math curriculum 2015-2016 - city charter high school -...

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Updated October 20, 2015

MATH CURRICULUM

Mathematics Curriculum The Mathematics curriculum helps students develop in-‐depth understanding of mathematical concepts, techniques and applications. Students use inquiry to explore open-‐ended situations, using methods employed by mathematicians and scientists in their work. Students routinely look for and articulate patterns, make, test and prove conjectures, and make connections among mathematical ideas.

City High delivers an integrated curriculum. Algebra and geometry, as well as the topics of probability, statistical reasoning, and discrete mathematics are addressed each year. All materials utilized in the course align with the Common Core State Standards for Mathematics.

9th Grade The freshman Mathematics class focuses on problem solving on one hand and strengthening and developing skills on the other . Each student has mathematics for three trimesters; two trimesters taught by a teacher who will loop with the class for four years, and one trimester taught by a teacher who stays with the class for grades 9 and 10. Content covered includes (1) functions, reasoning and problem-‐solving, (2)probability and expected value, (3) variables, graphs, linear functions and equations, (4) standard deviation, and (5) similar triangles and proportional reasoning.

By the end of 9th Grade Mathematics Class, students will (among other things) be able to:

• find, analyze, and generalize geometric and numeric patterns.

• calculate simple probability.

• calculate and use standard deviation to make predictions.

• write and solve a proportion representing a pair of similar triangles.

10th Grade In addition to dealing with geometry, statistics, quadratics and probability, by the end of 10th grade City High students will be ready to take the Keystone Algebra Exam. Each student has mathematics for three trimesters; two trimesters taught by a teacher who will loop with the class for four years, and one trimester taught by a teacher who stays with the class for grades 9 and 10. Content covered includes (1) area, volume, and the Pythagorean Theorem, (2) systems of equations and inequalities, (3) quadratic functions, (4) exponents and logs and (5) linear equations and slope.


• solve problems involving area of two-‐dimensional and volume of three-‐dimensional figures.

• explore triangles using trigonometry and the Pythagorean Theorem

• calculate the slope of a line using the formula (y2 – y1)/ (x2 – x1) and write its equation

• solve a system of equations and inequalities using various methods

• distribute monomials over polynomials and multiply two binomials of the form (Ax + C)(Bx + D)

• solve a quadratic equation by graphing or factoring.

• simplify expressions containing positive, negative and fractional exponents.

201 Stanwix Street

Suite 100

Pittsburgh, PA 15222

(412) 690-‐2489

Fax: (412) 690-‐2316

www.cityhigh.org



MATH CURRICULUM

201 Stanwix Street

Suite 100


(412) 690-‐2489

Fax: (412) 690-‐2316

www.cityhigh.org

11th Grade 11th graders engage with all topics from a traditional geometry course, as well as topics from Algebra II, Pre Calculus and Trigonometry . Students have the course for two trimesters. Content covered includes (1) two and three dimensional coordinate geometry, (2) matrix algebra, (3) rate of change, derivatives, and exponential growth, (4) permutations and combinations (5) circular functions and the physics of falling objects.


• derive and use the area formula for a circle.

• solve a system of three inequalities by using the feasible region and its corner points.

• calculate the slope of a line using the formula (y2 – y1)/ (x2 – x1)

• find the derivative of a function at a given point by calculating the slope of points near the given point.

• explain the significance of e.

• calculate the number of combinations or permutations for a given situation.

• convert degree measures to radians

• graph trigonometric functions.

12th Grade (As a senior, students take the 4th year of the Integrated Mathematics series or Calculus.) The senior Integrated Mathematics class is taught by one mathematics teacher. Students have the course for two trimesters. Content covered includes (1) periodic functions, vector components and complex numbers, (2) behavior of functions, (3) Central Limit Theorem and (4) derivatives and integrals.

By the end of the 12th Grade Mathematics Class, students will (among other things) be able to:

• use the quadratic formula.

• perform basic operations on imaginary and complex numbers.

• graph functions using principles of families of functions and determine the inverse of a function

• perform compositions of functions and calculate values of the function for a given expression.

• calculate margin of error and utilize the Central Limit Theorem

• calculate the derivative of simple polynomials and basic trigonometric functions

• explain derivative and integral and their relation to velocity and acceleration.

Calculus is taught by one mathematics teacher. A student has the course for two trimesters. The course covers most of the content addressed in a two-‐semester college calculus class. Units include (1) limits, (2) derivatives, (3)implicit differentiation, (4) integration, and (5) the Fundamental Theorem of Calculus



MATH CURRICULUM

By the end of the 12th grade Calculus Class, students will (among other things) be able to:

• explain the meaning of and evaluate limits

• explain the meaning of and calculate derivatives of a variety of functions

• sketch curves using maximum and minimum points, inflections points and concavity.

• explain the meaning of integration and take integrals of a variety of functions

• explain the Fundamental Theorem of Calculus and use it to find area under a curve

Mathematics Elective A mathematics elective is available for students who need a mathematics credit and/or have not demonstrated proficiency of the Keystone Algebra Exam. This one trimester course focuses on



Page 5

MATH CURRICULUM

Common Core State Standards for Mathematical Practice

Standard Dependent Independent

Make sense of problems and persevere in solving them.

Read, approach, and independently plan a pathway to work on a problem.

Encounter the idea of “struggle”, seek assistive resources, and use an alternative method provided.

Analyze constraints and make conjectures. Identify area of “struggle” and self select an alternative method.

Analyze mathematical

Relationships (tables, situations, graphs and equations).

Focus on metacognition, strengths and weaknesses, alternative methods, resources and support.

Monitor and evaluate the problem and the problem-‐solving process used.

Target “struggles” and carry out a corrective plan (within monitoring).

Reason abstractly and quantitatively.

Generate patterns, interact with algebraic rules, and begin to represent real world problems with symbols.

Contextualize responses by considering reasonableness of results, (mostly magnitude and whether rational numbers are appropriate or not).

Create appropriate symbolic representations of increasingly complex real world situations and interact with geometric formulas and measurements.

Contextualize responses by considering results, (magnitude, number type and unit).

Break complex situations into manageable pieces and reconstruct those using algebraic representations.

Contextualize responses by taking into account domain and range.

Deconstruct complex situations and create an appropriate representation of the problem at hand.

Contextualize responses by taking into account domain and range.

Construct viable arguments and critique the reasoning of others.

Move from explaining work as a series of numerical operations (I multiplied 3 x 5) to a set of math operations that make sense for the given context (I multiplied the rate of pay by the number of hours.)

Move from explaining the process to a classmate to listening and reacting to the way a classmate is thinking to connect their processes, correct errors in a meaningful way, and improve each others’ understanding.

Converse about math by noticing similarities, differences and connections, questioning, building upon each others’ ideas and enriching each others’ arguments.

Engage in critical, mathematically rich conversations with peers about the viability of a stated solution.

Model with mathematics.

State the models that

are used (table, graph, situation, equation), realize the goal is to move among them, make connections, and choose the most useful model for the problem.

Focus is on situations and tables.

Move among mathematical models, make connections, and choose the most useful model for the problem. Focus in on the more abstract models, equations and graphs.

Discuss efficiency of models.

Use all models effectively, and build a repertoire of algebraic equations/functions to serve in different math situations. Choose the most efficient model for a situation. Understand the limitations of models as they are used to make decisions.

Connect and blend models to solve math problems effectively and efficiently. Understand the limitations of models as they are used to make decisions.

201 Stanwix Street

Suite 100


(412) 690-‐2489

Fax: (412) 690-‐2316

www.cityhigh.org



Page 6

MATH CURRICULUM

Common Core State Standards for Mathematical Practice

Cont.

Standard

Cont. Dependent Independent

Use appropriate tools strategically.

Participate in discussions focused on the tools available (mental math, paper and pencil, manipulatives, geometry tools, scientific calculators, software, websites) and their usefulness. Use of a particular tool may be mandatory in some cases so students are exposed to its power and practice using it.

Decide what tool is most appropriate for a situation independently. Learning the usefulness of the graphing calculator is a priority.

Take advantage of more of the capabilities of the graphing calculator. Discuss the estimation skills required to use this device effectively, its limitations and the importance of evaluating results.

Use appropriate tools strategically, from independently deciding the most suitable tool, using it effectively and efficiently, dealing with its limitations, to evaluating the results.

Attend to precision.

Use variables, symbols, equations and units of measure consistently and appropriately.

Employ a systematic process (which may include defining variables, writing equations, using formulas and graphing with appropriate scaling of axes) so that the process is clearly communicated to others. Match the precision of the response with the situation.

Concentrate on the specificity of definitions, domain and range, constraints and possible cases. Match the precision of the response with the situation.

Employ a systematic process which shows a refinement with appropriate use of definitions, as well as consideration of domain and range, constraints and possible cases. Match the precision of the response with the situation.

Look for and make use of structure.

Use patterns to develop linear formulas. Cite similarities and differences as a method to make sense of new concepts and skills. Use problem-‐solving as a vehicle to develop new mathematics.

Explicitly discuss connections among math models. Use concept maps to organize math ideas, definitions and properties. Continue to use problem-‐solving as a vehicle to develop new mathematics.

Focus on metacognition to see new concepts and skills as fitting into an already developed schema in the mind. Perceive new content as a layer of sophistication upon already existing understandings of definitions, properties, functions, concepts and skills.

Intuitively assimilate and accommodate new mathematics into existing structures that are already understood and used.

Look for and Express regularity in repeated reasoning.

Realize repeated calculations can be generalized to create methods or shortcuts when solving problems and developing new mathematics. Specifically, develop an understanding of linear functions from noticing patterns.

Realize repeated calculations can be generalized to create methods or shortcuts when solving problems and developing new mathematics. Specifically, develop an understanding of quadratic functions and exponential functions by noticing patterns.

Realize repeated calculations can be generalized to create methods or shortcuts when solving problems and developing new mathematics. Specifically, use patterns to understand families of functions.

Realize the importance of intermediate results when solving a problem to help develop general methods. Use this idea to solve problems and create new mathematics.

201 Stanwix Street

Suite 100


(412) 690-‐2489

Fax: (412) 690-‐2316

www.cityhigh.org

math curriculum 2015-2016 - city charter high school -...

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