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Correlation of Interactive Mathematics Program (IMP), Years 1–4, 2nd Edition to Common Core State Standards (June 2010), Mathematics, High School DRAFT 1/24/11 S TANDARDS FOR M ATHEMATICAL P RACTICE The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy). Standard IMP Lessons 1. Make sense of problems and persevere in Throughout the IMP curriculum, students are assigned Problems of the solving them. Week (POWs), which are open-ended problems that cannot be solved easily in a short period of time. POWs help students develop Mathematically proficient students start by explaining thoughtfulness and perseverance, and force them to focus on their own to themselves the meaning of a problem and looking thinking processes. Students must explain and illustrate their strategies and for entry points to its solution. They analyze givens, solutions, and must justify their reasoning in clearly written reports. constraints, relationships, and goals. They make Representative IMP Year 1 Lesson: conjectures about the form and meaning of the Corey Camel (The Pit and the Pendulum) solution and plan a solution pathway rather than Corey has 3000 bananas to bring to a market which is 1000 miles away. Corey can only carry 1000 bananas per trip and eats one banana for every simply jumping into a solution attempt. They consider mile traveled. Out of the 3000 bananas, what is the most amount of analogous problems, and try special cases and simpler bananas that Corey can bring to market? Initially, students will say no forms of the original problem in order to gain insight bananas. But they are asked to make sense of the problem and delve deeper into its solution. They monitor and evaluate their into alternative solutions. To help students make headway to a solution, a progress and change course if necessary. Older mini-Corey Camel problem is presented to them on a smaller scale where students might, depending on the context of the they use pennies to find a solution. Students use simulations, pictures, problem, transform algebraic expressions or change tables of values, and alternative solutions to find an answer. Students work the viewing window on their graphing calculator to collaboratively, listen to each other’s solutions, and prove to each other that get the information they need. Mathematically their solution is the correct one. proficient students can explain correspondences between equations, verbal descriptions, tables, and Representative IMP Year 2 Lesson: Just Count the Pegs (Do Bees Build It Best?) graphs or draw diagrams of important features and Students recreate the problem of finding area on a geoboard similar to what relationships, graph data, and search for regularity or confronted Georg Alexander Pick as he formulated his eponymous trends. Younger students might rely on using concrete formula. Students look at different examples of polygons formed on a objects or pictures to help conceptualize and solve a geoboard, gather data, and construct a formula. Two different approaches problem. Mathematically proficient students check are considered and students are asked to support or refute the validity of the their answers to problems using a different method, two approaches. Students are also encouraged to come up with their own and they continually ask themselves, “Does this make approach as long as they can thoroughly support it. Although an acceptable sense?” They can understand the approaches of others answer would be Pick’s Formula, that is not the point of this POW. Rather, to solving complex problems and identify the teacher is looking at how students gathered data to solve this problem and how they support and defend their own findings while they examine correspondences between different approaches. the work of their peers and try to prove them wrong by counterexample. Representative IMP Year 3 Lesson: Let’s Make a Deal (Pennant Fever) The classic Monty Hall dilemma from the game show, “Let’s Make a Correlation of Interactive Mathematics to Common Core State Standards 1 of 37 Key Curriculum Press June 2010 S TANDARDS FOR M ATHEMATICAL P RACTICE Deal” is presented to the students. You can win a great prize or a prize that you could do without. Monty Hall shows you what’s behind one of the three curtains and asks you if you would like to change your original curtain or switch. Students are asked which strategy, switch or stay, is better and why. Students are introduced to the problem through a simulation. Then they are asked to examine the probabilities and mathematics behind the problem. The solution flies in the face of intuition as the probabilities support the “always switch” strategy. Students are asked to explain the better strategy and use mathematics and probability theory to support their choice. 2. Reason abstractly and quantitatively. Representative IMP Year 1 Lesson: An N-by-N Window (Patterns) Mathematically proficient students make sense of Students are given the task of finding a formula that will find the amount of quantities and their relationships in problem wood framing used in a 3 by 3 window. Then they explore windows of situations. They bring two complementary abilities to different dimensions and organize their data so that they can derive the bear on problems involving quantitative relationships: formula for an N by N Window. From the context of a situation, students the ability to decontextualize—to abstract a given are asked to decontextualize the problem so that they can derive the situation and represent it symbolically and manipulate formula for an N by N Window. the representing symbols as if they have a life of their own, without necessarily attending to their referents— Representative IMP Year 2 Lesson: and the ability to contextualize, to pause as needed Building the Best Fence (Do Bees Build It Best?) In previous activities, students investigated which dimensions for a during the manipulation process in order to probe into rectangular corral yielded the most area. They also investigated whether or the referents for the symbols involved. Quantitative not regular polygons had more area than non-regular polygons with the reasoning entails habits of creating a coherent same number of sides. In Building the Best Fence, students build upon representation of the problem at hand; considering the their prior work to generalize a formula to find the regular polygon that has units involved; attending to the meaning of quantities, the most area given the constraint of limited perimeter. not just how to compute them; and knowing and flexibly using different properties of operations and Representative IMP Year 3 Lesson: objects. Blue Book (Small World, Isn’t It?) Depreciation is often a misunderstood concept for high school students as they have not experienced depreciation in their everyday lives…until they own a car. The problem asks them to build a set of data regarding the depreciation of a car over a set number of years. After examining their data, students are asked to generalize a formula so that anyone can figure out how much a car depreciates after t number of years after its purchase. Correlation of Interactive Mathematics to Common Core State Standards 2 of 37 Key Curriculum Press June 2010 S TANDARDS FOR M ATHEMATICAL P RACTICE 3. Construct viable arguments and critique the Representative IMP Year 1 Lesson: reasoning of others. A Proof Gone Bad (Patterns) Students are asked to examine a proof written by a student regarding the Mathematically proficient students understand and use sum of interior angles of a polygon. There may be weaknesses in the stated assumptions, definitions, and previously student’s proof and the students are asked to find them. If they find those established results in constructing arguments. They weaknesses, they are asked to write a solid paragraph proof that supports make conjectures and build a logical progression of their version of the formula used to find the sum of interior angles. statements to explore the truth of their conjectures. They are able to analyze situations by breaking them Representative IMP Year 2 Lesson: into cases, and can recognize and use Who Gets A’s and Measles? (Is There Really a Difference?) counterexamples. They justify their conclusions, Two “scientific” studies are presented to students. One examines the behaviors of students and how those behaviors result in an A grade. The communicate them to others, and respond to the other study examines the attributes of people who contract measles. arguments of others. They reason inductively about Students are asked to comment on the studies, solicit additional data, making plausible arguments that take into information (if necessary) and conclude if the studies are useful. They are account the context from which the data arose. also asked to make improvements to the studies. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which Representative IMP Year 3 Lesson: is flawed, and—if there is a flaw in an argument— Proving Triples (Orchard Hideout) explain what it is. Elementary students can construct After learning about the Pythagorean Theorem, students find that there are arguments using concrete referents such as objects, unique sets of numbers that one can find as the measures of the sides of drawings, diagrams, and actions. Such arguments can right triangles: Pythagorean Triples. Students are asked to examine two sets of measurements and use them to determine if a triangle is a right triangle make sense and be correct, even though they are not or not. Students are then asked to write a proof regarding the multiplication generalized or made formal until later grades. Later, of each member of a Pythagorean Triple by a constant and whether or not it students learn to determine domains to which an will result in the measurements for a right triangle. argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. Correlation of Interactive Mathematics to Common Core State Standards 3 of 37 Key Curriculum Press June 2010 S TANDARDS FOR M ATHEMATICAL P RACTICE 4. Model with mathematics. Representative IMP Year 1 Lesson: Ox Expressions and Ox Expressions at Home (Overland Trail) Mathematically proficient students can apply the Students are given a list of variables and their definition. Students are mathematics they know to solve problems arising in asked to model real life situations with the variables such as “the amount of everyday life, society, and the workplace. In early water consumed in a day by a family” or “the number of people in a wagon grades, this might be as simple as writing an addition train.” The students are also asked to make sense of expressions written equation to describe a situation. In middle grades, a with variables. In some cases, the expressions that they are given are non- student might apply proportional reasoning to plan a sense and students must explain why putting two variables together doesn’t school event or analyze a problem in the community. necessary make for a sensible expression. By high school, a student might use geometry to solve a design problem or use a function to describe how Representative IMP Year 2 Lesson: one quantity of interest depends on another. Rock ‘n’ Rap and A Rock ‘n’ Rap Variation (Cookies) This linear programming problem gives the students several constraints Mathematically proficient students who can apply regarding the sales of rock CDs and rap CDs. Students use multiple what they know are comfortable making assumptions representations to model the situation and display the feasible region. and approximations to simplify a complicated Given a profit function, the students then find how much of each type of situation, realizing that these may need revision later. CD they should sell to maximize profit. Justification of their answer is vital They are able to identify important quantities in a to explaining why their combination of CDs results in the most profit. practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, Representative IMP Year 3 Lesson: flowcharts and formulas. They can analyze those Matrices in the Oven (Meadows or Malls?) relationships mathematically to draw conclusions. Given constraints about ingredients used to make various types of cookies, They routinely interpret their mathematical results in students are asked to model the constraints in the form of matrices. This problem provides a real life context that gives meaning to the rows and the context of the situation and reflect on whether the columns in a matrix. results make sense, possibly improving the model if it Students then use their knowledge of matrix operations to find the total has not served its purpose. amount of each ingredient used. Students must explain how they calculated the numbers they listed in their final matrix. Correlation of Interactive Mathematics to Common Core State Standards 4 of 37 Key Curriculum Press June 2010 S TANDARDS FOR M ATHEMATICAL P RACTICE 5. Use appropriate tools strategically. Representative IMP Year 1 Lesson: Sublett’s Cutoff Revisited ( Overland Trail) Mathematically proficient students consider the The graphing calculator is used to quickly plot data and allow students to available tools when solving a mathematical problem. graph various linear function that will fit the data best. Students have been These tools might include pencil and paper, concrete using paper and pencil methods up to this point and will discover that models, a ruler, a protractor, a calculator, a technology will allow them to explore more efficiently and deeper than spreadsheet, a computer algebra system, a statistical their previous experiences. package, or dynamic geometry software. Proficient After finding the line of best fit, they use their function to make predictions students are sufficiently familiar with tools regarding water consumption over a period of time. appropriate for their grade or course to make sound decisions about when each of these tools might be Representative IMP Year 2 Lesson: helpful, recognizing both the insight to be gained and Parabolas and Equations I and III (Fireworks) Students use the graphing calculator to explore families of functions. They their limitations. For example, mathematically begin with the simple function for a parabola and then investigate what proficient high school students analyze graphs of parts of the function make the graph narrower, wider, inverted, and functions and solutions generated using a graphing translated in any direction. Using the investigative approach, students soon calculator. They detect possible errors by strategically discover what each parameter does in y = a(x)^2 +k using estimation and other mathematical knowledge. When making mathematical models, they know that Representative IMP Year 3 Lesson: technology can enable them to visualize the results of Zooming Free-for-All (Small World, Isn’t it?) varying assumptions, explore consequences, and Students are laying the foundation for their study of the derivative by compare predictions with data. Mathematically investigating the slope of a line tangent to a point on a graph. By using the proficient students at various grade levels are able to ZOOM feature of the graphing calculator, students are “linearlizing” the function until it appears straight. This approximates the tangent line at the identify relevant external mathematical resources, point. Using the graphical approach, students understand that the rate of such as digital content located on a website, and use change at a point is the slope of the tangent line. them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. Correlation of Interactive Mathematics to Common Core State Standards 5 of 37 Key Curriculum Press June 2010 S TANDARDS FOR M ATHEMATICAL P RACTICE 6. Attend to precision. Representative IMP Year 1 Lesson: You’re the Storyteller: From Rules to Situations (Overland Trail) Mathematically proficient students try to Students are given a set of equations and must create a context that the communicate precisely to others. They try to use clear equation could represent. Precision is important in this activity as students definitions in discussion with others and in their own must create a situation, clearly state what the variable represents (including reasoning. They state the meaning of the symbols they units), and then find the number that will make the given equation true. choose, including using the equal sign consistently Clear communication of their variables and their meaning is vital to this and appropriately. They are careful about specifying activity. units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They Representative IMP Year 2 Lesson: calculate accurately and efficiently, express numerical Picturing Pictures (Cookies) answers with a degree of precision appropriate for the Given numerous constraints, students will ultimately display the feasible region for the problem on painting pictures. Students are asked to attend to problem context. In the elementary grades, students precision by identifying their variables, labeling their axes and showing the give carefully formulated explanations to each other. proper scaling on the axes. After making the graphical representation, By the time they reach high school they have learned students are asked to identify points that lie in the feasible region and the to examine claims and make explicit use of profit made from those points. Lastly, students are asked to write a profit definitions. function. Representative IMP Year 3 Lesson: How Many More People? (Small World, Isn’t it?) In order to explore population growth, students must graphically represent population data over time. Students are asked to graph this data on an appropriate scale of axes. Using their graph or using algebra, they then find the average increases over different periods of time. They are then asked to look at different intervals of time in order to compare growth rates. Precision in graphing and calculating the average increase is used throughout the activity. Correlation of Interactive Mathematics to Common Core State Standards 6 of 37 Key Curriculum Press June 2010 S TANDARDS FOR M ATHEMATICAL P RACTICE 7. Look for and make use of structure. Representative IMP Year 1 Lesson: Degree Discovery and Polygon Angles (Patterns) Mathematically proficient students look closely to Students explore the sum of interior angles of various polygon using a discern a pattern or structure. Young students, for protractor and various pattern blocks. After collecting data from various example, might notice that three and seven more is the polygons, they look for patterns to help them generalize the sum of interior same amount as seven and three more, or they may angles given the number of sides. A justification of their formula is also sort a collection of shapes according to how many necessary as several students will derive different forms of the same sides the shapes have. Later, students will see 7 × 8 formula. equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive Representative IMP Year 2 Lesson: property. In the expression x2 + 9x + 14, older Continuing the Pattern (All About Alice) students can see the 14 as 2 × 7 and the 9 as 2 + 7. Students explore the various powers of two in this activity. Throughout this unit, they have worked with positive exponents and see the growth with a They recognize the significance of an existing line in base of two. They now explore negative exponents and examine the a geometric figure and can use the strategy of drawing various patterns that emerge with negative exponents. Students explain an auxiliary line for solving problems. They also can how to find the result of a negative exponent using fractions as their step back for an overview and shift perspective. They results. can see complicated things, such as some algebraic expressions, as single objects or as being composed of Representative IMP Year 3 Lesson: several objects. For example, they can see Which is Which? & Formulas for nPr and nCr (Pennant Fever) 5 – 3(x – y)2 as 5 minus a positive number times a Combinations and permutations are explored in these activities. Students square and use that to realize that its value cannot be examine their previous work with combinations and permutations and are more than 5 for any real numbers x and y. asked to explain the difference between the two. They are to use the proper notation for both. After looking at the patterns in their work, they are to find a general formula for permutations in terms of n and r. They are also asked to find a general equation expressing the relationship between permutations and combinations. Correlation of Interactive Mathematics to Common Core State Standards 7 of 37 Key Curriculum Press June 2010 S TANDARDS FOR M ATHEMATICAL P RACTICE 8. Look for and express regularity in repeated Representative IMP Year 1 Lesson: reasoning. An N by N Window & More About Windows (Shadows) In these two activities, students examine their work for various sized Mathematically proficient students notice if windows and the calculation for the amount of wood needed to frame the calculations are repeated, and look both for general windows. They first begin with square window frames and generalize a methods and for shortcuts. Upper elementary students formula. They then move onto rectangular window frames and find a might notice when dividing 25 by 11 that they are formula for an M by N window. By examining their repeated work for repeating the same calculations over and over again, square and rectangular window frames, students will be able to derive a formula that works for both types of window frames. and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line Representative IMP Year 2 Lesson: Don’t Fence Me In & More Fencing, Bigger Corrals & Building the Best through (1, 2) with slope 3, middle school students Fence (Do Bees Build It Best?) might abstract the equation (y – 2)/(x – 1) = 3. Students build to the general formula to find the area of any regular Noticing the regularity in the way terms cancel when polygon by investigating these three activities. Much like their experience expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and in year 1, they begin with a simple polygon (a quadrilateral) and work their (x – 1)(x3 + x2 + x + 1) might lead them to the general way through polygons of increasing number of sides. By examining their formula for the sum of a geometric series. As they work with these various polygons, students derive the area formula for a work to solve a problem, mathematically proficient regular polygon. students maintain oversight of the process, while attending to the details. They continually evaluate the Representative IMP Year 3 Lesson: reasonableness of their intermediate results. Squaring the Circle & Using the Squared Circle & Hexagoning the Circle & Octagoning the Circle & Polygoning the Circle (Orchard Hideout) Students explore the case of the circumscribed polygon about a circle. Beginning with the square and working up to a polygon on n sides, students calculate the perimeter and area of the circumscribed polygon in terms of the radius of the circle. After examining their work, students will generalize a formula for the perimeter and area of any sided polygon. N UMBER AND Q UANTITY Standard IMP Lessons The Real Number System Extend the properties of exponents to rational exponents. 1. Explain how the definition of the meaning IMP Year 2, All About Alice: of rational exponents follows from extending A Half Ounce of Cake the properties of integer exponents to those values, allowing for a notation for radicals in It’s in the Graph terms of rational exponents. For example, we Stranger Pieces of Cake define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 All Roads Lead to Rome must equal 5. Correlation of Interactive Mathematics to Common Core State Standards 8 of 37 Key Curriculum Press June 2010 N UMBER AND Q UANTITY 2. Rewrite expressions involving radicals IMP Year 2, All About Alice: and rational exponents using the properties Stranger Pieces of Cake of exponents. All Roads Lead to Rome Use properties of rational and irrational numbers. 3. Explain why the sum or product of two This standard is not addressed in IMP, but a supplementary rational numbers is rational; that the sum of activity will be developed for the Year 2 unit, All About Alice. a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Quantities* Reason quantitatively and use units to solve problems. 1. Use units as a way to understand problems IMP Year 1, The Overland Trail: and to guide the solution of multi-step The Search for Dry Trails problems; choose and interpret units consistently in formulas; choose and Previous Travelers interpret the scale and the origin in graphs Who Will Make It? and data displays. Standard addressed throughout Years 1-4 2. Define appropriate quantities for the IMP Year 1, The Overland Trail: purpose of descriptive modeling. Travel on the Trail Moving Along Standard addressed throughout Years 1-4 3. Choose a level of accuracy appropriate to IMP Year 1, The Pit and the Pendulum: limitations on measurement when reporting Close to the Law quantities. Penny Weight Revisited IMP Year 2, Do Bees Build it Best?: Falling Bridges The Complex Number System Perform arithmetic operations with complex numbers. 1. Know there is a complex number i such IMP Year 4, The Diver Returns: that i2 = –1, and every complex number has Imagine a Solution the form a + bi with a and b real. Complex Numbers and Quadratic Equations Correlation of Interactive Mathematics to Common Core State Standards 9 of 37 Key Curriculum Press June 2010 N UMBER AND Q UANTITY 2. Use the relation i2 = –1 and the IMP Year 4, The Diver Returns: commutative, associative, and distributive Complex Components properties to add, subtract, and multiply complex numbers. 3. (+) Find the conjugate of a complex IMP Year 4, The Diver Returns: number; use conjugates to find moduli and Complex Conjugation quotients of complex numbers. Represent complex numbers and their operations on the complex plane. 4. (+) Represent complex numbers on the IMP Year 4, The Diver Returns: complex plane in rectangular and polar form Complex Components (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number. 5. (+) Represent addition, subtraction, This standard is not addressed in IMP, but a supplementary multiplication, and conjugation of complex activity will be developed for the Year 4 unit, The Diver Returns. numbers geometrically on the complex plane; use properties of this representation for computation. For example, (1 – √3i)3 = 8 because (1 – √3i) has modulus 2 and argument 120°. 6. (+) Calculate the distance between This standard is not addressed in IMP, but a supplementary numbers in the complex plane as the activity will be developed for the Year 4 unit, The Diver Returns. modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints. Use complex numbers in polynomial identities and equations. 7. Solve quadratic equations with real IMP Year 4, The Diver Returns: coefficients that have complex solutions. Complex Numbers and Quadratic Equations 8. (+) Extend polynomial identities to the This standard is not addressed in IMP, but a supplementary complex numbers. For example, rewrite activity will be developed for the Year 4 unit, The Diver Returns. x2 + 4 as (x + 2i)(x – 2i). 9. (+) Know the Fundamental Theorem of This standard is not addressed in IMP, but a supplementary Algebra; show that it is true for quadratic activity will be developed for the Year 4 unit, The Diver Returns. polynomials. Vector and Matrix Quantities Represent and model with vector quantities. Correlation of Interactive Mathematics to Common Core State Standards 10 of 37 Key Curriculum Press June 2010 N UMBER AND Q UANTITY 1. (+) Recognize vector quantities as having IMP Year 4, The Diver Returns: both magnitude and direction. Represent Absolutely Complex vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v). 2. (+) Find the components of a vector by IMP Year 4, The Diver Returns: subtracting the coordinates of an initial point Swimming Pointers from the coordinates of a terminal point. 3. (+) Solve problems involving velocity and IMP Year 4, The Diver Returns: other quantities that can be represented by Vector Velocities vectors. Swimming Pointers Perform operations on vectors. 4. (+) Add and subtract vectors. IMP Year 4, The Diver Returns: Complex Components 4a. Add vectors end-to-end, component- IMP Year 4, The Diver Returns: wise, and by the parallelogram rule. Complex Components Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes. 4b. Given two vectors in magnitude and IMP Year 4, The Diver Returns: direction form, determine the magnitude and Vector Velocities direction of their sum. 4c. Understand vector subtraction v – w as IMP Year 4, The Diver Returns: v + (–w), where –w is the additive inverse of Vector Velocities w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise. 5. (+) Multiply a vector by a scalar. This standard is not addressed in IMP, but a supplementary activity will be developed for the Year 4 unit, The Diver Returns. 5a. Represent scalar multiplication This standard is not addressed in IMP, but a supplementary graphically by scaling vectors and possibly activity will be developed for the Year 4 unit, The Diver Returns. reversing their direction; perform scalar multiplication component-wise, e.g., as c(vx, vy) = (cvx, cvy). Correlation of Interactive Mathematics to Common Core State Standards 11 of 37 Key Curriculum Press June 2010 N UMBER AND Q UANTITY 5b. Compute the magnitude of a scalar This standard is not addressed in IMP, but a supplementary multiple cv using ||cv|| = |c|v. Compute the activity will be developed for the Year 4 unit, The Diver Returns. direction of cv knowing that when |c|v ≠ 0, the direction of cv is either along v (for c > 0) or against v (for c < 0). Perform operations on matrices and use matrices in applications. 6. (+) Use matrices to represent and IMP Year 3, Meadows or Malls?: manipulate data, e.g., to represent payoffs or Inventing an Algebra incidence relationships in a network. Flying Matrices 7. (+) Multiply matrices by scalars to IMP Year 3, Meadows or Malls?: produce new matrices, e.g., as when all of Fresh Ingredients the payoffs in a game are doubled. 8. (+) Add, subtract, and multiply matrices IMP Year 3, Meadows or Malls?: of appropriate dimensions. Inventing an Algebra Back and Forth 9. (+) Understand that, unlike multiplication IMP Year 3, Meadows or Malls?: of numbers, matrix multiplication for square Things We Take for Granted matrices is not a commutative operation, but still satisfies the associative and distributive properties. 10. (+) Understand that the zero and identity IMP Year 3, Meadows or Malls?: matrices play a role in matrix addition and Solving the Simplest multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse. 11. (+) Multiply a vector (regarded as a This standard is not addressed in IMP, but a supplementary matrix with one column) by a matrix of activity will be developed for the Year 3 unit, Meadows or suitable dimensions to produce another Malls?. vector. Work with matrices as transformations of vectors. 12. (+) Work with 2 × 2 matrices as a This standard is not addressed in IMP, but a supplementary transformations of the plane, and interpret activity will be developed for the Year 3 unit, Meadows or the absolute value of the determinant in Malls?. terms of area. A LGEBRA Correlation of Interactive Mathematics to Common Core State Standards 12 of 37 Key Curriculum Press June 2010 A LGEBRA Standard IMP Lessons Seeing Structure in Expressions Interpret the structure of expressions. 1. Interpret expressions that represent a IMP Year 1, Patterns: quantity in terms of its context.★ Marcella’s Bagels Border Varieties Standard addressed throughout Years 1-4 1a. Interpret parts of an expression, such as IMP Year 1, The Overland Trail: terms, factors, and coefficients. Moving Along Fair Share on Chores Standard addressed throughout Years 1-4 1b. Interpret complicated expressions by IMP Year 1, The Pit and the Pendulum: viewing one or more of their parts as a single Penny Weight Revisited entity. For example, interpret P(1+r)n as the product of P and a factor not depending on IMP Year 3, High Dive: P. Planning for Formulas 2. Use the structure of an expression to IMP Year 1, The Overland Trail: identify ways to rewrite it. For example, see More Scrambled Equations and Mystery Bags x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored IMP Year 2, Fireworks: as (x2 – y2)(x2 + y2). Factors of Research Write expressions in equivalent forms to solve problems. 3. Choose and produce an equivalent form of IMP Year 1, The Overland Trail: an expression to reveal and explain More Fair Share on Chores properties of the quantity represented by the expression. 3a. Factor a quadratic expression to reveal IMP Year 2, Fireworks: the zeros of the function it defines. Factoring Let’s Factor! 3b. Complete the square in a quadratic IMP Year 2, Fireworks: expression to reveal the maximum or Squares and Expansions minimum value of the function it defines. Correlation of Interactive Mathematics to Common Core State Standards 13 of 37 Key Curriculum Press June 2010 A LGEBRA 3c. Use the properties of exponents to IMP Year 3, Small World, Isn’t It?: transform expressions for exponential The Generous Banker functions. For example the expression 1.15t can be rewritten as (1.151/12)12t ≈ 1.01212t to The Limit of Their Generosity reveal the approximate equivalent monthly interest rate if the annual rate is 15%. 4. Derive the formula for the sum of a finite IMP Year 2, All About Alice: geometric series (when the common ratio is More About Rallods not 1), and use the formula to solve problems. For example, calculate mortgage IMP Year 3, Small World, Isn’t It?: payments. Summing the Sequences – Part II Arithmetic with Polynomials and Rational Expressions Perform arithmetic operations on polynomials. 1. Understand that polynomials form a IMP Year 2, Fireworks: system analogous to the integers, namely, A Summary of Quadratics and Other Polynomials they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Understand the relationship between zeros and factors of polynomials. 2. Know and apply the Remainder Theorem: This standard is not addressed in IMP. For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x). 3. Identify zeros of polynomials when IMP Year 2, Fireworks: suitable factorizations are available, and use Make Your Own Intercepts the zeros to construct a rough graph of the function defined by the polynomial. Use polynomial identities to solve problems. 4. Prove polynomial identities and use them This standard is not addressed in IMP. to describe numerical relationships. For example, the polynomial identity (x2 + y2)2 = (x2 – y2)2 + (2xy)2 can be used to generate Pythagorean triples. 5. (+) Know and apply the Binomial IMP Year 3, Pennant Fever: Theorem for the expansion of (x + y)n in The Binomial Theorem and Row Sums powers of x and y for a positive integer n, where x and y are any numbers, with The Whys of Binomial Expansion coefficients determined for example by Pascal’s Triangle. Correlation of Interactive Mathematics to Common Core State Standards 14 of 37 Key Curriculum Press June 2010 A LGEBRA Rewrite rational expressions. 6. Rewrite simple rational expressions in This standard is not addressed in IMP. different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system. 7. (+) Understand that rational expressions This standard is not addressed in IMP, but a supplementary form a system analogous to the rational activity will be developed for the Year 3 unit, High Dive. numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions. Creating Equations* Create equations that describe numbers or relationships. 1. Create equations and inequalities in one IMP Year 1, Overland Trail: variable and use them to solve problems. The Big Buy Include equations arising from linear and quadratic functions, and simple rational and IMP Year 2, Cookies: exponential functions. How Many of Each Kind? Standard addressed throughout Years 1-4 2. Create equations in two or more variables IMP Year 2, Cookies to represent relationships between quantities; A Charity Rock graph equations on coordinate axes with labels and scales. Big State U 3. Represent constraints by equations or IMP Year 2, Cookies: inequalities, and by systems of equations Profitable Pictures and/or inequalities, and interpret solutions as viable or nonviable options in a modeling You Are What You Eat context. For example, represent inequalities IMP Year 3, Meadows or Malls?: describing nutritional and cost constraints on combinations of different foods. Eastside Westside Story 4. Rearrange formulas to highlight a quantity IMP Year 1, Shadows: of interest, using the same reasoning as in More Triangles for Shadows solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R. Reasoning with Equations and Inequalities Understand solving equations as a process of reasoning and explain the reasoning. Correlation of Interactive Mathematics to Common Core State Standards 15 of 37 Key Curriculum Press June 2010 A LGEBRA 1. Explain each step in solving a simple IMP Year 1, The Overland Trail: equation as following from the equality of The Mystery Bags Game numbers asserted at the previous step, starting from the assumption that the original More Mystery Bags equation has a solution. Construct a viable argument to justify a solution method. 2. Solve simple rational and radical IMP Year 2, Do Bees Build it Best: equations in one variable, and give examples Add to Simply Square Roots showing how extraneous solutions may arise. Solve equations and inequalities in one variable. 3. Solve linear equations and inequalities in IMP Year 1, The Overland Trail: one variable, including equations with More Fair Share for Hired Hands coefficients represented by letters. IMP Year 2, Cookies: Investigating Inequalities 4. Solve quadratic equations in one variable. IMP Year 2, Fireworks: Another Rocket Profiting from Widgets 4a. Use the method of completing the square IMP Year 2, Fireworks: to transform any quadratic equation in x into Squares and Expansions an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic The Quadratic Formula formula from this form. 4b. Solve quadratic equations by inspection IMP Year 2, Fireworks: (e.g., for x2 = 49), taking square roots, Square It! completing the square, the quadratic formula and factoring, as appropriate to the initial Squares and Expansions form of the equation. Recognize when the IMP Year 2, Do Bees Build It Best?: quadratic formula gives complex solutions and write them as a ± bi for real numbers a Impossible Rugs and b. Solve systems of equations. 5. Prove that, given a system of two This standard is not addressed in IMP, but a supplementary equations in two variables, replacing one activity will be developed for the Year 2 unit, Cookies. equation by the sum of that equation and a multiple of the other produces a system with the same solutions. Correlation of Interactive Mathematics to Common Core State Standards 16 of 37 Key Curriculum Press June 2010 A LGEBRA 6. Solve systems of linear equations exactly IMP Year 2, Cookies: and approximately (e.g., with graphs), Going Out For Lunch focusing on pairs of linear equations in two variables. Set It Up 7. Solve a simple system consisting of a IMP Year 2, Cookies: linear equation and a quadratic equation in Algebra Pictures two variables algebraically and graphically. For example, find the points of intersection between the line y = –3x and the circle x2 + y2 = 3. 8. (+) Represent a system of linear equations IMP Year 3, Meadows or Malls?: as a single matrix equation in a vector Inventing an Algebra variable. 9. (+) Find the inverse of a matrix if it exists IMP Year 3, Meadows or Malls?: and use it to solve systems of linear Finding an Inverse equations (using technology for matrices of dimension 3 × 3 or greater). Inverses and Equations Represent and solve equations and inequalities graphically. 10. Understand that the graph of an equation IMP Year 1, The Overland Trail: in two variables is the set of all its solutions From Rules to Graphs plotted in the coordinate plane, often forming a curve (which could be a line). Graphing Calculator In-Outs 11. Explain why the x-coordinates of the IMP Year 1, The Overland Trail: points where the graphs of the equations y = Graphing Free-for-All f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions Graphs in Search of Equations I & II approximately, e.g., using technology to Graphing Summary graph the functions, make tables of values, or find successive approximations. Include Standard addressed throughout Years 1-4 cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.★ 12. Graph the solutions to a linear inequality IMP Year 2, Cookies: in two variables as a half-plane (excluding Picturing Cookies – Part I the boundary in the case of a strict inequality), and graph the solution set to a Picturing Cookies – Part II system of linear inequalities in two variables Picturing Pictures as the intersection of the corresponding half- planes. Correlation of Interactive Mathematics to Common Core State Standards 17 of 37 Key Curriculum Press June 2010 F UNCTIONS Standard IMP Lessons Interpreting Functions Understand the concept of a function and use function notation. 1. Understand that a function from one set IMP Year 1, Patterns: (called the domain) to another set (called the Inside Out range) assigns to each element of the domain exactly one element of the range. If f is a Another In-Outer function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). 2. Use function notation, evaluate functions IMP Year 1, Overland Trail: for inputs in their domains, and interpret All Four, One – Linear Functions statements that use function notation in terms of a context. Getting the Gold 3. Recognize that sequences are functions, IMP Year 1, Patterns: sometimes defined recursively, whose Keep It Going domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1. Interpret functions that arise in applications in terms of the context. 4. For a function that models a relationship IMP Year 1, The Overland Trail: between two quantities, interpret key Family Comparisons by Algebra features of graphs and tables in terms of the quantities, and sketch graphs showing key IMP Year 2, Cookies: features given a verbal description of the Big State U relationship. Key features include: intercepts; intervals where the function is IMP Year 3, High Dive: increasing, decreasing, positive, or negative; Free Fall relative maximums and minimums; symmetries; end behavior; and periodicity.★ Standard addressed throughout Years 1-4 5. Relate the domain of a function to its IMP Year 1, The Overland Trail: graph and, where applicable, to the Fair Share on Chores quantitative relationship it describes. For example, if the function h(n) gives the IMP Year 2, Do Bees Build It Best?: number of person-hours it takes to assemble Don’t Fence Me In n engines in a factory, then the positive integers would be an appropriate domain for IMP Year 3, Small World, Isn’t It?: the function.★ Growing Up Correlation of Interactive Mathematics to Common Core State Standards 18 of 37 Key Curriculum Press June 2010 F UNCTIONS 6. Calculate and interpret the average rate of IMP Year 1, The Overland Trail: change of a function (presented symbolically Travel on the Trail or as a table) over a specified interval. Estimate the rate of change from a graph.★ Following Families on the Trail Analyze functions using different representations. 7. Graph functions expressed symbolically IMP Year 1, The Overland Trail: and show key features of the graph, by hand Straight-Line Reflections in simple cases and using technology for more complicated cases.★ IMP Year 1, The Pit and the Pendulum: Graphing Free-for-All 7a. Graph linear and quadratic functions and IMP Year 2, Cookies: show intercepts, maxima, and minima. Finding Linear Graphs IMP Year 2, Fireworks: Victory Celebration Standard addressed throughout Years 1-4 7b. Graph square root, cube root, and IMP Year 1, The Overland Trail: piecewise-defined functions, including step Graph Sketches functions and absolute value functions. IMP Year 1, The Pit and the Pendulum: Graphing Summary 7c. Graph polynomial functions, identifying IMP Year 2, Fireworks: zeros when suitable factorizations are Another Rocket available, and showing end behavior. IMP Year 4, The World of Functions: The End of the Function 7d. (+) Graph rational functions, identifying IMP Year 4, The World of Functions: zeros and asymptotes when suitable Approaching Infinity factorizations are available, and showing end behavior. 7e. Graph exponential and logarithmic IMP Year 2, All About Alice: functions, showing intercepts and end Graphing Alice behavior, and trigonometric functions, showing period, midline, and amplitude. Taking Logs to the Axes IMP Year 3, High Dive: Sand Castles Correlation of Interactive Mathematics to Common Core State Standards 19 of 37 Key Curriculum Press June 2010 F UNCTIONS 8. Write a function defined by an expression IMP Year 2, Fireworks: in different but equivalent forms to reveal Here Comes Vertex Form and explain different properties of the function. 8a. Use the process of factoring and IMP Year 2, Fireworks: completing the square in a quadratic function Finding Vertices and Intercepts to show zeros, extreme values, and symmetry of the graph, and interpret these in Pens and Corrals in Vertex Form terms of a context. 8b. Use the properties of exponents to IMP Year 2, All About Alice: interpret expressions for exponential A Wonderland Lost functions. For example, identify percent rate of change in functions such as y = (1.02)t, y Inflation, Depreciation, and Alice = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and IMP Year 3, Small World, Isn’t It?: classify them as representing exponential growth or decay. Return to A Crowded Place 9. Compare properties of two functions each IMP Year 2, Fireworks: represented in a different way (algebraically, Quadratics Choices graphically, numerically in tables, or by verbal descriptions). For example, given a A Quadratic Summary graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Building Functions Build a function that models a relationship between two quantities. 1. Write a function that describes a IMP Year 1, Patterns: relationship between two quantities.★ Squares and Scoops Another In-Outer Standard addressed throughout Years 1-4 1a. Determine an explicit expression, a IMP Year 1, Patterns: recursive process, or steps for calculation The Garden Border from a context. 1b. Combine standard function types using IMP Year 4, The World of Functions: arithmetic operations. For example, build a Slide That Function function that models the temperature of a cooling body by adding a constant function The Cost of Pollution to a decaying exponential, and relate these functions to the model. Correlation of Interactive Mathematics to Common Core State Standards 20 of 37 Key Curriculum Press June 2010 F UNCTIONS 1c. (+) Compose functions. For example, if IMP Year 4, The World of Functions: T(y) is the temperature in the atmosphere as Cozying Up to Composition a function of height, and h(t) is the height of a weather balloon as a function of time, then Order Among the Functions T(h(t)) is the temperature at the location of the weather balloon as a function of time. 2. Write arithmetic and geometric sequences IMP Year 1, Patterns: both recursively and with an explicit What’s Next? formula, use them to model situations, and translate between the two forms.★ Diagonals Illuminated IMP Year 3, Small World, Isn’t It?: Planning the Platforms Build new functions from existing functions. 3. Identify the effect on the graph of IMP Year 1, The Pit and the Pendulum: replacing f(x) by f(x) + k, k f(x), f(kx), and Graphing Free-for-All f(x + k) for specific values of k (both positive and negative); find the value of k given the IMP Year 2, Fireworks: graphs. Experiment with cases and illustrate Parabolas and Equations I, II, and III an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. 4. Find inverse functions. IMP Year 2, All About Alice: Alice on a Log Taking Logs to the Axes IMP Year 4, The World of Functions: An Inventory of Inverses 4a. Solve an equation of the form f(x) = c for IMP Year 4, The World of Functions: a simple function f that has an inverse and Linear Functions in Verse write an expression for the inverse. For example, f(x) =2 x3 for x > 0 or f(x) = (x+1)/(x–1) for x ≠ 1. 4b. (+) Verify by composition that one IMP Year 4, The World of Functions: function is the inverse of another. An Inventory of Inverses 4c. (+) Read values of an inverse function IMP Year 3, The World of Functions: from a graph or a table, given that the An Inventory of Inverses function has an inverse. Correlation of Interactive Mathematics to Common Core State Standards 21 of 37 Key Curriculum Press June 2010 F UNCTIONS 4d. (+) Produce an invertible function from a This standard is not addressed in IMP. non-invertible function by restricting the domain. 5. (+) Understand the inverse relationship IMP Year 2, All About Alice: between exponents and logarithms and use Alice on a Log this relationship to solve problems involving logarithms and exponents. Taking Logs to the Axes Linear and Exponential Models Construct and compare linear and exponential models and solve problems. 1. Distinguish between situations that can be IMP Year 2, All About Alice: modeled with linear functions and with Alice in Wonderland exponential functions. Graphing Alice 1a. Prove that linear functions grow by equal IMP Year 1, The Overland Trail: differences over equal intervals, and that Following Families on the Trail exponential functions grow by equal factors over equal intervals. IMP Year 2, All About Alice: A Wonderland Lost A New Kind of Cake 1b. Recognize situations in which one IMP Year 1, The Overland Trail: quantity changes at a constant rate per unit Fort Hall Businesses interval relative to another. 1c. Recognize situations in which a quantity IMP Year 2, All About Alice: grows or decays by a constant percent rate A Wonderland Lost per unit interval relative to another. IMP Year 3, Small World, Isn’t It? Comparative Growth 2. Construct linear and exponential IMP Year 1, The Pit and the Pendulum: functions, including arithmetic and So Little Data, So Many Rules geometric sequences, given a graph, a description of a relationship, or two input- IMP Year 3, Small World, Isn’t It?: output pairs (include reading these from a Planning the Platforms table). 3. Observe using graphs and tables that a IMP Year 2, All About Alice: quantity increasing exponentially eventually Rallods in Rednow Land exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. Correlation of Interactive Mathematics to Common Core State Standards 22 of 37 Key Curriculum Press June 2010 F UNCTIONS 4. For exponential models, express as a IMP Year 3, Small World, Isn’t it?: logarithm the solution to abct = d where a, c, Return to a Crowded Place and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. Interpret expressions for functions in terms of the situation they model. 5. Interpret the parameters in a linear or IMP Year 1, The Overland Trail: exponential function in terms of a context. Water Conservation IMP Year 2, All About Alice: Measuring Meals for Alice Trigonometric Functions Extend the domain of trigonometric functions using the unit circle. 1. Understand radian measure of an angle as IMP Year 4, How Much? How Fast? the length of the arc on the unit circle Trying a New Angle subtended by the angle. Different Angles 2. Explain how the unit circle in the IMP Year 3, High Dive: coordinate plane enables the extension of Extending the Sine trigonometric functions to all real numbers, interpreted as radian measures of angles What’s Your Cosine? traversed counterclockwise around the unit circle. 3. (+) Use special triangles to determine IMP Year 4, How Much? How Fast?: geometrically the values of sine, cosine, Different Angles tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosines, and tangent for x, π + x, and 2π – x in terms of their values for x, where x is any real number. 4. (+) Use the unit circle to explain IMP Year 4, How Much? How Fast?: symmetry (odd and even) and periodicity of Different Angles trigonometric functions. Model periodic phenomena with trigonometric functions. 5. Choose trigonometric functions to model IMP Year 3, High Dive: periodic phenomena with specified Graphing the Ferris Wheel amplitude, frequency, and midline.★ Ferris Wheel Graph Variations Correlation of Interactive Mathematics to Common Core State Standards 23 of 37 Key Curriculum Press June 2010 F UNCTIONS 6. (+) Understand that restricting a This standard is not addressed in IMP, but a supplementary trigonometric function to a domain on which activity will be developed for the Year 4 unit, The World of it is always increasing or always decreasing Functions. allows its inverse to be constructed. 7. (+) Use inverse functions to solve IMP Year 3, High Dive: trigonometric equations that arise in Not So Spectacular modeling contexts; evaluate the solutions using technology, and interpret them in A Practice Jump terms of the context.★ Prove and apply trigonometric identities. 8. Prove the Pythagorean identity sin2(θ) + IMP Year 3, High Dive: cos2(θ) = 1 and use it to calculate Pythagorean Trigonometry trigonometric ratios. More Pythagorean Trigonometry 9. (+) Prove the addition and subtraction IMP Year 4, As the Cube Turns: formulas for sine, cosine, and tangent and The Sine of a Sum use them to solve problems. Sum Tangents M ODELING Modeling Standards Modeling is best interpreted not as a collection of isolated topics but rather in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). G EOMETRY Standard IMP Lessons Congruence Experiment with transformations in the plane Correlation of Interactive Mathematics to Common Core State Standards 24 of 37 Key Curriculum Press June 2010 G EOMETRY 1. Know precise definitions of angle, circle, IMP Year 1, Patterns: perpendicular line, parallel line, and line Pattern Block Investigations segment, based on the undefined notions of point, line, distance along a line, and IMP Year 1, Shadows: distance around a circular arc. More About Angles IMP Year 3, Orchard Hideout: The Distance Formula Defining Circles 2. Model transformations in the plane using, IMP Year 4, As the Cube Turns: e.g., transparencies and geometry software; Flipping Points describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus stretch in a specific direction). 3. Given a rectangle, parallelogram, IMP Year 4, As the Cube Turns: trapezoid, or regular polygon, describe the Further Flips rotations and reflections that carry it onto itself. 4. Develop definitions of rotations, IMP Year 4, As the Cube Turns: reflections and translations in terms of An Animated POW Write-up angles, circles, perpendicular lines, parallel lines and line segments. 5. Given a specified rotation, reflection or IMP Year 4, As the Cube Turns: translation and a geometric figure, construct Further Flips the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Construct a sequence of transformations that will carry a given figure onto another. Understand congruence in terms of rigid motions 6. Use geometric descriptions of rigid IMP Year 1, Shadows: motions to transform figures and to predict Are Angles Enough? the effect of a rigid motion on a figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. Correlation of Interactive Mathematics to Common Core State Standards 25 of 37 Key Curriculum Press June 2010 G EOMETRY 7. Explain using rigid motions the meaning IMP Year 1, Shadows: of congruence for triangles as the equality of Why Are Triangles Special? all corresponding pairs of sides and all corresponding pairs of angles. Are Angles Enough? 8. Explain how the criteria for triangle IMP Year 1, Shadows: congruence (ASA, SAS, and SSS) follow Triangular Data from the definition of congruence. Prove geometric theorems 9. Prove theorems about lines and angles. IMP Year 1, Shadows: Theorems include: vertical angles are More About Angles congruent; when a transversal crosses parallel lines, alternate interior angles are Inside Similarity congruent and corresponding angles are A Parallel Proof congruent; points on a perpendicular bisector of a line segment are exactly those IMP Year 3, Orchard Hideout: equidistant from the segment’s endpoints. Equally Wet 10. Prove theorems about triangles. IMP Year 1, Shadows: Theorems include: measures of interior Very Special Triangles angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the Triangles Versus Other Polygons segment joining midpoints of two sides of a Some of these theorems not proved in IMP, but a supplementary triangle is parallel to the third side and half activity will be developed for the Year 1 unit, Shadows. the length; the medians of a triangle meet at a point. 11. Prove theorems about parallelograms. IMP Year 1, Shadows: Theorems include: opposite sides are Angles, Angles, Angles congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each Fit Them Together other and conversely, rectangle are Some of these theorems not proved in IMP, but a supplementary parallelograms with congruent diagonals. activity will be developed for the Year 1 unit, Shadows. Make geometric constructions 12. Make formal geometric constructions IMP Year 3, Orchard Hideout: with a variety of tools and methods (compass Equally Wet and straightedge, string, reflective devices, paper folding, dynamic geometric software, Only Two Flowers etc). Copying a segment; copying an angle; A Perpendicularity Proof bisecting a segment; bisecting an angle; constructing perpendicular lines, including On Patrol the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Correlation of Interactive Mathematics to Common Core State Standards 26 of 37 Key Curriculum Press June 2010 G EOMETRY 13. Construct an equilateral triangle, a This standard is not addressed in IMP, but a supplementary square and a regular hexagon inscribed in a activity will be developed for the Year 3 unit, Orchard Hideout. circle. Similarity, Right Triangles, and Trigonometry Understand similarity in terms of similarity transformations 1. Verify experimentally the properties of IMP Year 1, Shadows: dilations: Draw the Same Shape How to Shrink It? Make It Similar 1a. A dilation takes a line not passing IMP Year 1, Shadows: through the center of the dilation to a parallel Draw the Same Shape line, and leaves a line passing through the center unchanged. How to Shrink It? 1b. The dilation of a line segment is longer IMP Year 1, Shadows: or shorter in the ratio given by the scale Draw the Same Shape factor. How to Shrink It? The Statue of Liberty’s Nose 2. Given two figures, use the definition of IMP Year 1, Shadows: similarity in terms of similarity Ins and Outs of Proportion transformations to decide if they are similar; explain using similarity transformations the Similar Problems meaning of similarity for triangles as the equality of all pairs of angles and the proportionality of all pairs of sides. 3. Use the properties of similarity IMP Year 1, Shadows: transformations to establish the AA criterion Triangles Versus Other Polygons for similarity of triangles. Prove theorems involving similarity 4. Prove theorems about triangles using IMP Year 1, Shadows: similarity transformations. Theorems Inside Similarity include: a line parallel to one side of a triangle divides the other two proportionally, IMP Year 2, Do Bees Build It Best?; and conversely; the Pythagorean theorem Pythagorean Proof proved using triangle similarity. Pythagoras by Proportion Correlation of Interactive Mathematics to Common Core State Standards 27 of 37 Key Curriculum Press June 2010 G EOMETRY 5. Use triangle congruence and similarity IMP Year 1, Shadows: criteria to solve problems and to prove Mirror Madness relationships in geometric figures. A Shadow of a Doubt Define trigonometric ratios and solve problems involving right triangles 6. Understand that by similarity, side ratios IMP Year 1, Shadows: in right triangles are properties of the angles Right Triangle Ratios in the triangle, leading to definitions of trigonometric ratios for acute angles. Homemade Trig Tables 7. Explain and use the relationship between IMP Year 1, Shadows: the sine and cosine of complementary Your Opposite Is My Adjacent angles. 8. Use trigonometric ratios and the IMP Year 1, Shadows: Pythagorean Theorem to solve right triangles The Tree and the Pendulum in applied problems. Sparky and the Dude IMP Year 2, Bees: Leslie’s Floral Angles (+) Apply trigonometry to general triangles 9. Derive the formula A = ½ ab sin(C) for This standard is not addressed in IMP, but a supplementary the area of a triangle by drawing an auxiliary activity will be developed for the Year 3 unit, Orchard Hideout. line from a vertex perpendicular to the opposite side. 10. Prove the Laws of Sines and Cosines and IMP Year 2, Do Bees Build It Best?; use them to solve problems. Beyond Pythagoras Comparing Sines 11. Understand and apply the Law of Sines IMP Year 2, Do Bees Build It Best?; and the Law of Cosines to find unknown Beyond Pythagoras measurements in right and non-right triangles (e.g., surveying problems, resultant Comparing Sines forces). Circles Understand and apply theorems about circles 1. Prove that all circles are similar. This standard is not addressed in IMP, but a supplementary activity will be developed for the Year 3 unit, Orchard Hideout. Correlation of Interactive Mathematics to Common Core State Standards 28 of 37 Key Curriculum Press June 2010 G EOMETRY 2. Identify and describe relationships among IMP Year 3, Orchard Hideout: inscribed angles, radii, and chords. Include Inscribed Angles the relationship between central, inscribed and circumscribed angles; inscribed angles More Inscribed Angles on a diameter are right angles; the radius of Angles In and Out a circle is perpendicular to the tangent where the radius intersects the circle. 3. Construct the inscribed and circumscribed IMP Year 3, Orchard Hideout circles of a triangle, and prove properties of The Inscribed Circle angles for a quadrilateral inscribed in a circle. Medians and Altitudes 4. (+) Construct a tangent line from a point This standard is not addressed in IMP, but a supplementary outside a given circle to the circle. activity will be developed for the Year 3 unit, Orchard Hideout. Find arc lengths and areas of sectors of circles 5. Derive using similarity the fact that the This standard is not addressed in IMP, but a supplementary length of the arc intercepted by an angle is activity will be developed for the Year 4 unit, How Much, How proportional to the radius, and define the Fast?. radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Expressing Geometric Properties with Equations Translate between the geometric description and the equation for a conic section 1. Derive the equation of a circle of given IMP Year 3, Orchard Hideout: center and radius using the Pythagorean Completing the Square and Getting a Circle Theorem; complete the square to find the center and radius of a circle given by an equation. 2. Derive the equation of a parabola given a IMP Year 3, Orchard Hideout: focus and directrix. What’s A Parabola? 3. (+) Derive the equations of ellipses and IMP Year 3, Orchard Hideout: hyperbolas given two foci for the ellipse, and Ellipses and Hyperbola by Points and Algebra two directrices of a hyperbola. Use coordinates to prove simple geometric theorems algebraically Correlation of Interactive Mathematics to Common Core State Standards 29 of 37 Key Curriculum Press June 2010 G EOMETRY 4. Use coordinates to prove simple geometric IMP Year 3, Orchard Hideout: theorems algebraically. For example, prove Proving with Distance – Part I or disprove that a figure defined by four given points in the coordinate plane is a Proving with Distance – Part II rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2). 5. Prove the slope criteria for parallel and This standard is not addressed in IMP, but a supplementary perpendicular lines and use them to solve activity will be developed for the Year 3 unit, Orchard Hideout. geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). 6. Find the point on a directed line segment IMP Year 1, Shadows: between two given points that divide the Trying Triangles segment in a given ratio. 7. Use coordinates to compute perimeters of IMP Year 3, Orchard Hideout: polygons and areas for triangles and Sprinkler in the Orchard rectangles, e.g. using the distance formula.★ Daphne’s Dance Floor Geometric Measurement and Dimension Explain volume formulas and use them to solve problems 1. Give an informal argument for the IMP Year 2, Do Bees Build It Best?: formulas for the volume of a cylinder, Which Holds More? pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and Back on the Farm informal limit arguments. Shedding Light on Prisms 2. (+) Given an informal argument using This standard is not addressed in IMP. Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. 3. Use volume formulas for cylinders, IMP Year 2, Do Bees Build It Best?: pyramids, cones and spheres to solve Back on the Farm problems.★ IMP Year 3, Orchard Hideout: Cylindrical Soda Knitting Visualize relationships between two-dimensional and three-dimensional objects Correlation of Interactive Mathematics to Common Core State Standards 30 of 37 Key Curriculum Press June 2010 G EOMETRY 4. Identify the shapes of two-dimensional IMP Year 2, Do Bees Build It Best?: cross-sections of three-dimensional objects, Flat Cubes and identify three-dimensional objects generated by rotations of two-dimensional A Voluminous Task objects. IMP Year 4, As the Cube Turns: An Animated POW Modeling with Geometry Apply geometric concepts in modeling situations 1. Use geometric shapes, their measures and IMP Year 3, Orchard Hideout: their properties to describe objects (e.g., Orchard Growth Revisited modeling a tree trunk or a human torso as a cylinder).★ 2. Apply concepts of density based on area IMP Year 3, Small World, Isn’t It?: and volume in modeling situations (e.g., What a Mess! persons per square mile, BTUs per cubic foot).★ The Growth of the Oil Slick 3. Apply geometric methods to solve design IMP Year 2, Do Bees Build it Best?: problems (e.g., designing an object or Not a Sound structure to satisfy constraints or minimize cost; working with typographic grid systems Possible Patches based on ratios).★ S TATISTICS AND P ROBABILITY Standard IMP Lessons Interpreting Categorical and Quantitative Data Summarize, represent, and interpret data on a single count or measurement variable 1. Represent data with plots on the real IMP Year 1, The Game of Pig: number line (dot plots, histograms, and box What Are the Chances? plots). Rollin’, Rollin’, Rollin’ Waiting for a Double Correlation of Interactive Mathematics to Common Core State Standards 31 of 37 Key Curriculum Press June 2010 S TATISTICS AND P ROBABILITY 2. Use statistics appropriate to the shape of IMP Year 1, The Pit and the Pendulum: the data distribution to compare center Data Spread (median, mean) and spread (interquartile range, standard deviation) of two or more The Best Spread different data sets. 3. Interpret differences in shape, center, and IMP Year 1, The Pit and the Pendulum: spread in the context of the data sets, Making Friends with Standard Deviation accounting for possible effects of extreme data points (outliers). Deviations 4. Use the mean and standard deviation of a IMP Year 1, The Pit and the Pendulum: data set to fit it to a normal distribution and Penny Weight Revisited to estimate population percentages. Recognize that there are data sets for which Can Your Calculator Pass This Soft Drink Test? such a procedure is not appropriate. Use Standard Deviation Basics calculators, spreadsheets and tables to estimate areas under the normal curve. Summarize, represent, and interpret data on two categorical and quantitative variables 5. Summarize categorical data for two IMP Year 2, Is There Really a Difference?: categories in two-way frequency tables. What Would You Expect? Interpret relative frequencies in the context of the data (including joint, marginal and Who’s Absent? conditional relative frequencies). Recognize Big and Strong possible associations and trends in the data. 6. Represent data on two quantitative IMP Year 1, The Overland Trail: variables on a scatter plot and describe how Previous Travelers the variables are related. Sublette’s Cutoff 6a. Use a model function fitted to the data to IMP Year 1, The Overland Trail: solve problems in the context of the data. Who Will Make It? Use given model functions or choose a function suggested by the context. Emphasize The Basic Student Budget linear and exponential models. IMP Year 3, Small World, Isn’t It?: California and Exponents 6b. Informally assess the fit of a model This standard is not addressed in IMP. function by plotting and analyzing residuals. 6c. Fit a linear function for scatter plots that IMP Year 1, The Overland Trail: suggest a linear association. Previous Travelers Sublette’s Cutoff Revisited Interpret linear models Correlation of Interactive Mathematics to Common Core State Standards 32 of 37 Key Curriculum Press June 2010 S TATISTICS AND P ROBABILITY 7. Interpret the slope (rate of change) and the IMP Year 1, The Overland Trail: intercept (constant term) of a linear fit in the Fort Hall Businesses context of the data. Moving Along 8. Compute (using technology) and interpret This standard is not addressed in IMP, but a supplementary the correlation coefficient of a linear fit. activity will be developed for the Year 1 unit, The Overland Trail. 9. Distinguish between correlation and IMP Year 1, The Game of Pig: causation. Coincidence or Causation? IMP Year 2, Is There Really a Difference?: Late in the Day Making Inferences and Justifying Conclusions Understand and evaluate random processes underlying statistical experiments 1. Understand that statistics is a process for IMP Year 1, The Pit and the Pendulum: making inferences about population Pendulum Variations parameters based on a random sample from that population. IMP Year 2, Is There Really a Difference? Two Different Differences 2. Decide if a specified model is consistent IMP Year 1, The Game of Pig: with results from a given data-generating Waiting for a Double process, e.g. using simulation. For example, a model says a spinning coin falls heads up Expecting the Unexpected with probability 0.5. Would a result of 5 tails IMP Year 2, Is There Really a Difference?: in a row cause you to question the model? Loaded or Not? Make inferences and justify conclusions from sample surveys, experiments and observational studies 3. Recognize the purposes of and differences IMP Year 2, Is There Really a Difference?: among sample surveys, experiments and Samples and Populations observational studies; explain how randomization relates to each. Who Gets A’s and Measles? 4. Use data from a sample survey to estimate IMP Year 2, Is There Really a Difference?: a population mean or proportion; develop a Try This Case margin of error through the use of simulation models for random sampling. Fair Dice Correlation of Interactive Mathematics to Common Core State Standards 33 of 37 Key Curriculum Press June 2010 S TATISTICS AND P ROBABILITY 5. Use data from a randomized experiment to IMP Year 2, Is There Really a Difference?: compare two treatments; justify significant Loaded or Not? differences between parameters through the use of simulation models for random The Spoon or the Coin? assignment. Random but Fair 6. Evaluate reports based on data. IMP Year 2, Is There Really a Difference?: Bad Research On Tour with Chi-Square Conditional Probability and the Rules of Probability Understand independence and conditional probability and use them to interpret data 1. Describe events as subsets of a sample IMP Year 1, The Game of Pig: space (the set of outcomes) using What Are the Chances? characteristics (or categories) of the outcomes, or as unions, intersections, or Rug Games complements of other events (“or,” “and,” Portraits of Probabilities “not”). 2. Understand that two events A and B are IMP Year 1, The Game of Pig: independent if the probability of A and B Mystery Rugs occurring together is the product of their probabilities, and use this characterization to Martian Basketball determine if they are independent. 3. Understand the conditional probability of IMP Year 1, The Game of Pig: A given B as P(A and B)/P(B), and interpret The Theory of One-and-One independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. Correlation of Interactive Mathematics to Common Core State Standards 34 of 37 Key Curriculum Press June 2010 S TATISTICS AND P ROBABILITY 4. Construct and interpret two-way IMP Year 2, Is There Really a Difference?: frequency tables of data when two categories Data, Data, Data are associated with each object being classified. Use the two-way table as a sample Samples and Populations space to decide if events are independent and A Difference Investigation to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science and English. Estimate the probability that a randomly selected student from your class will favor science given that the student is a boy. Do the same for other subjects and compare the results. 5. Recognize and explain the concepts of IMP Year 2, Is There Really a Difference? conditional probability and independence in Quality of Investigation everyday language and everyday situations. For example, compare the chance of being Who Gets A’s and Measles? unemployed if you are female with the chance of being female if you are unemployed. Use the rules of probability to compute probabilities of compound events in a uniform probability model 6. Find the conditional probability of A IMP Year 1, The Game of Pig given B as the fraction of B’s outcomes that Streak-Shooting Shelly also belong to A and interpret the answer in terms of the model. Little Pig Strategies 7. Apply the Addition Rule, This standard is not addressed in IMP, but a supplementary P(A or B) = P(A) + P(B) – P(A and B), and activity will be developed for the Year 1 unit, The Game of Pig. interpret the answer in terms of the model. 8. (+) Apply the general Multiplication Rule This standard is not addressed in IMP. in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model. 9. (+) Use permutations and combinations to IMP Year 3, Pennant Fever: compute probabilities of compound events Who’s on First? and solve problems. Five for Seven What’s for Dinner? Using Probability to Make Decisions Calculate expected values and use them to solve problems Correlation of Interactive Mathematics to Common Core State Standards 35 of 37 Key Curriculum Press June 2010 S TATISTICS AND P ROBABILITY 1. Define a random variable for a quantity of IMP Year 1, The Game of Pig: interest by assigning a numerical value to The Theory of Two-Dice Sums each event in a sample space; graph the corresponding probability distribution using Pointed Rugs the same graphical displays as for data The Theory of One-and-One distributions. 2. Calculate the expected value of a random IMP Year 1, The Game of Pig: variable; interpret it as the mean of the Waiting for a Double probability distribution. Expecting the Unexpected Rollin’, Rollin’, Rollin’ 3. Develop a probability distribution for a IMP Year 1, The Game of Pig: random variable defined for a sample space Pig Tails in which theoretical probabilities can be calculated; find the expected value. For Little Pig Strategies example, find the theoretical probability The Game of Pig distribution for the number of correct answers obtained by guessing on all five questions of multiple-choice test where each question has four choices, and find the expected grade under various grading schemes. 4. Develop a probability distribution for a IMP Year 1: The Pit and the Pendulum: random variable defined for a sample space Standard Pendulum Data and Decisions in which probabilities are assigned empirically; find the expected value. For Pendulum Variations example, find a current data distribution on IMP Year 2, Is There Really a Difference?: the number of TV sets per household in the United States and calculate the expected Delivering Results number of sets per household. How many TV Paper or Plastic? sets would you expect to find in 100 randomly selected households? Two Different Differences Revisited Use probability to evaluate outcomes of decisions 5. Weigh the possible outcomes of a decision IMP Year 1, The Game of Pig: by assigning probabilities to payoff values Spinner Give and Take and finding expected values. Correlation of Interactive Mathematics to Common Core State Standards 36 of 37 Key Curriculum Press June 2010 S TATISTICS AND P ROBABILITY 5a. Find the expected payoff for a game of IMP Year 1, The Game of Pig: chance. For example, find the expected Mia’s Cards winnings from a state lottery ticket or a game at a fast-food restaurant. Aunt Zena at the Fair The Lottery and Insurance – Why Play? 5b. Evaluate and compare strategies on the IMP Year 1, The Game of Pig: basis of expected values. For example, A Fair Rug Game? compare a high-deductible versus a low- deductible automobile insurance policy Simulating the Carrier using various, but reasonable, chances of Another Carrier Dilemma having a minor or a major accident. 6. Use probabilities to make fair decisions IMP Year 1, The Game of Pig: (e.g., drawing by lots, using a random Spins and Draws number generator). A Fair Rug Game 7. Analyze decisions and strategies using IMP Year 1, The Game of Pig: probability concepts (e.g. product testing, What’s on Back? medical testing, pulling a hockey goalie at the end of a game). A Fair Deal for the Carrier? Correlation of Interactive Mathematics to Common Core State Standards 37 of 37 Key Curriculum Press June 2010

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