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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.




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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.




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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.




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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.




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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.




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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.




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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.




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                                           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


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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.



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                                        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).




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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

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                                                     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.




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                                                    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.
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                                                    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
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                                                   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.



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                                                     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
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                                                 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

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                                                 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




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                                                 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.



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                                                   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.



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                                                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
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                                                   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.

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                                                  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




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                                                   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.




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                                                  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




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                                                  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




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                                                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




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                                     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

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                                    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




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                                     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.




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                                  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

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                                  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
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                                   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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