High School Model Integrated Mathematics Course 3
This final course in the model integrated mathematics sequence opens with a more sophisticated and formal look at reasoning and proof in mathematics. Such reasoning is applied in numeric, algebraic, and everyday situations as well as through geometric propositions, such as the theorems in this course relating circles and their associated lines and angles.
More about the course 
A. Reasoning and Proof
Extending the fundamentals of mathematical reasoning introduced in Integrated Mathematics Course 1, students will formalize their understanding of mathematical logic and proof. In this course, reasoning is applied to numeric as well as geometric properties.
Successful students will:
A1 Use geometric examples to illustrate the relationships among undefined terms, axioms/postulates, definitions, theorems, and various methods of reasoning.
- Analyze and illustrate the effect of changing a definition or an assumption.
- Analyze the consequences of using alternative definitions; apply this especially to definitions of geometric objects.
- Demonstrate the effect that changing an assumption has on the validity of a conclusion.
A2 Present and analyze direct and indirect proofs using paragraphs or two-column or flow-chart formats.
A3 Establish simple facts about rational and irrational numbers using logical arguments and examples.
Examples: Explain why, if r and s are rational, then both r + s and rs are rational (for example, both ¾ and 2.3 are rational in 
, which is the ratio of two integers, hence rational); give examples to show that, if r and s are irrational, then r + s and rs could be either rational or irrational (for example, 
is irrational whereas 
is rational); show that a given interval on the real number line, no matter how small, contains both rational and irrational numbers.
A4 Given a degree of precision, determine a rational approximation for an irrational number.
Example: Use arithmetic methods to determine that 
when two-decimal precision is desired.
B. Geometric Reasoning and Proof
Work with circles provides opportunities to prove and apply important and more complex geometric theorems than those encountered in previous courses in the integrated course sequence. Geometric reasoning is extended to three dimensions, assisting students in developing better spatial sense and analysis skills. An optional section applying possible changes to the parallel postulate of Euclidean geometry may be used to introduce students to a very practical example of non-Euclidean space.
Successful students will:
B1 Know and apply the definitions and properties of a circle and the radius, diameter, chord, tangent, secant, and circumference of a circle.
B2 Recognize, verify, and apply statements about the properties of a circle.
- Recognize and apply the fact that a tangent to a circle is perpendicular to the radius at the point of tangency.
- Recognize, verify, and apply the relationships between central angles, inscribed angles, and circumscribed angles and the arcs they define.
Example: Show that a triangle inscribed on the diameter of a circle is a right triangle.
- Recognize, verify, and apply the relationships between inscribed and circumscribed angles of a circle and the arcs and segments they define.
Example: Prove that if a radius of a circle is perpendicular to a chord of the circle, then it bisects the chord.
B3 Determine the length of line segments and arcs, the magnitude of angles, and the area of shapes that they define in complex geometric drawings.
Examples: Determine the amount of glass in a semi-circular transom; identify the coverage of an overlapping circular pattern of irrigation; determine the distance for line of sight on the earth’s surface.
B4 Interpret and use locus definitions to generate two- and three-dimensional geometric objects.
Examples: The locus of points in the plane equidistant from two fixed points is the perpendicular bisector of the line segment joining them; the parabola defined as the locus of points equidistant from the point (5, 1) and the line y = –5 is 
; the locus of points in space equidistant from a fixed point is a sphere.
B5 Analyze cross-sections of basic three-dimensional objects and identify the resulting shapes.
- Describe all possible results of the intersection of a plane with a cube, prism, pyramid, or sphere.
B6 Describe the characteristics of the three-dimensional object traced out when a one- or two-dimensional figure is rotated about an axis.
B7 Analyze all possible relationships among two or three planes in space and identify their intersections.
- Identify a physical situation that illustrates two distinct parallel planes; identify a physical situation that illustrates two planes that intersect in a line.
- Demonstrate that three distinct planes may be parallel; two of them may be parallel to each other and intersect with the third, resulting in two parallel lines; or none may be parallel, in which case the three planes intersect in a single point, a single line, or by pairs in three parallel lines.
B8 Recognize that there are geometries other than Euclidean geometry, in which the parallel postulate is not true.
B9 Analyze and interpret geometry on a sphere. [OPTIONAL ENRICHMENT UNIT]
- Identify the parallel postulate as key in Euclidean geometry and analyze the effect of changes to that postulate
- Know and apply the definition of a great circle.
A great circle of a sphere is the circle formed by the intersection of the sphere with the plane defined by any two distinct, non-diametrically opposite points on the sphere and the center of the sphere.
Example: Show that arcs of great circles subtending angles of 180 degrees or less provide shortest routes between points on the surface of a sphere.
Since the earth is nearly spherical, this method is used to determine distance between distant points on the earth.
- Use latitude, longitude, and great circles to solve problems relating to position, distance, and displacement on the earth’s surface.
Displacement is the change in position of an object and takes into account both the distance and direction it has moved.
Example: Given the latitudes and longitudes of two points on the surface of the Earth, find the distance between them along a great circle and the bearing from one point to the other.
Bearing is the direction or angle from one point to the other relative to North = 0°. A bearing of N31°E means that the second point is 31° East of a line pointing due North of the first point.
- Interpret various two-dimensional representations for the surface of a sphere (e.g., two-dimensional maps of the Earth), called projections, and explain their characteristics.
Common projections are Mercator (and other cylindrical projections), Orthographic and Stereographic (and other Azimuthal projections), pseudo-cylindrical, and sinusoidal. Each projection has advantages for certain purposes and has its own limitations and drawbacks.
- Describe geometry on a sphere as an example of a non-Euclidean geometry in which any two lines intersect.
In spherical geometry, great circles are the counterpart of lines in Euclidean geometry. All great circles intersect. An angle between two great circles is either of the two angles formed by the intersecting planes defined by the great circles.
Examples: Show that, on a sphere, all lines intersect—that is, the parallel postulate does not hold true in this context; identify and interpret the intersection of lines of latitude with lines of longitude on a globe; recognize that the sum of the degree measures of the interior angles of a triangle on a sphere is greater than 180°.
C. Iteration and Its Applications
Recursive thinking is an important mathematical idea that naturally connects to the study of sequences and series. Sequences and series is included here as an optional topic and may be omitted if time or other constraints make its inclusion difficult.
Successful students will:
C1 Analyze, interpret, and describe relationships represented iteratively and recursively including those produced using a spreadsheet.
Examples: Recognize that the sequence defined by "First term = 5; each term after the first is six more than the preceding term" is the sequence whose first seven terms are 5, 11, 17, 23, 29, 35, and 41; recognize that the result of repeatedly squaring a number between –1 and 1 appears to approach zero while the result of repeatedly squaring a number less than –1 or greater than 1 appears to continue to increase; determine empirically how many steps are needed to produce four-digit accuracy in square roots by iterating the operations divide and average.
C2 Generate and describe sequences having specific characteristics; use calculators and spreadsheets effectively to extend sequences beyond a relatively small number of terms.
- Generate and describe the factorial function or the Fibonacci sequence recursively.
- Write a quadratic function in factored form, f(x) = a(x – r)(x – s) to identify the zeros of the function.
- Generate and describe arithmetic sequences recursively; identify arithmetic sequences expressed recursively.
Arithmetic sequences are those in which each term differs from its preceding term by a constant difference. To describe an arithmetic sequence, both the starting term and the constant difference must be specified.
Example: a1 = 5, an+1 = an + 2 describes the arithmetic sequence 5, 7, 9, 11, . . .
- Generate and describe geometric sequences recursively; identify geometric sequences expressed recursively.
Geometric sequences are those in which each term is a constant multiple of the term that precedes it. To describe a geometric sequence both the starting term and the constant multiplier (often called the common ratio) must be specified.
Example: a1 = 3, an+1 = -2an describes the geometric sequence 3, -6, 12, -24, . . .
C3 Represent, derive, and apply sequences and series. [OPTIONAL ENRICHMENT UNIT]
- Know and use subscript notation to represent the general term of a sequence and summation notation to represent partial sums of a sequence.
- Derive and apply the formulas for the general term of arithmetic and geometric sequences.
- Derive and apply formulas to calculate sums of finite arithmetic and geometric series.
- Derive and apply formulas to calculate sums of infinite geometric series whose common ratio r is in the interval (–1, 1).
- Model, analyze, and solve problems using sequences and series.
Examples: Determine the amount of interest paid over five years of a loan; determine the age of a skeleton using carbon dating; determine the cumulative relative frequency in an arithmetic or geometric growth situation.
D. Piecewise-Linear and Exponential Functions
Linear, proportional, reciprocal, quadratic, power, and polynomial functions have been studied in previous courses. This course rounds out the function toolkit with the introduction of piecewise-linear and exponential functions and their applications.
Successful students will:
D1 Identify key characteristics of absolute value, step, and other piecewise-linear functions and graph them.
- Interpret the algebraic representation of a piecewise-linear function; graph it over the appropriate domain.
- Write an algebraic representation for a given piecewise-linear function.
- Determine vertex, slope of each branch, intercepts, and end behavior of an absolute value graph.
- Recognize and solve problems that can be modeled using absolute value, step, and other piecewise-linear functions.
Examples: Postage rates, cellular telephone charges, tax rates.
D2 Graph and analyze exponential functions and identify their key characteristics.
- Describe key characteristics of the graphs of exponential functions and relate these to the coefficients in the general form f(x) = abx + c for b > 0, b ≠ 1.
Examples: Know that, if b > 1, exponential functions are increasing and that they approach a lower limit if a > 0 and an upper limit if a < 0 as x decreases; know that, if 0 < b < 1, exponential functions are decreasing and that they approach a lower limit if a > 0 and an upper limit if a < 0 as x increases.
- Explain and illustrate the effect that a change in a parameter has on an exponential function (a change in a, b, or c for f(x) = abx + c).
D3 Demonstrate the effect of compound interest, decay, or growth using iteration.
Examples: Using a spreadsheet, enter the amount of a loan, the monthly interest rate and the monthly payment in a spreadsheet. The formula (loan amount) · (1 + monthly interest rate) - (monthly payment) gives the amount remaining monthly on the loan at the end of the first month and the iterative "fill down" command will show the amount remaining on the loan at the end of each successive month; a similar process using past data about the yearly percent increase of college tuition and annual inflation rate will provide an estimate of the cost of college for a newborn in current dollar equivalents.
- Identify the diminishing effect of increasing the number of times per year that interest is compounded and relate this to the notion of instantaneous compounding.
D4 Determine the composition of simple functions, including any necessary restrictions on the domain. [OPTIONAL ENRICHMENT UNIT]
- Know the relationship among the identity function, composition of functions, and the inverse of a function, along with implications for the domain.
D5 Determine and identify key characteristics of inverse functions. [OPTIONAL ENRICHMENT UNIT]
- Analyze characteristics of inverse functions.
- Identify the conditions under which the inverse of a function is a function.
- Determine whether two given functions are inverses of each other.
- Explain why the graph of a function and its inverse are reflections of one another over the line y = x.
- Determine the inverse of linear and simple non-linear functions, including any necessary restrictions on the domain.
- Determine the inverse of a simple polynomial or simple rational function.
D6 Identify characteristics of logarithmic functions; apply logarithmic functions. [OPTIONAL ENRICHMENT UNIT]
- Identify a logarithmic function as the inverse of an exponential function.
If xy = z, x > 0, x ≠ 1, y an integer and z > 0, then y is the logarithm to the base x of z. The logarithm y = logxz is one of three equivalent forms of expressing the relation xy = z (the other being
).Examples: If 5a = b, then log5b = a.
- Know and use the definition of logarithm of a number and its relation to exponents.
Examples: log232 = log225 = 5; if x = log103, then 10x = 3.
- Prove basic properties of logarithms using properties of exponents (or the inverse exponential function).
- Use properties of logarithms to manipulate logarithmic expressions in order to extract information.
- Use logarithms to express and solve equations and problems.
Example: Explain why the number of digits in the binary representation of a decimal number N is approximately the logarithm to base 2 of N.
- Solve logarithmic equations; use logarithms to solve exponential equations.
Examples: log(x – 3) + log(x – 1) = 0.1; 5x = 8.
E. Characteristics and Transformations of Function and Equation Families
Students are expected to refresh their knowledge of all function relationships and deepen their understanding by distinguishing among them and identifying the result when simple coordinate transformations are applied. Building on prior experience with linear, simple polynomial, power, and exponential equations, students will solve rational and radical equations.
Successful students will:
E1 Distinguish among the graphs of linear, exponential, power, polynomial, or rational functions by their key characteristics.
Be aware that it can be very difficult to distinguish graphs of these various types of functions over small regions or particular subsets of their domains. Sometimes the context of an underlying situation can suggest a likely type of function model.
- Decide whether a given exponential or power function is suggested by the graph, table of values, or underlying context of a problem.
- Distinguish between the graphs of exponential growth functions and those representing exponential decay.
- Distinguish among the graphs of power functions having positive integral exponents, negative integral exponents, and exponents that are positive unit fractions (
).Power functions having exponents that are positive unit fractions are called root or radical functions.
- Identify and explain the symmetry of an even or odd power function.
- Where possible, determine the domain, range, intercepts, asymptotes, and end behavior of linear, exponential, power, polynomial, or rational functions.
Range is not always possible to determine with precision.
E2 Distinguish among linear, exponential, polynomial, rational, and power expressions; equations; and functions by their symbolic form.
- Identify linear, exponential, polynomial, rational, or power expressions, equations, or functions by their general form and the position of the variable.
Examples: f(x) = 3x is an exponential function because the variable is in the exponent while f(x) = x3 has the variable in the position of a base and is a power function; f(x) = x3 – 5 is a polynomial function but not a power function because of the added constant.
- Distinguish among power expressions, equations, and functions by the type of exponent.
Examples: f(x) = 3x5 is a polynomial function because the exponent is a positive integer, f(x) = 3x-5 is a rational or reciprocal function because the exponent is a negative integer, and f(x) = 3x1/5 is a radical function because the exponent is a unit fraction.
E3 Solve simple rational and radical equations in one variable.
- Use algebraic, numerical, graphical, and/or technological means to solve radical and rational equations.
- Know which operations on an equation produce an equation with the same solutions and which may produce an equation with fewer or more solutions (lost or extraneous roots) and adjust solution methods accordingly.
E4 Recognize and solve problems that can be modeled using exponential or power functions; interpret the solution(s) in terms of the context of the problem.
- Use exponential functions to represent growth functions, such as f(x) = anx (a > 0 and n > 1), and decay functions, such as f(x) = an-x (a > 0 and n > 1).
Exponential functions model situations where change is proportional to quantity (e.g., compound interest, population grown, radioactive decay).
- Use power functions to represent quantities arising from geometric contexts such as length, area, and volume.
Examples: The relationships between the radius and area of a circle, between the radius and volume of a sphere, and between the volumes of simple three-dimensional solids and their linear dimensions.
- Use the laws of exponents to determine exact solutions for problems involving exponential or power functions where possible; otherwise approximate the solutions graphically or numerically.
E5 Explain, illustrate, and identify the effect of simple coordinate transformations on the graph of a function.
- Interpret the graph of y = f(x – a) as the graph of y = f(x) shifted |a| units to the right (a > 0) or the left (a < 0).
- Interpret the graph of y = f(x) + a as the graph of y = f(x) shifted |a| units up (a > 0) or down (a < 0).
- Interpret the graph of y = f(ax) as the graph of y = f(x) expanded horizontally by a factor of
or compressed horizontally by a factor 
and reflected over the y-axis if a < 0. - Interpret the graph of y = af(x) as the graph of y = f(x) compressed vertically by a factor of or expanded vertically by a factor of and reflected over the x-axis if a < 0.
- Relate the algebraic properties of a function to the geometric properties of its graph.
Examples: The graph of
has vertical asymptotes at x = 1 and x = –1 while the graph of 
has a vertical asymptote at x = –2 but a hole at (2, ¼); the graph of 
is the same as the graph of 
translated five units to the left and 2 units down.
F. Mathematical Modeling with Data
Now that students have amassed experience with various function prototypes and with the effect of transformations on them, they would benefit from engaging in a project collecting and analyzing data. They will need to understand the differences among the major types of statistical studies. For the purposes of applying what they have learned about functions, a project that generates bivariate data would be most effective. As time permits, an optional section on transformation of data may be included to provide students with an introduction to how statisticians generally develop models for real data.
Successful students will:
F1 Describe the nature and purpose of sample surveys, experiments, and observational studies, relating each to the types of research questions they are best suited to address.
- Identify specific research questions that can be addressed by different techniques for collecting data.
- Critique various methods of data collection used in real-world problems, such as a clinical trial in medicine, an opinion poll, or a report on the effect of smoking on health.
- Explain why observational studies generally do not lead to good estimates of population characteristics or cause-and-effect conclusions regarding treatments.
F2 Plan and conduct sample surveys, observational studies, or experiments.
- Recognize and explain the rationale for using randomness in research designs; distinguish between random sampling from a population in sample surveys and random assignment of treatments to experimental units in an experiment.
Random sampling is how items are selected from a population so that the sample data can be used to estimate characteristics of the population; random assignment is how treatments are assigned to experimental units so that comparisons among the treatment groups can allow cause-and-effect conclusions to be made.
- Use simulations to analyze and interpret key concepts of statistical inference.
Key concepts of statistical inference include margin of error and how it relates to the design of a study and to sample size; confidence interval and how it relates to the margin of error; and p-value and how it relates to the interpretation of results from a randomized experiment.
F3 Determine, interpret, and compare linear models for data that exhibit a linear trend.
- Identify and evaluate methods of determining the goodness of fit of a linear model.
Examples: A linear model might pass through the most points, minimize the sum of the absolute deviations, or minimize the sum of the square of the deviations.
- Use a computer or a graphing calculator to determine a linear regression equation (least-squares line) as a model for data that suggest a linear trend.
- Use and interpret a residual plot or correlation coefficient to evaluate the goodness of fit of a regression line.
- Note the effect of outliers on the position and slope of the regression line; interpret the slope and y-intercept of the regression line in the context of the relationship being modeled.
F4 Apply transformations to data that exhibit curvature to analyze the underlying pattern of growth and its characteristics. [OPTIONAL ENRICHMENT UNIT]
- Apply transformations of data for the purpose of “linearizing” a scatter plot that exhibits curvature.
Examples: Apply squaring, square root, reciprocal, and logarithmic functions to input data, output data, or both; evaluate which transformation produces the strongest linear trend.
- Interpret the results of specific transformations in terms of what they indicate about the trend of the original data.
- Estimate the rate of exponential growth or decay by fitting a regression model to appropriate data transformed by logarithms.
- Estimate the exponent in a power model by fitting a regression model to appropriate data transformed by logarithms.
- Analyze how linear transformations of data affect measures of center and spread, the slope of a regression line, and the correlation coefficient.
- Use transformation techniques to select, interpret, and apply mathematical functions to summarize and model data; include models involving the functions and relationships found in all three model integrated courses.