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APS DISTRICT HIGH SCHOOL MATHEMATICS CURRICULUM FRAMEWORK Course Title: Interactive Math Program 2 (IMP 2) Course Number: 36044 Department: Mathematics ADS Number: 20024133 Prerequisites: Successful completion of IMP Year 1 or equivalent Length of Course: One Year Credit/PRI Area: .50 per Sem/Mathematics Grade Level(s): 10 - 11 Important Notes: The Interactive Mathematics Program (IMP) is an alternative 4-year curriculum for high school mathematics. The IMP curriculum includes the same topics covered in the standard high school curriculum (Algebra I, Geometry, Algebra II, Pre-Calculus), but in an integrated format. It also includes topics in probability, statistics, discrete mathematics, and matrix algebra. The curriculum emphasizes the use of critical thinking, problem solving, communication, collaboration, and technology. The IMP is closely aligned with the National Council of Teachers of Mathematics (NCTM) Principles and Standards documents. COURSE DESCRIPTION: The Interactive Mathematics Program (IMP) is a progressive four-year integrated, problem-centered mathematics curriculum. Graphing calculators and computer technology are used to enhance student understanding. The student explores open-ended situations in a way that closely resembles mathematics and scientists in their work. In the second year of IMP, the student works with powerful mathematical ideas, including the chi-square, the Pythagorean Theorem, and linear programming and learns a variety of approaches to solving equations. The student applies statistical tools to analyze two different sets of data, discovers geometric relationships, employs graphical reasoning to solve problems, and examines exponents. Problem contexts include statistical comparison of populations, the geometry of a honeycomb, and the optimizing of profits from a small business. References in parentheses following each performance standard refer to and are aligned with the State Mathematics Standards (MA), Albuquerque Public Schools District Mathematics Standards (APS) located at www.aps.edu, and the APS Language Arts Standards. Power standards are in italics. INTERACTIVE MATH PROGRAM 2 (IMP 2) 6.1.25 Albuquerque Public Schools STRATEGIES: The “Illustrations” column in the Program of Studies provides exemplars of the performance standards, strategies, and best practices based on current educational research. ASSESSMENTS: The “Illustrations” column also incorporates a variety of assessments and “check for” items, suggested by APS mathematics teachers. Assessments include: authentic and performance-based assessment, cooperative learning, on-going teacher observations, checklists, rubrics, formal and informal writing, self and peer assessments, small group and full class discussions, oral and multimedia presentations, projects, demonstrations, and portfolios/notebooks. SUGGESTED TEXTBOOKS AND INSTRUCTIONAL MATERIALS: Graphing calculators The Geometer's Sketchpad® software Fathom Dynamic Statistics™ software Teaching Handbook for the Interactive Mathematics Program A Teacher-to-Teacher Guide by Lori Green IMPressions - a newsletter published by Key Curriculum Press® each fall and spring for the Interactive Mathematics Program. You can request your free subscription online. The following is available from Key Curriculum Press at http://www.keypress.com. IMP Year 2 IMP Year 2 Student Textbook Solve It! Teacher’s Guide Is There Really a Difference? Teacher's Guide Do Bees Build It Best? Teacher's Guide Cookies Teacher's Guide All About Alice Teacher's Guide IMP Year 2 Calculator Guide for the TI-81, TI-82, and TI-83 Applicable to All IMP Years Classroom Manipulatives Kit It's All Write: A Writing Supplement for High School Mathematics SUGGESTED WEBSITES: http://www.nctm.org - the National Council of Teachers of Mathematics site http://www.mathimp.org – Interactive Mathematics Resource Center site http://www.keypress.com – Key Curriculum Press site Approved by HSCA: April, 2005 INTERACTIVE MATH PROGRAM 2 (IMP 2) 6.2.25 Albuquerque Public Schools STRAND I: GUIDING PRINCIPLES CONTENT STANDARD: The student identifies guidelines that provide focused, aligned, and sustained efforts to ensure that all students have access to high quality mathematics education. BENCHMARK: The student examines and integrates equity, curriculum, teaching, learning, assessment, and technology principles that establish the foundation for developing the student’s capability to mathematically reason and solve problems. GRADE PERFORMANCE STANDARDS ILLUSTRATIONS 10 - 11 The following Guiding Principles are extracted from the State of New NOTE: The student meets standards through creative classroom Mexico Mathematics Curriculum Frameworks document adopted in June, instruction, authentic assessments, high expectations, problem solving, 2002. multiple learning opportunities to apply skills, and technology support. The development and delivery of a successful mathematics program provides rich mathematical experiences accessible to ALL students. 1. Equity: Excellence in mathematics requires equity, including high expectations and strong support for all students. The IMP curriculum is structured around learning activities that meet the 2. Curriculum: A curriculum is more than a set of activities—it must be standards identified in this document. These activities vary from year to coherent, focused on important mathematical content, and clearly year. The following activities are specifically designed for the IMP Year 2 articulated across grades. curriculum and provide the basis for most of the illustrations used in the following strands. 3. Teaching: Effective mathematics teaching requires understanding what students know and need to learn and then challenging and supporting Solve It! them to learn it well. The focus of this unit is on using equations to represent real-life situations and on developing the skills to solve these equations. The student begins 4. Learning: Students must learn mathematics with understanding, actively with situations used in the first year of the curriculum and develops acquiring new knowledge from experience and prior knowledge. algebraic representations of problems. In order to find solutions to the equations that occur, the student explores the concepts of equivalent 5. Assessment: Multiple and varied assessments, both summative and expressions and equivalent equations. Using these concepts the student formative, should support the learning and furnish useful information to develops principles such as distributive property for working with both teachers and students. algebraic expressions and equations and he/she learns methods that he/she can use to solve any linear equations. The student also explores the 6. Technology: Technology is essential; it influences the mathematics that is relationships between an algebraic expression, a function, an equation, and taught and enhances student learning. a graph, and examines how to use graphs to solve nonlinear equations. Is There Really a Difference? In this unit, the student collects data and compares different population groups to one another. The student concentrates on this question: INTERACTIVE MATH PROGRAM 2 (IMP 2) 6.3.25 Albuquerque Public Schools GRADE PERFORMANCE STANDARDS ILLUSTRATIONS 10 - 11 If a sample from one population differs in some respect from a sample from a different population, how reliably can you infer that the overall populations differ in that respect? The student begins by making double bar graphs of some classroom data and explores the process of making and testing hypotheses. The student realizes that there is variation even among different samples from the same population and sees the usefulness of the concept of a null hypothesis as he/she examines this variation. The student builds on his/her understanding of standard deviation from the Year I unit The Pit and the Pendulum and learns that the chi-square statistic can give him/her the probability of seeing differences of a certain size in samples when the populations are really the same. The student’s work in this unit culminates in a two-week project in which he/she proposes a hypothesis about two populations that he/she thinks really differ in some respect. The student then collects sample data about the two populations and analyzes his/her data by using bar graphs, tables, and the chi-square statistic. Do Bees Build It Best? In this unit the student works on this problem: Bees store their honey in honeycombs that consist of cells they make out of wax. What is the best design for a honeycomb? To analyze this problem the student begins by learning about area and the Pythagorean Theorem. Then using the Pythagorean Theorem and trigonometry, the student finds a formula for the area of a regular polygon with fixed perimeter and finds that the larger the number of sides, the larger the area of the polygon. The student then turns his/her attention to volume and surface area focusing on prisms that have a regular polygon as the base. The student finds that for such prisms – he/she also wants the honeycomb cells to fit together –the mathematical winner in terms of maximizing volume for a given surface area is a regular hexagonal prism, which is essentially the choice of the bees. Cookies The focus of this unit is on graphing systems of linear inequalities and solving systems of linear equations. Although the central problem is one in linear programming, the student learns how to manipulate equations and how to reason using graphs. The student begins by considering a classic type of linear programming problem in which he/she is asked to maximize the profits of a bakery that makes plain and iced cookies. The student is constrained by the amount of INTERACTIVE MATH PROGRAM 2 (IMP 2) 6.4.25 Albuquerque Public Schools GRADE PERFORMANCE STANDARDS ILLUSTRATIONS 10 - 11 ingredients he/she has on hand and the amount of oven and labor time available. First the student works toward a graphical solution of the problem. He/She sees how the linear function can be maximized or minimized by studying the graph. Because the maximum or minimum point he/she is looking for is often the intersection of two lines, the student investigates a method for solving two equations in two unknowns. The student then returns to work in groups on the cookie problem, each group presenting both a solution and a proof that its solution does maximize profits. Finally, each group invents its own linear programming problem and makes a presentation of the problem and its solution to the class. All About Alice This unit starts with a model based on Lewis Carroll’s Alice in Wonderland, a story in which Alice’s height is doubled or halved by eating or drinking certain foods she finds. The student discusses this situation and comes up with the basic principles for working with exponents – positive, negative, zero, and even fractional – and an introduction to logarithms. Building on the work with exponents, the student discusses scientific notation and the manipulation of numbers written in scientific notation. INTERACTIVE MATH PROGRAM 2 (IMP 2) 6.5.25 Albuquerque Public Schools STRAND II: PROCESS STANDARDS CONTENT STANDARD: The student understands and uses mathematical processes. BENCHMARK: The student uses problem solving, reasoning and proof, communication, connections, and representations as appropriate in all mathematical experiences. GRADE PERFORMANCE STANDARDS ILLUSTRATIONS 10 - 11 The following State of New Mexico and APS Mathematics standards align NOTE: Illustrations include suggested activities for attaining each with the National Council of Teachers of Mathematics Standards (NCTM). performance standard. Key features to look for while assessing student performance are described in each unit. Processes of Problem Solving 1. Builds new mathematical knowledge through problem solving. 1 – 6. See Strand III, the illustration for performance standard #5. 2. Solves problems that arise in mathematics and other contexts. 3. Applies and adapts a variety of appropriate strategies to solve problems (APS – I.2). 4. Monitors and reflects on the process of problem solving (APS – I.4). 5. Draws on diverse knowledge and methods to solve problems (APS – I.3). 6. Generalizes problems (APS – I.3, I.4). Reasoning and Proof 7. Recognizes reasoning and proof as fundamental aspects of mathematics. 7 – 10. See Strand IV, the illustration for performance standards #6, #7. 8. Makes and investigates mathematical conjectures (APS – I.5). 9. Develops and evaluates mathematical arguments and proofs (APS – I.5). 10. Selects and uses various types of reasoning and methods of proof. Communication 11. Organizes and consolidates their thinking through communication. 11 – 14. The student describes the steps he/she took in attempting to solve the problem named below and explains his/her reasoning. He/She states the 12. Communicates their mathematical thinking coherently and clearly to solution as clearly as possible and writes it in a way that is convincing to peers, teachers, and others (APS – I.9). someone else—even someone who initially disagrees with his/her solution. [The Standard POW Write-up from Solve It!] 13. Analyzes and evaluates the mathematical thinking and strategies of organization of thinking others (APS – I.10). INTERACTIVE MATH PROGRAM 2 (IMP 2) 6.6.25 Albuquerque Public Schools GRADE PERFORMANCE STANDARDS ILLUSTRATIONS 10 - 11 14. Uses the language of mathematics to express mathematical ideas analysis precisely. effective communication appropriate language usage Connections 15. Recognizes and uses connections among mathematical ideas. 15 – 23. The student responds to the following scenario: Speedy is the star runner for her country’s track team. Among other things, 16. Understands how mathematical ideas interconnect and build on one she runs the last 400 meters leg of the 1600-meter relay race. A sports another to produce a coherent whole. analyst recently studied the film of a race in which she competed. The analyst came up with this formula to describe the distance Speedy had run 2 17. Recognizes and applies mathematics in contexts outside of mathematics at a given time in the race m(t) = 0.1t + 3t. In this formula, m(t) gives (APS – I.15). the number of meters Speedy had run after t seconds on the race, with both time and distance measured from the beginning of here 400-meter segment 18. Prepares mathematically for future careers (APS – I.14). of the race. (This formula might not be very accurate, but you are to work on this activity as if it were completely correct.) The student: 19. Identifies how seemingly different mathematical situations may be uses the formula for m(t) to fill in several rows of an In-Out table to essentially the same (e.g., the intersection of 2 lines is the same as the show how far Speedy had run at different times of the race. solution to a system of linear equations) (APS – I.13). uses the table to make a graph that represents this situation. He/She may need to add more information to the table in order to obtain a Representation graph that shows the entire time she is running. 20. Creates and uses representations to organize, record, and communicate writes an equation using the variable t to get the answer to the mathematical ideas. question “How long does it take for Speedy to run her first 200 meters?” 21. Selects, applies, and translates among mathematical representations to graphs the function m(t) = 0.1t2 + 3t on the graphing calculator and solve problems. uses the graph to get an approximate solution. [Where’s Speedy from Solve It!] 22. Uses representations to model and interpret physical, social, and accuracy mathematical phenomena. all required components 23. Uses a variety of mathematical representations that can be used mathematical applications purposefully and appropriately interchangeably in all four years use of technology (e.g., pictures, written symbols, oral language, real-world situations, and problem-solving strategies manipulative models) (APS - I.16). Collaboration 24. Shares ideas and works cooperatively with others (APS – I.8). 24 – 26. See Strand V, the illustration for performance standards #14, #16. 25. Subdivides a task so that group members can work independently on different parts of it (APS – I.1). 26. Describes methods used to approach a problem (APS – I.4). INTERACTIVE MATH PROGRAM 2 (IMP 2) 6.7.25 Albuquerque Public Schools STRAND III: ALGEBRA, FUNCTIONS, AND GRAPHS CONTENT STANDARD: The student understands algebraic situations and applications. BENCHMARKS: A. The student represents and analyzes mathematical situations and structures using algebraic symbols. B. The student understands patterns, relations, functions, and graphs. C. The student uses mathematical models to represent and understand quantitative relationships. D. The student analyzes change in various contexts. GRADE PERFORMANCE STANDARDS ILLUSTRATIONS 10 - 11 Benchmark A: The student represents and analyzes mathematical situations and structures using algebraic symbols. 1. Classifies numbers and members of the following sets (NM - I.A.1): 1, 3. The student learns in this activity that eating an ounce of this cake rationals doubles Alice’s height. He/She determines by what her height multiplies irrationals if Alice eats half an ounce of cake. (Hint: Keep in mind that eating half an ounce of cake and then eating another half ounce should have the same effect as eating one ounce of cake.) [A Half Ounce of Cake from All About Alice] applications accuracy 2. Simplifies numerical expressions using the order of operations, including 2, 3. The student solves a variety of problems similar to the following: exponents (NM - I.A.2). Solve each of these equations, explaining each step clearly. 8w – 3(2w – 9) = 7(w + 2) 3. Evaluates the numerical value of expressions of one or more variables that (x + 2)2 = 16 are (NM - I.A.3): [Homework 21: More Scrambled Equations and Mystery Bags from Solve polynomial , It!] rational, and accuracy radical. documentation of work clear explanations understanding of key concepts (e.g., order of operations) p/q 4. Simplifies algebraic monomial expressions raised to a power 4, 10. The student gives a general way of defining 2 for any fraction p/q [e.g., (5xy2 )3] and algebraic binomial [e.g., (5x2 + y)2] expressions raised and explains his/her ideas. to a power (NM - I.A.4). [Homework 10: Stranger Pieces of Cake from All About Alice] clarity in communication insights accuracy INTERACTIVE MATH PROGRAM 2 (IMP 2) 6.8.25 Albuquerque Public Schools GRADE PERFORMANCE STANDARDS ILLUSTRATIONS 10 - 11 5. Represents and analyzes relationships using written and verbal 5, 16, 19, 20, 22, 23, 28, 29, 31. The student solves the following problem: expressions, tables, equations, and graphs, and describes the connections among those representations (NM - I.A.6): Hassan paints both watercolors and pastels. Each type of picture takes him translates from verbal expression to algebraic formulae about the same amount of time to paint. He figures he has time to do a total (e.g., “Set up the equations that represent the data in the of at most 16 pictures. The materials for each pastel cost him $5, and the following equation: John’s father is 23 years older than John. materials for each watercolor cost him $15. He has $180 to spend on John is 4 years older than his sister Jane. John’s mother is 3 materials. He makes a profit of $40 on each pastel and a profit of $100 on years younger than John’s father. John’s mother is 9 times as each watercolor. He/She: old as Jane. How old are John, Jane, John’s mother, and expresses Hassan’s constraints as inequalities, using p to represent John’s father?”), the number of pastels he does and w to represent the number of given data in a table, constructs a function that represents these watercolors. data (linear only), and makes a graph that shows Hassan’s feasible region. In other words, given a graph, constructs a function that represents the graph the graph should show all the combinations of watercolors and (linear only). pastels that satisfy his constraints. for at least five points on your graph, finds the profit that Hassan makes for that combination. writes an algebraic expression to represent Hassan’s profit in terms of p and w. [Homework 7: Picturing Pictures from Cookies] multiple representations all required components accuracy 6. Knows, explains, and uses equivalent representations for the same real 6. The student applies a variety of number representations in real-world number including (NM I.A.7): situations similar to the following: integers, decimals, The mass of the earth is 5.98 x 1024 kg. The mass of the sun is percents, 1.99 x 1030 kg. Approximately how many earths would it take to have ratios, the same mass as the sun? [Big Numbers from All About Alice] scientific notation, applications numbers with integer exponents, connections inverses (reciprocal), and accuracy prime factoring. 7. Simplifies algebraic expressions using the distributive property 7, 9. The student finds an equivalent expression without parentheses for (NM -I.A.8). (r + 4)(r + 3) and explains his/her work. [Homework 14: One Each Way from Solve It!] accuracy use of appropriate property effective communication INTERACTIVE MATH PROGRAM 2 (IMP 2) 6.9.25 Albuquerque Public Schools GRADE PERFORMANCE STANDARDS ILLUSTRATIONS 10 - 11 8. Explains and uses the concept of absolute value (NM - I.A.9). 8, 25. The student determines how far he/she is from mile marker 47 if he/she is parked at mile marker 23 and later at mile marker 7. 9. Knows, explains, and uses equivalent representations for algebraic He/She explains his/her answers. expressions (NM - I.A.10). clear explanation 10. Simplifies square roots and cube roots with monomial radicands that are accuracy perfect squares or perfect cubes (e.g., 9a2x4) (NM - I.A.11). 11. Calculates powers and roots of real numbers, both rational and irrational 11, 30. The student determines how many ounces of base-10 cake Alice (NM - I.A.12). should eat to grow to 50 times her size. (For each ounce of base-10 cake Alice eats, her size increases 10 times.) [Homework 14: Alice on a Log from All About Alice] accurate calculations 12. Solves formulas for specified variables and 12. The student solves the equation for m in terms of W and v. The radical equations involving one radical: (NM - I.A.13). equation for kinetic energy of a moving object is given by the formula 2 W = 1/2mv , where W represents the kinetic energy, m represents the mass of the object, and v is the object’s velocity. [Homework 26: More Variable Solutions from Solve It!] manipulation of variables accuracy 13. Factors polynomials, difference of squares and perfect square trinomials, 13. The student uses the area model to factor the polynomial expression and the sum and difference of cubes (NM - I.A.14). 2 x – 9. [Homework 15: The Distributive Property and Mystery Lots from Solve It!] correct factorization understanding of patterns 14. Manipulates simple expressions with positive and negative exponents 14, 30, 33. The student by now has seen other ways to make sense of (NM - I.A.16). exponents that are not positive integers (i.e., the Alice story; the extension of a numerical pattern that starts with positive integer exponents; the definition that is consistent with the additive law of B C B+C exponents, which says A gA = A ; making a graph of the equation y A using positive integer values for x and using the x graph to estimate y for negative values of x.). Building on that -2 prior knowledge, he/she explains the meaning of 3 using as many of the four ways that make sense. [All Roads Lead to Rome from All About Alice] application of concepts INTERACTIVE MATH PROGRAM 2 (IMP 2) 6.10.25 Albuquerque Public Schools GRADE PERFORMANCE STANDARDS ILLUSTRATIONS 10 - 11 ability to use negative exponents accuracy 15. Uses the four basic operations (+, -, x, ÷) with (NM I.A.17): 15, 26, 33. The student solves the following problem: linear expressions. Yolanda jogged one mile to a lake, twice around the lake, and then jogged Benchmark B: The student understands patterns, relations, functions, one mile home. Altogether she jogged a total of six miles. How far is it and graphs. around the lake? Now suppose Yolanda jogged one mile to the lake, went 16. Translates among tabular, symbolic, and graphical representations of n times around the lake, and then back home. Create an In-Out table with functions (NM - I.B.4). the In being the number of times Yolanda went around the lake and the Out being the total distance she jogged. Find a rule for your In-Out table, using N for the In and d for the out. In other words, write an equation for d as a function of N, d = f(N). accurate computations use of a table accurate equation problem-solving strategies 17. The student finds the area of a square given the formula A(s) s 2 , 17. Explains and uses function notation (NM - I.B.5). where s represents the length of a side. [Homework 16: Don’t Fence Me In from Do Bees Build It Best?] manipulation of formulas use of correct notation accuracy 18, 21, 24, 27. For each of these pairs of equations, the student finds the point 18. Identifies the independent and dependent variables from an application of intersection of his/her graphs by a method other than graphing or guess- problem (e.g., height of a child) (NM - I.B.7). and-check and then verfifies the solution by graphing or by substituting the values into the pair of equations. 19. Describes the concept of a graph of an equation (NM - I.B.8). y = 3x and y = 2x + 5 Benchmark C: The student uses mathematical models to represent and 7x – 3y = –2 and 2y + 3 = 3x understand quantitative relationships. [Get the Point from Cookies] 20. Models real-world phenomena using linear and quadratic equations and understanding of systems linear line qualities (e.g., apply algebraic techniques to solve rate multiple methods of solving problems, work problems, and percent mixture problems; solves problems documentation if work that involve discounts, markups, commissions, and profit and computes accuracy simple and compound interest; applies quadratic equations to model throwing a baseball in the air ) (NM - I.C.1). INTERACTIVE MATH PROGRAM 2 (IMP 2) 6.11.25 Albuquerque Public Schools GRADE PERFORMANCE STANDARDS ILLUSTRATIONS 10 - 11 21. Uses a variety of computational methods (e.g., mental arithmetic, paper and pencil, technological tools) (NM - I.C.2). 22. Expresses the relationship between two variables using a table with a finite set of values and graph the relationship (NM - I.C.3). 23. Expresses the relationship between two variables using an equation and a graph (NM - I.C.4): graphs a linear equation and linear inequality in two variables, solves linear inequalities and equations in one variable, solves systems of linear equations in two variables and graph the solutions, and uses the graph of a system of equations in two variables to help determine the solution. 24. Solves applications involving systems of equations (NM - I.C.5). 25. Evaluates numerical and algebraic absolute value expressions (NM - I.C.6). 26. Creates a linear equation from a table of values containing co-linear data (NM - I.C.7). 27. Determines the solution to a system of equations in two variables from a given graph (NM - I.C.8). 28. Generates an algebraic sentence to model real-life situations (NM - I.C.9). 29. Writes an equation of the line that passes through two given points (NM - I.C.10). 30. Understands and uses NM - I.C.11): such operations as taking the inverse, finding the reciprocal, taking a root, and raising to a fractional power and the rules of exponents . 31. Verifies that a point lies on a line, given an equation of the line, and derives linear equations by using the point-slope formula (NM - I.C.12). Benchmark D: The student analyzes changes in various contexts. INTERACTIVE MATH PROGRAM 2 (IMP 2) 6.12.25 Albuquerque Public Schools GRADE PERFORMANCE STANDARDS ILLUSTRATIONS 10 - 11 32. Analyzes the effects of parameter changes on these functions (NM I.D.1): 32, 35, 36. The student investigates linear functions and straight-line graphs linear (e.g., changes in slope or coefficients), by answering the following questions: quadratic (e.g., f[x-a] changes coefficients and constants), and How do you change an equation in order to change the “slant” of its exponential [e.g., changes caused by increasing x(x + c) or (a x)]. graph? How do you change an equation in order to shift the whole graph up or down? 33. Solves routine two- and three-step problems relating to change using How do two linear functions give horizontal lines? parallel lines? concepts such as (NM - I.D.2): perpendicular lines? two lines that are mirror images of each other? exponents, Why? [Get It Straight from Solve It!] factoring, understanding of slope ratio, clear communication proportion, accuracy average, and percent. 34. Calculates the percentage of increase and decrease of a quantity 34, 36. The student determines the percent increase in the area of a square if (NM - I.D.3). the length of its side is increased 10%? [Homework 17: More Fencing, Bigger Corals from Do Bees Build It 35. Estimates the rate of change of a function or equation by finding the slope Best?] between two points on the graph (NM - I.D.5). accuracy 36. Evaluates the estimated rate of change in the context of the problem (NM I.D.6). TOPICS FOR FURTHER STUDY TOPICS FOR FURTHER STUDY Solving Equations, Inequalities and Systems See Strand I, Guiding Principles for descriptions of the curricular units As the student encounters ever more sophisticated mathematical situations, designed for IMP 2. Through these units, the student performs specific tasks he/she will need to be able to generate and solve a variety of equations, that meet the standards. inequalities, and systems. He/She begins by studying more complex linear and quadratic equations and systems. The student (NM IV.1): solves two-by-two linear quadratic and quadratic-quadratic systems Cookies solves and graphs equations and inequalities involving absolute value Cookies Logarithms and Exponential Functions Logs and exponential functions provide tools for more sophisticated modeling and applications for understanding real-life phenomena. Higher mathematics requires regular and successful use of logs and exponents to move beyond INTERACTIVE MATH PROGRAM 2 (IMP 2) 6.13.25 Albuquerque Public Schools GRADE PERFORMANCE STANDARDS ILLUSTRATIONS 10 - 11 polynomials.) The student (NM IV.2): operates with logs and exponential functions on the basis of their inverse relationship All About Alice uses exponential functions and common and natural logs to understand real-life situations (e.g., half-life, amortization, logistic growth) All About Alice uses logs and exponents to solve equations All About Alice INTERACTIVE MATH PROGRAM 2 (IMP 2) 6.14.25 Albuquerque Public Schools STRAND IV: GEOMETRY AND TRIGONOMETRY CONTENT STANDARD: The student understands geometric concepts and applications. BENCHMARKS: A. The student analyzes characteristics and properties of two- and three-dimensional geometric shapes and develops mathematical arguments about geometric relationships. B. The student specifies locations and describes spatial relationships using coordinate geometry and other representational systems. C. The student applies transformations and uses symmetry to analyze mathematical situations. D. The student uses visualization, spatial reasoning, and geometric modeling to solve problems. GRADE PERFORMANCE STANDARDS ILLUSTRATIONS 10 - 11 Benchmark A: The student analyzes characteristics and properties of two- and three-dimensional geometric shapes and develops mathematical arguments about geometric relationships. 1. Interprets and draws two-dimensional objects and finds the area and 1. The student considers only regular polygons for the shape of a coral as perimeter of basic figures [e.g., rectangles, circles, triangles, other he/she completes the following activity. polygons (e.g., rhombi, parallelograms, trapezoids)] (NM - II.A.1). Choose a value greater than 5 for the number of sides. What would the area of the coral be if you built it in the shape of a regular polygon with this many sides. You have only 300 feet of fencing. Repeat Question 1 for another regular polygon with more than five sides. Generalize the process. That is, suppose you have a polygon with n sides with a perimeter of 300 feet. Develop a formula for the area of the coral in terms of n. [Building the Best Fence from Do Bees Build It Best?] completion of all steps of the activity analysis individual participation 2. Finds the area and perimeter of a geometric figure composed of a 2. The student finds the area and the perimeter of a standard 400 meter combination of two or more rectangles, triangles, and/or semicircles with running track with 100-meter straightaways. just edges in common (NM - II.A.2). accuracy understanding of area and perimeter 3. Finds and uses measures of sides and interior and exterior angles of 3. The student responds to the following scenario: triangles and polygons to classify figures [e.g., scalene, isosceles, and equilateral triangles; rectangles (square and nonsquare); other convex A rug designer decided to make a rug consisting of three separate square polygons] (NM - II.A.3). pieces sewn together at their corners, with an empty triangular space between them. Use different size grid squares to construct tri-square rugs. INTERACTIVE MATH PROGRAM 2 (IMP 2) 6.15.25 Albuquerque Public Schools GRADE PERFORMANCE STANDARDS ILLUSTRATIONS 10 - 11 For each rug, determine which of the following three categories your rug fits into and justify your answer: the areas of the two smaller squares is smaller than the area of the largest square, the areas of the two smaller squares is greater than the area of the largest square, and the areas of the two smaller squares is equal to the area of the largest square. Look for a pattern in your results. Focus specifically on the triangular space between the squares. [Tri-Square Rug Games from Do Bees Build It Best?] completion of all steps of the activity analysis individual participation 4. Interprets and draws three-dimensional objects and finds the surface area 4, 11. For each prism, the student finds the value of each of these and volume of basic figures (e.g., spheres, rectangular solids, prisms, measurements: height of the prism, area of the base, volume of the polygonal cones), and calculates the surface areas and volumes of these prism, perimeter of the base, lateral surface area of the prism. figures as well as figures constructed from unions of rectangular solids [Shedding Light on Prisms from Do Bees Build It Best?] and prisms with faces in common, given the formulas for these figures all required components (NM - II.A.4). accuracy understanding of three-dimensional objects 5. Demonstrates an understanding of simple aspects of a logical argument 5. The student responds to the fvollowing scenario: (NM - II.A.5): identifies the hypothesis and conclusion in logical deduction and A pharmaceutical company wants to advertise its new product as an anti- uses counterexamples to show that an assertion is false and acne medicine. The Food and Drug Administration (FDA) is opposed to recognizes that a single counterexample is sufficient to refute an that plan. assertion. What is the company’s hypothesis about its product? What null hypothesis might the FDA want to test? [Homework 5: Questions Without Answers from Is There Really a Difference?] insightful response to questions understanding of logical argument reasoning 6. Demonstrates an understanding of inductive and deductive reasoning, 6, 7. The student recreates one of the better-known proofs of the Pythagorean explains the difference between inductive and deductive reasoning, and Theorem. identifies and provides examples of each (NM - II.A.6): [Pythagorean Proof from Do Bees Build It Best?] INTERACTIVE MATH PROGRAM 2 (IMP 2) 6.16.25 Albuquerque Public Schools GRADE PERFORMANCE STANDARDS ILLUSTRATIONS 10 - 11 for inductive reasoning, demonstrates understanding that individual participation showing a statement is true for a finite number of examples does understanding of the Pythagorean Theorem not show it is true for all cases unless the cases verified are all cases and for deductive reasoning, proves simple theorems. 7. Writes geometric proofs (including proofs by contradiction), including (NM - II.A.7): theorems involving the properties of parallel lines cut by a transversal line and the properties of quadrilaterals, theorems involving complementary, supplementary, and congruent angles, theorems involving congruence and similarity, and the Pythagorean Theorem (tangram proof). Benchmark B: The student specifies locations and describes spatial relationships using coordinate geometry and other representational systems. 8. Given two linear equations, determines whether the lines are parallel, 8. The student investigates linear functions and straight-line graphs and perpendicular, or coincide (NM - II.B.3). answers the following questions: How do two linear functions give horizontal lines? parallel lines? perpendicular lines? two lines that are mirror images of each other? Why? [Get It Straight from Solve It!] response to questions analysis clarity in communication 9. Uses basic geometric ideas (e.g., the Pythagorean Theorem, area, and 9. The student responds to the following scenario: perimeter of objects) in the context of the Euclidean Plane, calculates the perimeter of a rectangle with integer coordinates and sides parallel to the Peter bought a very special pen as a birthday present for an artist friend. coordinate axes and with sides not parallel. (NM - II.B.4). He has a sturdy box that he wants to use to mail the pen. The box is 4 inches wide, 2 inches deep, and 8 inches high. The pen is 10 inches long, so Peter knows he will have to place it in the box along the “long” diagonal. Will the pen fit in this box? Explain your reasoning orally or in writing. [Homework 25: Pythagoras and the Box from Do Bees Build It Best?] support for argument effective communication application of geometric concepts INTERACTIVE MATH PROGRAM 2 (IMP 2) 6.17.25 Albuquerque Public Schools GRADE PERFORMANCE STANDARDS ILLUSTRATIONS 10 - 11 Benchmark C: The student applies and uses symmetry to analyze mathematical situations. 10. Describes the effect of rigid motions on figures in the coordinate plane 10. The student makes some tessellation shapes and uses them to create a and space that include rotations, translations, and reflections tessellation picture. (NM – II.C.1): [POW 9: Tessellation Pictures from Do Bees Build It Best?] determines whether a given pair of figures on a coordinate plane creativity represents the effect of a translation, reflection, rotation, and/or insights dilation and application of symmetrical concepts sketches the planar figure that is the result of a given transformation of this type. Benchmark D: The student uses visualization, spatial reasoning, and geometric modeling to solve problems. 11. Solves problems involving the perimeter, circumference, area, volume, and surface area of common geometric figures (e.g., “Determine the surface area of a can of height h and radius r. How does the surface area change when the height is changed to 3h? How does the surface area change when the radius is changed to 3r? How does the surface area change when both h and r are doubled?”) (NM - II.D.3). 12. Solves problems using the Pythagorean Theorem (e.g., “Given the length 12. The student uses the problems in this assignment to show that the of a ladder and the distance of the base of the ladder from a wall, Pythagorean Theorem can be used to find a variety of different lengths. determine the distance up the wall to the top of the ladder”) (NM – IID.4). Ex: Bonny was doing one of her favorite trick billiard shots. Her shot started at one corner of an 8 foot x 5 foot table, hit the exact center of the back cushion, and rebounded into the other corner. How far did her billiard ball travel? [Homework 14: The Power of Pythagoras from Do Bees Build It Best?] application of the Pythagorean Theorem accuracy INTERACTIVE MATH PROGRAM 2 (IMP 2) 6.18.25 Albuquerque Public Schools GRADE PERFORMANCE STANDARDS ILLUSTRATIONS 10 - 11 13. Understands and uses elementary relationships of basic trigonometric 13, 14. The student uses one of the trigonometric ratios – sine, cosine, or functions defined by the angles of a right triangle (e.g., “What is the tangent - to find the value of h. radius of a circle with an inscribed regular octagon with the length of use of trigonometric ratios each side being exactly 2 feet?”). (NM - II.D.5). accuracy 14. Uses trigonometric functions to solve for the length of the second leg of a right triangle given the angles and the length of the first leg. (e.g., “A h surveyor determines that the angle subtended by a two-foot stick at right angles to his transit is exactly one degree. What is the distance from the transit to the base of the measuring stick?”) (NM - II.D.6). 43° 2.7 m INTERACTIVE MATH PROGRAM 2 (IMP 2) 6.19.25 Albuquerque Public Schools STRAND V: DATA ANALYSIS AND PROBABILITY CONTENT STANDARD: The student understands how to formulate questions, analyze data, and determine probabilities. BENCHMARKS: A. The student formulates questions that can be addressed with data and collects, organizes, and displays relevant data to answer them. B. The student selects and uses appropriate statistical methods to analyze data. C. The student develops and evaluates inferences and predictions that are based on data. D. The student understands and applies basic concepts of probability. GRADE PERFORMANCE STANDARDS ILLUSTRATIONS 10 - 11 Benchmark A: The student formulates questions that can be addressed with data and collects, organizes, and displays relevant data to answer them. 1. Understands the differences between the various methods of data 1 – 5, 12. The student learns that to test a hypothesis about a population, collection (NM - III.A.1). he/she needs to pick a sample that is likely to represent the population accurately. He/She applies that idea in considering the following: 2. Knows the characteristics of a well-designed and well-conducted survey: A member of the City Council wants to know what people in the city (NM - III.A.2): think about a proposed new park in the center of town. The differentiates between sampling and census and councilmember picks names out of the city phone book at random differentiates between a biased and an unbiased sample. and calls them to get a sample of opinions. Do you think names picked from the phone book form a good sample 3. Knows the characteristics of a well-designed and well-conducted of the city’s population? If not, what might be a better way to get a experiment (NM - III.A.3): representative sample. Explain your answer. differentiates between an experiment and an observational Give an example of a conclusion that the council member might study and reach based on a survey of people picked at random from the phone recognizes sources of bias in poorly designed experiments. book that might not be true about the city’s population in general. [Homework 1: Samples and Populations from Is There Really a 4. Understands the role of randomization in well-designed surveys and Difference? experiments (NM - III.A.4). collection of data all required components Benchmark B: The student selects and uses appropriate statistical insights/analysis methods to analyze data. effective communication 5. Understands the meaning of measurement data and categorical data, and understanding of sampling of the term “variable” (NM - III.B.1). 2 6. Understands the meaning of “univariate” (i.e., one variable) and 6. The student uses a 2-by-2 table to compute a statistic that compares two “bivariate” (i.e., two variable) data (NM - III.B.2). populations in terms of whether they possess a given characteristic. Such a table is said to have one degree of freedom. If more than one characteristic is being compared, the degrees of freedom increase and the corresponding probabilities change. [Degrees of Freedom from Is There Really a Difference?] INTERACTIVE MATH PROGRAM 2 (IMP 2) 6.20.25 Albuquerque Public Schools GRADE PERFORMANCE STANDARDS ILLUSTRATIONS 10 - 11 completion of task accuracy 7. For univariate data, displays the distribution and describes its shape using 7, 9. The student makes a frequency bar graph of the class’s coin flip results appropriate summary statistics, and understands the distinction between a and answers the following questions. The graph should show how often statistic and a parameter (NM - III.B.3): heads occurred in each 100-flip set. constructs and interprets frequency tables, histograms, stem and What percentage of the results had exactly 50 heads? leaf plots, and box and whisker plots, What percentage of the results had 55 or more heads? calculates and applies measures of central tendency (mean, About 90% of the results were between what two results? median, and mode) and measures of variability (range, quartiles, [Homework 9: Coin Flip Graph from Is There Really a Difference?] standard deviation), and all required components compares distributions of univariate data using back-to-back response to questions stem and leaf plots and parallel box and whisker plots. accuracy 8. For bivariate data, displays a scatter plot and describe its shape 8. The student creates scatter plots for two different sample populations and (NM – IIIB.4): refers to them to determine if a sample from one population differs in fits a linear model to a set of data using technological tools and some respect from a sample from a different population and how reliably describes and interprets the relationship/correlation between two can one infer that the overall populations differ in that respect? variables using technological tools. [Unit question for Is There Really a Difference?] interpretation Benchmark C: The student develops and evaluates inferences and reasonable conclusion predictions that are based on data. 2 9. Compares and draws conclusions between two or more sets of univariate 9, 10. The student calculates the statistic for appropriate data and applies data using basic data analysis techniques and summary statistics the related probability to determine if there is a difference between (NM - III.C.1). two populations with respect to some criteria. He/She considers the following experiment and then responds to the questions that follow. 10. Draws conclusions concerning the relationships among bivariate data (NM - III.C.2): As a service to his community, Buck conducted an experiment, using a makes predictions from a linear pattern in data, driving simulator to test people’s reflexes. He had some participants use determines the strength of the relationship between two sets of the simulator while sober and others do so while intoxicated beyond the data by examining the correlation, and legal limit. Buck recorded the number of participants in each category who understands that correlation does not imply a cause-and-effect “crashed” and the number who did not. relationship . INTERACTIVE MATH PROGRAM 2 (IMP 2) 6.21.25 Albuquerque Public Schools GRADE PERFORMANCE STANDARDS ILLUSTRATIONS 10 - 11 Crashed Didn’t Crash Sober 52 125 Drunk 66 88 What do you think Buck’s hypothesis and null hypothesis were for this experiment? 2 Use the statistic to find the probability that his sample of sober drivers and drunk drivers would have this large a difference in driving performance if alcohol did not affect driving ability. [Homework 22: Reaction Time from Is There Really a Difference?] response to questions reasonable conclusions computational accuracy comparisons understanding of statistical concepts 2 11. Uses simulations to explore the variability of sample statistics from a 11, 15, 16. The student finds the statistic for each of the three samples of 2 known population and construct sampling distributions (NM - III.C.3). coin flips given below and then uses the information in A Probability Chart to find the probability that he/she would get a result that far or 12. Understands how sample statistics reflect the values of population further from what he/she would expect if the coin were fair. parameters and uses sampling distributions as the basis for informal 25 heads and 35 tails inference (NM - III.C.4). 220 heads and 240 tails 995 heads and 1025 tails [Homework 17: A Collection of Coins from Is There Really a Difference?] understanding of sampling understanding of probability computational accuracy 13. Evaluates published reports that are based on data by examining the 13. The student finds an article in a newspaper or magazine that reports on a design of the study, the appropriateness of the data analysis, and the particular research study or survey, summarizes the main ideas or validity of conclusions (NM - III.C.5). conclusions of the study, and identifies information he/she would like to have about the study that was not included in the article. [Homework 3: Quality of Investigation from Is There Really a Difference?] reading analysis all required components effective communication INTERACTIVE MATH PROGRAM 2 (IMP 2) 6.22.25 Albuquerque Public Schools GRADE PERFORMANCE STANDARDS ILLUSTRATIONS 10 - 11 relevant information Benchmark D: The student understands and applies basic concepts of probability. 14. Explains the concept of a random variable (NM - III.D.1). 14, 16. In this experiment, the student works with partners to generate 40 random numbers. One partner handles the graphing calculator and reports 15. Understands the concept of probability as relative frequency the random numbers while the other partner records the result. With each (NM - III.D.2). random number, the person with the calculator says “bad” if the digit in the tenths place is 0, 1, or 2, and “good” if the tenths digit is 3, 4, 5, 6, 7, 8, or 16. Uses simulations to compute the expected value and probabilities of 9. The partner keeps a tally of how many good and bad tippers there are random variables in simple cases (NM - III.D.3). and announces when 40 results have been tallied. (The hypothesis is that 70% of adults are good tippers in restaurants.) [Random but Fair from Is There Really a Difference?] teamwork/collaboration experimental process record keeping understanding of randomness 17. Distinguishes between independent and dependent events (NM III.D.4). 17. The student responds appropriately to the following scenario: A medical researcher was trying to understand a measles epidemic in a city. She used a computer to examine the records of 500 patients of various doctors. These records showed each person’s sex, blood type, cholesterol level, blood pressure, weight, height, and medical history. Which of these factors do you think might have an affect on whether or not a person contracts measles? Explain your reasoning. [Homework 2: Who Gets A’s and Measles? from Is There Really a Difference?] response to question reasoning/analysis clear communication understanding of concepts INTERACTIVE MATH PROGRAM 2 (IMP 2) 6.23.25 Albuquerque Public Schools STRAND VI: LITERACY CONTENT STANDARD: The student communicates mathematical principles through reading, writing, speaking, and research opportunities. BENCHMARK: The student demonstrates through a variety of writing and speaking requirements proficiency in reading comprehension, specialized vocabulary, and reasoning. GRADE PERFORMANCE STANDARDS ILLUSTRATIONS 10 - 11 The following performance standards are aligned with 10th grade APS Note: The very nature of IMP courses, as the name suggests, requires the Language Arts Standards. student to interact; to engage in activities that enhance his/her vocabulary of mathematics; to communicate symbolically, orally, and in written formats; and to think critically through problem-solving situations. Through consistent integration of the mathematical processes, the student works collaboratively with other students, requiring whole or small group discussions; listens to other’s viewpoints whether it be via print, technology, or guest speaker; displays data in an organized fashion; and makes connections. Consequently, literacy strategies are integrated and reflected in every strand. The following citations illustrate specific examples of these strategies although numerous opportunities are presented throughout the year and throughout the curriculum. 1. Asks critical questions prompted by texts and researches answers for a 1 – 3. All strands and more specifically Strand V, the illustration for broader understanding (APS – LA I.1). performance standard #13. 2. Makes generalizations about text that are supported by specific references in the text (APS – LA II.2). 3. Reads critically and independently to draw conclusions from research (APS – LA I.3). 4. Develops increased competence in using the writing process to create a 4 – 7. All strands and illustrations where the student explains his/her final product (APS – LA III.1). reasoning can be done orally or in writing. 5. Develops increased competence and fluency in using elements of effective writing (APS – LA III.1). 6. Develops increased competence in using a variety of technology to present information appropriate for the intended purpose and audience (APS – LA III.3). INTERACTIVE MATH PROGRAM 2 (IMP 2) 6.24.25 Albuquerque Public Schools GRADE PERFORMANCE STANDARDS ILLUSTRATIONS 10 - 11 7. Develops increased competence in using writing conventions (APS – LA III.4). 8. Clearly articulates a position and develops arguments using a variety of 8 – 10. All strands but more specifically see the illustrations in Strand III. methods (APS – LA IV.4). 9. Analyzes an instance of public speaking or media presentation (APS – LA V.1). 10. Responds reflectively through dialogue and discussion to written and visual texts (APS – LA V.2). 11. Uses systematic strategies to organize and record information 11, 12. See Strand V, the illustration for performance standard #13. (APS – LA VI.1). 12. Uses a variety of information resources to critically interpret and evaluate experiences and ideas (APS – LA VI.2). INTERACTIVE MATH PROGRAM 2 (IMP 2) 6.25.25 Albuquerque Public Schools