Mathematics

The foundation of Christchurch Mathematics begins with a simple question: What is Mathematics? Everyone knows the word and relates an image or experience to it, but to truly define mathematics is difficult, perhaps impossible. The study of Mathematics encompasses much more than the concepts, subjects and textbooks traditionally taught in most schools.

At Christchurch, we aim to help each student explore this question deeply and to discover his/her own answer. We help our students truly feel and taste the beauty of numbers, to fully understand the roots and foundation of mathematical truths, to gain a deep appreciation for mathematical ways of thinking, and from there to refine and expand one’s individual understanding and expression of this elegant and powerful language.

Our approach to Mathematics revolves around the following concepts and values. The work and practice of Math gives us the playground in which to explore these skills and concepts so that we can make them our own and use them in the realm of Math, but also carry them out across many topics of learning and discovery.

A Universal Language
Absolute Truth
The Ability to Prove and Demonstrate
The Real “Real World Applications” of Math
A Feel for Numbers and Relativity
Patterns, Art, and Abstraction
Spatial Math and Thinking
Knowing vs. Memorization
The True Role of Technology

“Rather than telling students how to solve new kinds of problems, we want them to be able to explore and “discover”solutions on their own.” (Retrieved March 1, 2018, https://mathigon.org/about). In short, this is our goal.

Through this process, we hope our students will become more confident in abstract thinking and analysis, number sense and spatial math, and have pride in the process required to gain these skills. It is our hope that when our students are presented with a particular narrative founded in numbers, they will be empowered to see the truth behind the numbers and from this construct their own narrative.

The mathematics requirements for CCS students begin with the traditional curriculum included in a “college preparatory” pathway. That is, students must first pass the basic level of Algebra 2, which is the national standard for secondary mathematics preparation for college entrance and standardized assessments. Following this threshold, students will explore practical or theoretical applications of quantitative reasoning. These post-college-prep options include a wide range of topics, projects, and opportunities to engage with quantitative reasoning.

*Foundations of Algebra

Foundation of Algebra: One of the most important fundamental values that we have at Christchurch School is to “meet students where they are.” We understand that sometimes learning gaps can occur. This discrepancy between what a student has learned and what a student is expected to know can grow in a way that will make learning more difficult the longer it is not addressed. Therefore, we created this course with the aim of teaching foundational algebra concepts while addressing the learning needs of the individual student. Our goal is that by the end of the course that every student would have the skill set necessary to be successful in an Algebra 1 course.

To address learning gaps, we must embrace the fact that there is not a “one size fits all” approach to do this. We must find a way to identify a student’s deficiency and provide the tools necessary to address these problem areas. Because of this, students will be required to purchase an ALEKS software license. ALEKS is a powerful piece of EdTech that uses machine learning and artificial intelligence to diagnose student knowledge gaps and to plot a learning course to address those gaps while teaching course content. All student assessments and homework activities assigned on ALEKS will be personalized.

However, there is value in classroom instruction. Therefore, the class will receive regular face-to-face instruction on topics that will prepare students for Algebra 1. Students will begin with transformation geometry and the study of rigid transformations, congruency, and similarity. Next, they will study linear relationships in their multiple representations while building on their ability to work with equations in one or two variables. From here, students will be introduced to concepts in probability and statistics, properties of exponents, subsets of the real number system, and the Pythagorean Theorem.

Here at CCS, teachers are encouraged to teach through activity and investigation whenever possible. Therefore, many of the concepts will be introduced and investigated using a problem-based model. Students will be forced out of their comfort zone and encouraged to extend their knowledge to solve problems based on the topics of their current unit.

While this course is almost like a ‘pre-Algebra’ course, we know that students’ experience with math is varied, and it may be necessary for a student to take this course even if they have prior credit for a similar course. This ensures that they are on track to move forward successfully in the core sequence at Christchurch School.
*Algebra I

Algebra 1 is the first required course in our “college preparatory” pathway. It serves as the foundation for all subsequence math courses and is the gateway to a new way of thinking about and solving problems. These concepts will help students understand how to model relationships between two quantities. Students will explore equations, inequalities, linear systems, polynomials & factoring, exponents, exponential functions, and quadratic functions. Teachers will also facilitate data talks to promote data literacy. These data talks will happen a minimum of once every 2 weeks.

CCS encourages teachers to teach through activity and investigation whenever possible. Therefore, many concepts will be introduced and investigated using a problem-based model. Students will be forced out of their comfort zone and encouraged to extend their knowledge to solve problems based on the topics of their current unit.

We know that students’ experience with math is varied and they may need to take this course even if they have prior credit for a similar course. This ensures they are on track to move forward successfully in the college preparatory sequence at Christchurch School. Students might be asked to take a placement test to show content mastery.
Geometry

Geometry is the second required course in our “college preparatory” pathway. The fundamental purpose of the Geometry course is to formalize and extend students’ geometric experiences from the middle grades. Students explore more complex geometric situations and deepen their explanations of geometric relationships, moving towards formal mathematical arguments. In this course, students will experiment and construct general ideas that will help them prove different concepts. This is a major central theme of Geometry: to justify reasoning or to prove what you know. Therefore, students will learn to justify their reasoning through formal geometric proofs. Other topics covered may include but are not limited to include perimeter/area, trigonometry, transformations, congruence, similarity, and probability.

CCS encourages teachers to teach through activity and investigation whenever possible. Therefore, many concepts will be introduced and investigated using a problem-based model. Students will be forced out of their comfort zone and encouraged to extend their knowledge to solve problems based on the topics of their current unit.

We know that students’ experience with math is varied and it may be necessary for them to take this course even if they have prior credit for a similar course. This ensures they are on track to move forward successfully in the college preparatory sequence at Christchurch School. Students who take this course must have credit in Algebra 1 or prove their understanding of Algebra 1 through examination. This ensures they are on track to move forward successfully in the college preparatory sequence at Christchurch School.
*Algebra II and Trigonometry

Algebra 2 - As the final course in our “college preparatory” pathway, Algebra 2 serves as the culmination of all things learned throughout the previous courses. Students will apply and extend their understandings from previous courses by focusing on finding connections between multiple representations of functions, transformations of different function families, finding zeros of polynomials, connecting them to graphs and equations of polynomials, and modeling periodic phenomena with trigonometry. Students will also be introduced to more complex equations and what it means to have complex/imaginary solutions. A central theme of Algebra 2 is to be able to apply and model real-world situations with the concepts that they learn.

Here at CCS, teachers are encouraged to teach through activity and investigation whenever possible. Therefore, many of the concepts will be introduced and investigated using a problem-based model. Students will be forced out of their comfort zone and encouraged to extend their knowledge to solve problems based on the topics of their current unit.

We know that students’ experience with math is varied, and it may be necessary for a student to take this course even if they have prior credit for a similar course. This ensures that they are on track to move forward successfully in the college preparatory sequence at Christchurch School. Students who take this course must have credit in Algebra 1 & Geometry or prove their understanding of both through examination.
*Honors QR Investigations and Calculus I

Honors QR Investigations and Calculus 1 is an open-ended and project-oriented course that gives students the space and structure to investigate the mathematics and applications of calculus. Calculus, the study of change, is evident in countless abstract and concrete applications. However, this course is structured so that students will experience a deep exploration of calculus topics that would go far beyond what a traditional course would go. We believe that this exploration would allow for a better understanding of calculus and its applications.

The honors track is designed for students to dig deeper into the abstract spaces of calculus in applied areas. Students will be challenged to use current events, local and global phenomena, and their own passions to direct their work and research. There is no set textbook for the course, but many resources will be used as students find their own applications and develop their own content experiences. As the course content is emergent in nature, each section and year will be different in content.

Prerequisite: Successful completion of Honors QR Analysis of Change

Honors QR Investigations and Calculus 1 carries Dual Enrollment credit through

Virginia Wesleyan University
*Honors QR Investigations and Calculus II

Honors QR Investigations and Calculus 2 is an open-ended and project-oriented course that gives students the space and structure to investigate the mathematics and applications of calculus. Calculus, the study of change, is evident in countless abstract and concrete applications. However, this course is structured so that students will experience a deep exploration of calculus topics that would go far beyond what a traditional course would go. We believe that this exploration would allow for a better understanding of calculus and its applications.

The honors track is designed for students to dig deeper into the abstract spaces of calculus in applied areas. Students will be challenged to use current events, local and global phenomena, and their own passions to direct their work and research. There is no set textbook for the course, but many resources will be used as students find their own applications and develop their own content experiences. As the course content is emergent in nature, each section and year will be different in content.

Prerequisite: Successful completion of Honors QR Calculus 1
Advanced Algebra II with Trigonometry

Advanced Algebra II with Trigonometry is a rigorous accelerated course that is designed for students that are highly motivated and are ready to engage with mathematical concepts at a deeper level. This course builds upon the foundation of Algebra I and Geometry and covers all standard Algebra II topics, including linear equations, systems and inequalities, quadratic equations and inequalities, radicals, complex and imaginary numbers, exponential functions, polynomials and polynomial equations, rational expressions and equations, and logarithmic functions. Additionally, a significant portion of the course is devoted to trigonometric functions, their graphs, identities, and applications, as well as solving trigonometric equations.

Graphing calculators are utilized to enhance conceptual understanding and analytical skills; however, students are expected to demonstrate proficiency with and without calculators.

Prerequisite: Successful completion of Algebra I and Geometry with a B in both courses
Advanced Quantitative Reasoning A - Topics in Data Science

Advanced Quantitative Reasoning A-Topics in Data Science will introduce students to the world of quantitative thinking. Quantitative thinking involves using numbers and data to understand and solve problems. It's essentially the ability to translate the world around you into mathematical terms and use those terms to reason logically and make informed decisions. Students in this course will learn how to accurately represent quantities, analyze data, problem-solve using learned mathematical concepts, use logical reasoning, and communicate ideas effectively through data talks and other collaborative activities. For most of the course, students will learn these skills organically by investigating certain topics in data science. However, some topics will be introduced in a more traditional lecture style. The curriculum will also introduce students to the main ideas in data science through free tools such as Google Sheets, Python, Data Commons, and Tableau. Most units are project-based, where students will demonstrate their understanding through artifacts created over the course of the unit. By the end of the course, students will be well-equipped to think quantitatively and translate that thinking into action. They will have had practice real-world problems using data analysis techniques, make data-driven decisions and communicate their findings to a variety of audiences.

Prerequisite: Incoming placement or successful completion of Algebra II or Advanced Algebra II
Advanced Quantitative Reasoning B - Advanced Topics in Data Science

Advanced Quantitative Reasoning B- Advanced Topics in Data Science is a deeper dive into quantitative thinking and data science topics. Students will explore topics in liberal arts mathematics, such as voting systems, personal finance, and number theory, while also exploring advanced data science topics, such as data prioritization and machine learning. Students will continue to strengthen their ability to communicate ideas effectively through data talks and collaborative work. This combination equips students to understand the underlying mathematical principles and leverage data analysis techniques to solve real-world problems. By the end of the course, students will be well-prepared to critically evaluate information, make data-driven decisions, and contribute meaningfully to fields that rely heavily on quantitative analysis.

Prerequisite: Successful completion of Advanced QR-A
Algebra I

Algebra I is a foundational, college-preparatory mathematics course that introduces students to abstract reasoning and the use of symbolic language to model and solve real-world problems. As the first step in the high school math sequence, this course lays the groundwork for future study in mathematics and science, and provides students with tools to think logically, analyze patterns, and communicate mathematical ideas.

Students will study core algebraic topics including linear equations and inequalities, systems of equations, polynomials and factoring, exponents and exponential functions, and quadratic relationships. Students will be challenged to move beyond procedural steps and develop strong conceptual reasoning and ability to apply their knowledge in real-world contexts.

Prerequisite: Teacher Recommendation
Algebra II with Trigonometry

Algebra II with Trigonometry is the third required course in our “college preparatory” pathway. As one of two choices students can take as their final course in the “college preparatory” pathway, Algebra II is the culmination of all things learned throughout the previous courses. Students will apply and extend their understandings from previous courses by focusing on finding connections between multiple representations of functions, transformations of different function families, finding zeros of polynomials, connecting them to graphs and equations of polynomials, and modeling periodic phenomena with trigonometry, time permitting. Students will also be introduced to more complex equations and what it means to have complex/imaginary solutions. A central theme of Algebra II is applying and modeling real-world situations with the concepts they learn. Completing this course will make students eligible for the Advanced Quantitative Reasoning A course. Students who desire to take Honors Analysis of Change will have additional requirements to complete.

CCS encourages teachers to teach through activity and investigation whenever possible. Therefore, many concepts will be introduced and investigated using a problem-based model. Students will be forced out of their comfort zone and encouraged to extend their knowledge to solve problems based on the topics of their current unit.

We know that students’ experience with math is varied and it may be necessary for a student to take this course even if they have prior credit for a similar course. This ensures they are on track to move forward successfully in the college preparatory sequence at Christchurch School. Students who take this course must have credit in Algebra 1 & Geometry or prove their understanding of both through examination.
Honors Calculus I

Honors Calculus is an open-ended and project-oriented course that gives students the space and structure to investigate the mathematics and applications of calculus. Calculus, the study of change, is evident in countless applications, both abstract and concrete. The honors track is designed for students to dig deeper into the abstract spaces of calculus in applied areas. Students will be challenged to use current events, local and global phenomena, and their own passions to direct their work and research. There is no set textbook for the course, but many resources will be used as students find their own applications and develop their own content experiences. As the course content is emergent in nature, each section and year will be different in content. Students will be engaging with quantitative reasoning through challenges or topics such as data sciences and data analytics, economics, policy, current events, sociology, research methodologies, statistics, data visualizations, and general engagement with quantities and a quantitative approach to problem solving.

Prerequisite: Successful completion of Honors QR Analysis of Change or a Precalculus course with B or greater
Honors Calculus II

Honors Calculus II is an honors course for students who have successfully completed Honors Quantitative Reasoning – Investigations and Calculus A (Honors Calculus I). The course is a mixture of parts of a typical college Calculus II course and part of a college differential equations course. It is designed to enrich existing mathematical skills and knowledge while also exploring new mathematical topics and realizing the numerous applications that require advanced calculus. The course includes a review and expansion of calculus topics previously covered in Honors Calculus I, such as optimization, basic integration, infinite series, and basic differential equations. A deep exploration of logarithmic and exponential functions at the calculus level is included to allow for the study of a wider variety of integration and more advanced differential equations. Students will learn applications of integration and how to identify and solve various kinds of differential equations, including first order separable, homogeneous, linear, and (if time permits) Bernoulli differential equations. Students increase their appreciation for the art of mathematics throughout the course, while mastering some of the most commonly utilized applications of calculus in the modern world. The students will also continue work from previous CCS math courses that focus on enriching numerical logic and problem solving skills, while also realizing the strong connection between the age of enlightenment, calculus, choice, chaos, and how this is useful knowledge to help fully understand our modern world.

Prerequisite: Successful completion of Honors QR Analysis of Change – Investigations and Calculus A or equivalent Calculus 1 course with a B or greater
Honors Calculus II

Honors Calculus II is an honors course for students who have successfully completed Honors Quantitative Reasoning – Investigations and Calculus A (Honors Calculus I). The course is a mixture of parts of a typical college Calculus II course and part of a college differential equations course. It is designed to enrich existing mathematical skills and knowledge while also exploring new mathematical topics and realizing the numerous applications that require advanced calculus. The course includes a review and expansion of calculus topics previously covered in Honors Calculus I, such as optimization, basic integration, infinite series, and basic differential equations. A deep exploration of logarithmic and exponential functions at the calculus level is included to allow for the study of a wider variety of integration and more advanced differential equations. Students will learn applications of integration and how to identify and solve various kinds of differential equations, including first order separable, homogeneous, linear, and (if time permits) Bernoulli differential equations. Students increase their appreciation for the art of mathematics throughout the course, while mastering some of the most commonly utilized applications of calculus in the modern world. The students will also continue work from previous CCS math courses that focus on enriching numerical logic and problem solving skills, while also realizing the strong connection between the age of enlightenment, calculus, choice, chaos, and how this is useful knowledge to help fully understand our modern world.

Prerequisite: Successful completion of Honors QR Analysis of Change – Investigations and Calculus A or equivalent Calculus 1 course with a B or greater
Honors Precalculus

Honors Pre-Calculus bridges the gap between algebra's rudimentary topics and calculus's advanced topics. Calculus is a cornerstone of mathematics in science, engineering, economics, and many other fields. Through a blend of conceptual understanding, practical skills development, and real-world applications, students will gain the mathematical fluency needed to excel in their future studies. Students will begin with topics that will strengthen their understanding of advanced algebraic topics such as functions and algebraic manipulation. While exploring algebra, students will be introduced to the concepts of limits, exploring function end-behavior using this lens. Furthermore, students will learn trigonometry topics, trigonometry ratios, and their graphical representation, and how to solve their equations. By the end of the course, students will be well-equipped to apply these concepts independently and see how they seamlessly flow together to lay the foundation for understanding and applying the powerful techniques of calculus. As an honors-level course, students will be expected to engage with a more rigorous curriculum, including deeper exploration of concepts, solving more challenging problems, and potentially covering additional topics not found in the standard course. Honors QR Analysis of Change carries Dual Enrollment credit through Virginia Wesleyan University.

Prerequisite: Incoming placement or successful completion of Algebra II with examination or Advanced Algebra II
Honors Precalculus

Honors Pre-Calculus bridges the gap between algebra's rudimentary topics and calculus's advanced topics. Calculus is a cornerstone of mathematics in science, engineering, economics, and many other fields. Through a blend of conceptual understanding, practical skills development, and real-world applications, students will gain the mathematical fluency needed to excel in their future studies. Students will begin with topics that will strengthen their understanding of advanced algebraic topics such as functions and algebraic manipulation. While exploring algebra, students will be introduced to the concepts of limits, exploring function end-behavior using this lens. Furthermore, students will learn trigonometry topics, trigonometry ratios, and their graphical representation, and how to solve their equations. By the end of the course, students will be well-equipped to apply these concepts independently and see how they seamlessly flow together to lay the foundation for understanding and applying the powerful techniques of calculus. As an honors-level course, students will be expected to engage with a more rigorous curriculum, including deeper exploration of concepts, solving more challenging problems, and potentially covering additional topics not found in the standard course. Honors QR Analysis of Change carries Dual Enrollment credit through Virginia Wesleyan University.

Prerequisite: Incoming placement or successful completion of Algebra II with examination or Advanced Algebra II
Honors QR Analysis of Change (Precalculus with Limits)

Honors QR Analysis of Change (Precalculus with Limits) is designed to bridge the gap between algebra's rudimentary topics and calculus's advanced topics. Calculus is a cornerstone of mathematics in science, engineering, economics, and many other fields. Through a blend of conceptual understanding, practical skills development, and real-world applications, students will gain the mathematical fluency needed to excel in their future studies. Students will begin with topics that will strengthen their understanding of advanced algebraic topics such as functions and algebraic manipulation. While exploring algebra, students will be introduced to the concepts of limits, exploring function end-behavior using this lens. Furthermore, students will learn trigonometry topics, trigonometry ratios, and their graphical representation, and how to solve their equations. By the end of the course, students will be well-equipped to apply these concepts independently and see how they seamlessly flow together to lay the foundation for understanding and applying the powerful techniques of calculus.

As an honors-level course, students will be expected to engage with a more rigorous curriculum, including deeper exploration of concepts, solving more challenging problems, and potentially covering additional topics not found in the standard course.

Prerequisite: Incoming placement or successful completion of Algebra II with examination or Advanced Algebra II. Honors QR Analysis of Change carries Dual Enrollment credit through Virginia Wesleyan University.
Honors QR Investigations and Calculus A

Honors QR Investigations and Calculus A is an open-ended and project-oriented course that gives students the space and structure to investigate the mathematics and applications of calculus. Calculus, the study of change, is evident in countless applications, both abstract and concrete. The honors track is designed for students to dig deeper into the abstract spaces of calculus in applied areas. Students will be challenged to use current events, local and global phenomena, and their own passions to direct their work and research. There is no set textbook for the course, but many resources will be used as students find their own applications and develop their own content experiences. As the course content is emergent in nature, each section and year will be different in content. Students will be engaging with quantitative reasoning through challenges or topics such as data sciences and data analytics, economics, policy, current events, sociology, research methodologies, statistics, data visualizations, and general engagement with quantities and a quantitative approach to problem solving.

Prerequisite: Successful completion of Honors QR Analysis of Change or a Precalculus course. Honors QR Investigations and Calculus A carries Dual Enrollment credit through Virginia Wesleyan University.
Honors QR Investigations and Calculus B

Honors QR Investigations and Calculus B is an honors course for students who have successfully completed Honors Quantitative Reasoning – Investigations and Calculus A. The course is a mixture of parts of a typical college Calculus II course and part of a college differential equations course. It is designed to enrich existing mathematical skills and knowledge while also exploring new mathematical topics and realizing the numerous applications that require advanced calculus. The course includes a review and expansion of calculus topics previously covered in Honors Calculus I, such as optimization, basic integration, infinite series, and basic differential equations. A deep exploration of logarithmic and exponential functions at the calculus level is included to allow for the study of a wider variety of integration and more advanced differential equations. Students will learn applications of integration and how to identify and solve various kinds of differential equations, including first order separable, homogeneous, linear, and (if time permits) Bernoulli differential equations. Students increase their appreciation for the art of mathematics throughout the course, while mastering some of the most commonly utilized applications of calculus in the modern world. The students will also continue work from last previous CCS math courses that focus on enriching numerical logic and problem solving skills, while also realizing the strong connection between the age of enlightenment, calculus, choice, chaos, and how this is is useful knowledge to help fully understand our modern world.

Prerequisite: Successful completion of Honors QR Analysis of Change – Investigations and Calculus A or equivalent Calculus 1 course. Honors QR Investigations and Calculus B carries Dual Enrollment credit through Virginia Wesleyan University.
Precalculus

Pre-Calculus focuses the use of functions to represent data and make estimates or predictions about future occurrences, an introduction to trigonometric ratios and their use in problem solving, probability and statistics. Students will explore, trigonometric ratios, probability including permutations, combinations and counting principle, polynomial functions and applications of linear, exponential, quadratic, polynomial, radical, rational and logarithmic functions. This course will also focus on trigonometric inverses and identities along with graphical analysis of said functions. Emphasis will be placed on reasoning, authentic problem solving, integration of scientific concepts, and the effective use of technology (including a required Ti-Nspire handheld). Pre-requisite: successful completion of the Integrated Math sequence or equivalent.
Precalculus

Pre-Calculus focuses the use of functions to represent data and make estimates or predictions about future occurrences, an introduction to trigonometric ratios and their use in problem solving, probability and statistics. Students will explore, trigonometric ratios, probability including permutations, combinations and counting principle, polynomial functions and applications of linear, exponential, quadratic, polynomial, radical, rational and logarithmic functions. This course will also focus on trigonometric inverses and identities along with graphical analysis of said functions. Emphasis will be placed on reasoning, authentic problem solving, integration of scientific concepts, and the effective use of technology (including a required Ti-Nspire handheld).

Pre-requisite: successful completion of the Integrated Math sequence or equivalent.
Probability and Statistics

Probability and Statistics offers a rigorous, inquiry-driven introduction to statistical reasoning, designed for students interested in conducting meaningful research. It has a strong emphasis on research-based investigation and data interpretation. Students will engage in a full cycle of research including formulating hypotheses, collecting data, applying appropriate statistical techniques, analyzing data, and presenting findings.

Topics include descriptive statistics, experimental design, data visualization, probability theory, discrete and continuous distributions, sampling techniques, hypothesis testing, confidence intervals, correlation, and linear regression. The statistics curriculum emphasizes students developing analytic skills to make informed, data-driven decisions in a variety of contexts. There is an emphasis placed on the ethical use of data, evaluating sources, and distinguishing between correlation and causation.

Students will utilize graphing calculators, spreadsheets, and statistical software to facilitate the exploration of real-world data sets; however, students are expected to demonstrate proficiency with and without calculators.

Prerequisite Successful Completion of Algebra II or Algebra II with Trigonometry