Wayne RESA
Curriculum, OS/MAISA 

Common Core Initiative

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Overarching Questions and Enduring Understandings

How can algebraic and geometric ideas be used to explore and connect representations of quadratic relations from the conic sections?

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Graphic Organizer
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Unit Abstract

This unit is an extension of the Quadratic Units from Algebra 1. Students will build on prior experiences with quadratic functions (parabolas) graphing, factoring, completing the square, solving and transforming parabolas graphically and algebraically this time encountering equations with imaginary solutions (no real solutions) since not all solutions to quadratic equations are real numbers. Therefore, the study of complex numbers will be included in this unit. They will also explore quadratic relations, more commonly known as conic sections, by making connections between algebraic and geometric relationships.


Conic sections are all considered quadratic curves because the highest power of any variable is two. Slicing a double napped cone (two cones with a shared apex) with planes at various angles will yield a parabola, circle, ellipse, or hyperbola. Students explore the locus of points that determine the four conic sections and use the distance formula along with geometric descriptions to generate equations for each of the conic sections. Observing the equations (both standard and transformation form) and the resulting graphs should include identifying type of conic section shown and all of its special properties.

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Expectations/Standards
MI: Mathematics
MI: HS: Num/Quantity
The Complex Number System
HSN-CN.A. Perform arithmetic operations with complex numbers.
HSN-CN.A.1. Know there is a complex number i such that i² = –1, and every complex number has the form a + bi with a and b real.
HSN-CN.A.2. Use the relation i² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
HSN-CN.A.3. (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
HSN-CN.C. Use complex numbers in polynomial identities and equations.
HSN-CN.C.7. Solve quadratic equations with real coefficients that have complex solutions.
HSN-CN.C.8. (+) Extend polynomial identities to the complex numbers. For example, rewrite x² + 4 as (x + 2i)(x – 2i).
HSN-CN.C.9. (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
MI: HS: Algebra
Seeing Structure in Expressions
HSA-SSE.B. Write expressions in equivalent forms to solve problems.
HSA-SSE.B.3a. Factor a quadratic expression to reveal the zeros of the function it defines.
HSA-SSE.B.3b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
Reasoning with Equations & Inequalities
HSA-REI.B. Solve equations and inequalities in one variable.
HSA-REI.B.4a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)² = q that has the same solutions. Derive the quadratic formula from this form.
HSA-REI.B.4b. Solve quadratic equations by inspection (e.g., for x² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.
HSA-REI.C. Solve systems of equations.
HSA-REI.C.7. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = –3x and the circle x² + y² = 3.
MI: HS: Functions
Interpreting Functions
HSF-IF.C. Analyze functions using different representations.
HSF-IF.C.7a. Graph linear and quadratic functions and show intercepts, maxima, and minima.
HSF-IF.C.8a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
MI: HS: Geometry
Expressing Geometric Properties with Equations
HSG-GPE.A. Translate between the geometric description and the equation for a conic section
HSG-GPE.A.1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.
HSG-GPE.A.2. Derive the equation of a parabola given a focus and directrix.
HSG-GPE.A.3. (+) Derive the equations of ellipses and hyperbolas given two foci for the ellipse, and two directrices of a hyperbola.
© Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.
Unit Level Standards

There are standards listed in this section for two reasons.

  1. The standards have been modified to be appropriate for this unit. Text in gray font is part of the Michigan K-12 standard but does not apply to this unit. Text in brackets denotes a modification that has been made to the standard.
  2. The standards contain content that is developed and/or utilized across multiple units.

Modified For this Unit

Creating Equations

HSA-CED.A. Create equations that describe numbers or relationships.

HSA-CED.A.1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

 

Geometric Measurement & Dimensions

HSG-GMD.B. Visualize the relation between two-dimensional and three-dimensional objects

HSG-GMD.B.4. Identify cross-sectional shapes of slices of three-dimensional objects [cones], and identify three-dimensional objects generated by rotations of two-dimensional objects.

 

Developed and/or utilized across multiple units

Seeing Structure in Expressions

HSA-SSE.B. Write expressions in equivalent forms to solve problems.

HSA-SSE.B.3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

 

Creating Equations

HSA-CED.A. Create equations that describe numbers or relationships.

HSA-CED.A.2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

 

Reasoning with Equations & Inequalities

HSA-REI.D. Represent and solve equations and inequalities graphically.

HSA-REI.D.10. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

 

Interpreting Functions

HSF-IF.A. Understand the concept of a function and use function notation.

HSF-IF.A.1. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

 

HSF-IF.B. Interpret functions that arise in applications in terms of the context.

HSF-IF.B.4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.★

 

HSF-IF.B.5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.★

HSF-IF.B.6. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

 

HSF-IF.C. Analyze functions using different representations.

HSF-IF.C.7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

 

HSF-IF.C.8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

 

HSF-IF.C.9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

 

Building Functions

HSF-BF.A. Build a function that models a relationship between two quantities.

HSF-BF.A.1. Write a function that describes a relationship between two quantities.

HSF-BF.A.1a. Determine an explicit expression, a recursive process, or steps for calculation from a context.

HSF-BF.A.1b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.

 

HSF-BF.B. Build new functions from existing functions.

HSF-BF.B.3. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

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Essential Questions
Essential/Focus Questions

 

  1. What shapesresultfrompassing a plane through a cone?
  2. How are the equation and the properties of a circle and ellipse similar and different?
  3. Given the equation of a conic section, how can you identify whether the graph will be a circle, ellipse, parabola, or hyperbola?
  4. How can you use a line for the directrix and a point for the focus to sketch a resulting parabola?
  5. What methods are available to convert an equationin the general conic form into a form more suitable for graphing?
Content (Key Concepts)

relationship of circles, ellipses, and hyperbolas to cones

circle

ellipse

parabola

hyperbola

locus of points

completing the square

discriminant

symmetry, lines and axes of graphs of conic sections

major axis andminor axis

transverse axis and conjugate axis

asymptotes

focus, foci

vertex, vertices

distance formula

directrix

eccentricity

i =√-1

 

complex number (a + bi, where a and b are Real Numbers)

conjugate of a complex number

 

 

Standard form

Ax2 + Bxy + Cy2 + Dx+Ey + F = 0

 

Transformation form

(x-h)​2+ (y-k)2= r​2

 

 

(x-h)2+ (y-k)​2​ = 1

a​2 b​2

 

(x-h)2_ ​(y-k)2 = 1

a2 b2

 

x = a(y-k)​2​+ h or y = a(x-h)​2​+k

 

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Unit Assessment Tasks
Skills (Intellectual Processes)

Standards for Mathematical Practice

Students have opportunities to:

  • make sense of problems and persevere in solving them by building new knowledge of conic sections through problem solving;
  • model with mathematics to develop an understanding of how mathematical ideas interconnect and build on one another to produce a coherent whole; and
  • use appropriate tools strategically particularlytechnology based tools to recognize and use connections among mathematical ideas.
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Lesson Plan Sequence
Lesson Plans (Sequence)
Resources
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Wayne RESA