Mathematics Course Overview: Functions, Algebra, and Trigonometry, Study notes of Mathematics

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Integrated Math Course 3

Escondido Union High School District

Curriculum is designed for Del Lago Academy Campus of Applied Sciences

Escondido Union High School District

Integrated Math 3

Course Length: Year-Long Grade Level: 11 - 12

UC/CSU Requirement:

Meets UC “c” Mathematics requirement as a new EUHSD Course

Meets UC “c” Mathematics requirement as a new EUHSD Weighted

Honors Mathematics Course With Honors Extension Activities

Graduation Requirement:

Meets EUHSD 20 Mathematics Credits

Course Number Semester 1: 1487- INTEGRATED MATH 3 A

• INTEGRATED MATH 3 HONORS A

Course Number Semester 2: 1488- INTEGRATED MATH 3 B

• INTEGRATED MATH 3 HONORS B

Transcript Abbreviation: 1487- INT MATH 3 A

1488- INT MATH 3 B

• INTEGRATED MATH 3 H A

-INTEGRATED MATH 3 H B

Number of Credits: 10 Mathematics Per Semester

Prerequisite/s Required: N/A

Prerequisite/s Recommended: N/A

Board Approval Date Curriculum: 6/17/14 (updated change in credits effective 16-17 school year)

Board Approval Date Textbook:

The Escondido Union High School District’s Courses 1 and 2 were completed in November 2012 by a team of teachers. The

Escondido Union High School District’s Course 3 was completed in 2013/14 by a team of Del Lago mathematics teachers. The team

utilized an extensive list of resources to design the document.

In addition to the regular college preparatory course of study, Scholars at Del Lago Academy may elect to participate in a rigorous

Honors pathway. The Honors curriculum was developed in conjunction with the regular course of study and all Honors activities and

requirements are outlined in BLUE font so that the reader can distinguish the Honors requirements from the traditional course of

study.

learning provides opportunity for collaboration, communication, and a robust learning environment.

This course is the third of an integrated and investigative mathematics program designed to use patterns, modeling, and conjectures to build

student understanding and competency in mathematics. The overarching goal of this course is to teach students how to extend their current

mathematical knowledge and use this to build upon concepts taught in this third course of study. The students will be expected to learn through

collaboration, collection of data, experimentation, and conjectures.

This course aligns perfectly with the five goals of the UC Mathematics requirement. The students will enhance their understanding of

mathematical sense making, make and test conjectures and justify conclusions, use mathematical models to represent real-world data, be able to

provide clear and concise answers, and have computational and symbolic fluency. All five of these goals are embedded in both the curriculum and

the core pedagogical beliefs of the Math Department.

Integrated Math 3 Course Description

This is the third course of an integrated and investigative mathematics program designed to use patterns, modeling, and conjectures to

build student understanding and competency in mathematics. The students will be expected to learn through collaboration, collection

of data, experimentation, and conjectures. Technology tools will also play an important role in learning. By using technology to

collect and model data, students will be able to make conjectures about the data and develop a robust understanding of the

mathematical principles involved. This course aligns perfectly with the five goals of the UC Mathematics requirement. The students

will learn mathematical sense making, make and test conjectures and justify conclusions, use mathematical models to represent real-

world data, be able to provide clear and concise answers, and have computational and symbolic fluency. All five of these goals are

embedded in both the curriculum and the core pedagogical beliefs of the Math Department.

The Honors Integrated Course 3 program at Del Lago academy is open to any student wishing to challenge him or herself in meeting

additional Honors curriculum requirements. The Honors scholar will be expected to complete all additional Honors extension

activities outlined within the curriculum document in addition to the traditional course of study.

Functions Overview

Interpreting Functions

• Understand the concept of a function and use function notation
• Interpret functions that arise in applications in terms of the

context

• Analyze functions using different representations

Building Functions

• Build a function that models a relationship between two

quantities

• Build new functions from existing functions

Linear, Quadratic, and Exponential Models

• Construct and compare linear, quadratic, and exponential

models and solve problems

• Interpret expressions for functions in terms of the situation they

model

Trigonometric Functions

• Extend the domain of trigonometric functions using the unit

circle

• Model periodic phenomena with trigonometric functions
• Prove and apply trigonometric identities

Mathematics | High School—Modeling

Modeling links classroom mathematics and statistics to everyday life, work, and decision-making. Modeling is the process of choosing and using

appropriate mathematics and statistics to analyze empirical situations, to understand them better, and to improve decisions. Quantities and their

relationships in physical, economic, public policy, social, and everyday situations can be modeled using mathematical and statistical methods.

When making mathematical models, technology is valuable for varying assumptions, exploring consequences, and comparing predictions with

data. A model can be very simple, such as writing total cost as a product of unit price and number bought, or using a geometric shape to describe a

physical object like a coin. Even such simple models involve making choices. It is up to us whether to model a coin as a three-dimensional

cylinder, or whether a two-dimensional disk works well enough for our purposes. Other situations—modeling a delivery route, a production

schedule, or a comparison of loan amortizations—need more elaborate models that use other tools from the mathematical sciences. Real-world

situations are not organized and labeled for analysis; formulating tractable models, representing such models, and analyzing them are

appropriately a creative process. Like every such process, this depends on acquired expertise as well as creativity.

Geometry Overview

Congruence

• Experiment with transformations in the plane
• Understand congruence in terms of rigid motions
• Prove geometric theorems
• Make geometric constructions

Similarity, Right Triangles, and Trigonometry

• Understand similarity in terms of similarity transformations
• Prove theorems involving similarity
• Define trigonometric ratios and solve problems involving right

triangles

• Apply trigonometry to general triangles

Circles

• Understand and apply theorems about circles
• Find arc lengths and areas of sectors of circles

Expressing Geometric Properties with Equations

• Translate between the geometric description and the equation

for a conic section

• Use coordinates to prove simple geometric theorems algebraically

Geometric Measurement and Dimension

• Explain volume formulas and use them to solve problems
• Visualize relationships between two-dimensional and three-

dimensional objects

Modeling with Geometry

• Apply geometric concepts in modeling situations

Statistics and Probability Overview

Interpreting Categorical and Quantitative Data

• Summarize, represent, and interpret data on a single count or

measurement variable

• Summarize, represent, and interpret data on two categorical and

quantitative variables

• Interpret linear models

Making Inferences and Justifying Conclusions

• Understand and evaluate random processes underlying

statistical experiments

• Make inferences and justify conclusions from sample surveys,

experiments and observational studies

Conditional Probability and the Rules of Probability

• Understand independence and conditional probability and use

them to interpret data

• Use the rules of probability to compute probabilities of

compound events in a uniform probability model

Using Probability to Make Decisions

• Calculate expected values and use them to solve problems
• Use probability to evaluate outcomes of decisions

situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

(^5) CCSS.Math.Practice.MP5 Use appropriate tools strategically

Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.

(^6) CCSS.Math.Practice.MP6 Attend to precision

Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.

(^7) CCSS.Math.Practice.MP7 Look for and make use of structure

Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x^2 + 9 x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3( x – y ) 2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.

(^8) CCSS.Math.Practice.MP8 Look for and express regularity in repeated reasoning

Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, 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 through (1, 2) with slope 3, middle school students might abstract the equation ( y – 2)/( x – 1) = 3. Noticing the regularity in the way terms cancel when expanding ( x – 1)( x + 1), ( x – 1)( x^2 + x + 1), and ( x – 1)( x^3 + x 2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

Integrated Course 3 Assessment Overview:

Required Assessments:

• 3 Benchmarks per semester
• Final Exam

Additional Assessments:

• Assessment for Learning- Feedback given in many forms such as direct teacher conversations and revision suggestions, mini-quizzes,

peer grading rubrics, and self-grading will help students to improve their quality of work and deepen their understanding.

• Assessment of Learning- Small quizzes, larger unit tests, and benchmark exams will assess what students have learned. Small quizzes

and unit tests will be made up of both highly contextualized problems and more procedural problems. The contextualized problems will

assess students’ ability to make sense of problems, reason abstractly, and construct viable arguments as well as modeling with

mathematics. Procedural problems will focus on assessing students’ use of tools, attending to precision, making use of structure, and

expressing regularity in repeated reasoning. The benchmark exams will be common assessments that will provide data to further drive

changes in teaching strategies in order to enhance student success.

• Writing to Learn- Students will keep a mathematics journal to write about their thinking during investigations and projects. Students will

be required to justify their reasoning during any investigation and be able to summarize their conclusions. Writing about the mathematics

will encourage students to construct viable arguments and critique the reasoning of others.

• Prior Assessment Data- Data such as that provided by state testing, NWEA scores, and Benchmark assessment data given by the middle

schools will help teachers to focus teaching in order to fill holes in students’ learning while keeping on track with new curriculum.

• Diagnostic Assessment- data provided by the Math Diagnostic Testing Project will provide teachers with individualized and whole class

data in order to diagnose the needs of their students for success in the course.

• Performance Assessments- Assessments through performance on cross-curricular projects will be used frequently throughout the year.

There will be smaller projects and key assignments during the year in which students will show their mastery of the content through a

project that allows students to display their learning using multiple representations. There will be one large interdisciplinary project per

semester as well as the smaller projects. The first large project will focus on the human body. The Exercise and Nutrition, Humanities,

and math teachers will collaborate to help students keep track of nutrition and health goals. The mathematics content involved in this

project will consist of linear functions, slope, graphing, and statistics. Each performance assessment will go through many revision

processes in order to instill an ethic of excellence in each student. An oral and written component of each performance assessment will be

part of the criteria for judging.

• Informal Daily Assessments- will be used daily by teachers through warm ups, journaling, questioning, and observations. Informal

observations are important for teachers to conduct daily to get the pulse of the classroom and see how students are internalizing the

content and procedures.

Instructional Strategies

The instructional methods and strategies listed below support the delivery of our Integrated Course Two. With emphasis on group work and

investigative activities, the Standards of Mathematical Practice will be applied throughout our curriculum.

• Anticipatory Set/Investigation Introduction- Through use of videos, manipulatives, tools, pictures, and other media, teachers will

introduce the environment, whether cultural, financial or otherwise, which leads to each investigation. The introduction will set the stage

for students to make sense of the problem and persevere in solving it as well as modeling the mathematics involved.

• Investigative & Hands-on Activities - The purpose of investigative activities is for students to learn by doing. Students will apply

mathematics they know to solve problems by modeling. Mathematically proficient students who can apply what they know will be

comfortable making assumptions and approximations to simplify a complicated situation, realizing it may need revision later.

Furthermore, students will reason abstractly and quantitatively to make sense of quantities and their relationships in problem solving.

• Collaborative Group Work - Students will often work in a collaborative setting to investigate, model, and solve problems. A culture of

participation and exploration will encourage all students to learn. Opting out of learning will not be an option. Students are encouraged

to construct viable arguments and critique the reasoning of others when discussing their conjectures.

• Questioning for Understanding and Metacognition- Teachers will use questioning strategies to lead students to deeper understanding of

both content and their own thinking processes. Questions that induce student reflection will be implemented informally, through teacher-

led questioning, and formally through math journals, exit slips, mini quizzes, etc. Students will reason abstractly, construct viable

arguments, and look for and express regularity in repeated reasoning.

• Monitor and Adjust- Teachers will monitor students’ progress using both formal and informal assessments and adjust pace and focus of

class accordingly. Teachers will anticipate likely student responses and be prepared to facilitate students’ learning via his or her own ways

of thinking about a problem. Teachers will help students make adjustments in thinking so that students can make sense of the problem and

persevere in solving it.

• Daily Assignments & Guided Practice - Daily class assignments and classroom practice will vary daily giving student’s opportunities to

practice what they’ve learned in class. This instructional approach provides students the opportunity to look for and express regularity in

repeated reasoning by practicing mathematical strategies learned in class.

• Direct Instruction & Note-making - To help with procedural fluidity students will need to take notes during periods of direct instruction.

This instructional strategy will help teachers model how to look for and make use of structure while attending to precision.

• Independent Practice - Homework for extended practice will be assigned when appropriate. Students will use this opportunity to enhance

their learning and practice. Nightly homework may be an extension of an investigation, more practice with similar scenarios, or practice

with basic symbolic skills. Students will use appropriate tools strategically and attend to precision while extending their learning.

• Project-based Learning - Students will explore real-world problems and utilize the tools necessary to make sense of problems and

persevere in solving them. In order to solve real-world problems students will be required to reason abstractly and quantitatively to

construct viable arguments for their conclusions. Cross-curricular projects will integrate topics and topics and make learning meaningful.

• Using Graphing Calculators to assist with mathematical problems - Students will learn to use graphing calculators appropriately and

efficiently in their daily assignments. They will be able to use technological tools to explore and deepen their understanding of concepts.

Unit Unit Overview and Topics Essential Learning’s

(I can… statements)

Key Investigations and

Assignments

Mathematical

Practices

A-CED1, 1.1

• create equations and inequalities in one variable including ones with absolute value and use them to solve problems in and out of context
• judge the validity of an argument according to properties of real numbers, exponents, and logarithms

Functions: F-IF 1, 2, 3, 4, 5,7a, 7c, 7d, 7e*, 8b, 9

• understand the definition of function, function notation and the graphical representation of f as the graph of the equation y=f(x)
• use function notation, evaluate functions for certain inputs, and interpret statements that use function notation in context
• recognizes that sequences are functions whose domain is the set of integers
• interpret key features of functions such as intercepts, extrema, intervals for increasing and decreasing, etc. in graphs and tables
• relate the domain of a function to its graph and quantitative relationship in context
• graph linear and quadratic functions and show intercepts, maxima, and minima
• graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior
• graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior

2. Students are required to create a

web 2.0 presentation utilizing all of the key aspects of the CC standards for speaking and listening and demonstrate their project outcomes to a group of their peers and community leaders. The learning will focus on applying the Golden Rule to nature, art, and architecture. (Presentation).

Honors Extension Activity 2 Students will complete the following problem sets designed to extend their understanding of recursion and iteration by exploring the iterative nature of fractal geometry and using an iteration process to explore the Fibonacci sequence through the investigation of sunflower growth patterns.

Lesson 1: pg. 475 (15-20) Lesson 2: pg. 507 (20-25) Lesson 3: pg. 529 (19-22)

Unit Unit Overview and Topics Essential Learning’s

(I can… statements)

Key Investigations and

Assignments

Mathematical

Practices

3. graph exponential and logarithmic functions, showing intercepts and end behavior
4. use the properties of exponents to interpret expressions for exponential functions
5. compare properties of two functions given different representations

F-BF 1a, 2

• determine an explicit expression, a recursive process, or steps for calculation from a context
• write arithmetic and geometric sequences both recursively and explicitly

F-LE 1a, 1b, 1c, 2, 5*

• prove that linear functions grow by equal differences and exponential by equal factors over equal intervals
• recognize situations in which one quantity changes at a constant rate per unit interval relative to another
• recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another
• construct linear and exponential functions given graphs, descriptions, or two coordinate pairs
• interpret the parameters in a linear or exponential function in terms of a context

Unit Unit Overview and Topics Essential Learning’s

(I can… statements)

Key Investigations and

Assignments

Mathematical

Practices

Algebra: 1a, 1b, 3c, 3d, 3f*

• interpret parts of an expression such as terms, factors, and coefficients
• interpret expressions by viewing one part as a single entity
• use the properties of exponents to transform equivalent expressions for exponential functions
• prove simple laws of logarithms
• understand and use the properties of logarithms to simplify logarithmic numeric expressions and identify their approximate value

A-CED 1, 1.1, 2, 3, 4*

• create equations and inequalities in one variable including ones with absolute value and use them to solve problems in and out of context
• judge the validity of an argument according to properties of real numbers, exponents, and logarithms
• create equations in two or more variables to represent relationships between quantities; graph equations with appropriate labels and scales on the coordinate plane
• represent constraints by equations or inequalities, systems of equations or inequalities, and interpret solutions as viable or nonviable in context
• solve for a particular variable in a formula

A-REI 1, 3, 10

• explain each step in solving a simple equation and construct a viable argument to justify solution method

Unit Unit Overview and Topics Essential Learning’s

(I can… statements)

Key Investigations and

Assignments

Mathematical

Practices

• solve linear equations and inequalities in one variable, including equations with coefficients represented by letters
• understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane

Functions: F-IF 7e*, 8b, 9

• graph exponential and logarithmic functions, showing intercepts and end behavior
• use the properties of exponents to interpret expressions for exponential functions
• compare properties of two functions given different representations

F-BF 4a, 4b, 4c, 4d, 5+

• write an expression for the inverse of a function of the form f(x)=c
• verify that a function is an inverse by using the composition of functions
• read values of an inverse function from a graph or a table
• produce an invertible function by restricting the domain
• understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents

F-TF 6+, 7+

• understand that restricting the domain on a trigonometric function allows its inverse to be constructed