Math-II

Summer / Fall Workshop - IN PROGRESS

Winter/Spring 2025 - From beginning of December to beginning of May, 2025. 

Outline

Part I: Content Domains for Subject Matter Understanding and Skill  in Mathematics  

GEOMETRY (SMR Domain 3)  

Candidates demonstrate an understanding of the foundations of geometry as outlined in the California  Common Core Content Standards for Mathematics (Grade 7, Grade 8, and High School). Candidates  demonstrate a depth and breadth of conceptual knowledge to ensure a rigorous view of geometry and  its underlying structures. They demonstrate an understanding of axiomatic systems and different forms  of logical arguments. Candidates understand, apply, and prove theorems relating to a variety of topics  in two- and three-dimensional geometry, including coordinate, synthetic, non-Euclidean, and  transformational geometry.  

0001 Plane Euclidean Geometry (SMR 3.1)  

 a. Apply the Parallel Postulate and its implications and justify its equivalents (e.g., the  Alternate Interior Angle Theorem, the angle sum of every triangle is 180 degrees)  

 b. Demonstrate knowledge of complementary, supplementary, and vertical angles  

 c. Prove theorems, justify steps, and solve problems involving similarity and congruence   

d. Apply and justify properties of triangles (e.g., the Exterior Angle Theorem, concurrence  theorems, trigonometric ratios, triangle inequality, Law of Sines, Law of Cosines, the  Pythagorean Theorem and its converse)  

 e. Apply and justify properties of polygons and circles from an advanced standpoint (e.g.,  derive the area formulas for regular polygons and circles from the area of a triangle)  

 f. Identify and justify the classical constructions (e.g., angle bisector, perpendicular bisector,  replicating shapes, regular polygons with 3, 4, 5, 6, and 8 sides)  

0002 Coordinate Geometry (SMR 3.2)  

 a. Use techniques in coordinate geometry to prove geometric theorems   

b. Model and solve mathematical and real-world problems by applying geometric concepts to  two-dimensional figures 

 c. Translate between the geometric description and the equation for a conic section   

d. Translate between rectangular and polar coordinates and apply polar coordinates and  vectors in the plane  

0003 Three-Dimensional Geometry (SMR 3.3)  

 a. Demonstrate knowledge of the relationships between lines and planes in three dimensions  (e.g., parallel, perpendicular, skew, coplanar lines)  

 b. Apply and justify properties of three-dimensional objects (e.g., the volume and surface area  formulas for prisms, pyramids, cones, cylinders, spheres) 

 c. Model and solve mathematical and real-world problems by applying geometric concepts to  three-dimensional figures  

0004 Transformational Geometry (SMR 3.4)  

 a. Demonstrate knowledge of isometries in two- and three-dimensional space (e.g., rotation,  translation, reflection), including their basic properties in relation to congruence   

b. Demonstrate knowledge of dilations (e.g., similarity transformations or change in scale  factor), including their basic properties in relation to similarity, volume, and area  

PROBABILITY AND STATISTICS (SMR Domain 4)  

Candidates demonstrate an understanding of statistics and probability distributions as outlined in the  California Common Core Content Standards for Mathematics (Grade 7, Grade 8, and High School).  Candidates demonstrate a depth and breadth of conceptual knowledge to ensure a rigorous view of  probability and statistics and their underlying structures. They solve problems and make inferences  using statistics and probability distributions. 

0005 Probability (SMR 4.1)  

 a. Prove and apply basic principles of permutations and combinations   

b. Illustrate finite probability using a variety of examples and models (e.g., the fundamental  counting principles, sample space)  

c. Use and explain the concepts of conditional probability and independence   

d. Compute and interpret the probability of an outcome, including the probabilities of  compound events in a uniform probability model   

e. Use normal, binomial, and exponential distributions to solve and interpret probability  problems  

f. Calculate expected values and use them to solve problems and evaluate outcomes of  decisions  

0006 Statistics (SMR 4.2)  

 a. Compute and interpret the mean and median of both discrete and continuous distributions   

b. Compute and interpret quartiles, range, interquartile range, and standard deviation of both  discrete and continuous distributions  

c. Select and evaluate sampling methods appropriate to a task (e.g., random, systematic,  cluster, convenience sampling) and display the results  

d. Apply the method of least squares to linear regression  

e. Apply the chi-square test  

f. Interpret scatter plots for bivariate data to investigate patterns of association between two  quantities (e.g., correlation), including the use of linear models  

g. Interpret data on a single count or measurement variable presented in a variety of formats  (e.g., dot plots, histograms, box plots)  

h. Demonstrate knowledge of P-values and hypothesis testing  

i. Demonstrate knowledge of confidence intervals  

Part II: Subject Matter Skills and Abilities  

Applicable to the Content Domains in Mathematics  

Candidates for Single Subject Teaching Credentials in mathematics use inductive and deductive  reasoning to develop, analyze, draw conclusions, and validate conjectures and arguments. As they  reason both abstractly and quantitatively, they use counterexamples, construct proofs using  contradictions, construct viable arguments, and critique the reasoning of others. They create multiple  representations of the same concept. They know the interconnections among mathematical ideas, use  appropriate tools strategically, and apply techniques and concepts from different domains and sub 

domains to model the same problem. They explain mathematical interconnections with other  disciplines. They are able to communicate their mathematical thinking clearly and coherently to  others, orally, graphically, and in writing. They attend to precision, including the use of precise  language and symbols.  

Candidates make sense of routine and complex problems, solving them by selecting from a variety of  strategies. They look for and make use of structure while demonstrating persistence and reflection in  their approaches. They analyze problems through pattern recognition, look for and express regularity  in repeated reasoning, and use analogies. They formulate and prove conjectures, and test conclusions  for reasonableness and accuracy. They use counterexamples to disprove conjectures.  

Candidates select and use different representational systems (e.g., coordinates, graphs). They  understand the usefulness of transformations and symmetry to help analyze and simplify problems.  They model with mathematics to analyze mathematical structures in real contexts. They use spatial  reasoning to model and solve problems that cross disciplines.