, and enhancing their metacognitive skills (i.e., What did I learn? How did I approach the problem? How could we do it differently?)
5.
Build on prior knowledge. Children today enter the school environment with prior knowledge of mathematics as well as other concepts. These children have more knowledge than we when we entered school. This is due to many technological advances in our society such as computers, calculators, televisions, and CDs. We as teachers should and must build on this pre-school knowledge.
6.
Stimulate the students' curiosity. Have students use discovery-oriented, inquiry-based and problem-centered approaches to investigate and understand mathematical content, thereby promoting the students' intrinsic motivation to learn mathematics.
7.
Use calculators to develop conceptual understanding. Don’t just count on calculators to explore number concepts and patterns. Use them to help students focus on problem solving.
8.
Make math authentic. Integrate mathematics with other content areas and to the real world in a meaningful way. Students who have experiences that link mathematics to other subject areas are able to apply previous knowledge to new situations.
9.
Provide students with challenging opportunities. Challenge students with many types of problem solving experiences including non-routine problems and open-ended problems. Present students with problems that they don't already know how to solve, and encourage them to make conjectures about what the solution might be and how it might be obtained.
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10.
Maximize opportunities for meaningful learning. Plan most lessons so that they involve more than one content strand/outcome and take place over more than one day. This will give students enough time to explore a situation thoroughly and solidify learning.
11.
Establish a talking/listening environment. Require students to talk and write about their understanding of mathematical concepts and to listen to others describe their mathematical thinking. Students should be able to construct, explain, justify and apply a variety of problem-solving strategies.
12.
Encourage students to work together in small groups. Structure the group activity so that all students participate appropriately.
13.
Develop and use many different approaches to solving a problem. Encourage the students to do the same. Teach mathematical ideas, whenever possible, through posing a problem, setting up a situation, or asking a question.
14.
Make mathematical discussion a daily part of the classroom activity. Have students identify and explain key mathematical concepts and model situations using oral, written, concrete, pictorial and graphical methods. Encourage student-student as well as student-teacher dialogue.
15.
Develop student's mental math competence. Require students to use mental math strategies and estimation to determine the reasonableness of answers. Estimation of sums and differences should be part of the computational process from the very first activity with any sort of computation.
16.
Ask good questions often (open ended ones with a variety of responses, or those which require reasons or conclusions). Have students ask questions of their own and encourage them to respond to each other’s questions.
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Course: 8th Grade Mathematics Unit: Representing Algebraic Relationships Big Idea
Skills and Standards
- Algebraic relationships can be modeled using tables, graphs, and equations; and these models can be constructed to represent situations and predict outcomes.
4.3.8A1 Represent and analyze relationships among variable quantities and solve problems using patterns, functions, and algebraic concepts and processes. Students will be able to recognize, describe, extend, and create patterns involving whole numbers, rational numbers, and integers.
Content • •
•
•
Collecting and analyzing data Working with graphs, tables, and equations Examining linear and nonlinear relationships Identifying inverse relationships
4.3.8C2 Use patterns, relations, symbolic algebra and linear functions to model situations. 4.3.8D2 Solve simple linear equations informally, graphically, and using formal algebraic methods. 4.3.8D3 Solve simple linear inequalities.
Lessons / Activities Use a real world situation to develop skill in collecting and recording data in tables and graphs Develop strategies for making predictions based on data tables or graph models Explore the process of writing an equation given the graph of a line Explore the meaning of slope and y-intercept in relation to a set of data Develop strategies to write an equation of a line given the slope and y-intercept, the slope and the coordinates of a point on the line, or the coordinates of two points on the line Develop strategies for making predictions from tables and graph models Recognize and distinguish between linear and non-linear relationships Identifying and describing the characteristics of inverse relationships Using ideas about rates of change to sketch graphs for, and to match graphs to, given situations Using ideas about rates of change to create stories that fit given graphs Explore symbolic representations for several graph models
Sample Assessment Items • Mr. Block bought a flowering bush for his garden. When he planted it, the bush had three flowers. One week later the plant had nine flowers. Two weeks later, it had twenty-seven flowers. If the plant keeps flowering at the same rate, which of the following expressions represents the number of flowers on the bush nine weeks after Mr. Block planted it? A. 37
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B. 38
C. 39 D. 310
Which equation satisfies the relationship in the chart below? x -1 2 3
y -6 3 6
A. y = 4x - 5 B. y = - 2x C. y = 3 x - 3 D. y = 2x - 1
Cross Content Connections Language Arts Literacy: Discussion (small group and whole class), Writing solutions to open-ended problems Data collection, analyzing data, and graphic representation of data are used in virtually every field of study. Examples of connections include: Tide tables, weather charts, and data from meteorology studies (science) Demographic distributions and trend analysis (social sciences) Resources Thinking with Mathematical Models - Connected Mathematics, Boston, Pearson/Prentice Hall, 2002 The following activity can be accessed at http://illuminations.nctm.org/ActivityDetail.aspx?ID=146 Line of Best Fit. This activity allows the user to enter a set of data, plot the data on a coordinate grid, and determine the equation for a line of best fit. The following activity can be accessed at http://www.prenhall.com/divisions/esm/app/calc_v2/ Graphing Calculator Help. A website provided through Prentice Hall publishing that assists the user in the functions and use of a graphing calculator.
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Course: 8th Grade Mathematics
Unit: Understanding the Pythagorean Theorem
Big Idea
Skills and Standards
Lessons / Activities
Sample Assessment Items
- The relationship of the areas of the squares on the sides of a right triangle is such that the sum of the areas of the two smaller squares equals the area of the largest square. The principles of the Pythagorean Theorem can be used in determining the distance between two points whose coordinates are known.
4.2.8A2 Understand and apply the Pythagorean Theorem.
4.3.8C2 Use patterns, relations, symbolic algebra and linear functions to model situations.
Using coordinates for specifying locations Using coordinates to specify direction and distance Develop strategies for connecting properties of geometric shapes, such as parallel sides, to coordinate representations Finding areas of polygons and the length of a line segment drawn on a dot grid Begin to develop an understanding of the concept of square root Develop an understanding of the Pythagorean Theorem through exploration Using the Pythagorean Theorem to find areas of squares drawn on a dot grid Using the Pythagorean Theorem to find the distance between two points on a grid Relate areas of squares to the lengths of the sides Develop through investigation the special properties of a 30-60-90 right triangle Using the properties of special right triangles to solve real world problems Begin to connect decimal and fractional representations of rational numbers Develop strategies for estimating lengths of hypotenuses of right triangles Reviewing the concept of the slope of a line Making connections between the concept of slope and the idea of irrational numbers Develop strategies for using slopes to test whether lines are parallel or perpendicular
To keep a fence from falling, a farmer ties a rope to the top of the fence and nails it to the ground. If the fence is 5 yards high and the rope is 7 yards long, to the nearest tenth, what is the distance from the nail to the fence?
5 yd
7 yd
A. 4.9 yards B. 6.2 yards C. 8.6 yard D. 9.8 yard
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On a baseball diamond the bases are 90 ft apart. What is the distance from home plate to second base in a straight line? Explain your reasoning.
(Tip: You may want to draw a picture to help you.)
Content
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Relationships between coordinates, slope, distance, and area
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Strategies for finding the distance between two points on a coordinate grid
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Extend understanding of the properties of irrational numbers
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Properties and applications of the Pythagorean Theorem
Cross Content Connections Language Arts Literacy: Discussion (small group and whole class), Writing solutions to open-ended problems
Visual Arts: Examining and understanding areas, patterns, transformations, and symmetries in irregular figures.
Science: Working with scientific formulas and expressions that involve formulating, reading, and interpreting symbolic rules.
Resources Looking for Pythagoras - Connected Mathematics, Boston, Pearson/Prentice Hall, 2002 The following activity can be accessed at http://illuminations.nctm.org/ActivityDetail.aspx?ID=146 Proof without Words: Pythagorean Theorem. This activity offers a dynamic, geometric “visual” proof of the Pythagorean Theorem and why c2 = a2 + b2. The following activity can be accessed at http://www.prenhall.com/divisions/esm/app/calc_v2/ Graphing Calculator Help. A website provided through Prentice Hall publishing that assists the user in the functions and use of a graphing calculator.
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Course: 8th Grade Mathematics Big Idea - There exists algebraic relationships in which variables grow and decay exponentially. -Growth patterns can model a relationship that is exponential or linear. Content •
•
•
Building and analyzing exponential models Reasoning with and about exponential relationships Exploring the significance of shapes of graphs and patterns in tables
•
Exploring rates of growth
•
Recognizing and describing situations that can be modeled with exponential functions
Unit: Exponential Relationships
Skills and Standards 4.1.8B2 Use exponentiation to find whole number powers of numbers. 4.3.8A1 Recognize, describe, extend, and create patterns involving whole numbers, rational numbers, and integers. 4.3.8B2 Recognize and describe the difference between linear and exponential growth, using tables, graphs, and equations. 4.3.8C2 Use patterns, relations, symbolic algebra and linear functions to model situations.
Lessons / Activities
Sample Assessment Items
•
Becoming familiar with the understanding of basic experimental growth patterns Recognizing exponential patterns in tables, graphs, and equations Developing strategies for solving problems involving exponential growth Expressing a number that is the product of identical factors in exponential form and standard form Recognizing patterns of exponential growth in tables and equations Developing methods of comparing and contrasting exponential growth to linear growth Reasoning with and solving problems involving exponents and exponential growth Determining the growth factor in a given exponential model Creating representations of an exponential population model given sample population data Investigating increases in the value of an asset due to compound growth Recognizing patterns of exponential decay in tables, graphs, and equations Using knowledge of exponents to write equations for models of exponential decay Begin to describe the effects of varying the values of a and b in the equation y = a(bx) on the graph of that equation
Joseph folds a sheet of paper in half, forming two sections. He folds it in half again, giving him four sections. He continues to fold the paper in half for as long as he can. a. Construct a table comparing the number of folds to the total number of sections. b. Construct a graph representing the information in your table. c. Describe the relationship between the number of folds and the total number of sections created by the folds.
Cross Content Connections Language Arts Literacy: Discussion (small group and whole class), Writing solutions to open-ended problems
Science: Working with science activities that involve the half-life of elements, the expansion of the universe, etc.
Resources Growing, Growing, Growing - Connected Mathematics, Boston, Pearson/Prentice Hall, 2002 The following activity can be accessed at http://www.hhs.helena.k12.mt.us/Campbell/growth.htm . Exponential Growth. This is a collection of websites about exponential growth and decay, as well as information on HIV and the AIDS virus. The AIDS and HIV data and information contained in these websites models exponential growth and decay through population data. The following activity can be accessed at http://www.prenhall.com/divisions/esm/app/calc_v2/. Graphing Calculator Help. A website provided through Prentice Hall publishing that assists the user in the functions and use of a graphing calculator.
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Course: 8th Grade Mathematics
Big Idea -A quadratic function can be recognized as an equation written as a product of two linear factors. -There are characteristics of tables, graphs, and equations that define them as quadratic functions.
Content •
Analyzing quadratic relationships by examining tables, graphs, and equations
•
Comparing characteristics of tables and graphs for quadratic relationships with those for linear and exponential relationships and using those characteristics to make
Unit: Quadratic Relationships
Skills and Standards 4.1.8 Construct and use concrete, pictorial, symbolic, and graphical models to represent problem situations and effectively apply processes of mathematical modeling in mathematics and other areas. 4.2.8 Identify and explain key mathematical concepts and model situations using geometric and algebraic methods.
Lessons / Activities Begin to develop an awareness of quadratic functions and how to recognize them from patterns in tables and graphs Describing patterns in tables of quadratic functions and predict subsequent entries Recognizing the characteristic shape of the graph of a quadratic function and observe such features as lines of symmetry, maximum points, vertex, and intercepts Using tables and graphs of quadratic relationships to answer questions about a situation Developing an understanding of equivalent expressions Recognizing a quadratic function from an equation, written as a product of two linear factors, or in expanded form as ax2 + bx + c Observing and describing patterns or regularity and change in data Exploring equations that model different situations Using tables, graphs, and equations to find and interpret maximum or minimum values Comparing quadratic relationships to linear and cubic relationships
Sample Assessment Items • A ship conducting oceanographic research drops anchor offshore Honiara, the capitol of the Solomon islands in the South Pacific. When the anchor is tossed into the water, the depth in feet, D, it descends in t seconds is given by the equation D = -4t2 + 12t. a.
b.
c.
If it takes the anchor 10 seconds to reach the bottom, how deep is the water where the ship has dropped anchor? If the ship moves to another location and the anchor takes 8.5 seconds to reach the bottom, how deep is the water in that spot? If the ship anchors in the harbor of Honiara, where the water is 72 feet deep (that is, D = -72), how long will it take for the anchor to reach the bottom when it is dropped?
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predictions •
Understanding the significance of x and y intercepts, maximum and minimum points, and lines of symmetry
Cross Content Connections
Language Arts Literacy: Discussion (Small group and whole class), writing solutions to open-ended problems Science: Working with science activities that involve the formulas written the quadratic from or what involve maximum and/or minimum values.
Resources Frogs, Fleas, and Painted Cubes - Connected Mathematics, Boston, Pearson/Prentice Hall, 2002 The following activity can be accessed at http://www.explorelearning.com/index.cfm?method=cResource.dspDetail&ResourceID=85 Reflections of Quadratic Functions This is a website that is an online simulation that explores and compares the graphs of quadratic functions. The following activity can be accessed at http://www.prenhall.com/divisions/esm/app/calc_v2/ Graphing Calculator Help. A website provided through Prentice Hall publishing that assists the user in the functions and use of a graphing calculator.
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Course: 8th Grade Mathematics
Unit: Algebraic Reasoning
Big Idea
Skills and Standards
- The properties of real numbers are used in symbolic sentences to communicate reasoning. - There is an appropriate order of operations in evaluating algebraic expressions.
4.3.8A1 Recognize, describe, extend, and create patterns involving whole numbers, rational numbers, and integers.
Content • •
•
Making sense of symbols Using the appropriate order of operations in evaluating expressions Writing symbolic sentences, using parentheses and properties of real numbers, to communicate effectively
•
Reasoning with equivalent expressions
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Solving linear
•
Descriptions using tables, verbal and symbolic rules, graphs, simple equations, or expressions.
4.3.8D4 Create, evaluate, and simplify algebraic expressions involving variables. • Order of operations, including appropriate use of parentheses • Distributive property • Substitution of a number for a variable • Translation of a verbal phrase or sentence into an algebraic expression, equation, or inequality, and vice versa
Lessons / Activities
Sample Assessment Items
•
Developing an understanding of the conventional order of operations rules by examining symbolic expressions involving addition, subtraction, multiplication, division, and exponents Evaluating expressions by applying the rules of order of operations Exploring the distributive property by simplifying and comparing expressions Examining tables and graphs to determine the equivalency of two or more expressions Using contextual clues to interpret symbolic expressions Solving a variety of real world problems using the distributive and commutative properties Solving simple quadratic equations symbolically Use the properties of real numbers to write equivalent expressions Developing systematic methods of solving equations, including linear equations, quadratic equations, and equivalent expressions Introduction to symbolic solutions for equations of the form y = ax2 + bx Writing expressions for surface area
Mr. Fern is in charge of the recycling program at Metropolis Middle School. He keeps track of the number of aluminum cans collected at the school athletic events and the number of people who attend each event. Mr. Fern has found that, in general, the more people who attend an event, the more cans he can expect to collect. From his data, Mr. Fern developed the equation C = 2.5(p - 40) – 100, where C is the expected number of cans that will be collected and p is the number of people attending the event.
A. If 100 people attend a basketball game, how many cans would Mr. Fern expect to collect? B. If 150 people attend a softball match, how many cans would Mr. Fern expect to collect? C. Batina works with Mr. Fern in the recycling program. She says that, based on her analysis of the data, the equation for predicting the number of cans collected should be C = 2.5(p – 80). Are Mr. Fern’s and Batina’s expressions for the number of can’s equivalent? Explain your answer.
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and quadratic equations •
Modeling and solving problems
Cross Content Connections Resources
Language Arts Literacy: Discussion (small group and whole class). Writing solutions to open-ended problems
Science: Working with science activities that involve the half-life or elements, the expansion of the universe, etc.
Say It with Symbols - Connected Mathematics, Boston, Pearson/Prentice Hall, 2002 The following activity can be accessed http://www.funbrain.com/algebra/ Operation Order Algebra Game This website is an interactive website where students can practice and learn additional skills in the order of operations, along with other math concepts. The following activity can be accessed at http://www.prenhall.com/divisions/esm/app/calc_v2/ Graphing Calculator Help. A website provided through Prentice Hall publishing that assists the user in the functions and use of a graphing calculator
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Course: 8th Grade Mathematics
Unit: Symmetry and Transformations
Big Idea
Skills and Standards
- There are unique properties that exist in a symmetric design. Symmetry transformations of figures include reflections, translations, and rotations. -Transformational geometry can be used to describe motions, patterns, and designs in the real world.
4.2.8B1 Understand and apply transformations. •
•
•
Content •
•
•
•
Recognizing symmetry in designs Creating designs with reflectional, rotational, or translational symmetries.
•
Lessons / Activities
Finding the image, given the preimage, and viceversa
Sequence of transformations needed to map one figure onto another
Reflections, rotations, and translations result in images congruent to the pre-image Dilations (stretching/ shrinking) result in images similar to the pre-image
4.2.8B2 2. Use iterative procedures to generate geometric patterns. •
Describing rigid motions in words and with coordinate rules
Fractals (e.g., the Koch Snowflake)
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Self-similarity
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Composing symmetry transformations
Construction of initial stages
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Patterns in successive stages (e.g., number of
Explore and identify several designs to understand reflectional, rotational, and translational symmetry Examine various tools and procedures for testing symmetry and making symmetric figures Design shapes that have specified symmetries Explore the relationships between figures and their images under reflections, rotations, and translations and begin to write precise rules for finding images under each type or transformation Examine combinations of reflections by reflecting a figure over intersecting lines and parallel lines Writing directions for drawing figures composed of line segments Develop coordinate rules for specifying rotations of 90º, 180º, 270º, and 360º Examining a combination of reflections to determine if there is equivalency to a single transformation Explore combinations of symmetry transformations of an equilateral triangle and a square and create tables showing the results of every combination of two symmetry transformations for these figures
Sample Assessment Items
•
WOW
A. Does the word above have reflectional symmetry? If so, draw all the lines of symmetry. B. Change one letter in the word to form a word with rotational symmetry. C. How could you check that your word in part B has rotational symmetry? D. Where is the center of rotation for your word? What is the angle of rotation?
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triangles in each stage of Sierpinski's Triangle) •
4.2.8C2 Use a coordinate grid to model and quantify transformations (e.g., translate right 4 units).
Cross Content Connections
Language Arts Literacy: Discussion (small group and whole class), writing solutions to open-ended problems.
Science: Working with science activities that involve phenomena in nature that include symmetrical designs and structure.
Visual Arts: Relating to and appreciating the symmetrical designs in art and nature. Resources Kaleidoscopes, Hubcaps, and Mirrors - Connected Mathematics, Boston, Pearson/Prentice Hall, 2002. The following activity can be accessed at http://illuminations.nctm.org/ActivityDetail.aspx?ID=24. Mirror Tool. An interactive website that contains an activity that allows
the user to investigate symmetry. You can rotate, flip, or reflect a figure across a line. The following activity can be accessed at http://www.shodor.org/interactivate/activities/index.html. Math Activities. This website is an interactive website of math activities that are designed for either group or individual exploration into middle school math concepts. The following activity can be accessed at http://www.prenhall.com/divisions/esm/app/calc_v2/ . Graphing Calculator Help. A website provided through Prentice Hall publishing that assists the user in the functions and use of a graphing calculator.
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Course: 8th Grade Mathematics
Unit: Data and Statistics
Big Idea
Skills and Standards
- The process of statistical investigation can be used to explore problems. There are a variety of graphs with unique elements that impact the presentation of information. -Outcomes are affected by the manner of representation of sample data and bias.
4.4.8A1 Select and use appropriate representations for sets of data, and measures of central tendency (mean, median, and mode).
Lessons / Activities
Sample Assessment Items
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Type of display most appropriate for given data
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Box-and-whisker plot, upper quartile, lower quartile
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Scatter plot
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Calculators and computer used to record and process information
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Finding the median and mean (weighted average) using frequency data.
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Effect of additional data on measures of central tendency
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4.4.8A2 Make inferences and formulate and evaluate arguments based on displays and analysis of data.
Analyze data from real world phenomena Explore relationships among data using scatter plots Compare data using tables, stem and leaf plots, histograms, and box and whiskers plots Explore the influence of extreme values on calculations of data using mean and median Examine situations that use data from samples and that use samples from populations and begin to make distinctions between samples and populations Consider various sampling strategies in the context of creating a survey Analyze data from a sample to make predictions about a population Explore and consider the differences among convenience samples, voluntary-response samples, and random samples Explore techniques for choosing samples randomly from a population Apply knowledge about samples to address real world situations Use data from real world samples to estimate a characteristic found in a population
Marci works on the yearbook staff at Metropolis Middle School. Of the 92 businesses in the downtown area, 41 purchased advertising space in the yearbook last year. A. Suppose Marci wants to investigate why businesses did not advertise in the yearbook last year. Describe a sampling strategy she could use to call 10 businesses. B. Suppose Marci wants to investigate how satisfied advertisers are with yearbook ads. Describe a sampling strategy she could use to call 10 businesses. C. Suppose Marci wants to investigate how likely a typical downtown business is to advertise in the upcoming yearbook. Describe a sampling strategy she could use to call 10 businesses.
Content •
Using the
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process of statistical investigation to explore problems •
Composing and decomposing graphs and recognizing the elements of graphs, the interrelationship s among graphical elements, and the impact of these elements on the presentation of information in a graph
•
Describing the shape of the data in a graph, including such elements as clusters, gaps, outliers, symmetry or skew, what is typical, the spread in the data, and the single or multiple peaks in the data
•
Distinguishing between a sample and a population
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Cross Content Connections Language Arts Literacy: Discussion (small group and whole class), writing solutions to open-ended problems
Science, Social Studies, and other subjects: Working with learning activities that involve a variety of data contexts and graphs, including line plots, bar graphs, circle graphs, stem and leaf plots, histograms, box and whiskers plots, and scatter plots. Resources
Samples and Populations - Connected Mathematics, Boston, Pearson/Prentice Hall, 2002. The following activity can be accessed at http://www.shodor.org/interactivate/activities/index.html. Math Activities. This website is an interactive website of math activities that are designed for either group or individual exploration into middle school math concepts. The following activity can be accessed at http://www.prenhall.com/divisions/esm/app/calc_v2/ . Graphing Calculator Help. A website provided through Prentice Hall publishing that assists the user in the functions and use of a graphing calculator.
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Modifications/Differentiating Instruction The suggestions for adaptation and enrichment that are listed below apply to all of the instructional units. It is important to note that adaptations should be made for any students according to their needs in order to make the explorations more accessible to them. It is also important to note that enrichment activities are not for an elite group of students, but that any student may excel at any time and benefit from a specific enrichment or extension. The same student may at different times need either adaptations or enrichment. Adaptations for Special Needs, ELL’s, and others -See CMP2 Special Needs Handbook for recommendations -See Implementing and Teaching Guide pp. 87-101 -Brainstorming- writing ideas on board -Word walls in English & native language -Rebus Techniques -Modeling -Visual/Verbal Prompts -Graphic organizers- tree chart, Venn diagram, etc. -Using informal math language before formal terminology
Enrichment: “Going Further” questions within the explorations See Extension problems within the unit
Student-generated assessments and scoring rubricshttp://www.figurethis.org/ Figure This! Math Challenges for the Family. This interactive website offers interesting math challenges that middle-school students can do at home with their families or in small groups.
Instructional Materials: Primary instruction program Connected Mathematics 2* Lappan, Fey, et al. Pearson Prentice Hall 2006 Supporting programs Springboard Middle School Mathematics College Entrance Examination Board 2004 Question Quest LL Teach, Inc. 2002
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*About the Connected Mathematics Program The Connected Mathematics Program is the primary instructional program at grades six through eight. It is a standards-based, problem-centered curriculum. The role of the teacher in a problem-centered curriculum differs from the traditional role, in which the teacher explains ideas thoroughly and demonstrates procedures so students can quickly and accurately duplicate these procedures. A problem-centered curriculum is best suited to an inquiry model of instruction. The teacher and students investigate a series of problems; through discussion of solution methods, embedded mathematics, and appropriate generalizations students grow in their ability to become reflective learners. Teachers have a crucial role to play in establishing the expectations for discussion in the classroom and for orchestrating discourse on a daily basis. The Connected Mathematics materials are designed to help students and teachers build an effective pattern of instruction in the classroom. A community of mutually supportive learners works together to make sense of the mathematics through: the problems themselves; the justification the students are asked to provide on a regular basis; student opportunities to discuss and write about their ideas. To help teachers think about their teaching, the Connected Mathematics Program uses a three-phase instructional model, which contains a Launch of the lesson, an Exploration of the central problem, and a Summary of the new learning. The Launch of a lesson is typically done as a whole class; yet during this launch phase of instruction students are sometimes asked to think about a question individually before discussing their ideas as a whole class. The launch phase is also the time when the teacher introduces new ideas, clarifies definitions, reviews old concepts, and connects the problem to past experiences of the students. It is critical that, while giving students a clear picture of what is expected, the teacher is careful not to reveal too much and lower the challenge of the task to something routine, or limit the rich array of strategies that may evolve from an open launch of the problem. In the Explore phase, students may work individually, in pairs, in small groups, or occasionally as a whole class to solve the problem. As they work, they gather data, share ideas, look for patterns, make conjectures, and develop problem-solving strategies. The teacher's role during this phase is to move about the classroom, observing individual performance and encouraging on-task behavior. The teacher helps students persevere in their work by asking appropriate questions and providing confirmation or redirection where needed. For students who are interested in deeper investigation, the teacher may provide extra challenges related to the problem. These challenges are provided in the Teacher's Guide. Substantive whole-class discussion most often occurs during the Summarize phase when individuals and groups share their results. Led by the teacher's questions, the students investigate ideas and strategies and discuss their thoughts. Questioning by other students and the teacher, challenges students' ideas, driving the development of important concepts. Working together, the students synthesize information, look for generalities, and extract the strategies and skills involved in solving the problem. Since the goal of the summarize phase is to make the mathematics in the problem more explicit, teachers often pose, toward the end of the summary, a quick problem or two to be done individually as a check of student progress. Connected Mathematics is different from traditional programs. Because important concepts are embedded within problems rather than explicitly stated and demonstrated in the student text, the teacher plays a critical role in helping students develop appropriate understanding, strategies, and skills. It is the teachers' thoughtful reflections on student learning that will create a productive classroom environment. Teachers who have experienced success with Connected Mathematics have made two noteworthy suggestions: 26
(i)
(ii)
The teacher should work through each investigation prior to the initiation of instruction. Teachers who invest time in doing the problems in at least two different ways will be better equipped to Launch the investigation, facilitate the Exploration and Summary of the problem, and know what mathematics assessment is appropriate. The teacher should engage in ongoing professional conversations about the mathematics in the Connected Mathematics Program they are using, sharing strategies for improving student achievement.
The format of the student books is also much different from traditional mathematics texts. The student pages are uncluttered and have few nonessential features. Because students develop strategies and understanding by solving problems, the books do not contain worked-out examples that demonstrate solution methods. Since it is also important that students develop understanding of mathematical definitions and rules, the books contain few formal definitions and rules. These non-consumable student books should be kept in a three-ring binder during instruction and collected when instruction has been completed. It is essential that the teacher develops and maintains a notebook management system. The "Implementing and Teaching Guide” provides strategies to assist the teacher with the purposes and organizational format for student notebooks.
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Suggested Timeline
Unit
Connected Mathematics Module
Duration
Representing Algebraic Relationships
Thinking with Mathematical Models
September-October (6 weeks)
Understanding the Pythagorean Theorem
Looking for Pythagoras
October-November (6 weeks)
Exponential Relationships
Growing, Growing, Growing
November-December (5 weeks)
Quadratic Relationships
Frogs, Fleas, and Painted Cubes
January (5 weeks)
Algebraic Reasoning
Says It with Symbols
February-March (6 weeks)
Symmetry and Transformations
Kaleidoscopes, Hubcaps, and Mirrors
March-April (6 weeks)
Data and Statistics
Samples and Populations
May - June (6 weeks)
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Technology in the Connected Mathematics Project Connected Mathematics was developed with the belief that calculators should always be available to students, and that students should know when and how to use them. In grade 6, students need standard, four-function calculators. In grades 7 and 8, students need graphing calculators with table and statistical-display capabilities. Students use four-function calculators to simplify complicated calculations and explore patterns in computations. Graphing calculators are used to investigate functions and as a tool for solving problems. Students use graphing calculators to explore the shape and features of graphs of linear, exponential, and quadratic functions as well as the patterns of change in the tables of such functions. And in addition to using symbolic solution methods, students use graphing-calculator tables and graphs to solve equations. Although computers are not required for any of the investigations, there are optional computer activities that enhance some of the investigations and give students an opportunity to practice their skills. For example, the Coin Game is provided to accompany the unit How Likely Is It? This game can simulate large numbers of coin tosses and graph the results, allowing students to observe that experimental results approach theoretical results over the long run. In addition to the CMP computer activities, the commercially available software Geo-logo is used in some optional geometry investigations to explore shapes and transformations. This program is referred to as Turtle Math in the student books. Supporting Websites:
National Council of Teachers of Mathematics http://www.nctm.org Contains news and information of interest to math teachers
•
New Jersey Department of Education http://www.state.nj.us/education/ Information about state department regulations, including state testing program.
•
www.everydaymath.uchicago.edu Everyday Mathematics website Information, teacher resources, professional development opportunities, and more for the Everyday Mathematics program
•
NCTM Illuminations www.illuminations.nctm.org Provides Standards-based resources, including online games and activities, including the Factor Game and the Product Game, as well as standards-based lessons
•
http://www.figurethis.org/. Figure This! Math Challenges for the Family This interactive website offers interesting math challenges that middle-school students can do at home with their families or in small groups.
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•
NJ PEP www.njpep.org This is NJDOE’s online professional development site. It provides a wealth of information about professional development opportunities, state assessments, and standards.
•
Math Goodies http://www.mathgoodies.com Interactive math lessons, as well as homework help, puzzles, calculators.
•
Math Forum http://mathforum.org/ A center for teachers, students, parents, and citizens at all levels who have an interest in mathematics education (includes lesson plans, open-ended problems with multiple solutions, homework helper ask Dr. Math, and more)
•
National Library of Virtual Manipulatives http://nlvm.usu.edu/en/nav/vlibrary.html A variety of online math manipulatives for all grade levels
•
Math Solutions
www.mathsolutions.com
Founded by Marilyn Burns, includes professional development opportunities, resources for teachers and administrators, and a newsletter addressing various issues around mathematics teaching and learning
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Assessment in the Connected Mathematics Program Assessment in Connected Mathematics is an extension of the learning process, as well as an opportunity to check what students can do. For this reason, the assessment is multidimensional, giving students many ways to demonstrate how they are making sense of the mathematics. The Standards for Curriculum and Evaluation in School Mathematics (NCTM, 1989) and the Assessment Standards for School Mathematics (NCTM, 1995) provide guidelines that describe mathematics education in schools, not only in terms of mathematical objectives, but in terms of the methods of instruction, the processes used by students in learning and doing mathematics, and the students' disposition towards mathematics. The Connected Mathematics assessment component attempts to capture this broad perspective of mathematics education by collecting data concerning these three dimensions of student learning:
Content knowledge: Assessing content knowledge involves determining what students know and what they are able to do. Mathematical disposition: A student's mathematical disposition is healthy when he or she responds well to mathematical challenges and sees himself or herself as a learner and inventor of mathematics. Disposition also includes confidence, expectations, and metacognition (reflecting on and monitoring one's own learning). Work habits: A student's work habits are good when he or she is willing to persevere, contribute to group tasks, and follow tasks to completion. These valuable skills are used in nearly every career. To assess work habits, it is important to ask questions, such as "Are the students able to organize and summarize their work?" and "Are the students progressing in becoming independent learners?"
The NCTM Principles and Standards 2000 reinforces the CMP philosophy on assessment. Its Assessment Principle states: Assessment should support the learning of important mathematics and furnish useful information to both teachers and students.
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Preparation for Standardized Assessment The New Jersey Assessment of Skills and Knowledge (NJASK) and Grade Eight Proficiency Assessment (GEPA) NJASK is a state test given to students at grades three through seven and GEPA is the state test given to students at grade eight to measure whether they have gained the knowledge and skills identified in the New Jersey Core Curriculum Content Standards (NJCCCS). Prior to 2006, there were NJASK assessments for grades 3 and 4 only. Beginning in 2006, there are NJASK assessments for grades 5, 6, and 7 as well. The results of NJASK and GEPA are also used to satisfy federal requirements under the No Child Left Behind Act (NCLB). While the best way to prepare for NJASK, GEPA, or any standardized assessment is high quality instruction, it is also critical that students understand its structure, question formats, and scoring and receive instruction in test-taking strategies.
About the Mathematics Section of NJASK and GEPA There are four clusters assessed on NJASK and GEPA. These clusters correspond with the NJCCCS. I. Number and Numerical Operations II. Geometry and Measurement III. Patterns and Algebra IV. Data Analysis, Probability, and Discrete Math Each multiple choice question is worth 1 point and should take an average of 1-2 minutes to complete. Each open-ended question is worth 3 points and should take an average of 7-8 minutes to complete. Calculators are permitted for all questions on GEPA and all questions on NJASK except those multiple choice questions in the no-calculator section.
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Open-Ended Problem Solving and Scoring For students to be successful in responding to open-ended problems, they must have experiences in formulating complete and accurate responses to open-ended questions, and understand the process by which they will be assessed. A suggested procedure for presenting an open-ended task is given below. When simulating a testing situation, students should strive to provide a comprehensive response to each open-ended problem in 7-8 minutes.
Step 1
Introduce the problem. (For the first problem, you may want to conduct a discussion of what constitutes a complete solution.) Students work in groups to read and discuss the solution of the problem.
Step 2
Each group comes to consensus on a solution that would earn a score of three points using the New Jersey Holistic Scoring Guide.
Step 3
As a class, discuss the solution and alternate methods of solution, and make sure students understand the rubric. Groups self assess using the scoring rubric and/or exchange work to assess each other.
Step 4
Discuss and justify scores, and discuss how scores could be improved. Class generates a “perfect” solution, which can be used as a model. Students may also score anonymous student work samples, discussing how the score could be improved if it has not earned a score of 3.
Step 5
Assess understanding through quiz, individual work, or other method of choice.
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A sample open-ended problem and task-specific scoring rubric appear below. For additional sample items see the unit outlines and Question Quest. Sample Problem – “Parent Quilt” The students of Smith Street Middle School are planning to make a quilt in honor of their parents. The quilt will be rectangular, with a section for each grade (grade 6, grade 7, and grade 8). The number of students in each grade is shown below: Number of Grade
6 7
8 • •
Students
72 108 60
On your answer paper, draw a rectangle like the one shown here. The rectangle will represent the “parent quilt.” Divide it into sections for the three grades. Explain why your division is fair. Task-Specific Scoring Rubric for “Parent Quilt”
3-point response: The student provides a diagram in which the area is divided into reasonably proportional areas (about 30% for grade 6, 45% for grade 7, and 25% for grade 8), and provides clear explanation / justification for all decisions. Minor arithmetic errors are acceptable, as long as the student uses a mathematically correct method of solution (for example: the student might make an error in calculating the portion of the area for one grade, but provides a diagram which is accurate based on that information.) 2-point response: The student provides correct numerical responses and diagram, but fails to provide explanation or justification. 1-point response: The student only begins to provide a solution (e.g., finds only the portion for grade 6, and answers no other parts). 0-point response: The student fails to provide an answer, or the answer is inappropriate.
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Multiple Choice Questions
For students to be successful in correctly responding to multiple-choice assessment items, they must have experience in using their mathematical knowledge in combination with efficient strategies to select the best response to a question and understand the process by which they will be assessed. A suggested procedure for developing accuracy and efficiency in responding to multiple choice questions is given below. When simulating a testing situation, students should strive to respond to a multiple-choice question in 1-2 minutes. Step 1 Step 2 Step 3 Step 4 Step 5
Introduce the problem. For the first problem, you may want to model your thinking process by talking aloud about how you eliminated incorrect choices and how you determined the correct choice. Students work independently to determine their solution of the problem. Group members justify their choice of answer and discuss why they eliminated the distractors. Share the correct answer and discuss in small groups what errors students might have made that would have caused them to select the distractors as being correct. Call upon several students to share their solution strategies. Emphasize use of efficient methods, as time is a factor. Stress the importance of answering the question that is asked, making connections between the distractors and assuming the question to be answered. Assess understanding through quiz, individual work, or other method of choice.
Suggestions for students: Solve first; then choose the answer. The wrong choices represent common mistakes, so be sure to check your calculations. When you’re not sure: 9 Eliminate unreasonable choices. 9 Eliminate any two choices that have the same meaning. 9 Try substituting answer choices in the problem to see which one works. Suggestions for teachers: 9When discussing multiple choice items in class, ask students why they chose incorrect responses. 9When practical, have students explain how each wrong response was designed. “What would a student do wrong to get this answer?”
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Sample multiple choice item appears below. For additional sample items see the unit outlines and Question Quest. Sample Items: Maria plans to buy a new calculator that sells for $168. They have been marked down 25%. She sees an advertisement stating that all marked down items will be reduced an additional 30% off the sale price. What will Maria pay, including 6% sales tax on her purchase? a. $80.14
b.$85.68
`
c. $93.49
d. $102.48
The correct response is c, $93.49. What error might cause a student to choose a? (The student probably found 55% off instead of reducing the first percentage, 25%, and then reducing 30% from the sale price.) What error might cause a student to choose b? (Probably the student found 49% off by combining 25% + 30% – 6%.) What error might cause a student to choose d? (The student probably combined all the percentages to get 61%, and then just found 61% of the original price.) I want to pay the pizza delivery person for the pizza, which cost $16.00, plus 15% tip. I have $10's, $5's, and $1's. What should I give the pizza delivery person to pay for the bill and tip, so I get the least amount of change back? a. $10, $10
b. $10, $5, $1
c. $5, $5, $5, $1, $1, $1, $1
d. $10, $5, $1, $1, $1
The correct response is c. What error in reasoning is demonstrated by each incorrect response?
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Appendix
A: B:
D:
E:
New Jersey Core Curriculum Content Standards Cumulative Progress Indicators for Grades K-2 Assessment B.1: Purposes of Assessment Based on Standards B.2: New Jersey Mathematics Holistic Scoring Rubric B.3: Kid-Friendly Rubric for Mathematics Assignments C: Accountable Talk Questioning D.1: Questioning: What Research Tells Us D.2: Bloom’s Taxonomy D.3: Questions to Encourage Accountable Talk D.4: Questions to Encourage Critical Thinking Professional Resources for K-8 Teachers of Mathematics
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Appendix A New Jersey Core Curriculum Content Standards for Mathematics
4.1
Number and Numerical Operations All students will develop number sense and will perform standard numerical operations and estimations on all types of numbers in a variety of ways.
4.2
Geometry and Measurement All students will develop spatial sense and the ability to use geometric properties, relationships, and measurement to model, describe, and analyze phenomena.
4.3
Patterns and Algebra All students will represent and analyze relationships among variable quantities and solve problems involving patterns, functions, and algebraic concepts and processes.
4.4
Data Analysis, Probability, and Discrete Mathematics All students will develop an understanding of the concepts and techniques of data analysis, probability, and discrete mathematics, and will use them to model situations, solve problems, and analyze and draw appropriate inferences from data.
4.5
Mathematical Processes All students will use mathematical processes of problem solving, communication, connections, reasoning, representations, and technology to solve problems and communicate mathematical ideas.
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New Jersey Core Curriculum Content Standards for Mathematics Cumulative Progress Indicators for Grade 8
Building upon knowledge and skills gained in preceding grades, by the end of Grade 8, students will: A.
Number Sense
1.
Extend understanding of the number system by constructing meanings for the following (unless otherwise noted, all indicators for grade 8 pertain to these sets of numbers as well):
Rational numbers Percents Exponents Roots Absolute values Numbers represented in scientific notation
2.
Demonstrate a sense of the relative magnitudes of numbers.
3.
Understand and use ratios, proportions, and percents (including percents greater than 100 and less than 1) in a variety of situations.
4.
Compare and order numbers of all named types.
5.
Use whole numbers, fractions, decimals, and percents to represent equivalent forms of the same number.
6.
Recognize that repeating decimals correspond to fractions and determine their fractional equivalents.
5/7 = 0. 714285714285… = 0.
7.
Construct meanings for common irrational numbers, such as p (pi) and the square root of 2.
B.
Numerical Operations
1.
Use and explain procedures for performing calculations involving addition, subtraction, multiplication, division, and exponentiation with integers and all number types named above with:
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Pencil-and-paper Mental math Calculator
2.
Use exponentiation to find whole number powers of numbers.
3.
Find square and cube roots of numbers and understand the inverse nature of powers and roots.
4.
Solve problems involving proportions and percents.
5.
Understand and apply the standard algebraic order of operations, including appropriate use of parentheses.
C.
Estimation
1.
Estimate square and cube roots of numbers.
2.
Use equivalent representations of numbers such as fractions, decimals, and percents to facilitate estimation.
3.
Recognize the limitations of estimation and assess the amount of error resulting from estimation.
STANDARD 4.2 (GEOMETRY AND MEASUREMENT) ALL STUDENTS WILL DEVELOP SPATIAL SENSE AND THE ABILITY TO USE GEOMETRIC PROPERTIES, RELATIONSHIPS, AND MEASUREMENT TO MODEL, DESCRIBE AND ANALYZE PHENOMENA. Building upon knowledge and skills gained in preceding grades, by the end of Grade 8, students will: A.
Geometric Properties
1.
Understand and apply concepts involving lines, angles, and planes.
Complementary and supplementary angles Vertical angles Bisectors and perpendicular bisectors Parallel, perpendicular, and intersecting planes Intersection of plane with cube, cylinder, cone, and sphere
2.
Understand and apply the Pythagorean theorem.
3.
Understand and apply properties of polygons.
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4.
Quadrilaterals, including squares, rectangles, parallelograms, trapezoids, rhombi Regular polygons Sum of measures of interior angles of a polygon Which polygons can be used alone to generate a tessellation and why
Understand and apply the concept of similarity.
Using proportions to find missing measures Scale drawings Models of 3D objects
5.
Use logic and reasoning to make and support conjectures about geometric objects.
B.
Transforming Shapes
1.
Understand and apply transformations.
2.
Finding the image, given the pre-image, and vice-versa Sequence of transformations needed to map one figure onto another Reflections, rotations, and translations result in images congruent to the pre-image Dilations (stretching/shrinking) result in images similar to the pre-image
Use iterative procedures to generate geometric patterns.
Fractals (e.g., the Koch Snowflake) Self-similarity Construction of initial stages Patterns in successive stages (e.g., number of triangles in each stage of Sierpinski’s Triangle)
C.
Coordinate Geometry
1.
Use coordinates in four quadrants to represent geometric concepts.
2.
Use a coordinate grid to model and quantify transformations (e.g., translate right 4 units).
D.
Units of Measurement
1.
Solve problems requiring calculations that involve different units of measurement within a measurement system (e.g., 4’3" plus 7’10" equals 12’1").
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2.
Use approximate equivalents between standard and metric systems to estimate measurements (e.g., 5 kilometers is about 3 miles).
3.
Recognize that the degree of precision needed in calculations depends on how the results will be used and the instruments used to generate the measurements.
4.
Select and use appropriate units and tools to measure quantities to the degree of precision needed in a particular problem-solving situation.
5.
Recognize that all measurements of continuous quantities are approximations.
6.
Solve problems that involve compound measurement units, such as speed (miles per hour), air pressure (pounds per square inch), and population density (persons per square mile).
E.
Measuring Geometric Objects
1.
Develop and apply strategies for finding perimeter and area.
Geometric figures made by combining triangles, rectangles and circles or parts of circles Estimation of area using grids of various sizes Impact of a dilation on the perimeter and area of a 2-dimensional figure
2.
Recognize that the volume of a pyramid or cone is one-third of the volume of the prism or cylinder with the same base and height (e.g., use rice to compare volumes of figures with same base and height).
3.
Develop and apply strategies and formulas for finding the surface area and volume of a three-dimensional figure.
4.
Volume - prism, cone, pyramid Surface area - prism (triangular or rectangular base), pyramid (triangular or rectangular base) Impact of a dilation on the surface area and volume of a three-dimensional figure
Use formulas to find the volume and surface area of a sphere.
STANDARD 4.3 (PATTERNS AND ALGEBRA) ALL STUDENTS WILL REPRESENT AND ANALYZE RELATIONSHIPS AMONG VARIABLE QUANTITIES AND SOLVE PROBLEMS INVOLVING PATTERNS, FUNCTIONS, AND ALGEBRAIC CONCEPTS AND PROCESSES. Building upon knowledge and skills gained in preceding grades, by the end of Grade 8, students will: A.
Patterns
1.
Recognize, describe, extend, and create patterns involving whole numbers, rational numbers, and integers.
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Descriptions using tables, verbal and symbolic rules, graphs, simple equations or expressions Finite and infinite sequences Arithmetic sequences (i.e., sequences generated by repeated addition of a fixed number, positive or negative) Geometric sequences (i.e., sequences generated by repeated multiplication by a fixed positive ratio, greater than 1 or less than 1) Generating sequences by using calculators to repeatedly apply a formula
B.
Functions and Relationships
1.
Graph functions, and understand and describe their general behavior.
Equations involving two variables Rates of change (informal notion of slope)
2.
Recognize and describe the difference between linear and exponential growth, using tables, graphs, and equations.
C.
Modeling
1.
Analyze functional relationships to explain how a change in one quantity can result in a change in another, using pictures, graphs, charts, and equations.
2.
Use patterns, relations, symbolic algebra, and linear functions to model situations.
Using concrete materials (manipulatives), tables, graphs, verbal rules, algebraic expressions/equations/inequalities Growth situations, such as population growth and compound interest, using recursive (e.g., NOW-NEXT) formulas (cf. science standard 5.5 and social studies standard 6.6)
D.
Procedures
1.
Use graphing techniques on a number line.
2.
Solve simple linear equations informally, graphically, and using formal algebraic methods.
3.
Absolute value Arithmetic operations represented by vectors (arrows) (e.g., "-3 + 6" is "left 3, right 6")
Multi-step, integer coefficients only (although answers may not be integers) Using paper-and-pencil, calculators, graphing calculators, spreadsheets, and other technology
Solve simple linear inequalities.
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4.
Create, evaluate, and simplify algebraic expressions involving variables.
5.
Order of operations, including appropriate use of parentheses Distributive property Substitution of a number for a variable Translation of a verbal phrase or sentence into an algebraic expression, equation, or inequality, and vice versa
Understand and apply the properties of operations, numbers, equations, and inequalities.
Additive inverse Multiplicative inverse Addition and multiplication properties of equality Addition and multiplication properties of inequalities
STANDARD 4.4 (DATA ANALYSIS, PROBABILITY, AND DISCRETE MATHEMATICS) ALL STUDENTS WILL DEVELOP AN UNDERSTANDING OF THE CONCEPTS AND TECHNIQUES OF DATA ANALYSIS, PROBABILITY, AND DISCRETE MATHEMATICS, AND WILL USE THEM TO MODEL SITUATIONS, SOLVE PROBLEMS, AND ANALYZE AND DRAW APPROPRIATE INFERENCES FROM DATA. Building upon knowledge and skills gained in preceding grades, by the end of Grade 8, students will: A.
Data Analysis
1.
Select and use appropriate representations for sets of data, and measures of central tendency (mean, median, and mode).
Type of display most appropriate for given data Box-and-whisker plot, upper quartile, lower quartile Scatter plot Calculators and computer used to record and process information Finding the median and mean (weighted average) using frequency data. Effect of additional data on measures of central tendency
2.
Make inferences and formulate and evaluate arguments based on displays and analysis of data.
3.
Estimate lines of best fit and use them to interpolate within the range of the data.
4.
Use surveys and sampling techniques to generate data and draw conclusions about large groups.
B.
Probability
1.
Interpret probabilities as ratios, percents, and decimals.
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2.
Determine probabilities of compound events.
3.
Explore the probabilities of conditional events (e.g., if there are seven marbles in a bag, three red and four green, what is the probability that two marbles picked from the bag, without replacement, are both red).
4.
Model situations involving probability with simulations (using spinners, dice, calculators and computers) and theoretical models.
Frequency, relative frequency
5.
Estimate probabilities and make predictions based on experimental and theoretical probabilities.
6.
Play and analyze probability-based games, and discuss the concepts of fairness and expected value.
C.
Discrete Mathematics—Systematic Listing and Counting
1.
Apply the multiplication principle of counting.
Permutations: ordered situations with replacement (e.g., number of possible license plates) vs. ordered situations without replacement (e.g., number of possible slates of 3 class officers from a 23 student class) Factorial notation Concept of combinations (e.g., number of possible delegations of 3 out of 23 students)
2.
Explore counting problems involving Venn diagrams with three attributes (e.g., there are 15, 20, and 25 students respectively in the chess club, the debating team, and the engineering society; how many different students belong to the three clubs if there are 6 students in chess and debating, 7 students in chess and engineering, 8 students in debating and engineering, and 2 students in all three?).
3.
Apply techniques of systematic listing, counting, and reasoning in a variety of different contexts.
D.
Discrete Mathematics—Vertex-Edge Graphs and Algorithms
1.
Use vertex-edge graphs and algorithmic thinking to represent and find solutions to practical problems.
Finding the shortest network connecting specified sites Finding a minimal route that includes every street (e.g., for trash pick-up) Finding the shortest route on a map from one site to another Finding the shortest circuit on a map that makes a tour of specified sites Limitations of computers (e.g., the number of routes for a delivery truck visiting n sites is n!, so finding the shortest circuit by examining all circuits would overwhelm the capacity of any computer, now or in the future, even if n is less than 100)
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STANDARD 4.5 (MATHEMATICAL PROCESSES) ALL STUDENTS WILL USE MATHEMATICAL PROCESSES OF PROBLEM SOLVING, COMMUNICATION, CONNECTIONS, REASONING, REPRESENTATIONS, AND TECHNOLOGY TO SOLVE PROBLEMS AND COMMUNICATE MATHEMATICAL IDEAS.
Strands and Cumulative Progress Indicators At each grade level, with respect to content appropriate for that grade level, students will: A.
Problem Solving
1.
Learn mathematics through problem solving, inquiry, and discovery.
2.
Solve problems that arise in mathematics and in other contexts (cf. workplace readiness standard 8.3).
Open-ended problems Non-routine problems Problems with multiple solutions Problems that can be solved in several ways
3.
Select and apply a variety of appropriate problem-solving strategies (e.g., "try a simpler problem" or "make a diagram") to solve problems.
4.
Pose problems of various types and levels of difficulty.
5.
Monitor their progress and reflect on the process of their problem solving activity.
B.
Communication
1.
Use communication to organize and clarify their mathematical thinking.
Reading and writing Discussion, listening, and questioning
2.
Communicate their mathematical thinking coherently and clearly to peers, teachers, and others, both orally and in writing.
3.
Analyze and evaluate the mathematical thinking and strategies of others.
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4.
Use the language of mathematics to express mathematical ideas precisely.
C.
Connections
1.
Recognize recurring themes across mathematical domains (e.g., patterns in number, algebra, and geometry).
2.
Use connections among mathematical ideas to explain concepts (e.g., two linear equations have a unique solution because the lines they represent intersect at a single point).
3.
Recognize that mathematics is used in a variety of contexts outside of mathematics.
4.
Apply mathematics in practical situations and in other disciplines.
5.
Trace the development of mathematical concepts over time and across cultures (cf. world languages and social studies standards).
6.
Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.
D.
Reasoning
1.
Recognize that mathematical facts, procedures, and claims must be justified.
2.
Use reasoning to support their mathematical conclusions and problem solutions.
3.
Select and use various types of reasoning and methods of proof.
4.
Rely on reasoning, rather than answer keys, teachers, or peers, to check the correctness of their problem solutions.
5.
Make and investigate mathematical conjectures.
6.
Counterexamples as a means of disproving conjectures Verifying conjectures using informal reasoning or proofs.
Evaluate examples of mathematical reasoning and determine whether they are valid.
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E.
Representations
1.
Create and use representations to organize, record, and communicate mathematical ideas.
Concrete representations (e.g., base-ten blocks or algebra tiles) Pictorial representations (e.g., diagrams, charts, or tables) Symbolic representations (e.g., a formula) Graphical representations (e.g., a line graph)
2.
Select, apply, and translate among mathematical representations to solve problems.
3.
Use representations to model and interpret physical, social, and mathematical phenomena.
F.
Technology
1.
Use technology to gather, analyze, and communicate mathematical information.
2.
Use computer spreadsheets, software, and graphing utilities to organize and display quantitative information.
3.
Use graphing calculators and computer software to investigate properties of functions and their graphs.
4.
Use calculators as problem-solving tools (e.g., to explore patterns, to validate solutions).
5.
Use computer software to make and verify conjectures about geometric objects.
6.
Use computer-based laboratory technology for mathematical applications in the sciences.
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Appendix B.1 Assessment Based on Standards
Assessment is an integral component of the educational process. When properly designed and administered, assessments provide important information to help guide and inform instruction. When assessments are embedded in the curriculum they serve to determine the students’ initial understandings, to monitor student progress, and to collect information and provide feedback. Assessment is the means by which we determine whether or not students are making progress toward achieving the New Jersey Core Curriculum Content Standards in Mathematics. These performance standards define students’ academic responsibilities and, by implication, the teaching responsibilities of the school. How do we determine whether students have lived up to their academic responsibilities? We assess their work (is it “good enough?”) by comparison with the standards. The primary purpose of mathematics assessment in the Plainfield Public Schools is to improve student learning. The District uses a variety of evaluation and assessment tools (both formative and summative) to determine the effectiveness of the curriculum and instruction, to provide clear expectations for student learning, and to assist in educational planning. Assessment takes place in a variety of formats and situations. In addition to the New Jersey Ask, GEPA and HSPA, Standards Proficiency Assessment (SPA), and New Standards Reference Exam, (summative evaluation tools), teachers are utilizing end of unit assessments, portfolios, district created Targeted Assessment Process (TAP) tests, teacher observations, and anecdotal records (formative evaluation tools) to help them make informed instructional decisions. Both types of assessment, formative and summative, are essential to effective instruction and enhanced student learning.
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Appendix B.2 NEW JERSEY MATHEMATICS HOLISTIC SCORING RUBRIC
3-point response The response shows complete understanding of the problem’s mathematical concepts. The student executes procedures completely and gives relevant responses to all parts of the task. The response contains a few minor errors, if any. The response contains a clear, effective explanation detailing how the problem was solved so that the reader does not need to infer how and why decisions were made.
2-point response The response shows nearly complete understanding of the problem’s mathematical concepts. The student executes nearly all procedures and gives relevant responses to most parts of the task. The response may have minor errors. The explanation detailing how the problem was solved may not be clear causing the reader to make some inferences.
1-point response The response shows limited understanding of the problem’s mathematical concepts. The response and procedures may be incomplete and/or may contain major errors. An incomplete explanation of how the problem was solved contributes to questions as to how and why decisions were made.
0-point response The response shows limited to no understanding of the problem’s mathematical concepts. The procedures, if any, contain major errors. There may be no explanation of the solution or the reader may not be able to understand the explanation. The reader may not be able to understand how and why decisions were made.
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Appendix B.3 The Kid-Friendly rubric below is used as a guide to help students to understand the New Jersey Mathematics Holistic Scoring Rubric. Kid-Friendly Rubric for Mathematics Assignments Score
Clearly Stated/Correct Response to the problem Correct response with correct label
3
What mathematics did you use in your solution?
Your answer shows… • If your first idea didn’t work, you tried something else. • You really understand the math in the problem.
How did you represent the mathematics?
There’s a helpful… • Set of calculations • Chart, table, or graph • Drawing or diagram • Written description …that shows the math
How well did you explain your solution?
A person who reads your work can see that… • You thought through the whole problem. • All your ideas are clear. • It is well-organized. … and your work is readable.
Your answer is right
2
Correct response with incorrect label
Your answer shows…
There is a helpful…
•
• • • •
•
1
0
If your first idea didn’t work, you tried something else. You understand the most important math in the problem.
Set of calculations Chart, table, or graph Drawing or diagram Written description
Your answer may be wrong
That shows the math, but it might be incomplete or have a mistake in it.
Incorrect response with correct label
Your answer shows…
There is a helpful…
•
You understand some of the math in the problem. You left out some important math, or you didn’t show much of the math you did use.
• • • •
Incorrect response with no label or incorrect response with no label written
Your answer doesn’t show…
There is…
•
• A blank space • Doodling or scribbling • A set of unrelated numbers …that does not show effort to solve this problem.
What mathematics you understand.
You did not use a mathematical approach.
Set of calculations Chart, table, or graph Drawing or diagram Written description but it can’t help solve this problem.
A person who reads your work can see that… • Some of your ideas are clear. • You organized some of your work … and your work is readable
A person who reads your work can see that… • You wrote down some of your ideas. • You tried to organize your ideas. … and your work is readable
A person who reads your work can see that… • There is little or no effort to solve the problem. … and the work may not be readable
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Appendix C Accountable Talk Talk is fundamental to learning. But not all talk sustains learning. For classroom talk to promote learning it must be accountable- accountable to the learning community, to mathematics, and to rigorous thinking. Talk should be relevant and make connections within and between concepts. Accountable talk relates to the current conversation in one of the following ways:
States difference or similarity to something already said Supports or contradicts something already said Extends a previously stated idea Proves a previously stated idea
Accountable talk means using evidence to support what is said by:
Referring to a definition Using data collected from an investigation Using data collected by someone else Giving an analogy Stating rules that are observed Looking at approaches that did not work and saying why they did not work Looking at different approaches and describing them
Accountable talk means taking action when you are confused by:
Keeping common sense engaged Restating the problem or idea in your own words Asking someone if your statement of the idea or problem makes sense Looking up the idea in resource books Listening to how someone else makes sense of the problem Asking your teacher at an appropriate time to listen to your confusion Seeking help from the teacher, a peer, a tutor, etc. outside of class.
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Appendix D QUESTIONING: WHAT RESEARCH TELLS US Twenty-five centuries after Socrates, educators still employ classroom questioning to develop embryonic ideas and to cement understanding. A well-timed question can prompt reflective thinking in ways a lecture never could. Classroom questioning is one of the most commonly-used teaching strategies, occupying thirty-five to fifty-five percent of teachers’ instructional time (www.nwrel.org). Questions are used to motivate student interest, review previous learning, assess understanding, stimulate critical thinking, and to prompt reflection. The “right” question can instigate critical thinking, just as surely as a “wrong” question can shut down thinking (www.exploratorium.edu). Pacing Research confirms that questions interspersed in a lesson promote fact retention more effectively than a segregated questioning episode at the end. However, question frequency is not correlated to understanding of complex material (www.nwrel.org). On the other hand, wait time, the period between posing a question and prompting a response, is strongly related to student engagement and critical thinking. After asking a question, teachers generally allow one second or less for students to begin a response, before they repeat or rephrase the question. When a student answers, the teacher reacts to the response or asks another question within one second. The fast pace of typical classroom questioning sessions leaves little opportunity for students to formulate complex responses or to reflect on their understanding. Cognitive Demand Questions from the bottom of Bloom’s Cognitive Taxonomy ask for recall of previously taught information or recitation of facts. There is little cognitive demand associated with such questions. High cognitive demand questions require the student to manipulate information to create new insights, and are frequently associated with Bloom’s analysis, synthesis and evaluation levels. (Schorr 2004) “State the formula for finding the area of a circle” is an example of a low cognitive demand question. It requires only recall of a previously memorized fact. On the other hand, an open-ended question such as, “How do you think the area of a circle with diameter n relates to the area of a square with side length n?” provokes analytic and evaluative thinking. In a typical classroom, the ratio of low cognitive demand to high cognitive demand classroom questions is about three to one. (www.nwrel.org) However, the level of cognitive demand is not inherently “good” or “bad.” In fact, research is inconclusive regarding the effects on student achievement of consistently asking questions at a particular cognitive level. Experienced teachers vary the cognitive demand and the format of classroom questions to maintain student interest and allow multiple entry points into the lesson. What Can Teachers Do to Improve Their Questioning? Teachers’ questioning behavior can be viewed in terms of its effect on students. When teachers use skilled questioning, they engage their students in an exploration of content. Carefully framed questions enable students to reflect on their understanding and consider new possibilities. The questions rarely require a simple yes/no response and may have many possible correct answers. Experienced teachers allow students time to think before they must respond to a question and encourage all students to participate. Teachers often probe a student’s answer, seeking clarification or elaboration through such questions as, ‘Could you give an example of that?’ or ‘Would you explain further what you mean?’ (Danielson 1996). Because the purpose of classroom questioning is to reveal students’ thinking and develop their critical analysis skills, the teacher’s lesson plans should include prepared questions carefully crafted to bring out the “big ideas” of the lesson. These focusing questions complement, rather than replace, the questions generated by the teacher as the lesson progresses.
The teacher should allow ample wait time for students to process questions. For lower cognitive demand questions, optimal wait time is three seconds. For high cognitive demand questions, there is no apparent upper limit to wait time (www.nwrel.org). After a student responds, the teacher should wait three to five seconds for the initial responder or other
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students to elaborate or offer additional comments. These practices result in five- to seven-fold increases in the length of student responses, and reduction in the rate of nonresponse from about 30% of questions to less than 5% (www.eiu.edu). Finally, the teacher must value the insights and contributions of all students. This means classroom questions can not be the “(. . .) private preserve of a few – the bright, the male, the English-speaking.” (www.cat.ilstu.edu). Van de Walle (2001) refers to the student’s disposition that he/she can understand mathematics as a function of the teacher’s practice of posing rigorous questions, then withdrawing to let the student solve in his own way and in his own time. Often, the teacher is torn between encouraging student autonomy and protecting students from frustration. However, the classroom can not become a place of intellectual inquiry unless all parties are encouraged and expected to play a role in the examination, analysis and communication of ideas. Conclusion Questioning is a powerful and versatile tool for teachers to motivate, direct and assess learning. Teachers use questions to launch lessons, scaffold student learning, focus or refocus student attention, encourage student self-assessment, extend the scope of a problem and to suggest further questions, as well as to assess students’ understanding. Research findings can help us improve our questioning technique, which profoundly affects student achievement. For more information on classroom questioning: Brualdi, Amy C. “Classroom Questions.” Practical Assessment, Research & Evaluation 6(6). 1998. November 15, 2004. Cotton, Kathleen. “Classroom Questioning.” School Improvement Research Series, NWREL, Danielson, Charlotte. Enhancing Professional Practice: A Framework for Teaching. Alexandria, VA: Association for Supervision and Curriculum Development, 1996. Metts, Sandra. “Suggestions for Classroom Discussion.” Center for the Advancement of Teaching. 2004. November 15, 2004. Rowe, Mary Budd. “Payoff from Pausing.” Eastern Illinois University Science Education. June 1995. November 5, 2004. Van De Walle, John A. Elementary and Middle School Mathematics: Teaching Developmentally, 4th ed. New York: Longman, 2001. Wolf, Dennis Palmer. “The Art of Questioning.” Institute for Inquiry. Winter 1987. November 5, 2004.
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Appendix D.2 Bloom’s Cognitive Taxonomy Classroom questions can be classified using Bloom’s Cognitive Taxonomy. The levels range from Knowledge (simplest) to Evaluation (most difficult). Raising the level of questioning encourages students’ critical thinking.
Cognitive Domain
Description
I
Knowledge
Recall of facts.
II
Comprehension
Understanding facts.
III
Application
IV
Analysis
Selection and use of a rule. Breaking a whole into components.
Example Verbs “The learner will be able to . . .” Cite, Define, Identify, Label, List, Match, Name, Quote, Recite, Reproduce Convert, Describe, Explain, Give Examples, Illustrate, Summarize, Translate Apply, Compute, Demonstrate, Employ, Predict, Prepare, Show, Solve Analyze, Diagram, Differentiate, Dissect, Distinguish, Examine, Outline
Sample Question What is the formula for finding area of a circle? In the formula A = π r 2 , what does the variable r represent? How many square inches of pizza are in a 16 inch (diameter) pie? Maria cut a circle into eight sectors and arranged them like this:
How did she determine that the area of the “parallelogram” is (πr ⋅ r) ? V
Synthesis
VI
Evaluation
Assembling parts into a logical whole. Judging the merit of an inference.
Create, Design, Develop, Extend, Invent, Plan, Project, Propose, Theorize, Write Appraise, Assess, Compare, Conclude, Contrast, Critique, Evaluate, Judge, Weigh
Given: Area of circle A = n Area of circle B = n Prove: Circle A ≅ circle B Would you prefer a cookie with a 3 inch radius or three 1 inch cookies? Why?
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Appendix D.3 Questions to Improve Accountable Talk in the Math Classroom 1. What are you doing? 2. Can you describe it to me? 3. Why are you doing that? 4. How does it fit your solution? 5. How does this help you understand? 6. Is this always true? 7. How is it defined? 8. Can you give me examples? 9. When is it true? Is it ever not true? 10. How will we know if we thought of all the cases? 11. How does this connect with other math ideas? 12. What is the same and what is different about two approaches that work?
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Appendix D.4 Questions to Encourage Critical Thinking •
What does that tell us?
•
Why do I need to do that?
•
What is the relationship? How do they work?
•
Can you help him/her out? Expand what he/she has said so far.
•
Do you agree with ________’s thinking? Why? Or Why not?
•
Does that make sense? Why?
•
Does that seem reasonable?
•
How did you figure that out?
•
Can you prove your answer is correct?
•
Can anyone restate that idea in your own words?
•
Can you say more about that idea?
•
Can you tell more about how you are thinking about this?
•
What do you know about the situation right now?
•
What do you think it means?
•
Does this remind you of other things you’ve done before?
•
Where could you start? What are your ideas?
•
Can you see a way to relate _________’s way of thinking about the problem to ___________’s way of thinking?
•
How is your strategy mathematically alike or different from __________’s strategy?
•
Are they using different strategies, or are their strategies similar in ways?
•
Why does (strategy “x”) work? Will it always work?
•
Now that you have more information, what do you think?
•
Why does strategy ”y” not work?
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*When you ask a question and students answer it; pause before you respond. This opens opportunities for other students to consider what was said. *Question both correct and incorrect answers. This encourages students to prove the confidence they have in their answers.
Adapted from work of Yvonne Grant
Here are some fundamental questions you can use to explore further student responses to any tasks. •
Tell me how you did that.
•
Why is that true?
•
How did you reach that conclusion?
•
Does that always work? Why or why not?
•
How could you prove that?
•
What assumptions are you making?
•
What would happen if…? What if no…?
•
Do you see a pattern? What is it? Explain.
•
How did you think about the problem?
•
How is your method like hers/his? How is it different?
•
How does this relate to…?
•
What ideas that we have learned before were useful in solving this problem?
When students are working together, you can ask questions like: •
What do you think about what ____________ has said?
•
Do you agree? Disagree? Why or why not?
•
Does anyone have the same answer but a different way to explain it?
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Appendix E Professional Resources for K – 8 Teachers of Mathematics Please see your math coordinator for information about borrowing these resources. Print and Video Materials Barnett, Carne, et al. Decimals, Ratios, and Percents: Hard to Teach and Hard to Learn., New Hampshire: Heinemann, 1994. Boaler, Jo and Humphreys, Cathy. Connecting Mathematical Ideas: Middle School Video Cases to Support Teaching and Learning. New Hampshire: Heinemann, 2005. Braddon, Hall, Taylor. Math Through Children’s Literature. Colorado: Teacher Ideas Press, 1993. Bresser, Rusty and Holtzman, Caren. Developing Number Sense Grades 3 – 6. California: Math Solutions Publications, 1999. Burns, Marilyn and Tank, Bonnie. A Collection of Math Lessons from Grades 1 – 3. California: Math Solutions Publications, 1990. Burns, Marilyn. A Collection of Math Lessons from Grades 3 - 6. California: Math Solutions Publications, 1999. Burns, Marilyn. About Teaching Mathematics: A K – 8 Resource. California: Math Solutions Publications, 2000. Burns, Marilyn. Math: Facing an American Phobia. California: Math Solutions Publications, 1998. Burns, Marilyn. What Are You Teaching My Child? (Videotape) Scholastic Inc., 1994 (also available in Spanish) Carpenter, Thomas, et al. Children’s Mathematics: Cognitively Guided Instruction. New Hampshire: Heinemann, 1999. Copley, Juanita. The Young Child and Mathematics. Washington, D.C.: National Association for the Education of Young Children, 2000. Corwin, Rebecca B. Talking Mathematics: Supporting Children’s Voices. New Hampshire: Heinemann, 1996. Dacey, Linda Schulman and Eston, Rebeka. Growing Mathematical Ideas in Kindergarten. California; Math Solutions Publications, 1999. Danielson, Charlotte. Enhancing Professional Practice: A Framework for Teaching. Virginia: Association for Supervision and Curriculum Development, 1996. 59
Danielson, Charlotte. Enhancing Student Achievement: A Framework for School Improvement. Virginia: Association for Supervision and Curriculum Development, 2002. Fosnot, Catherine and Dolk, Maarten. Young Mathematicians at Work: Constructing Fractions, Decimals, and Percents. New Hampshire: Heinemann, 2002. Fosnot, Catherine and Dolk, Maarten. Young Mathematicians at Work: Constructing Multiplication and Division. New Hampshire: Heinemann, 2001. Fosnot, Catherine and Dolk, Maarten. Young Mathematicians at Work: Constructing Number Sense, Addition, and Subtraction. New Hampshire: Heinemann, 2001. Ginsburg, Herbert P. et al. The Teacher’s Guide to Flexible Interviewing in the Classroom: Learning What Children Know About Math. Massachusetts: Allyn and Bacon, 1998. Glickman, Carl. Leadership for Learning: How to Help Teachers Succeed. Virginia; Association for Supervision and Curriculum Development, 2002. Goldberg, Mark. Lessons From Exceptional School Leaders. Virginia: Association for Supervision and Curriculum Development, 2001. Hiebert, James, et al. Making Sense: Teaching and Learning Mathematics with Understanding. New Hampshire: Heinemann, 1997. Kamii, Constance. Number in Preschool and Kindergarten. Washington, D.C.: National Association for the Education of Young Children, 1982. Lappan, Glena, et al. Getting to Know Connected Mathematics: An Implementation Guide. Illinois: Prentice Hall, 2002 (can be downloaded at www.phschool.com) Loucks-Horsley et al. Designing Professional Development for Teachers of Science and Mathematics. California; Corwin Press, Inc., 1998. Marzano, Robert J. What Works in Schools: Translating Research into Action. Virginia: Association for Supervision and Curriculum Development, 2003. Mokros, Jan, Russell, Susan Jo, Economopoulos, Karen. Beyond Arithmetic; changing Mathematics in the Elementary Classroom. New York: Dale Seymour Publications, 1995. Mokros, Jan. Beyond Facts and Flashcards: Exploring Math With Your Kids. New Hampshire: Heinemann, 1996. 60
Murray, Megan. Schools and Families: Creating a Math Partnership. Massachusetts: TERC, 2002. National Commission on Mathematics and Science Teaching for the 21st Century. “Before It’s Too Late: A Report to the Nation.” (with videotape highlights). Washington D.C., 2000. (can be downloaded at www.edgov/americacounts/). National Council of Teachers of Mathematics. Principles and Standards for School Mathematics. Virginia: NCTM, 2000. National Research Council. Adding It Up. Washington, D.C., National Academy APress, 2002. National Research Council. Everybody Counts: A Report to the Nation on the Future of Mathematics Education. Washington, D.C., National Academy Press, 1989 National Research Council. Helping Children Learn Mathematics. Washington, D.C., National Academy Press, 2002. Russell, Susan Jo, et al. Implementing the Investigations in Number, Data, and Space Curriculum. New Jersey: Dale Seymour Publications, 1998. Serra, Michael, et al. Discovering Mathematics: A Guide for Teachers (Discovering Algebra, Discovering Geometry, Discovering Advanced Algebra). California: Key Curriculum Press, 2004. Skinner, Penny. It All Adds Up: Engaging 8 – 12 Year Olds in Math Investigations. California: Math Solutions Publications, 1999. Stigler, James W. and Hiebert, James. The Teaching Gap. New York: The Free Press, 1999. Stein, Mary Kay, et al. Implementing Standards-Based Mathematics Instruction. New York: Teachers College Press, 2000. Stronge, James H. Qualities of Effective Teachers. Virginia: Association for Supervision and Curriculum Development, 2002. Sullivan, Peter and Liliburn, Pat. Good Questions for Math Teaching: Why Ask Them and What to Ask. California: Math Solutions Publications, 2002. West Lucy and Staub, Fritz C. Content-Focused Coaching: Transforming Mathematics Lessons. New Hampshire: Heinemann, 2003.
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Periodicals and Journals Educational Leadership (ASCD) Journal for Research in Mathematics Education (NCTM) Math Solutions Online (see www.mathsolutions.com) Mathematics Teaching in the Middle Schools (NCTM) Teaching Children Mathematics (NCTM)
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