In particular, it is expected that students will come to this course with a strong conceptual foundation in ratios, rates, and proportional relationships and an understanding of simple linear and non-linear patterns of growth and their representation in the coordinate plane. In addition, students should have a thorough knowledge of the key characteristics of basic geometric shapes and objects. Students entering Integrated Mathematics Course 1 should be prepared to develop a more formal approach to similarity and congruence including, as the course develops, the proofs of key theorems about congruence and similarity in triangles. Proportional functions also follow from this foundation and offer an opportunity to reinforce students’ experiences with linear relationships. An introduction to coordinate transformations of functions is applied here to linear, proportional, and simple reciprocal functions; this concept will be revisited again in Integrated Mathematics Course 3 when additional function types have been introduced. Following a discussion of propositional logic and geometric proof, the congruence and similarity theorems are verified, and basic geometric theorems are applied to geometric constructions and to the definitions of ratios in the trigonometry of right triangles. The course concludes with topics from algebra and discrete mathematics that involve reasoning about compound situations—that is, situations involving two or more events. The habits and tools of analysis and logical reasoning developed through geometric topics can and should be applied throughout mathematics. The closing unit in this course applies these tools to probability and probability distributions.

Appropriate use of technology is expected in all work. In Integrated Mathematics Course 1, this includes employing technological tools to assist students in forming and testing conjectures, creating graphs and data displays, and determining and assessing lines of fit for data. Geometric constructions should be performed using geometric software as well as classical tools, and technology should be used to aid three-dimensional visualization. Testing with and without technological tools is recommended.

How a particular subject is taught influences not only the depth and retention of the content of a course but also the development of skills in inquiry, problem solving, and critical thinking. Every opportunity should be taken to link the concepts of Integrated Mathematics Course 1 to those encountered in middle school mathematics as well as to other disciplines. Students should be encouraged to be creative and innovative in their approach to problems, to be productive and persistent in seeking solutions, and to use multiple means to communicate their insights and understanding.

The Major Concepts below provide the focus for Integrated Mathematics Course 1. They should be taught using a variety of methods and applications so that students attain a deep understanding of these concepts.

• Proportion, Scale, and Similarity
• Proportional Functions
• Fundamentals of Logic
• Geometric Relationships, Proof, and Constructions
• Linear Equations, Inequalities, and Systems
• Counting and Computing Probability for Compound Events

Maintenance Concepts should have been taught previously and are important foundational concepts that will be applied in this course. Continued facility with and understanding of the Maintenance Concepts is essential for success in the Major Concepts defined for this course.

• Ratios, Rates, and Proportions
• Functions and Coordinate Graphs
• Linear Relationships
• Systems of Linear Equations
• Basic Geometric Shapes and Objects and Their Characteristics
• Rigid Motions in the Plane
• Basic Probability

A. Proportion, Scale, and Similarity

Rates, ratios, and proportions are a major focus of a middle school curriculum. This course builds on that knowledge, extending proportions and scaling to arithmetic and geometric applications. The section following this one will extend these concepts to proportional functions in algebra.

Successful students will:

A1 Extend and apply understanding about rates and ratios, estimation, and measurement to derived measures, including weighted averages, using appropriate units and unit analysis to express and check solutions.

Derived measures are those achieved through calculations with measurement that can be taken directly.

1. Create and interpret scale drawings as a tool for solving problems.
2. Use unit analysis to clarify appropriate units in calculations.

   Example: The calculation for converting 50 feet per second to miles per hour can be checked using the unit calculation. yields the correct units since the units feet, seconds and minutes all appear in both numerator and denominator.

3. Identify applications that can be expressed using derived measures or weighted averages; use and identify potential misuses of derived measures or weighted averages.

   Examples: Percent change and density are examples of derived measures; grade averages, stock market indexes, the consumer price index, and unemployment rates are examples of weighted averages.

A2 Use ratios and proportional reasoning to apply a scale factor to a geometric object, a drawing, a three-dimensional space, or a model and analyze the effect.

A scale factor is a fixed positive real number, r, that multiplies the distances between any two points of a figure, resulting in a figure having the same shape.

1. Extend the concept of scale factor to relate the length, area, and volume of other figures and objects.

   Example: Compare the metabolic rate of a man with that of someone twice his size, assuming that the metabolic rate of the human body is proportional to the body mass raised to the ¾ power.

A3 Identify and use relationships among volumes of common solids.

1. Identify and apply the 3:2:1 relationship between the volumes of circular cylinders, hemispheres, and cones of the same height and circular base.
2. Recognize that the volume of a pyramid is one-third the volume of a prism of the same base area and height and use this to solve problems involving such measurements.

A4 Analyze, interpret, and represent origin-centered dilations and relate them to scaling and similarity.

An origin-centered dilation with scale factor r maps every point (x, y) in the coordinate plane to the point (rx, ry).

1. Interpret and represent origin-centered dilations of objects on the coordinate plane.

   Example: In the following figure, triangle A’B’C’ with A’(9,3), B’(12,6), and C’(15,0) is the dilation of triangle ABC with A(3,1), B(4,2), and C(5,0). The scale factor for this dilation is 3.

   

2. Explain why the image under an origin-centered dilation is similar to the original figure.
3. Show that an origin-centered dilation maps a line to a line with the same slope, that dilations map parallel lines to parallel lines (lines passing through the origin remain unchanged and are parallel to themselves), and that a dilation maps a figure into a similar figure.

A5 Identify and apply conditions that are sufficient to guarantee similarity of triangles.

Informally, two geometric objects in the plane are similar if they have the same shape. More formally, having the same shape means that one figure can be mapped onto the other by means of rigid transformations and/or an origin-centered dilation.

1. Identify two triangles as similar if the ratios of the lengths of corresponding sides are equal (SSS criterion), if the ratios of the lengths of two pairs of corresponding sides and the measures of the corresponding angles between them are equal (SAS criterion), or if two pairs of corresponding angles are congruent (AA criterion).
2. Apply the SSS, SAS, and AA criteria to verify whether or not two triangles are similar.
3. Apply the SSS, SAS, and AA criteria to construct a triangle similar to a given triangle using straightedge and compass or geometric software.
4. Identify the constant of proportionality and determine the measures of corresponding sides and angles for similar triangles.
5. Use similar triangles to demonstrate that the rate of change (slope) associated with any two points on a line is a constant.
6. Recognize, use, and explain why a line drawn inside a triangle parallel to one side forms a smaller triangle similar to the original one.

A6 Identify congruence as a special case of similarity; determine and apply conditions that guarantee congruence of triangles.

Informally, two figures in the plane are congruent if they have the same size and shape. More formally, having the same size and shape means that one figure can be mapped into the other by means of a sequence of rigid transformations.

1. Determine whether two plane figures are congruent by showing whether they coincide when superimposed by means of a sequence of rigid motions (translation, reflection, or rotation).
2. Identify two triangles as congruent if the lengths of corresponding sides are equal (SSS criterion), if the lengths of two pairs of corresponding sides and the measures of the corresponding angles between them are equal (SAS criterion), or if two pairs of corresponding angles are congruent and the lengths of the corresponding sides between them are equal (ASA criterion).
3. Apply the SSS, SAS, and ASA criteria to verify whether or not two triangles are congruent.
4. Apply the definition and characteristics of congruence to make constructions, solve problems, and verify basic properties of angles and triangles.

   Examples: Identify two triangles as congruent if two pairs of corresponding angles and their included sides are all equal (ASA criterion); verify that the bisector of the angle opposite the base of an isosceles triangle is the perpendicular bisector of the base; construct an isosceles triangle with a given base angle.

A7 Extend the concepts of similarity and congruence to other polygons in the plane.

1. Identify two polygons as similar if have the same number of sides and angles, if corresponding angles have the same measure, and if corresponding sides are proportional; identify two polygons as congruent if they are similar and their constant of proportionality equals one.
2. Determine whether or not two polygons are similar.
3. Use examples to show that analogues of the SSS, SAS, and AA criteria for similarity of triangles do not work for polygons with more than three sides.

B. Proportional Functions

Linear patterns of growth are a focus of the middle school curriculum. Description, analysis, and interpretation of lines should continue to be reinforced and extended, as students work in this course with functions that express direct proportions. The reciprocal functions introduced here should be linked back to student experience with proportions and with the simple exponential patterns of growth studied in middle school. These functions are strongly linked to the concepts of scaling and similarity addressed earlier in this course. Also included here is a first look at how changes in parameters affect the graph of a function.

Successful students will:

B1 Recognize, graph, and use direct proportional relationships.

A proportion is composed of two pairs of real numbers, (a, b) and (c, d), with at least one member of each pair non-zero, such that both pairs represent the same ratio. A linear function in which f(0) = 0 represents a direct proportional relationship. The function f(x) = kx, where k is constant, describes a direct proportional relationship.

1. Analyze the graph of a direct proportional relationship, f(x) = kx and identify its key characteristics.

   The graph of a direct proportional relationship is a line that passes through the origin (0, 0) whose slope is the constant of proportionality.

2. Compare and contrast the graphs of x = k, y = k, and y = kx, where k is a constant.
3. Recognize and provide a logical argument that if f(x) is a linear function, g(x) = f(x) – f(0) represents a direct proportional relationship.

   In this case, g(0) = 0, so g(x) = kx. The graph of f(x) = mx + b is the graph of the direct proportional relationship g(x) = mx shifted up (or down) by b units. Since the graph of g(x) is a straight line, so is the graph of f(x).

4. Recognize quantities that are directly proportional and express their relationship symbolically.

   Example: The relationship between length of the side of a square and its perimeter is directly proportional.

B2 Recognize, graph, and use reciprocal relationships.

A function of the form f(x) = k/x where k is constant describes a reciprocal relationship. The term “inversely proportional” is sometimes used to identify such relationships, however, this term can be very confusing since the word "inverse" is also used in the term "inverse function" (the function y = f^-1(x) with the property that f f^-1(x) = f^-1 f(x) = x, which describes the identity function).

1. Analyze the graph of reciprocal relationships, f(x) = k/x and identify its key characteristics.

   The graph of f(x) = k/x is not a straight line and does not cross either the x– or the y–axis (i.e., there is no value of x for which f(x) = 0, nor is there any value for f(x) if x= 0).

2. Recognize quantities that are inversely proportional and express their relationship symbolically.

   Example: The relationship between lengths of the base and side of a rectangle with fixed area is inversely proportional.

B3 Distinguish among and apply linear, direct proportional, and reciprocal relationships; identify and distinguish among applications that can be expressed using these relationships.

1. Identify whether a table, graph, formula, or context suggests a linear, direct proportional, or reciprocal relationship.
2. Create graphs of linear, direct proportional, and reciprocal functions by hand and using technology.
3. Distinguish practical situations that can be represented by linear, directly proportional, or inversely proportional relationships; analyze and use the characteristics of these relationships to answer questions about the situation.

B4 Create, interpret, and apply mathematical models to solve problems arising from contextual situations that involve linear relationships.

1. Distinguish relevant from irrelevant information, identify missing information, and find what is needed or make appropriate estimates.
2. Apply problem solving heuristics to practical problems: Represent and analyze the situation using symbols, graphs, tables, or diagrams; assess special cases; consider analogous situations; evaluate progress; check the reasonableness of results; and devise independent ways of verifying results.

B5 Explain and illustrate the effect of varying the parameters m and b in the family of linear functions and varying the parameter k in the families of directly proportional and reciprocal functions.

C. Fundamentals of Logic

This relatively short unit formalizes the vocabulary and methods of reasoning that form the foundation for logical arguments in mathematics. Examples should be taken from numeric and algebraic branches of mathematics as well as from everyday reasoning and argument. While this unit emphasizes the application of reasoning in a broad spectrum of contexts, the following unit will mainly apply logical thinking to geometric contexts.

Successful students will:

C1 Use mathematical notation, terminology, syntax, and logic; use and interpret the vocabulary of logic to describe statements and the relationship between statements.

1. Identify and give examples of definitions, conjectures, theorems, proofs, and counterexamples.
2. Describe logical statements using such terms as assumption, hypothesis, conclusion, converse, and contrapositive.

C2 Make, test, and confirm or refute conjectures using a variety of methods

1. Distinguish between inductive and deductive reasoning; explain and illustrate the importance of generalization in mathematics.

   Inductive reasoning is based on observed patterns and can be used in mathematics to generate conjectures, after which deductive reasoning can be used to show that the conjectures are true in all circumstances. Inductive reasoning cannot prove propositions; valid conclusions and proof require deduction.

2. Construct simple logical arguments and proofs; determine simple counterexamples.
3. Demonstrate through example or explanation how indirect reasoning can be used to establish a claim.
4. Recognize syllogisms, tautologies, and circular reasoning and use them to assess the validity of an argument.
5. Recognize and avoid flawed reasoning; recognize flaws or gaps in the reasoning used to support an argument.

   Example: The fact that A implies B does not imply that B implies A.

C3 Analyze and apply algorithms for searching, for sorting, and for solving optimization problems.

1. Identify and apply algorithms for searching, such as sequential and binary.
2. Describe and compare simple algorithms for sorting, such as bubble sort, quick sort, and bin sort.

   Example: Compare strategies for alphabetizing a long list of words; describe a process for systematically solving the Tower of Hanoi problem.

3. Know and apply simple optimization algorithms.

   Example: Use a vertex-edge graph (network diagram) to determine the shortest path needed to accomplish some task.

D. Geometric Relationships, Proof, and Constructions

Once students have gained experience with logic in multiple venues, geometry—partially because of its physical aspects—provides an excellent context in which to hone reasoning skills. This section identifies coordinate transformations as one example of generalization in mathematics. It applies generalization as well as inductive and deductive reasoning to establish similarity theorems (introduced earlier) and geometric constructions. This topic also offers the opportunity to reinforce the theorems about angles and triangles encountered in middle school.

Successful students will:

D1 Interpret, represent, and verify geometric relationships.

1. Use the Pythagorean theorem to determine slant height, surface area, and volume for pyramids and cones; justify the process through diagrams and logical reasoning.
2. Present and analyze geometric proofs using paragraphs or two-column or flow-chart formats.

   Example: Explain why, if two lines are intersected by a third line and the corresponding angles, alternate interior angles, or alternate exterior angles are congruent, then the two original lines must be parallel.

3. Use coordinates and algebraic techniques to interpret, represent, and verify geometric relationships in the plane.

   Examples: Given the coordinates of the vertices of a quadrilateral, determine whether it is a parallelogram; given a line segment in the coordinate plane whose endpoints are known, determine its length, midpoint, and slope; find an equation of a circle given its center and radius, and conversely, given an equation of a circle, find its center and radius.

D2 Analyze, execute, explain, and apply simple geometric constructions.

1. Perform and explain simple straightedge and compass constructions.
2. Apply properties of lines and angles to perform and justify basic geometric constructions.

   Example: Use properties of alternate interior angles to construct a line parallel to a given line.

3. Use geometric computer or calculator packages to create and test conjectures about geometric properties or relationships.

D3 Show how similarity of right triangles allows the trigonometric functions sine, cosine, and tangent to be properly defined as ratios of sides.

1. Know the definitions of sine, cosine, and tangent as ratios of sides in a right triangle and use trigonometry to calculate the length of sides, measure of angles and area of a triangle.
2. Derive, interpret and use the identity sin²θ + cos²θ = 1 for angles θ between 0° and 90°.

   This identity is a special representation of the Pythagorean theorem.

E. Linear Equations, Inequalities and Systems

Considering what happens when two or more conditions exist is the theme that ties together the ideas found in the final two sections of the course. Understanding the language and meaning of mathematical terms lays the foundation for the solution of systems of equations and inequalities. Linear systems provide another opportunity to reinforce the basics of linear functions and offer a myriad of opportunities for contextual problem solving.

Successful students will:

E1 Know the concepts of sets, elements, empty set, relations (e.g., belong to), and subsets, and use them to represent relationships among objects and sets of objects.

1. Recognize and use different methods to define sets (lists, defining property).
2. Perform operations on sets: union, intersection, complement.

   Example: Use Boolean search techniques to refine online bibliographic searches.

3. Create and interpret Venn diagrams to solve problems.
4. Identify whether a given set is finite or infinite; give examples of both finite and infinite sets.

E2 Use and interpret relational conjunctions ("and," "or," "not"), terms of causation ("if...then"), and equivalence ("if and only if").

1. Distinguish between the common uses of such terms in everyday language and their use in mathematics.
2. Relate and apply these operations to situations involving sets.

E3 Solve equations and inequalities involving the absolute value of a linear expression in one variable.

1. Use conjunctions and disjunctions to express equations and inequalities involving absolute value as compound sentences that do not involve absolute value.

   Examples: Rewrite the absolute value inequality |x - 18| ≥ 27 as the disjunction x - 18 ≥ 27 or x - 18 ≤ -27.

2. Graph the solution of a single-variable inequality involving the absolute value of a linear expression as an open or closed interval on the number line or as a union of two of them.

E4 Solve and graph the solution of a linear inequality in two variables.

1. Know what it means to be a solution of a linear inequality in two variables, represent solutions algebraically and graphically, and provide examples of ordered pairs that lie in the solution set.
2. Graph a linear inequality in two variables and explain why the graph is always a half-plane (open or closed).

E5 Solve systems of two or more linear inequalities in two variables and graph the solution set.

Example: The set of points (x, y) that satisfy all three inequalities 5x - y ≥ 3, 3x + y ≤ 10, and 4x - 3y ≤ 6 is a triangle, the intersection of three half-planes whose points satisfy each inequality separately.

E6 Solve systems of linear equations in two and three variables using algebraic procedures; describe the possible arrangements of the graphs of three linear equations in three variables and relate these to the number of solutions of the corresponding system of equations.

E7 Recognize and solve problems that can be modeled using a linear inequality or a system of linear equations or inequalities; interpret the solution(s) in terms of the context of the problem.

Example: Optimization problems that can be approached through linear programming.

F. Counting and Computing Probability for Compound Events

The final topic addressed in this integrated course extends the compound thinking developed earlier from algebraic contexts to those involving discrete events. Counting the number of ways a series of events can occur and applying prior knowledge of probability encourages students to see linkages across mathematical content areas. As with linear equations, inequalities, and systems, these topics have important contextual applications.

Successful students will:

F1 Represent and calculate probabilities associated with compound events.

1. Distinguish between dependent and independent events.
2. Use Venn diagrams to summarize information about compound events.
3. Represent bivariate categorical data in a two-way frequency table; show how such a table can be used effectively to calculate and study relationships among probabilities for two events.
4. Recognize probability problems that can be represented by geometric diagrams, on the number line, or in the coordinate plane; represent such situations geometrically and apply geometric properties of length or area to calculate the probabilities.
5. Use probability to interpret odds and risks and recognize common misconceptions.

   Examples: After a fair coin has come up heads four times in a row, explain why the probability of tails is still 50%; analyze the risks associated with a particular accident, illness, or course of treatment; assess the odds of winning the lottery or being selected in a random drawing.

F2 Construct and interpret discrete graphs and charts to represent contextual situations.

1. Construct and interpret network graphs and use them to diagram social and organizational networks.

   A graph is a collection of points (nodes) and the lines (edges) that connect some subset of those points; a cycle on a graph is a closed loop created by a subset of edges. A directed graph is one with one-way arrows as edges.

   Examples: Determine the shortest route for recycling trucks; schedule when contestants play each other in a tournament; illustrate all possible travel routes that include four cities; interpret a directed graph to determine the result of a tournament.

2. Construct and interpret decision trees to represent the possible outcomes of independent events.

   A tree is a connected graph containing no closed loops (cycles).

   Examples: Classification of quadrilaterals; repeated tossing of a coin; possible outcomes of moves in a game.

3. Construct and interpret flow charts.

F3 Determine the number of ways events can occur using permutations, combinations, and other systematic counting methods.

A permutation is a rearrangement of distinct items in which their order matters; a combination is a selection of a given number of distinct items from a larger number without regard to their arrangement (i.e., in which their order does not matter).

1. Know and apply organized counting techniques such as the Fundamental Counting Principle.

   The Fundamental Counting Principal is a way of determining the number of ways a sequence of events can take place. If there are n ways of choosing one thing and m ways of choosing a second after the first has been chosen, then the Fundamental Counting Principal says that the total number of choice patterns is
   n · m.

   Examples: How many different license plates can be formed with two letters and three numerals? If the letters had to come first, how many letters would be needed to create at least as many different license plate numbers? How many different subsets are possible for a set having six elements?

2. Distinguish between counting situations that do not permit replacement and situations that do permit replacement.

   Examples: How many different four-digit numbers can be formed if the first digit must be non-zero and each digit may be used only once? How many are possible if the first digit must be non-zero but digits can be used any number of times?

3. Distinguish between situations where order matters and situations where it does not; select and apply appropriate means of computing the number of possible arrangements of the items in each case.
4. Interpret and simplify expressions involving factorial notation; use factorial notation to express permutations and combinations.

   Examples: Interpret 6! as the product 6 · 5 · 4 · 3 · 2 · 1; recognize that = 15 · 14 · 13 = 2,730.

About the Benchmarks

Elementary (K–6) Strands and Grade Levels

Secondary (7–12) Strands

Secondary Model Course Sequences

Secondary Assessments and Tasks

Correlations to the Secondary Benchmarks

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A. Proportion, Scale and Similarity

B. Proportional Functions

C. Fundamentals of Logic

D. Geometric Relationships, Proof and Constructions

E. Linear Equations, Inequalities and Systems

F. Counting and Computing Probability for Compound Events

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