Download Integrated Math 2 Foundations 10_May 2016

FOLSOM CORDOVA UNIFIED SCHOOL DISTRICT
INTEGRATED MATH 2 FOUNDATIONS (10)
Date: April 2016
Proposed Grade Level(s): 10-12
Grading: A-F
Course Length: 1 year
Subject Area: 1 year
Credits: 5 per semester
Articulation Units: N/A
Prerequisite(s): IEP Team Recommendation
Intent to Pursue ‘A-G’ College Prep Status: No
COURSE DESCRIPTION:
This is a non-college preparatory course, structured around problems and investigations that build spatial
visualization skills, conceptual understanding of geometry topics, the algebra 1 topics not covered in Integrated
Math 1or Integrated Math 1(10), introductory statistical analysis, and an awareness of connections between
different ideas. The slower pace and reduced student to teacher ratio allows time to address individual IEP
(Individualized Education Plan) goals. Students are encouraged to investigate and conjecture to develop their
reasoning skills. Lessons are structured for students to collaborate actively by working collaboratively with
peers. The course will focus on the algebra I standards not completed in Integrated Math 1 or Integrated Math
1(10), and develop an understanding of foundational geometry standards that students will learn in Integrated
Math 2, without engaging in formal proofs. The big ideas of the course are presented in an integrated
algebra/geometry context. Any aspect of this class is subject to accommodations based on IEPs.
GENERAL GOALS/ESSENTIAL QUESTIONS:
The Integrated Math 2 Foundations (10) course has a focus on quadratic expressions, equations, and functions;
comparing their characteristics and behavior to those of linear and exponential relationships from Integrated
Math 1 or Integrated Math 1(10). This course also has a focus on geometric concepts that include congruence,
similarity, and polygon attributes and theorems. The Integrated Math 2 Foundations (10) course includes
standards from the conceptual categories of Number and Quantity, Algebra, Functions, Geometry, and Statistics
and Probability. Some standards are repeated in multiple higher mathematics courses, therefore instructional
notes, which appear in bold, indicate what is appropriate for study in this particular course. The scope of the IM
2 Foundations (10) course is exploring algebraic and geometric relationships, justification and similarity,
probability and introduction to trigonometry, factoring, quadratic functions, transformations and similarity,
exploring proof, polygons and circles, and functions.
For the Integrated Math 2 Foundations (10) course, instructional time should focus on seven critical areas: (1)
extend the laws of exponents to rational exponents; (2) compare key characteristics of quadratic functions with
those of linear and exponential functions; (3) create and solve equations and inequalities and graphs involving
linear, exponential, and quadratics and (4) establish criteria for similarity of triangles based on dilations and
proportional reasoning; (5) polygons and circles; (6) algebraic and geometric relationships, and (7) right
triangles.
Upon completion of the course, students can move into Integrated Math 2 or Integrated Math 2 (10), with a C or
better, or continue on a non-college preparatory path. This course will satisfy the Integrated Math 2 graduation
requirement for FCUSD.
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Essential questions include but are not limited to: How is thinking algebraically different from thinking
arithmetically? What are some examples of using algebra to solve geometry problems? What are the differences
between rational and irrational numbers? How do we approximate irrational numbers? What are the similarities
and differences between the images and pre-images generated by translations? What are some real-world
examples of quadratic functions? How do you decide the best method to solve a quadratic equation?
COMMON CORE STATE STANDARD READING COMPONENT:
The major principles of the California Common Core Standards for Mathematics places strong emphasis on
focus, coherence, and rigor. Students are to pursue with equal intensity conceptual understanding and
procedural skill and fluency, along with application. Students will read high school level math textbook
scenarios and real-world math problems to apply mathematical ideas and concepts. Students will read for
explicitly stated and/or implied information that will be used to ask questions and to draw conclusions.
Students will translate the context of reading material into math symbols, equations, graphs, diagrams, and other
mathematical models.
The curriculum has literacy strategies embedded within the text that assists students in the following:
Understanding math tasks; Communicating their understanding orally and through writing; Writing about math;
Building math vocabulary; and Building academic vocabulary.
COMMON CORE STATE STANDARD WRITING COMPONENT:
Students are to think across grades and link to major topics in each grade. Students will use productive
discussion in small and large groups to build conceptual understanding. Through writing, students explain,
assert and defend claims. Students will also use written explanations to support mathematical procedures.
COMMON CORE STATE STANDARD SPEAKING AND LISTENING COMPONENTS:
Students will use productive discussion in small and large groups to build conceptual understanding. Students
will engage in productive dialogue to make sense of information from a variety of sources: graphs, tables,
diagrams, figures, mathematical models, expressions and equations, and written explanations, to name a few.
Speaking and listening protocols for small and large groups will be used to facilitate student success in the
process of learning mathematics and will provide students opportunities to share ideas and critique the ideas of
others.
CTE INDUSTRY SECTOR / PATHWAY / STANDARDS:
N/A
DETAILED UNITS OF INSTRUCTION:
Chapters
Chapter 1
Exploring Algebraic and Geometric
Relationships
Sections
Section 1.1
Attributes of Polygons
More Attributes of Polygons
Section 1.2
1.2.1 Making Predictions and Investigating Results
1.2.2 Perimeters and Areas of Enlarging Patterns
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1.2.3 Area as a Product and a Sum
1.2.4 Describing a Graph
Chapter 2
Justification and Similarity
Section 1.3
1.3.1 Angle Pair Relationships
1.3.2 Angles Formed by Transversals
1.3.3 More Angles Formed by Transversals
1.3.4 Angles and Sides of a Triangle
Section 2.1
Triangle Congruence Theorems
Flowcharts for Congruence
Converses
Section 2.2
2.2.1 Dilations
2.2.2 Similarity
Chapter 3
Probability and Trigonometry
Chapter 4
Factoring and
More Trigonometry
Chapter 5
Section 2.3
2.3.1 Conditions for Triangle Similarity
2.3.2 Determining Similar Triangles
2.3.3 Applying Similarity
2.3.4 Similar Triangle Proofs
Chapter 3 Probability and Trigonometry
Section 3.1
Using an Area Model
Using a Tree Diagram
Probability Models
Unions, Intersections, and Complements
Expected Value
Section 3.2
3.2.1 Constant Ratios in Right Triangles
3.2.2 Connecting Slope Ratios to Specific Angles
3.2.3 Expanding the Trig Table
3.2.4 The Tangent Ratio
3.2.5 Applying the Tangent Ratio
Chapter 4 Factoring and More Trigonometry
Section 4.1
4.1.1 Introduction to Factoring Expressions
4.1.2 Factoring with Area Models
4.1.3 Factoring More Quadratics
4.1.4 Factoring Completely
4.1.5 Factoring Special Cases
Section 4.2
4.2.1 Sine and Cosine Ratios
4.2.2 Selecting a Trig Tool
4.2.3 Inverse Trigonometry
4.2.4 Trigonometric Applications
Section 5.1
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Quadratic Functions
Chapter 6
Transformations and Similarity
Investigating the Graphs of Quadratic Functions
Multiple Representations of Quadratic Functions
Zero Product Property
Writing Equations for Quadratic Functions
Completing the Quadratic Web
Section 5.2
5.2.1 Perfect Square Equations
5.2.2 Completing the Square
5.2.3 More Completing the Square
5.2.4 Introduction to the Quadratic Formula
5.2.5 Solving and Applying Quadratic Equations
5.2.6 Introducing Complex Numbers
Chapter 6 More Right Triangles
Section 6.1
Special Right Triangles
Pythagorean Triples
Special Right Triangles and Trigonometry
Radicals and Fractional Exponents
Section 6.2
6.2.1 At Your Service
6.2.2 Angles on a Pool Table
6.2.3 Shortest Distance Problems
6.2.4 The Number System and Deriving the Quadratic
Formula
Chapter 7
Proof and Conditional Probability
Chapter 8
6.2.5 Using Algebra to Find a Maximum
6.2.6 Analyzing a Game
Section 7.1
Explore-Conjecture-Prove
Properties of Rhombi
Two Column Proofs
More Geometric Proofs
Using Similar Triangles to Prove Theorems
Section 7.2
7.2.1 Conditional Probability and Independence
7.2.2 More Conditional Probability
7.2.3 Applications of Probability
Section8.1
8.1.1Constructing Triangle Centers
Polygons and Circles
Section 8.2
8.2.1 Angles of Polygons
8.2.2 Areas of Regular Polygons
Section 8.3
8.3.1 Area Ratios of Similar Figures
8.3.2 Ratios of Similarity
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Chapter 9
Modeling with Functions
Section 8.4
8.4.1 A Special Ratio
8.4.2 Arcs and Sectors
8.4.3 Circles in Context
Section 9.1
Modeling Nonlinear Data
Parabola Investigation
Graphing Form of a Quadratic Function
Transforming the Absolute Value Functions
Section 9.2
9.2.1 Quadratic Applications with Inequalities
9.2.2 Solving Systems of Equations
Section 9.3
9.3.1 Average Rate of Change and Projectile Motion
9.3.2 Comparing the Growth of Functions
9.3.3 Piecewise-Defined Functions
9.3.4 Combining Functions
Chapter 10
Circles and More
Chapter 11
Solids
Chapter 12
Counting and Closure
Section 9.4
9.4.1 Inverse Functions
Section 10 .1
10.1.1 Modeling Nonlinear Data
10.1.2 Completing the Square for Equations of Circles
The Geometric Definition of a Parabola
Section 10.2
10.2.1 Introduction to Chords
10.2.2 Angles and Arcs
10.2.3 Chords and Angles
10.2.4 Tangents
10.2.5 Tangents and Arcs
Section 11.1
Prisms and Cylinders
Volumes of Similar Solids
Ratios of Similarity
Section 11.2
11.2.1 Volume of a Pyramid
11.2.2 Surface Area and Volume of a Cone
11.2.3 Surface Area and Volume of a Sphere
Chapter 12 Counting and Closure
Section 12.1
The Fundamental Counting Principle
Permutations
Combinations
Categorizing Counting Problems
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Section 12.2
12.2.1 Using Geometry to Calculate Probabilities
12.2.2 Choosing a Model
12.2.3 The Golden Ratio
12.2.4 Some Challenging Probability Problems
INTERVENTION/SUPPORT/REVIEW Checkpoint Materials
Solving Problems with Linear and Exponential Relationships
Calculating Areas and Perimeters of Complex Shapes
Angle Relationships in Geometric Figures
Solving Proportions and Similar Figures
Calculating Probabilities
Factoring Quadratic Expressions
Applying Trigonometric Ratios and Pythagorean Theorem
The Quadratic Web
. Solving Quadratic Equations
. Angle Measures and Areas of Regular Polygons
. Circles, Arcs, Sectors, Chords, and Tangents
TEXTBOOKS AND RESOURCE MATERIALS:
Core Connections Integrated II, CPM (College Preparatory Mathematics), 2015
SUBJECT AREA CONTENT STANDARDS TO BE ADDRESSED:
The eight Standards for Mathematical Practice describe the attributes of mathematically proficient students and
expertise that mathematics educators at all levels should seek to develop in their students. Mathematical
practices provide a vehicle through which students engage with and learn mathematics with a focus on reading,
writing, and explaining.
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
The Content Standards in the Integrated Math 2 Foundations (10) course come from the following conceptual
categories: Number and Quantity, Algebra, Geometry, and Statistics and Probability. The standards are not
listed in the order in which they should be taught. Further, the standards provide content to be developed
throughout the school year through rich instructional practices.
Number and Quantity
The Real Number System
Extend the Properties of exponents to rational exponents.
N-RN 1: Explain how the definition of the meaning of rational exponents follows from extending the properties
of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.
N-RN 2: Rewrite expressions involving radicals and rational exponents using the properties of exponents.
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Use properties of rational and irrational numbers
N-RN 3: Explain why the sum or product of two rational numbers is rational; that the sum of a rational number
and an irrational is irrational; and that the product of a nonzero rational number and an irrational number is
irrational.
Algebra
Seeing Structure in Expressions
Interpret the structure of expressions.
A-SSE 2: Use the structure of an expression to identify ways to rewrite it.
Write expressions in equivalent forms to solve problems.
A-SSE 3a: Factor a quadratic expression to reveal the zeros of the function it defines.
A-SSE 3b: Complete the square in a quadratic expression to reveal the maximum or minimum value of the
function it defines.
A-SSE 3c: Use the properties of exponents to transform expressions for exponential functions.
Arithmetic with Polynomials and Rational Expressions
Perform arithmetic operations on polynomials
A-APR 1: Understand that polynomials form a system analogous to the integers, namely, they are closed under
the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
Reasoning with Equations and Inequalities
Solve equations and inequalities in one variable
A-REI 4a: Use the method of completing the square to transform any quadratic equation in x into an equation of
the form (𝑥 − 𝑝)2 = 𝑞 that has the same solutions. Derive the quadratic formula from this form.
A-REI 4b: Solve quadratic equations by inspection taking square roots, completing the square, the quadratic
formula, and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula
gives complex solutions and write them as a ± bi for real numbers a and b.
Solve systems of equations.
A-REI 7: Solve a simple system consisting of a linear equation and a quadratic equation in two variables,
algebraically and graphically.
FUNCTIONS
Interpreting Functions
Analyze functions using different representations.
IF-7b: Graph linear and quadratic functions and show intercepts, maxima, and minima.
IF-8a: Use the process of factoring and completing the square in a quadratic function to show zeros, extreme
values, and symmetry of the graph, and interpret these in terms of a context.
IF-8b: Use the properties of exponents to interpret expressions for exponential functions.
Building Functions
BF-4a: Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression
for the inverse.
Linear, Quadratic and Exponential Models
F-LE 6: Apply quadratic functions to physical problems, such as the motion of an object under the force of
gravity.
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GEOMETRY
Congruence
Prove Geometric Theorems
G-CO 9: Prove theorems about lines and angles.
G-CO 10: Prove theorems about triangles.
G-CO 11: Prove theorems about parallelograms.
Similarity, Right Triangles, and Trigonometry
Understand similarity in terms of similarity transformations.
G-SRT 1: Verify experimentally the properties of dilations given by a center and a scale factor:
G-SRT 1a: Dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line
passing through the center unchanged.
G-SRT 1b: The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
G-SRT 2: Given two figures, use the definition of similarity in terms of similarity transformations to
decide if they are similar; explain using similarity transformations the meaning of similarity for triangles
as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of
sides.
G-SRT 3: Use the properties of similarity transformations to establish the Angle-Angle criterion for two
triangles to be similar.
Prove theorems involving similarity.
G-SRT 4: Prove theorems about triangles.
G-SRT 5: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in
geometric figures.
Define trigonometric ratios and solve problems involving right triangles.
G-SRT 6: Understand that similarity, side ratios in right triangles are properties of the angles in the triangle,
leading to definitions of trigonometric ratios for acute angles.
G-SRT 7: Explain and use the relationship between the sine and cosine of complementary angles.
G-SRT 8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
G-SRT 8.1: Derive and use trigonometric ratios for special right triangles.
Circles
Understand and apply theorems about circles.
G-C 1: Prove that all circles are similar.
G-C 2: Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship
between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius
of a circle is perpendicular to the tangent where the radius intersects the circle.
G-C 3: Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a
quadrilateral inscribed in a circle.
G-C 4: Construct a tangent line from a point outside a given circle to the circle.
Expressing Geometric Properties with Equations
Translate between the geometric description and the equation for a conic section
G-GPE 1: Derive the equation of a circle of given center and radius using Pythagorean Theorem; complete the
square to find the center and radius of a circle given by an equation.
G-GPE 2: Derive the equation of a parabola given a focus and directrix.
Use coordinates to prove simple geometric theorems algebraically.
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G-GPE 4: Use coordinates to prove simple geometric theorems algebraically.
G-GPE 6: Find the point on a directed line segment between two given points that partitions the segment in a
given ratio.
Geometric Measurement and Dimension
Explain volume formulas and use them to solve problems.
G-GMD 1: Give an informal argument for the formulas for the circumference of a circle, area of a circle,
volume of a cylinder, pyramid, and cone.
G-GMD 3: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
G-GMD 5: Know that the effect of a scale factor k greater than zero on length, area, and volume is to multiply
each by 𝑘, 𝑘 2 , 𝑘 3 , respectively; determine length, area and volume measures using scale factors.
G-GMD 6: Verify experimentally that in a triangle, angles opposite longer sides are larger, sides opposite larger
angles are longer, and the sum of any two side lengths is greater than the remaining sides length; apply these
relationships to solve real-world and mathematical problems.
STATISTICS and PROBABILITY
Conditional Probability and the Rules of Probability
Understand independence and conditional probability and use them to interpret data.
S-CP 1: Describe events as subsets of a sample space using characteristics of the outcomes, or as unions,
intersections, or complements of other events.
S-CP 2: Understand that two events A and B are independent if the probability of A and B occurring together is
the product of their probabilities, and use this characterization to determine if they are independent.
S-CP 3: Understand the conditional probability of A given B as P(A and B)/P(B) and interpret 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.
S-CP 4: Construct and interpret two-way frequency tables of data when two categories are associated with each
object being classified. Use the two-way table as a sample space to decide if events are independent and to
approximate conditional probability.
S-CP 5: Recognize and explain the concepts of conditional probability and independence in everyday language
and everyday situations.
Use the rules of probability to compute probabilities of compound events in a uniform probability model.
S-CP 6: Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and
interpret the answer in terms of the model.
S-CP 7: Apply the Addition Rule P(A or B0 = P(A) + P(B) – P (A and B), and interpret the answer in terms of
the model.
S-CP 8: Apply the general Multiplication Rule in a uniform probability model.
S-CP 9: Use permutations and combinations to compute probabilities of compound events and solve problems.
Using Probability to Make Decisions
Use probability to evaluate outcomes of decisions.
S-MD 6: Use probabilities to make fair decisions.
S-MD 7: Analyze decisions and strategies using probability concepts.
DISTRICT ESLRS TO BE ADDRESSED:
When students exit a secondary mathematics course, they will be:
• Self-directed Learners who will be able to use notes and a textbook to assist them in continuing their
learning outside of the classroom setting.
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• Efficient Communicators who can explain mathematical concepts to others and use mathematics to organize
and explain data.
• Quality Producers who understand the importance of neat, organized, work that demonstrates their thinking
and understanding of the solution they’ve formed to solve a problem.
• Constructive Thinkers who are able to attack problems with organization, logic, and mathematical skills
they’ve developed in a systematic fashion.
• Collaborative Workers who can work in a variety of settings in culturally diverse groups. They will be able
to form and use study groups to strengthen their own understanding in addition to providing the same
service for classmates.
• Responsible Citizens who accept the consequences of their actions and who demonstrate their understanding
of their role in the learning process.
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