Nova Scotia

12. Quadratics

• 12.A3 demonstrate an understanding of the role of irrational numbers in applications

  • Classify rational and irrational numbers (10-A.8)

• 12.A4 demonstrate an understanding of the nature of the roots of quadratic equations

  • Using the discriminant (12-C.9)

• 12.A7 describe and interpret domains and ranges using set notation

  • Domain and range (12-A.1)

• 12.A9 represent non-real roots of quadratic equations as complex numbers

• 12.B1 demonstrate an understanding of the relationships that exist between arithmetic operations and the operations used when solving equations

  • Solve a quadratic equation using square roots (12-C.5)
  • Solve a quadratic equation by factoring (12-C.6)
  • Solve a quadratic equation by completing the square (12-C.7)
  • Solve a quadratic equation using the quadratic formula (12-C.8)
  • Solve equations with sums and differences of cubes (12-D.11)
  • Solve equations using a quadratic pattern (12-D.13)

• 12.B10 derive and apply the quadratic formula

  • Solve a quadratic equation using the quadratic formula (12-C.8)

• 12.B11 (advanced) analyze the quadratic formula to connect its components to the graphs of quadratic functions

• 12.C1 model real-world phenomena using quadratic functions

• 12.C3 sketch tables and graphs from descriptions and collected data

  • Graph a quadratic function (12-C.3)

• 12.C4 demonstrate an understanding of patterns that are arithmetic, power, and geometric, and relate them to corresponding functions

• 12.C8 describe and translate between graphical, tabular, written, and symbolic representations of quadratic relationships

  • Characteristics of quadratic functions (12-C.1)
  • Find the maximum or minimum value of a quadratic function (12-C.2)
  • Graph a quadratic function (12-C.3)
  • Match quadratic functions and graphs (12-C.4)

• 12.C9 translate between different forms of quadratic equations

• 12.C10 (advanced) determine the equation of a quadratic function using finite differences

• 12.C15 relate the nature of the roots of quadratic equations and the x-intercepts of the graphs of the corresponding functions

  • Characteristics of quadratic functions (12-C.1)
  • Match quadratic functions and graphs (12-C.4)

• 12.C22 solve quadratic equations

  • Solve a quadratic equation using square roots (12-C.5)
  • Solve a quadratic equation by factoring (12-C.6)
  • Solve a quadratic equation by completing the square (12-C.7)
  • Solve a quadratic equation using the quadratic formula (12-C.8)

• 12.C23 solve problems involving quadratic equations

• 12.C29 analyze tables and graphs to distinguish between linear, quadratic, and exponential relationships

  • Identify linear, quadratic and exponential functions from graphs (10-S.1)
  • Identify linear, quadratic and exponential functions from tables (10-S.2)

• 12.C31 analyze and describe the characteristics of quadratic functions

  • Characteristics of quadratic functions (12-C.1)
  • Find the maximum or minimum value of a quadratic function (12-C.2)

• 12.C32 demonstrate an understanding of how the parameter changes affect the graphs of quadratic functions

• 12.F1 analyze, determine, and apply scatter plots and determine the equations for curves of best fit, using appropriate technology

  • Outliers in scatter plots (12-AA.5)
  • Match correlation coefficients to scatter plots (12-AA.6)

12. Rate of Change

• 12.B4 (optional) calculate average rates of change

  • Average rate of change I (C-J.1)
  • Average rate of change II (C-J.2)

• 12.C16 interpret solutions to equations based on context

• 12.C17 (optional) demonstrate an understanding of the concept of rate of change in a variety of situations

• 12.C18 (optional) demonstrate an understanding that the slope of a line tangent to a curve is the instantaneous rate of change of the curve at the point of tangency

• 12.C27 (optional) approximate and interpret slopes of tangents to curves at various points on the curves, with and without technology

• 12.C28 explore and describe the dynamics of change depicted in tables and graphs

• 12.C30 (optional) describe and apply rates of change by analysing graphs, equations, and descriptions of linear and quadratic functions

12. Exponential Growth

• 12.A5 demonstrate an understanding of the role of real numbers in exponential and logarithmic expressions and equations

• 12.A7 describe and interpret domains and ranges using set notations

  • Domain and range of exponential and logarithmic functions (12-G.1)

• 12.B1 demonstrate an understanding of the relationships that exist between arithmetic operations and the operations used when solving equations

  • Solve exponential equations by rewriting the base (12-G.4)
  • Solve exponential equations using common logarithms (12-G.5)
  • Solve logarithmic equations I (12-G.6)
  • Solve logarithmic equations II (12-G.7)

• 12.B2 demonstrate an understanding of the recursive nature of exponential growth

  • Exponential growth and decay: word problems (12-G.11)

• 12.B12 apply real number exponents in expressions and equations

  • Convert between exponential and logarithmic form: rational bases (12-F.1)
  • Evaluate logarithms (12-F.2)
  • Evaluate logarithms: mixed review (12-F.9)
  • Evaluate exponential functions (12-G.2)

• 12.B13 solve systems of equations using inverse matrices

• 12.C2 model real-world phenomena using exponential functions

  • Exponential growth and decay: word problems (12-G.11)
  • Compound interest: word problems (12-G.12)
  • Continuously compounded interest: word problems (12-G.13)

• 12.C3 sketch tables and graphs from descriptions and collected data

  • Match exponential functions and graphs (12-G.3)

• 12.C4 demonstrate an understanding of patterns that are arithmetic, power, and geometric and relate them to corresponding functions

• 12.C11 describe and translate between graphical, tabular, written, and symbolic representations of exponential and logarithmic relationships

  • Match exponential functions and graphs (12-G.3)

• 12.C19 demonstrate an understanding, algebraically and graphically, that the inverse of an exponential function is a logarithmic function

  • Convert between exponential and logarithmic form: rational bases (12-F.1)

• 12.C24 solve exponential and logarithmic equations

  • Solve exponential equations by rewriting the base (12-G.4)
  • Solve exponential equations using common logarithms (12-G.5)
  • Solve logarithmic equations I (12-G.6)
  • Solve logarithmic equations II (12-G.7)

• 12.C25 solve problems involving exponential and logarithmic equations

• 12.C29 analyze tables and graphs to distinguish between linear, quadratic, and exponential relationships

  • Identify linear, quadratic and exponential functions from graphs (10-S.1)
  • Identify linear, quadratic and exponential functions from tables (10-S.2)

• 12.C33 analyze and describe the characteristics of exponential and logarithmic functions

  • Match exponential functions and graphs (12-G.3)

12. Circle Geometry

• 12.A5 demonstrate an understanding of the role of real numbers in exponential and logarithmic expressions and equations

• 12.F1 analyze, determine, and apply scatter plots and determine the equations for curves of best fit, using appropriate technology

  • Outliers in scatter plots (12-AA.5)
  • Match correlation coefficients to scatter plots (12-AA.6)

• 12.D1 develop and apply formulas for distance and midpoint

  • Midpoints (10-F.2)
  • Distance between two points (10-F.3)

• 12.E3 (advanced) write the equations of circles and ellipses in transformational form and as mapping rules to visualize and sketch graphs

• 12.E4 apply properties of circles

  • Find properties of circles (12-Q.4)
  • Write equations of circles in standard form (12-Q.5)

• 12.E5 apply inductive reasoning to make conjectures in geometric situations

• 12.E7 investigate, make, and prove conjectures associated with chord properties of circles

• 12.E8 (advanced) investigate, make, and prove conjectures associated with angle relationships in circles

• 12.E9 (advanced optional) investigate, make, and prove conjectures associated with tangent properties of circles

• 12.E11 write proofs using various axiomatic systems and assess the validity of deductive arguments

  • Proofs involving angles (11-T.4)
  • Proofs involving parallel lines I (11-T.5)
  • Proofs involving parallel lines II (11-T.6)
  • Proofs involving triangles I (11-T.13)
  • Proofs involving triangles II (11-T.14)

• 12.E12 demonstrate an understanding of the concept of converse

  • Converses, inverses and contrapositives (12-X.5)

• 12.E13 (advanced) analyze and translate between symbolic, graphical, and written representations of circles and ellipses

  • Write equations of circles in standard form (12-Q.5)
  • Write equations of ellipses in standard form (12-Q.9)
  • Convert equations of conic sections from general to standard form (12-Q.13)

• 12.E14 (advanced) translate between different forms of equations of circles and ellipses

  • Convert equations of conic sections from general to standard form (12-Q.13)

• 12.E15 (advanced) solve problems involving the equations and characteristics of circles and ellipses

  • Find properties of circles (12-Q.4)
  • Find properties of ellipses (12-Q.7)

• 12.E16 (advanced) demonstrate the transformational relationship between the circle and the ellipse

12. Probability

• 12.A6 develop an understanding of factorial notation and apply it to calculating permutations and combinations

• 12.B8 determine probabilities using permutations and combinations

  • Find probabilities using combinations and permutations (12-Y.4)

• 12.G1 develop and apply simulations to solve problems

• 12.G2 demonstrate an understanding that determining probability requires the quantifying of outcomes

  • Introduction to probability (12-Y.1)
  • Calculate probabilities of events (12-Y.2)

• 12.G3 demonstrate an understanding of the fundamental counting principle and apply it to calculate probabilities of dependent and independent events

  • Counting principle (11-Y.4)

• 12.G4 compare predicted and experimental results for familiar situations involving chance, using technology to extend the number of experimental trials

• 12.G5 (advanced) determine conditional probabilities

  • Find conditional probabilities (12-Y.7)
  • Independence and conditional probability (12-Y.8)
  • Find conditional probabilities using two-way frequency tables (12-Y.9)

• 12.G7 distinguish between situations that involve combinations and permutations

  • Find probabilities using combinations and permutations (12-Y.4)

• 12.G8 develop and apply formulas to evaluate permutations and combinations

  • Combinations and permutations (12-Y.3)
  • Find probabilities using combinations and permutations (12-Y.4)

• 12.G9 (advanced) demonstrate an understanding of binomial expansion and its connection to combinations

  • Pascal's triangle and the Binomial Theorem (12-D.15)
  • Binomial Theorem I (12-D.16)
  • Binomial Theorem II (12-D.17)

• 12.G10 (advanced) connect Pascal's Triangle with combinatorial coefficients

  • Pascal's triangle (12-D.14)