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UNIFYING CONCEPTS AND PROCESSES Standard 1: Students will understand and use Mathematics in Problem Solving. Benchmarks, Grades 9-12 STANDARD CORRELATION 1.1 Analyze problem-solving approaches to In the first year, IMP builds on students’ intuitive notions of and methods for investigate and understand mathematical problem solving o develop over the course of the curriculum an in-depth content. understanding of mathematical concepts and techniques and of the ways to apply them. Students routinely experiment with examples, look for and articulate patterns, make, test and prove conjectures, and make connections among mathematical ideas. Students are encouraged to write about these methods on a regular basis, including in their write-ups of Problems of the Week (POW’s). Here they reflect on the techniques that they applied to solve problems so their methods can be clearly understood by others. Additionally, the curriculum requires student to present their thinking in both a large class and small group environment. In these settings, students are exposed to different approaches to solving problems and learn to critically evaluate their efficiency, effectiveness, and accuracy. Various techniques for solving problems are suggested or presented by the teacher throughout the four years of the curriculum. Patterns HW2-Who‟s Who? As the Cube Turns HW17-How Did We Get Here? New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /1 STANDARD CORRELATION 1.2 Formulate problems from global The majority of the units in the IMP curriculum center on a unit problem. Students mathematical situations. read about a scenario, identify the problem to be solved, generate appropriate assumptions, and develop mathematics and methods to solve the problem. These problems are generated from a variety of settings, including made-up games, literature, science, world population, and a circus. The unit problems of The Game of Pig, Do Bees Build It Best?, Meadows or Malls?, and High Dive are good examples from each of the four years. Additional, smaller examples: The Game of Pig POW7-Making a Game Is There Really a Difference? POW6-Is There Really a Difference? Cookies • CW-Producing Programming Problems 1.3 Select the best strategies to solve a wide A basic strength of the IMP curriculum is the exposure students have to the many variety of problems in diverse contexts. possible ways to solve a wide variety of problems. The IMP units with their central problems and the smaller problems within each of the units also provide students with a diversity of contexts and challenges. Students are encouraged to write-up their methods for solving Problems of the Week (POW’s). Additionally, students make oral presentations to the whole class and in small group settings. In these settings, students can learn how others thought about solving problems, and recognize efficient or effective techniques. Various techniques for solving problems are suggested or presented by the teacher throughout the four years of the curriculum. These opportunities in the classroom allow students to develop a broad range of strategies and techniques for solving problems. Students must select a strategy to solve problems in the various contexts of the entire curriculum. Through this ongoing process of selection, reselection, and reflection, students find strategies that work best for them as well as best for various types of problems. New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /2 1.4 Verify and interpret results with respect to Again, this expectation is ongoing throughout the curriculum. Because the the original problem situation. curriculum is problem driven, students must determine the reasonableness of their solutions in the context of the original situation. No answer key at the back of the book exists. STANDARD CORRELATION 1.5 Analyze solutions and strategies for use in The IMP curriculum centers on modeling real phenomena with mathematics. mathematical modeling. Students analyze solutions and strategies throughout IMP. Three examples are given below. The Pit and the Pendulum This unit problem asks students to predict how long twelve swings of a thirty-foot pendulum will take. Students experiment with and explore measurement variation for the sake of developing an appropriate scale model. From tests of various scale models, students predict the period of the thirty-foot pendulum from trends in the periods of the smaller pendulums. Meadows or Malls? In this unit, students model a problem about land development with a system of linear equations, extending concepts of linear programming to problems with several variables. They develop and apply matrix algebra to solve the central unit problem, and graphing linear equations in three variables and recognizing that these graphs are planes in 3-space. High Dive In this unit, students determine when a circus diver must be released from a rotating Ferris Wheel in order to land in a cart of water wheeling past along a track. Students carefully reconsider the potential implications of the assumptions they made in modeling the original problem toward the end of the unit. • HW30-The Danger of Simplification New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /3 STANDARD CORRELATION 1.6 Use manipulatives, calculators, A tenet of the IMP classroom is that just as these tools are part of the resources computers, and other tools as appropriate mathematicians bring to the work they do, they are to be available to students. . in order to strengthen mathematical They are an integral part of the curriculum. Here is an example from one unit from thinking, understanding, and power to each of the four years. build upon foundational concepts. The Pit and the Pendulum In this unit, students must predict from a set of data. Since it is unlikely they have studied a wide variety of functions at this stage in their mathematics education, students do an activity called Graphing Free-for-All in which they explore a wide variety of graphs. This adds to their repertoire of functions from which to select for their model. Do Bees Build It Best? Students develop the Pythagorean Theorem in this unit. In the activity cited here, students compare areas as a step toward developing the Pythagorean theorem: CW-Tri-Square Rug Games Meadows or Malls? Students are able to access complex linear programming problems (such as the six variable unit problem) because they learn to use the calculator to make possible the complex algebraic processes that would otherwise be necessary. As the Cube Turns Students use the calculator in a game of figuring out one trig ratio if given another. In this investigation, students derive trigonometric identities. HW12-Double Trouble HW13-A Broken Button Additionally, in this unit, students pull together unit topics in an end of the unit, self-created animation project. POW5-An Animated POW New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /4 UNIFYING CONCEPTS AND PROCESSES Standard 2: Students will understand and use Mathematics in Communications. Benchmarks, Grades 9-12 STANDARD CORRELATION 2.1 Analyze personal mathematical thinking Each unit concludes with students assembling a portfolio. These portfolios always for validity and applicability to specific include a reflective piece on the mathematics studied during the unit. Many of problems. these reflections ask students to discuss what mathematics they learned and how it can be applied. A few examples highlight the role the portfolios play: Patterns HW2-Who‟s Who The Pit and the Pendulum HW27-Beginning Portfolios Cookies HW26-Continued Portfolio Selection Pennant Fever HW16-Which Is Which? High Dive HW8-Acceleration Variations and Sine Summary HW31-A Trigonometric Reflection As the Cube Turns HW17-How Did We Get Here? The Pollster’s Dilemma HW18-Confidence and Clarabell New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /5 STANDARD CORRELATION 2.2 Use drawings, discussions, reading, Students do this throughout all four years of the curriculum. Each student will writing, and listening to access, learn, and present to their groups and to the entire class regularly, whether it be a response to communicate mathematical ideas. a class activity or a prepared presentation of a larger, Problem of the Week. These presentations prompt questioning and discussion within the groups or the classroom. Additionally, students complete formal write-ups of these Problems of the Week eight to ten times per semester. Within these write-ups, students must communicate effectively using the language of mathematicians. A few examples are: Overland Trail • Days 15-18: Making Predictions With Graphs Cookies • Days 4-7: Picturing Cookies 2.3 Select the most economical and illustrative Students do this throughout the four years of the curriculum. Each student will method to communicate mathematical present to their groups and to the entire class regularly, whether it be a response to concepts, thoughts, and problem solutions a class activity or a prepared presentation of a larger Problem of the Week. including mathematical notation, charts, Additionally, students complete formal write-ups of these Problems of the Week slides, graphs, maps, drawings, pictures, eight to ten times per semester. Within these write-ups, students must sound recordings, video, e-mail, and communicate effectively using the language of mathematicians—notation, charts, others. graphs, etc. For example, in Orchard Hideout, students use circles and dots on graph paper to model different size orchards, creating diagrams to clarify their ideas. In Meadows or Malls?, they build physical models or use the corner of a room to physically represent the x, y, z planes. In As the Cube Turns, students approach the challenge of representing a three-dimensional object on a two-dimensional screen through the use of both physical materials and a story about two spiders and a thread of web that connects them. New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /6 STANDARD CORRELATION 2.4 Analyze mathematical ideas through the Students explore and make sense of mathematics and mathematical ideas with use of learning tools such as these tools daily, throughout the whole curriculum. Numerous and varied activities manipulatives, calculators, and in the curriculum require using manipulatives and graphing calculators. Graphing computers. calculators help students address a complex linear programming problem in Meadors or Malls? Students use the graphing calculator throughout As the Cube Turns. Some other examples are: Patterns • Days 13-14 The Chef‟s Hot and Cold Cubes Shadows • Classwork: Shadow Data Gathering Solve ItI • HW 7: You‟re the Jester 2.5 Describe the economy, power, and Again, this benchmark is supported daily throughout the curriculum. Only one elegance of mathematical notation and its example from each year is used to illustrate: role in the development of mathematical The Overland Trail ideas. CW-“The Overland Trail” Portfolio Solve It! HW24-A Distributive Summary Meadow or Malls HW37-Beginning Portfolios – Part II Know How CW-Knowing What You Know • HW4-What to Know When New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /7 STANDARD CORRELATION 2.6 Reading presentations of mathematics Students will critically analyze one another’s presentations and problem write-ups. with understanding. Below are a few specific examples: Patterns Day5: Students read each other’s problem statements of the POW1 write-up for clarity. Day7: Students read an example of a POW-style write-up. They note the style of the write-up and discuss reactions. Cookies Day18: Students are to read each others’ papers for the purpose of noting weakness in clarity. Fireworks Day2: Students read Quadratics and other Polynomials for understanding. Orchard Hideout Day1: Students read A Geometric Summary for understanding. Day19: Students read The Standard Equation of the Circle for understanding. Meadows or Malls? Day 10: Students read The Three Variable Coordinate System for understanding. Day27: Students read Matrix Basics for understanding. Day31: Students read Matrices and Linear Systems for understanding. High Dive Day6-Extending the Sine As the Cube Turns HW24-Flipping Points New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /8 STANDARD CORRELATION 2.6 continued Know How Students complete a project in which they learn about a new mathematical concept by relying on reading mathematical text. CW-Radian Measure HW2-The Law of Sines HW3-The Ellipse HW5-A Fractional Situation HW6-Complex Numbers CW-Know How Project The World of Functions Day12-Students read Approaching Infinity The Pollster’s Dilemma Day6-Students read The Central Limit Theorem New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /9 UNIFYING CONCEPTS AND PROCESSES Standard 3: Students will understand and use Mathematics in Reasoning. Benchmarks, Grades 9-12 STANDARD CORRELATION 3.1 Develop and test conjectures and This benchmark is one of the pedagogical foundations of the curriculum. Students mathematical arguments. develop and test conjectures and mathematical arguments daily throughout the four years of the curriculum. 3.2 Evaluate reasoning strategies to select the Students evaluate reasoning strategies in the context of solving problems daily. most appropriate reasoning method to Most classroom presentations of methods to solve problems include several solve a given problem. presentations of different techniques to solve the same problem. Class discussions clarify the variety of techniques and students are provided the opportunities to determine what works best for them. 3.3 Judge the validity of mathematical Students must develop and present mathematical arguments daily. They listen to, arguments. evaluate, and judge the validity of these mathematical arguments. The classroom structure relies on arguments being presented, discussed, refined, reworked, and/or discarded. 3.4 Construct and evaluate logical arguments. This benchmark is another one of the pedagogical foundations of the curriculum. Students must develop and present logical arguments daily. Students listen to, evaluate, and judge the validity of these logical arguments. The classroom structure relies on arguments being presented, discussed, refined, reworked, and/or discarded. Students learn to always make their assumptions explicit. Here are a few examples in which logic is explicitly considered in the curriculum: Patterns CW-A Proof Gone Bad All About Alice POW14-More from Lewis Carroll POW15-A Logical Connection Orchard Hideout Day3-The previous activities and homeworks generate a classroom discussion of “If…then…” statements and the notion of converse. Know How HW3-Now Let‟s Prove It! New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /10 STANDARD CORRELATION 3.5 Formulate counter examples to Again, this benchmark is supported in the pedagogical foundations of the understand mathematical reasoning. curriculum. Students must develop and present arguments daily. Students listen to, evaluate, and judge the validity of these arguments. The classroom structure relies on arguments being presented, discussed, refined, reworked, and/or discarded. The use of counterexamples plays an ongoing role in developing generalizations. Shadows CW-Is There a Counterexample? HW9-Triangular Counterexamples New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /11 UNIFYING CONCEPTS AND PROCESSES Standard 4: Students will understand and use Mathematical Connections. Benchmarks, Grades 9-12 STANDARD CORRELATION 4.1 Relate mathematical procedures in one Patterns representation to procedures in equivalent HW10-Pulling Out Rules representations. Solve It! HW16-Views of the Distributive Property HW30-A Solving Sampler Meadows or Malls? Students generate both graphical and symbolic methods/algorithms for solving linear programming problems in two variables. These methods are extended to three variable systems as well. The graphical model is relied on to generalize a symbolic method for any size linear programming problem. CW-What Wood Would Woody Want? HW15-The “More Cookies” Region and Strategy HW24-Constraints Without a Context CW-Eastside Westside Story HW35-Getting Ready for “Meadows or Malls?” CW-“Meadows or Malls” Revisited As the Cube Turns HW15-Comin‟ Round Again (and Again…) The World of Functions Students formally define arithmetic operations on symbolic forms of functions. Students relate these operations to the tabular and graphical forms of the functions. CW-The Arithmetic of Functions • HW18-The Arithmetic of Graphs New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /12 STANDARD CORRELATION 4.2 Compare and contrast equivalent This notion of equivalent representations of the same concept is the foundation for representations of the same concept. the development of most mathematical concepts throughout the four years of the curriculum. The Overland Trail CW-Out Numbered HW13-Situations, Graphs, Tables, and Rules HW14-Rules, Tables, and Graphs CW-Previous Travelers Solve It! CW-A Lot of Changing Sides HW13-Why Are They Equivalent? HW14-One Each Way CW-Distributing the Area HW15-The Distributive Property and Mystery Lots HW16-Views of the Distributive Property CW-Taking Some Out, Part I HW18-Subtracting Some Sums CW-Taking Some Out, Part II HW19-Randy, Sandy, and Dandy Return HW24-A Distributive Summary Pennant Fever HW6-Diagrams, Baseball, and Losing „Em All The World of Functions CW-Functions in Verse HW29-Beginning Portfolio Selection The Pollster’s Dilemma • HW11-A Distribution Example New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /13 STANDARD CORRELATION 4.3 Assess the relationship among Mathematical topics are inherently interrelated. This notion is evident to IMP mathematical topics. students as they develop mathematical understanding in the context of real problems. Because of this structure in which students learn and experience mathematics, students are always working in and around several of the various standards. Examples from each of the four years: The Pit and the Pendulum HW27-Beginning Portfolios Cookies • HW27-Cookies Portfolio. Students consider how key mathematical ideas were developed and how they worked together to solve the unit problem. Orchard Hideout HW24-“Orchard Hideout” Portfolio As the Cube Turns HW17-How Did We Get Here? The World of Functions HW24-Fish, Ladders, and Bacteria The Pollster’s Dilemma HW2-Pennant Fever Reflection • CW-Roberto and the Coin Revisited 4.4 Incorporate the use of technology into the Again, this benchmark is supported in the pedagogical foundations of the application of mathematical reasoning curriculum. These modern tools are always to be available to students just as they and problem solving to other disciplines. are available to mathematicians in their work. New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /14 STANDARD CORRELATION 4.5 Evaluate mathematical solutions for Each unit problem presents an opportunity for students to connect analysis and problems in daily life and in the greater results to daily life and the greater society. Students are encouraged to reflect on society. these connections in their portfolio and in end of the unit discussions. They are challenged to make decisions about real-life problem situations; for example. when they assume the role of consultant or elected representative. It requires students to go beyond simply finding numerical answers. The unit problems of the following units are particularly worth noting: The Pit and the Pendulum, Is There Really a Difference?, Cookies, Meadows or Malls?, Small World, Isn‟t It?, Pennant Fever, and Pollster‟s Dilemma. Additionally, certain activities stand out: Is There Really a Difference? HW12-Drug Dragnet: Fair or Foul? As the Cube Turns POW4-A Wider Windshield Wiper, Please The Pollster’s Dilemma CW-Let‟s Vote on It! • HW17-A Teaching Dilemma New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /15 NUMBER AND OPERATIONS CONCEPTS Standard 5: Students will understand and use numbers and number relationships. Benchmarks, Grades 9-12 STANDARD CORRELATION 5.1 Extend number-sense skills to include Do Bees Build it Best? irrational numbers. CW-Any Two Sides Work HW13-Make the Lines Count HW18-Simply Square Roots All About Alice HW9-It‟s in the Graph HW10-Stranger Pieces of Cake Fireworks HW5-Factoring Begun. In class discussion, students consider the significance of irrational roots and what “not factorable” means. Orchard Hideout HW1-Geometry and a Mini-Orchard CW-Polygoning the Circle. Ensuing class discussion makes sense of the transcendental number . Small World, Isn’t It? CW-Find That Base! HW28-The Limit of Their Generosity HW29-California‟s Population with e‟s 5.2 Apply number-sense skills within the real It is difficult to pinpoint specific examples of this benchmark because students number system. develop and are asked to draw upon their number-sense skills throughout the entire IMP curriculum. Because students are asked to make sense of problem situations for themselves, this number sense is the framework within which students create meaning. The IMP curriculum builds in the expectation that students must apply number-sense skills daily. Any item that is mentioned in Standard 5-7 would provide a good example for this benchmark. New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /16 STANDARD CORRELATION 5.2 continued POW6-SubDivvy and POW7-Crack the Code! in the Year 3 unit Meadows or Malls? are specific samples of how students experience and work with the properties of the real number system. The Pollster’s Dilemma • HW9-Gaps in the Table 5.3 Apply ratios, proportions, and percents in The Overland Trail more complex mathematical situations. • Day 17-28: Students consider ideas of direct variation, slope (informally), and rate-time-distance problems. The Pit and the Pendulum • POW13-Corey the Camel Shadows HW7-The Statue of Liberty‟s Nose HW8-Make it Similar HW10-Similar Problems HW11-From Top to Bottom HW16-Ins and Outs of Proportion HW22-Right Triangle Ratios Day23-25: Students generate the right triangle trigonometry definitions. Solve It! • HW26-More Variable Solutions Is There Really a Difference? CW-Measuring Weirdness HW12-Drug Dragnet: Fair or Foul? CW-Does Age Matter? HW16-Assigning Probabilities Do Bees Build It Best? HW17-More Fencing, Bigger Corrals HW27-Cereal Box Sizes All About Alice HW7-Continuing the Pattern New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /17 STANDARD CORRELATION 5.3 continued Small World, Isn’t It? Students are asked to analyze the derivative of exponential functions and note the proportionality properties that distinguish exponential functions from other families of functions. CW-Slippery Slopes HW20-How Does It Grow? CW-The Power of Powers HW23-The Power of Powers, Continued CW-Find That Base! As the Cube Turns CW-Fractional Snacks HW21-More Walking for Clyde CW-Monorail Delivery CW-And Fred Brings the Lunch HW25-Where‟s Bonita? • CW-Lunch in the Window New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /18 5.4 Analyze and interpret numerical Patterns relationships in one- and two-dimensional CW-The Chef‟s Hot & Cold Cubes graphs, both manually and using tools The Overland Trail such as graphing calculators and CW-Wagon Train Sketches and Situations computers. HW11-Graph Sketches CW-In Need of Numbers HW12-The Issues Involved (Students consider issues related to scaling axes and inferring form graphs.) CW-Out Numbered (Students generalize relationships from graphs.) CW-“Out of Action” and “Sublette‟s Cutoff” Revisited Solve It! • CW- The Graph Solves the Problem Is There Really a Difference? 2 Day17-18: Students develop and interpret a probability graph. Cookies HW2-Investigating Inequalities CW-Picturing Cookies-Part I • CW-Picturing Cookies-Part II New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /19 STANDARD CORRELATION 5.4 continued All About Alice HW9-It‟s in the Graph • CW-All Roads Lead to Rome Fireworks CW-The Ups and Downs of Quadratics Small World, Isn’t It? CW-ZOOOOOOOOM! HW15-On a Tangent HW21-The Significance of a Sign HW27-Comparing Derivatives High Dive Students define polar coordinates and develop expressions for rectangular coordinates in terms of polar coordinates. Example activities include: CW-Some Polar Practice • HW17-Polar Coordinates on the Ferris Wheel As the Cube Turns Students apply algebraic and trigonometric reasoning to translate and rotate points in two dimensions. Additionally, students will project points in 3-space to a two dimensional plane. For example: HW15-Comin‟ Round Again (and Again…) HW29-Mirrors in Space New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /20 NUMBER AND OPERATION CONCEPTS Standard 6: Students will understand and use number systems and number theory. Benchmarks, Grades 9-12 STANDARD CORRELATION 6.1 Use order relations within the real number Throughout the Patterns unit, students observe and generalize numerical patterns. system. For example: HW7-Extended Bagels CW-Consecutive Sums Note: this unit sets the stage for student thinking throughout the remaining four years of the IMP curriculum. The Overland Trail POW8-The Haybaler Problem POW11-High-Low Differences Solve It! HW6-The Mystery Bags Game Cookies POW12-Kick It! Throughout the All About Alice unit students compare relative sizes of very large and very small numbers using exponents, scientific notation, and logarithms. Some examples: HW14-Alice on a Log CW-Taking Logs to the Axes HW15-Base 10 Alice HW16-Warming Up to Scientific Notation CW-Big Numbers Fireworks CW-Factored Intercepts. Students recognize the role the zero property of multiplication plays in solving equations. 6.2 Apply number theory concepts to a The Pit and the Pendulum variety of problem situations. POW14-Eight Bags of Gold POW15-Twelve Bags of Gold New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /21 STANDARD CORRELATION 6.2 continued Shadows POW17-Cutting the Pie Solve It! POW1-A Digital Proof POW2-Tying the Knots POW3-Divisor Counting Is There Really a Difference? POW4-A Timely Phone Tree POW5-Punch the Clock Cookies POW12-Kick It! POW13-Shuttling Around Orchard Hideout POW9-A Marching Strip Meadows or Malls? POW6-SubDivvy POW7-Crack the Code! Small World, Isn’t It? POW9-Planning the Platforms POW10-Around King Arthur‟s Table High Dive POW1-The Tower of Hanoi Know How POW6-Which Weights Weigh What? World of Functions POW7-One Mile at a Time During this unit, students will analyze functions using ideas of finite differences, both formally and informally. CW-Linear Tables CW-Quadratic Tables CW-Quadratic Tables by Algebra New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /22 STANDARD CORRELATION 6.2 continued HW6-A General Quadratic CW-Exponential Tables • HW7-A Cubic Pattern 6.3 Identify how seemingly different Patterns mathematical situations may be HW22-Border Varieties essentially the same (for example, the The Overland Trail intersection of two lines is the same as the CW-More Fair Share on Chores solution to a system of linear equations). HW23-More Fair Share for Hired Hands CW-Water Conservation HW25-The Big Buy Shadows POW17-Cutting the Pie Solve It! HW12-Who‟s Right? HW16-Views of the Distributive Property Cookies CW-Profitable Pictures Fireworks HW4-Make Your Own Intercepts Orchard Hideout In this unit, students study coordinate geometry. They rely on the Pythagorean theorem to generate a wide variety of relationships, including the distance formula and general formula defining circles. Example activities include: HW4-In, On, or Out? HW6-The Distance Formula HW18-Defining Circles Meadows or Malls? HW36-Beginning Portfolios – Part I New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /23 STANDARD CORRELATION 6.3 continued Small World, Isn’t It? Students generate the number e in two seemingly unrelated contexts. First, x students look for the base (b) of an exponential function of the form f(x) = b so that f(x) = f(x). Next, students rediscover e in the context of compounding interest. Pennant Fever CW-Pascal‟s Triangle HW24-Hi There! HW25-Pascal‟s Triangle and the Combinatorial Coefficients CW-Combinations, Pascal‟s Way HW26-Binomials and Pascal – Part I HW27-Binomials and Pascal – Part II • HW28-A Pascal Portfolio As the Cube Turns • HW27-Spiders and Cubes 6.4 Compare and contrast the real number All About Alice system and its various subsystems with Students consider the nature of roots, logarithms and their irrationality. regard to structural characteristics. Meadows or Malls? Students examine whether matrix operations have certain properties such as associativity and commutativity. Students also identify the matrix identity for multiplication. The following activities highlight these student experiences: HW31-Solving the Simplest HW32-Things We Take for Granted Small World, Isn’t It? Students consider the nature of irrational and transcendental numbers. The World of Functions POW8-A Spin on Transitivity HW22-Order Among the Functions • HW26-An Inventory of Inverses New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /24 STANDARD CORRELATION 6.5 Develop and analyze algorithms. The IMP curriculum provides a structure for students to constantly develop algorithms for solving a variety of problems. Students analyze the accuracy and/or effectiveness of peer’s algorithms. Some specific examples of these: Shadows CW-Mirror Magic HW19-To Measure a Tree • CW-A Shadow of a Doubt Solve It! HW6-The Mystery Bags Game • Day13-21: Students generate the necessary algorithms for applying the Distributive Property within symbolic algebraic representations. Key examples include: CW-A Lot of Changing Sides HW13-Why Are They Equivalent? CW-Distributing the Area HW15-The Distributive Property and Mystery Lots CW-Taking Some Out, Part I HW18-Subtracting Some Sums CW-Taking Some Out, Part II • HW24-A Distributive Summary Cookies CW-Get the Point. Students develop an algorithm to algebraically solve for the point of intersection of two linear functions. POW13-Shuttling Around Fireworks CW-Who‟s Perfect HW6-More About Perfection. In these two activities, students generate a method to “complete the square.” New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /25 STANDARD CORRELATION 6.5 continued Meadows or Malls? Students generate algorithms to solve two variable linear programming problems. One algorithm is graphic in nature, the other relies purely on symbolic algebraic arguments. Students compare and contrast the benefits of each algorithm. Next, each is extended to three variable problems. Students see that the graphing becomes more complex. As students consider linear programming problems beyond three variables, they realize that only the algebraic algorithm is practical. Students conclude the unit by formulating a algorithm to solve any size linear programming problems relying on matrix algebra and calculator technology. Small World, Isn’t It? HW9-Points, Slopes, and Equations CW-A Basis for Disguise HW29-California Population with e‟s As the Cube Turns One primary focus of this unit is for students to develop a structural understanding of programming language. This begins with students reading and writing their own plain-language programs. They will translate these plain-language programs into the language of a graphing calculator. Some example activities include: HW2-Programming Without a Calculator HW3-Learning the Loops HW5-Moving‟ On HW6-Some Back and Forth CW-Swing That Line! HW19-What‟s Going on Here? New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /26 NUMBER AND OPERATION CONCEPTS Standard 7: Students will understand and use computation and estimation. Benchmarks, Grades 9-12 STANDARD CORRELATION 7.1 Develop, analyze, and explain methods The IMP curriculum provides a structure for students to constantly develop and for solving a variety of problem situations. explain their methods for solving a variety of problems. The development of these methods may be done individually or collaboratively. Explanations may be written or stated verbally in small groups, formal presentations, or as part of a classroom discussion. Students analyze the accuracy and/or effectiveness of peers’ methods, as well as examples presented in the curriculum itself. Some specific examples of these: Patterns CW-A Proof Gone Bad Shadows HW13-Inventing Rules HW16-Ins and Outs of Proportions Solve It! HW6-The Mystery Bags Game Do Bees Build It Best? CW-Going into the Gallery Cookies Students develop a set of steps to solve two variable linear programming problems. This is summarized by students in : HW23-Beginning Portfolio Selection Students must thoroughly write-up all POW’s. Students are expected to present and explain their thinking and methods for solving these problems. The example below is only one of many that illustrates this feature of IMP that pervades through all four years: Meadows or Malls? POW5-That‟s Entertainment. Again, this POW is an excellent example of this ongoing expectation throughout the four years of IMP. New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /27 STANDARD CORRELATION 7.1 continued High Dive This unit is marked by an overwhelming set of generalizations and equations that students develop (and justify) for solving problems. A sample: CW-Free Fall HW10-A Practice Jump HW11-Where Does He Land? • CW-Generalizing the Platform 7.2 Extend solutions of problems to formulate The Overland Trail predictions. CW-Previous Travelers HW17-The Basic Student Budget The Pit and the Pendulum This unit’s problem (predict the period of a thirty foot pendulum) asks students to not only develop proper small scale models of a pendulum, but also to collect data from them to formulate a prediction for the period of a thirty foot pendulum. The capstone activity is: CW-The Thirty-Foot Prediction Cookies POW13-Shuttling Around High Dive POW1-The Tower of Hanoi The Pollster’s Dilemma POW10-The King‟s Switches New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /28 STANDARD CORRELATION New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /29 7.3 Justify the reasonableness of solutions and IMP requires students to justify the reasonableness of their conclusions or predictions. predictions throughout its curriculum. IMP challenges students with big problems that take time to solve. Here are examples of smaller problems in the context of units: The Overland Trail CW-Who Will Make It? HW18-Out of Action The Pit and the Pendulum HW20-Mehrdad the Market Analyst CW-Bird Houses CW-The Thirty-Foot Pendulum Solve It! HW13-Why Are They Equivalent? Do Bees Build it Best? HW4-An Area Shortcut? HW19-Falling Bridges Orchard Hideout HW22-Big Earth, Little Earth Meadows or Malls? POW5-That‟s Entertainment. Students must complete a write-up of all the POW’s they do. Students are expected to justify and defend solutions they find. Again, this POW is an excellent example of this ongoing expectation throughout the four years of the IMP. Small World, Isn’t It? CW-Return to “Small World, Isn‟t It?” The World of Functions CW-Better Braking The Pollster’s Dilemma CW-The Pollster‟s Dilemma” Revisited New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /30 GEOMETRY AND MEASUREMENT CONCEPTS Standard 8: Students will have a foundation in geometric concepts. Benchmarks, Grades 9-12 STANDARD CORRELATION 8.1 Interpret and draw three-dimensional Do Bees Build it Best? objects. Students strengthen their three-dimensional spatial visualization skills. CW-Building the Biggest HW1-What to Put It In? HW21-Flat Cubes CW-Flat Boxes HW22-Not a Sound CW-A Voluminous Task HW23-Put Your Fist into It HW24-A Sculpture Garden CW-The World of Prisms‟ CW-Shedding Light on Prisms HW25-Pythagoras and the Box HW26-Back on the Farm CW-Which Holds More? HW27-Cereal Box Sizes CW-A Tessellating We Go Orchard Hideout HW20-Cylindrical Soda Meadows or Malls? HW10-What Do They Have in Common? CW-Just the Plane Facts HW15-The “More Cookies” Region and Strategy As the Cube Turns CW-A Return to the Third Dimension New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /31 STANDARD CORRELATION 8.2 Deduce properties of figures using Shadows transformations and using coordinates. POW16-Spiralaterals HW1-Shadows and Spiralaterals Students develop intuitive ideas about “same shape” and learn formal definitions of similarity and congruence. • CW-How to Shrink It? Students discover criteria for polygons to be similar and, in particular, for triangles to be similar. CW-Why are Triangles Special? CW-Are Angles Enough? HW11-From Top to Bottom HW15-Inside Similarity Do Bees Build it Best? POW8-Just Count the Pegs POW9-Tessellation Pictures POW10-Possible Patches Cookies Students see that setting a linear expression equal to different constants gives a family of parallel lines. HW8-Curtis and Hassan Make Choices Fireworks CW-The Ups and Downs of Quadratics Orchard Hideout Day6- Students identify symmetry within a unit coordinate circle. The also notice similar triangles. HW18-Defining Circles Meadows or Malls? HW24-Constraints Without a Context. In this activity, students must argue which combinations of constraints could not possibly create potential corners of a three-dimensional feasible region. High Dive CW-At Certain Points in Time New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /32 STANDARD CORRELATION 8.2 continued As the Cube Turns CW-Move That Line! HW10-Cornering the Cabbage • HW11-Goin‟ Round the Origin 8.3 Classify figures in terms of congruence Shadow: and similarity and apply these Students apply properties of similar triangles to physical situations. relationships. HW6-Draw the Same Shape CW-How to Shrink It? HW8-Make it Similar HW10-Similar Problems HW12-Very Special Triangles HW15-Inside Similarity HW18-Mirror Madness CW-More Triangles for Shadows CW-The Return of the Tree 8.4 Represent problem situations with The Game of Pig geometric models and apply properties of Students create area models to represent probability situations. figures. HW7-Portraits of Probabilities HW12-Pointed Rugs HW14-A Fair Rug Game? HW16-Streak Shooting Shelly CW-Martian Basketball CW-Little Pig Strategies HW23-Continued Little Pig Investigation • HW24-Even More Little Pig Investigation New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /33 STANDARD CORRELATION 8.4 continued Shadows Students develop geometric models of mirror reflections and lamp and sun shadows. CW-How Long is a Shadow? CW-Mirror Magic HW19-To Measure a Tree HW21-The Sun Shadow Problem An analysis of the properties of this model allows students to define a generalized relationship among the shadow model variables. • CW-A Shadow of a Doubt Students apply properties of similarity to solve problems. CW-A Shadow of a Doubt CW-More Triangles for Shadows CW-The Return of the Tree Do Bees Build it Best? Students model throughout the unit as a means for developing mathematics to solve unit problem. Orchard Hideout Again, students model throughout the unit as a means for developing mathematics to solve unit problem. A few examples: Day2- Students “coordinatize” the unit problem. POW2-Equally Wet POW3-On Patrol CW-Cable Ready Meadows or Malls? HW15-The “More Cookies” Region and Strategy HW16-Finishing Off the Cookies HW24-Constraints Without a Context New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /34 STANDARD CORRELATION 8.4 continued Pennant Fever CW-Choosing for Chores HW3-Baseball Probabilities HW4-Possible Outcomes CW-How Likely is All Wins? HW6-Diagrams, Baseball, and Losing „Em All CW-“Race for the Pennant!” Revisited High Dive CW-High Dive • HW1-The Ferris Wheel As the Cube Turns POW4-A Wider Windshield Wiper, Please HW10-Cornering the Cabbage The World of Functions CW-Return of the Shadow New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /35 8.5 Deduce properties of and relationships Patterns between figures from given assumptions. Students investigate the interior angle sum of polygons. CW-Pattern Block Angles CW-Degree Discovery HW17-Diagonally Speaking HW18-Polygon Angles HW19-An Angular Summary Shadows Students develop properties of triangles from given assumptions. HW9-Triangular Counterexamples POW18-Trying Triangles Do Bees Build it Best? Students create formulas for the areas of rectangles, triangles, parallelograms, and trapezoids. HW6-The Ins and Outs of Area HW7-Forming Formulas New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /36 STANDARD CORRELATION 8.5 continued Students discover that the ratio of the areas of similar figures is equal to the square of the ratio of their corresponding linear dimensions. HW17-More Fencing, Bigger Corrals CW-More Opinions About Corrals CW-Building the Best Fence Students develop principles relating the volume and surface area of a prism to the area and perimeter of its base. CW-The Ins and Outs of Boxes CW-Shedding Light on Prisms Students discover that the ratio of the surface areas of similar solids is equal to the square of the ratio of their corresponding linear dimensions, and that the ratio of the volumes of similar solids is equal to the cube of the ratio of their corresponding linear dimensions. HW27-Cereal Box Sizes Orchard Hideout Day 4-Students deduce the notion of tangent line to a circle in the context of line of sight through an orchard. CW-Lines of Sight for Radius Six Meadows or Malls? CW-Just the Plane Fact 8.6 Identify congruent and similar figures Shadow: using transformations. Students intuitively explore various transformations. HW15-Inside Similarity Orchard Hideout Day 6-Students identify symmetry within a unit coordinate circle and notice similar rotated or translated triangles. High Dive Throughout this unit, students use congruent triangles within a unit circle to develop trigonometric identities. New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /37 STANDARD CORRELATION 8.7 Analyze properties of Euclidean Orchard Hideout transformations and relate CW-The Other Gap transformations to vectors. CW-Lines of Sight for Radius Six High Dive CW-Testing the Definition HW15-What‟s Your Cosine? CW-The Ideal Skateboard HW27-Racing the River CW-One O‟Clock Without Gravity CW-Release at Any Angle As the Cube Turns Students derive formulas for the sine and the cosine of the sum of two angles. HW12-Double Trouble CW-The Sine of a Sum New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /38 GEOMETRY AND MEASUREMENT CONCEPTS Standard 9: Students will understand and use measurement. Benchmarks, Grades 9-12 STANDARD CORRELATION 9.1 Apply measurement as a tool in other This is an integral part of the IMP curriculum, and here are some examples: disciplines and in everyday problem Patterns situations. Students measure angles to solve problems and to generalize relationships. CW-A Protracted Engagement HW17-Diagonally Speaking CW-Degree Discovery Pit and the Pendulum Measurement is used throughout the unit to collect and analyze data as a means for developing mathematics to solve the unit problem. Do Bees Build it Best? Students use measurement concepts to find area, surface area, and volume. CW-Nailing Down Area HW2-Approximating Areas HW4-An Area Shortcut? CW-A Voluminous Task HW23-Put Your Fist Into It CW-Shedding Light on Prisms Orchard Hideout In this unit, students model an orchard with trees at lattice points. Upon introduction to the problem, students draw a model and use a straightedge and ruler to begin estimating coordinate distances. New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /39 STANDARD CORRELATION 9.2 Identify and use the appropriate units and This is an integral part of the IMP curriculum, and here are some examples: tools of measurement to the degree of Patterns accuracy required in particular problems. Students measure angles to solve problems and to generalize relationships. CW-A Protracted Engagement HW17-Diagonally Speaking CW-Degree Discovery Pit and the Pendulum Measurement is used throughout the unit to collect and analyze data as a means for developing mathematics to solve unit problem. Do Bees Build it Best? Students use measurement concepts to find area, surface area, and volume. CW-Nailing Down Area HW2-Approximating Areas HW4-An Area Shortcut? CW-A Voluminous Task HW23-Put Your Fist Into It CW-Shedding Light on Prisms Orchard Hideout In this unit, students model an orchard with trees at lattice points. Upon introduction to the problem, students draw a model and use a straightedge and ruler to begin estimating coordinate distances. New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /40 STATISTICS AND PROBABILITY CONCEPTS Standard 10: Students will understand and use statistics. Benchmarks, Grades 9-12 STANDARD CORRELATION 10.1 Construct and draw inferences from This is done throughout the four year curriculum. Students construct charts, tables, charts, tables, and graphs that summarize and graphs of empirical data generated in the classroom from experiments and data from inside and outside the school from purely mathematical patterns. Students analyze these charts, tables, and environment. graphs to generalize trends, often for predictions. A few examples: The Overland Trail Students graph tables of information and create lines of best fit to make predictions and estimates. CW-Previous Travelers The Pit and the Pendulum HW17-A Picture is Worth a Thousand Words HW20-Mehrdad the Market Analyst Is There Really a Difference? Days 1-3: double bar graphs CW-Graphing the Difference HW16-Assigning Probabilities POW6-Is There Really a Difference? Small World, Isn’t It? CW-Small World, Isn‟t It? The World of Functions CW-Brake! CW-Brake! Revisited HW20-“Small World, Isn‟t It?” Revisited The Pollster’s Dilemma CW-The Pollster‟s Dilemma CW-Let‟s Vote on It! • CW-What Does It Mean? New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /41 STANDARD CORRELATION 10.2 Use curve fitting to predict from data. The Overland Trail Students intuitively sketch lines of best fit, then develop these graphical estimates into algebraic representations. CW-Previous Travelers HW15-Broken Promises HW16-Sublette‟s Cutoff CW-Who Will Make It? HW17-The Basic Student Budget HW18-Out of Action CW-“Out of Action” and Sublette‟s Cutoff” Revisited The Pit and the Pendulum HW20-Mehrdad the Market Analyst CW-Bird Houses CW-The Thirty-Foot Prediction All About Alice HW9-It‟s in the Graph Small World, Isn’t It? CW-Small World, Isn‟t It? HW8-California, Here I Come! CW-Return to “Small World, Isn‟t It?” The World of Functions HW15-And Now…Back to Marcus Dunkalot CW-Let‟s Regress • CW-Better Braking New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /42 STANDARD CORRELATION 10.3 Apply measures of central tendency, The Pit and the Pendulum variability, and correlation. This unit’s main focus is normal distribution and standard deviation. Students develop and apply these concepts throughout the unit. HW7-What‟s Normal? CW-What‟s Rare? HW9-Penny Weight CW-Mean School Data CW-Data Spread HW11-Dinky and Minky Spread Data HW12-The Best Spread HW14-Penny Weight Revisited HW15-Can Your Calculator Pass This Soft Drink Test? HW16-Standard Pendulum Data and Decisions CW-Pendulum Variations HW18-Pendulum Conclusions Is There Really a Difference? CW-Bacterial Culture HW10-Decisions with Deviation The Pollster’s Dilemma CW-A Normal Poll CW-From Numbers to Proportions CW-Different p, Different HW16-Mean, Median, and Mode New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /43 STANDARD CORRELATION 10.4 Define sampling and recognize its role in Is There Really a Difference? statistical claims. • POW6-Is There Really a Difference Students consider the characteristics of a good sample. HW2-Who Gets A‟s and Measles? HW3-Quality of Investigation CW-Two Different Differences HW17-A Collection of Coins In the above activities, as well as those which follow, students develop tools for determining when differences in samples mean that the larger populations are likely to be different. CW-Loaded or Not? CW-Measuring Weirdness The Pollster’s Dilemma CW-The Pollster‟s Dilemma HW1-No Bias Allowed! CW-Sampling Seniors HW3-Throw Back the Little Ones HW4-Graphs of the Theory CW-The Theory of Polls HW5-Civics in Action HW13-Putting Your Formulas to Work HW14-Is Twice As Many Twice As Good CW-Let‟s Vote on It! CW-The Worst Case Scenario CW-“The Pollster‟s Dilemma” Revisited New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /44 10.5 Design an experiment to study a problem Is There Really a Difference? and use the correct statistical procedures POW6-Is There Really a Difference? to summarize and analyze the data. The Pollster’s Dilemma CW-Back to the Circus CW-Let‟s Vote on It! HW19-How Big? STANDARD CORRELATION 10.6 Analyze the effects of data The Pit and the Pendulum transformations on measures of central CW-Making Friends with Standard Deviation tendency and variability. HW13-Deviations Pollster’s Dilemma HW13-Putting Your Formulas to Work HW14-Is Twice As Many Twice As Good? CW-The Worst Case Scenario CW-Polling Puzzles • HW20-How Much Better is Bigger? New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /45 STATISTICS AND PROBABILITY CONCEPTS Standard 11: Students will understand and use probability. Benchmarks, Grades 9-12 STANDARD CORRELATION 11.1 Use simulations to estimate probability. The Game of Pig Use of simulations, including electronic simulations, to estimate probability is a major focus throughout this unit, especially evident in: HW15-A Sixty-Percent Solution HW17-Aunt Zena at the Fair (students use a calculator’s random number generator) HW19-The Carrier‟s Payment Plan Quandary CW-Using the Programmed Simulation CW-The Game of Little Pig Pennant Fever CW-Simulate a Deal The Pollster’s Dilemma CW-Sampling Seniors New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /46 STANDARD CORRELATION 11.2 Use experimental or theoretical The Game of Pig probability to represent and solve Use of experimental and theoretical probability to solve problems is a major focus problems involving uncertainty. throughout this unit, especially evident in: CW-The Gambler‟s Fallacy POW5-What‟s On Back? HW9-Rollin‟, Rollin‟ Rollin‟ CW-The Theory of Two-Dice Sums HW15-A Sixty-Percent Solution CW-The Theory of One-and-One HW19-The Carrier‟s Payment Plan Quandary HW21-Another Carrier Dilemma CW-Little Pig Strategies, and its series of activities concluding in CW-The Best Little Pig Pennant Fever Students solve probability problems throughout this unit, relying primarily on theoretical models. The Pollster’s Dilemma CW-The Theory of Three Person Polls CW-The Theory of Polls HW5-Civics in Action • HW13-Putting Your Formulas to Work New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /47 11.3 Apply the definition of a random variable The Game of Pig to experimental design. Students write a calculator program to create a simulation for CW-Using the Programmed Simulation. A random number (integer) is selected and stored into the calculator’s memory. The Pit and the Pendulum Students plan and perform controlled scientific experiments. HW3-If I Could Do It Over Again CW-Time is Relative HW4-What‟s Your Stride? CW-Pendulum Variations New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /48 STANDARD CORRELATION 11.4 Create and interpret discrete probability The Game of Pig distributions. Students make and interpret frequency bar graphs (histograms): HW3-Expecting the Unexpected HW9-Rollin‟, Rollin‟, Rollin‟ The Pit and the Pendulum Students collect, express, and analyze data using frequency bar graphs (histograms). CW-Time is Relative HW4-What‟s Your Stride? HW5-Pulse Analysis CW-The Standard Pendulum HW16-Standard Pendulum Data & Decisions Is There Really a Difference? Throughout this unit, students collect, express, and analyze data using double bar 2 graphs. Students also construct an empirical distribution chart, generalize the 2 curve, and interpret this curve as a probability chart for statistics. CW-Graphing the Difference CW-Random But Fair Pennant Fever HW29-Graphing the Games he Pollster’s Dilemma HW4-Graphs of the Theory HW5-Civics in Action HW12-Why Is That Batter Sneezing? New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /49 STANDARD CORRELATION 11.5 Describe the normal curve and apply its The Pit and the Pendulum properties to answer questions about data Students estimate the likelihood of events based on the properties of the normal that are assumed to be normally curve. distributed. HW10-An (AB)Normal Rug Students use standard deviation to decide whether a variation in experiment results is significant. HW16-Standard Pendulum Data & Decisions CW-Pendulum Variations HW18-Pendulum Conclusions Is There Really a Difference? CW-Bacterial Culture HW10-Decisions with Deviation The Pollster’s Dilemma CW-Deviations of Swinging HW6-Means and More in Middletown HW7-Gifts Aren‟t Always Free CW-Normal Areas HW8-More Middletown Musings HW17-A Teaching Dilemma New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /50 STANDARD CORRELATION 11.6 Solve enumeration and finite probability The Game of Pig problems. Students solve finite probability problems throughout this unit. Pennant Fever Students develop generalizable counting principles by first constructing organized lists, then correctly apply the multiplication principle and the concepts of permutation and combination. HW12-Top That Pizza! CW-Double Scoops HW13-Triple Scoops CW-More Cones for Johanna HW14-Cones from Bowls, Bowls from Cones CW-Bowls for Jonathan HW15-At the Olympics HW16-Which is Which? CW- Formulas for nPr and nCr Finite probability examples: POW12-Let‟s Make a Deal POW13-Fair Spoons High Dive POW2-Paving Patterns The Pollster’s Dilemma HW2-Pennant Fever Reflection HW3-Throw Back the Little Ones HW10-A Plus for the Community HW12-Why Is That Batter Sneezing? New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /51 FUNCTION AND ALGEBRA CONCEPTS Standard 12: Students will understand and use patterns and functions. Benchmarks, Grades 9-12 STANDARD CORRELATION New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /52 12.1 Model everyday experiences with a Patterns variety of functions. HW22-Border Varieties The Overland Trail CW-Wagon Train Sketches and Situations HW11-Graph Sketches CW-Previous Travelers CW-Who Will Make It? HW17-The Basic Student Budget HW18-Out of Action CW-Graphing Calculator In Outs CW-Getting the Gold The Pit and the Pendulum This unit problem is solved with a function generated by students. Shadows Again, students solve this unit problem by developing a symbolic relationship among four variables affecting shadow length. CW-Shadow Data Gathering CW-A Shadow of a Doubt Solve It! CW-Memories of Yesteryear HW2-Building a Foundation Cookies Students solve this unit problem, a profit dilemma at a bakery, by developing the appropriate system of linear inequalities to summarize constraints and then determine the maximum of a feasible region. Additionally: HW4-Inequality Stories HW21-Inventing Problems CW-Producing Programming Problems New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /53 STANDARD CORRELATION 12.1 continued • Fireworks HW1-A Corral Variation HW8-How Much Can They Drink? Small World, Isn’t It? In this unit, students create meaning to exponential growth and decay functions. Students generate several exponential functions for situations, most notably: HW21-Blue Book HW25-California and Exponents HW29-California Population with e‟s CW-Return to “Small World, Isn‟t It?” High Dive CW-At Certain Points in Time HW10-A Practice Jump CW-Cart Travel Time CW-First-Quadrant Platform The World of Functions HW3-More Families HW4-Story Sketches III CW-Return of the Shadow CW-Families Have Many Different Members HW16-Midnight Express HW17-In the Lead HW19-Name That Family! HW23-Cozying Up to Composition New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /54 STANDARD CORRELATION 12.2 Represent and analyze relationships using Throughout all four years of the curriculum, students effectively use the Rule of 3 tables, graphs, rules, and equations. to represent and analyze relationships Patterns HW5-Inside Out HW7-Extended Bagels HW10-Pulling Out Rules HW16-Another In-Outer HW20-Squares and Scoops HW23-Cutting Through the Layers The Overland Trail CW-Wagon Trains and Story Sketches HW11-Graph Sketches CW-In Need of Numbers HW12-The Issues Involved CW-Out Numbered HW13-Situations, Graphs, Tables, and Rules CW-Previous Travelers CW-Graphing Calculator In Outs CW-Fair Share on Chores CW-More Fair Share on Chores HW23-More Fair Share for Hired Hands CW-Water Conservation HW25-The Big Buy HW27-Catching Up at Auburn CW-Getting the Gold New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /55 New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /56 STANDARD CORRELATION New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /57 12.2 continued The Pit and the Pendulum HW21-So Little Data, So Many Rules CW-Graphing Free-for-All HW22-Graphs in Search of Equations I HW23-Graphs in Search of Equations II HW24-Graphing Summary HW25-Graphs in Search of Equations III Shadows CW-Looking for Equations POW17-Cutting the Pie Solve It! HW29-Functioning in the Math World Is There Really a Difference? CW-Graphing the Difference HW16-Assigning Probabilities Do Bees Build It Best? HW6-The Ins and Outs of Area POW8-Count the Pegs CW-The Ins and Outs of Boxes Cookies Students develop an understanding of graphs of inequalities and extend this to the notion that a feasible region is a picture of the common set of solutions to a system of equations. CW-Picturing Cookies-Part I CW-Picturing Cookies-Part II All About Alice HW1-Graphing Alice HW7-Continuing the Pattern HW9-It‟s in the Graph CW-Taking Logs to the Axes Fireworks HW8-How Much Can They Drink? New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /58 STANDARD CORRELATION 12.2 continued Meadows or Malls? Students consider three-dimensional graphical models depicting the relationships among three variables. HW10-What Do They Have in Common? HW11-Trying Out Triples HW12-More Cookies HW15-The “More Cookies” Region and Strategy Small World, Isn’t It? HW16-What‟s It All About? High Dive CW-Free Fall As the Cube Turns HW14-Oh, Say What You Can See The World of Functions CW-Rumble, Grumble HW21-The Composition of Functions • CW-The Cost of Pollution 12.3 Translate among tabular, symbolic, and The Overland Trail graphical representations of functions. CW-Out Numbered HW13-Situations, Graphs, Tables, and Rules HW14-Rules, Tables, and Graphs HW19-What We Needed CW-“Out of Action” and “Sublette‟s Cutoff” Revisited HW21-Biddy Mason • HW22-Fair Share for Hired Hands Solve It! HW11-Line It Up CW-Where‟s Speedy? HW28-Swinging Pendulum • HW29-Functioning in the Math World Cookies • HW17-A Reflection on Money New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /59 12.3 continued All About Alice HW7-Continuing the Pattern • CW-Taking Logs to the Axes The World of Functions CW-Linear Tables CW-Quadratic Tables HW8-Mystery Tables CW-Brake! Revisited HW13-Fitting Mia‟s Bird Houses Again HW14-Mystery Tables II HW28-Transforming Graphs, Tables, and Situations 12.4 Model a variety of problem situations Several units in IMP focus on a type of function and give students many using the same type of function. opportunities to write and use the same type of function in a variety of problem situations. There is an emphasis in The Overland Trail and Cookies on linear functions, in Shadows on direct and inverse variation, in Fireworks on quadratic function, in Small World, Isn‟t It? on exponential functions, and in High Dive on cyclic functions. Highlights are listed below. The Overland Trail Students develop the notion that linear functions all have a constant rate of change. Shadows HW10-Similar Problems CW-Mirror Magic Cookies Students repeatedly graph linear functions in the context of solving linear programming problems. All About Alice CW-Taking Logs to the Axes Fireworks • HW9-Corrals and Pens Again. Students consider quadratics in a variety of forms in two new situations. New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /60 STANDARD CORRELATION 12.4 continued Small World, Isn’t It? Students create symbolic, tabular, and graphical models of a variety of exponential situations, including: CW-How Much for Broken Eggs? HW17-Small, But Plentiful HW19-The Forgotten Account HW22-Blue Book HW26-Calfifornia and Exponents HW29-California Population with e‟s CW-Return to “Small World, Isn‟t It?” High Dive Students consider various sinusoidal situations and develop symbolic and graphical models to represent the situation. The World of Functions HW1-Story Sketches HW2-Story Sketches II HW5-Back to the Basics HW9-Bigger Means Smaller HW11-An Average Drive CW-What Will It Be Worth? The Pollster’s Dilemma CW-Graphing Distributions 12.5 Analyze the effects of parameter changes As stated above, several units focus on a variety of types of functions. The fourth on the graph of functions. year unit, The World of Functions, summarizes and formalizes the effects of parameter changes. Highlights are listed below. The Overland Trail Students note how the magnitude and sign of the coefficient in a linear function affects the graph, days 15-21. New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /61 STANDARD CORRELATION New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /62 12.5 continued The Pit and the Pendulum Students do an introductory investigation into families of functions. CW-Graphing Free-for-All HW24-Graphing Summary CW-The Thirty-Foot Prediction Solve It! HW11-Line It Up CW-Get It Straight Cookies These two activities and following classroom discussions analyze families of parallel lines. CW-Profitable Pictures CW-Hassan‟s a Hit! All About Alice Day2, HW1,9 • CW-Taking Logs to the Axes Fireworks CW-The Ups and Downs of Quadratics Small World, Isn’t It? CW-Tweaking the Function High Dive HW4-Graphing the Ferris Wheel HW5-Ferris Wheel Graph Variations CW-The “Plain” Sine Graph HW7-More Beach Adventures The World of Functions CW-Double Dose of Functions HW27-Slide That Function • HW28-Transforming Graphs, Tables, and Situations New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /63 STANDARD CORRELATION 12.6 Explore and represent everyday situations The Overland Trail using the sine and cosine functions. HW20-More Graph Sketches High Dive HW4-Graphing the Ferris Wheel HW5-Ferris Wheel Graph Variations HW6-Sand Castles HW12-Carts and Periodic Functions As the Cube Turns HW11-Goin‟ Round the Origin HW12-Double Trouble HW14-Oh, Say What You Can See HW15-Comin‟ Round Again (and Again…) The World of Functions HW2-Story Sketches II HW4-Story Sketches III New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /64 FUNCTIONS AND ALGEBRA CONCEPTS Standard 13: Students will understand and apply algebraic concepts. Benchmarks, Grades 9-12 STANDARD CORRELATION 13.1 Represent situations that involve variable This notion is fundamental to the development of symbolic mathematics in the IMP quantities with expressions, equations, curriculum. The key units of this development are noted below, along with a few inequalities, and matrices. example activities. Patterns HW11-Add It Up (summation notation) Overland Trail CW-The Family HW4-Family Constraints HW6-Shoe Laces CW-Ox Expressions HW9-Ox Expressions at Home HW26-Water for One More The Pit and the Pendulum CW-Bird Houses CW-The Thirty Foot Prediction Shadows The student’s goal in this unit is to model the shadow situation symbolically. Additionally: HW10-Similar Problems HW18-Mirror Madness CW-A Shadow of a Doubt Solve It! CW-Memories of Yesteryear HW2-Building a Foundation HW10-What Will it Answer? New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /65 STANDARD CORRELATION New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /66 13.1 continued Do Bees Build it Best? HW7-Forming Formulas CW-Tri-Square Rug Games CW-Building the Best Fence CW-Shedding Light on Prisms • HW25-Pythagoras and the Box Cookies CW-Feasible Diets HW7-Picturing Pictures CW-Big State U All About Alice CW-Alice in Wonderland HW15-Base 10 Alice Fireworks HW1-A Corral Variation Orchard Hideout HW4-In, On, or Out? Following classroom discussion students create a symbolic relationship for points in, on, or outside a circle of radius r centered at the origin. HW6-The Distance Formula HW7-A Snack in the Middle HW18-Defining Circles Meadows or Malls? HW2-Heavy Flying Throughout the unit students meet this benchmark. However, this unit is marked by the shift into matrices to represent and solve linear programming problems. Key activities include: CW-Inventing an Algebra CW-Flying Matrices HW28-Matrices in the Oven HW29-Fresh Ingredients New Mexico Mathematics Unifying Concepts and Processes CW-“Meadows or Malls?” Revisited Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /67 STANDARD CORRELATION 13.1 continued Small World, Isn’t it? CW-What a Mess! HW13-The Growth of the Oil Slick HW22-The Sound of a Logarithm Pennant Fever CW-Formulas for nPr and nCr HW30-Binomial Probabilities As the Cube Turns HW15-Comin‟ Round Again (and Again…) CW-Rotation Matrix in Three Dimensions The World of Functions CW-Quadratic Tables by Algebra HW6-A General Quadratic The Pollster’s Dilemma • CW-The Search is On 13.2 Use tables and graphs as tools to interpret Students use tables and graphs to interpret expressions, equations, and inequalities expressions, equations, and inequalities. throughout the four years of the IMP. Students are often directed to use tables and graphs by the curriculum itself. They become part of the student strategy for solving problems. Furthermore, many algebraic generalizations are developed through tabular or graphical analyses. A few key examples are noted below. The Overland Trail Students build the connections among tables, graphs, rules, and the original situation. HW13-Situations, Graphs, Tables, and Rules • HW14-Rules, Tables, and Graphs Solve It! CW-The Graph Solves the Problem CW-The Graphing Calculator Solver Cookies Students interpret and solve two-variable linear programming problems by generating a graph of the feasible region defined by the constraints presented. New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /68 STANDARD CORRELATION 13.2 continued All About Alice HW9-It‟s in the Graph CW-All Roads Lead to Rome CW-Taking Logs to the Axes Small World, Isn’t It? HW27-Comparing Derivatives High Dive The unit problem cannot be solved using symbolic manipulation. Many students will solve the equation they’ve developed by determining the point of intersection on a graph. CW-Moving Cart, Turning Ferris Wheel CW-“High Dive” Concluded The World of Functions CW-Functions in Verse HW25-Linear Functions in Verse • CW-Double Dose of Functions New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /69 STANDARD CORRELATION New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /70 13.3 Operate on expressions and matrices and Overland Trail solve equations and inequalities. HW8-To Kearney by Equation Shadows HW13-Inventing Rules CW-The Sun Shadow Solve It! HW7-You‟re the Jester HW9-More Letters, Numbers, and Mystery Bags HW25-All by Itself HW26-More Variable Solutions HW27-A Mixed Bag Do Bees Build It Best? HW18-Simply Square Roots Cookies HW13-Getting on Good Terms HW18-More Linear Systems CW-“How Many of Each Kind?” Revisited HW19-A Charity Rock HW20-Back on the Trail Fireworks • CW-Fireworks Height Revisited Meadows or Malls? Students develop the necessary techniques and understanding of matrix operations to solve matrix equations of any size. HW31-Solving the Simplest CW-Finding an Inverse HW33-Inverses and Equations High Dive HW2-As the Ferris Wheel Turns HW9-Not So Spectacular CW-Moving Cart, Turning Ferris Wheel HW25-Falling Time for Vertical Motion New Mexico Mathematics Unifying Concepts and Processes CW-High Dive Concluded Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /71 STANDARD CORRELATION 13.3 continued The World of Functions • CW-Back to the Corral 13.4 Identify the implications and usefulness of Shadows mathematical abstractions and symbolism. HW21-The Sun Shadow Problem Solve It! HW30-A Solving Sampler Is There Really a Difference? HW25-Beginning Portfolio Selection Do Bees Build it Best? HW28-A Portfolio of Formulas All About Alice HW17-An Exponential Portfolio Meadows or Malls? HW37-Beginning Portfolios – Part II Small World, Isn’t It? HW30-Beginning Portfolios – Part I 13.5 Use matrices to solve linear systems. In the unit Meadows or Malls? students learn to express linear systems in terms of matrices and develop the matrix operations required to understand the role of matrices in the solution process. Some examples are: CW-Finding an Inverse CW-Calculators Again CW-“Meadows or Malls?” Revisited 13.6 Apply trigonometry to problem situations Shadows involving triangles. CW-The Tree and the Pendulum HW23-Smokey and the Dude CW-The Sun Shadow Do Bees Build It Best? HW9-More Gallery Measurements HW10-Shadows and Sailboats HW20-Leslie‟s Floral Angles New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /72 13.6 continued Orchard Hideout CW-Hexagoning the Circle HW12-Octagoning the Circle CW-Polygoning the Circle CW-Cable Ready CW-The Other Gap CW-Lines of Sight for Radius Six CW-Hiding in the Orchard High Dive Students will use right triangle trigonometry throughout this unit. Some examples include: HW2-As the Ferris Wheel Turns HW3-A Clear View CW-The Ideal Skateboard HW27-Racing the River As the Cube Turns HW11-Goin‟ Round the Origin HW12-Double Trouble CW-The Sine of a Sum HW13-A Broken Button HW14-Oh, Say What You Can See HW15-Comin‟ Round Again (and Again…) HW19-What‟s Going on Here? CW-Follow That Point! HW31-One Turn of a Cube CW-Rotation Matrix in Three Dimensions Know How CW-Radian Measure HW2-The Law of Sines The World of Functions • HW24-Fish, Ladders, and Bacteria New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /73 STANDARD CORRELATION 13.7 Determine maximum and minimum The Overland Trail points of a graph and interpret the results HW2-Hats for Families in problem situations. Do Bees Build it Best? HW16-Don‟t Fence Me In (determine maximum value) Cookies In the context of solving linear programming problems, students determine the maximum or minimum of a linear function over a region. For example: CW-Profitable Pictures HW8-Curtis and Hassan CW-Big State U CW-“How Many of Each Kind” Revisited CW-Producing Programming Problems Fireworks HW9-Corrals and Pens Again Small World, Isn’t It? HW21-The Significance of a Sign High Dive HW5-Ferris Wheel Graph Variations 13.8 Describe limiting processes by examining The Pit and the Pendulum infinite sequences, series, and areas under HW10-An (AB)Normal Rug (estimate area under curve) curves. Do Bees Build It Best? CW-Building the Best Fence All About Alice HW7-Continuing the Pattern in Alice Orchard Hideout Day11-14: Students analyze the proportionality constant between the area of a polygon and the radius of an inscribed circle, following the trend in this constant of proportionality as the number of sides increases towards infinity. This work deepens or makes clearer the meaning of for students. This analysis is also done for the ratio between radius and perimeter. New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /74 STANDARD CORRELATION 13.8 continued Small World, Isn’t It? CW-Find That Base! CW-The Generous Banker HW28-The Limit of Their Generosity The World of Functions CW-Don‟t Divide That! CW-The End of the Function HW12-Creating the Ending You Want The Pollster’s Dilemma CW-Normal Areas New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /75 13.9 Evaluate the logic of algebraic procedures. The Overland Trail CW-Ox Expressions HW9-Ox Expressions at Home Shadows HW13-Inventing Rules HW16-Ins and Outs of Proportion Solve It! Students develop generalizable techniques to solve equations and see that algebraic methods will solve any linear equation. HW6-The Mystery Bag Gam CW-Scrambling Equations HW21-More Scrambled Equations and Mystery Bags CW-Old Friends and New Friends HW22-New Friends Visit Your Home HW23-From One Variable to Two Students develop the Distributive Property. HW12-Who‟s Right? CW-A Lot of Changing Sides HW13-Why Are They Equivalent? HW14-One Each Way CW-Distributing the Area HW15-The Distributive Property and Mystery Lots HW16-Views of the Distributive Property • CW-A Distributive Summary STANDARD CORRELATION New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /76 13.9 continued Students understand subtraction of sums in parentheses. CW-Taking Some Out, Part I HW18-Subtracting Some Sums HW19-Randy, Sandy, and Dandy Return Cookies HW2-Investigating Inequalities CW-My Simplest Inequality HW3-Simplifying Cookies CW-Get the Point All About Alice Students develop meaning of exponential and logarithmic properties and laws via contextual situations, algebraic laws, graphing, and number patterns. CW-Here Goes Nothing CW-Many Meals for Alice HW5-In Search of the Law CW-A Half Ounce of Cake HW11-Confusion Reigns CW-All Roads Lead to Rome Fireworks Students develop and discuss meaningful methods to “complete the square”. CW-Who‟s Perfect? • HW6-More About Perfection Meadows or Malls? Day18-19: Students develop the notion of solving systems by Gaussian elimination. Day 32-33: After students develop elimination methods to solve systems more easily, they develop the necessary ideas regarding multiplicative inverse and the identity matrix in order to solve matrix equations. High Dive HW13-Planning for Formulas Know How HW5-A Fractional Situation The Pollster’s Dilemma CW-The Search is On New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /77 New Mexico Mathematics Unifying Concepts and Processes Correlation with Interactive Mathematics Program, Year 1-4 Key Curriculum Press /78

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