PUSD Math News – Mathematics 2
Module 8: Circles and Other Conics
Module 8 Overview – Circles and Other
Conics
(Standards: G.PE.1, G.PE.2)
Student and Teacher materials can be found
at Mathematics Vision Project
http://www.mathematicsvisionproje(ct.org/
(Curriculum>Secondary Mathematics
Two>Module 8: Circles and Other Conics)
In this module, students derive the equation of a
circle using the Pythagorean Theorem. They also
use previously learned skills by completing the
square to find the center and radius of a circle
given an equation. Students learn to write the
equation of a circle given various information. In
addition to circles students deepen their
understanding of quadratics by deriving the
equation of a parabola given a focus and directrix.
They then connect the equations of parabolas to
prior work with quadratic functions and learn to
write the equation of a parabola with a vertical
directrix. Students use material from this and
previous modules to construct an argument that
all parabolas are similar.
In Module 8 students continue to develop
proficiency in the Standards of Mathematical
Practices. They use definitions to derive equations
(MP2). Students also make conjectures about
circles and parabolas and back these up with
evidence from their work (MP3). In addition they
make connections between the geometry and the
algebra involved in circles and parabolas (MP7).
Scan the QR code below to take you
directly to the PUSD Secondary Math
Resources webpage for
Mathematics 2, Module 8: Circles and
Other Conics
You will find the student text, newsletter,
standards for the module, homework help
links and more!
https://goo.gl/2VurSs
Vocabulary and Major Mathematical
Concepts
Note: Section numbers followed by an H will be
addressed in the Honors Mathematics 2 course.
Prerequisite Concepts and Skills:
 Quadratic functions
 Completing the square
 Pythagorean Theorem
 Distance formula
 Midpoint
 Transformations of a graph
 Geometric transformations
 Similarity
 Equations of lines in point-slope form
PUSD Math News – Mathematics 2
Module 8: Circles and Other Conics
Asymptotes (8.8H) – a line or curve that
approaches a given curve arbitrarily closely, as
illustrated in the diagram below.
Ellipse (8.7H) – the set of all points (x,y) whose
sum of distances from two distinct points, called
foci, is constant. In the diagram below, the sum of
the lengths of each set of colored lines is this
constant.
Imagehttp://jwilson.coe.uga.edu/emt668/emat6680.f99/Kim/emat6690/instr
uctional%20unit/hyperbola/Hyperbola/Image135.gif
Center (8.1) – in this module center refers to the
center of a circle, ellipse or hyperbola.
Conic section (8.8H) – a curve that can be
derived by taking slices of a double-napped cone
(two cones balanced perfectly on their vertices).
Image- http://cseligman.com/text/history/ellipses.htm
Expanded form (8.2) – the form obtained by
multiplying out the factored form of an
expression.
Factored form (8.2) – an expression written as
the product of its factors.
Focus/Foci (8.4) – a point from which distances
are measured in forming a conic section; the
point at which these distances converge. The
plural of focus is foci.
Image- http://2012books.lardbucket.org/books/advancedalgebra/section_11/ff781607ff288130c23a6d43496bea82.png
Directrix (8.4) – a line from which distances are
measured in forming a conic section.
Hyperbola (8.8H) – all points found by keeping
the difference of the distances from two points,
called foci, constant.
Image- http://mathworld.wolfram.com/images/eps-gif/Directrix_620.gif
Image- http://images.slideplayer.com/31/9691197/slides/slide_3.jpg
PUSD Math News – Mathematics 2
Module 8: Circles and Other Conics
Line/Axis of symmetry (8.4) – the line that cuts
an image in half.
Main Topics
Note: Section numbers followed by an H will be
addressed in the Honors Mathematics 2 course.
Section in student text – Task done in class
Related Homework Help Videos
8.1 – Deriving the equation of a circle using
the Pythagorean Theorem
Image- http://image.mathcaptain.com/cms/images/67/parabolaimage.png
Parabola(8.5) – the set of points in a plane
where any point is at an equal distance from: a
fixed point (the focus ), and. a fixed straight line
(the directrix ).
Pythagorean triples (8.1) – a set of positive
integers a, b and c that fits the rule a2 + b2 = c2.
Standard form (8.3) – for conic sections,
standard form is when the equation is written in
one of the following forms:
Special products and factors
https://goo.gl/d99bkD
Pythagorean triples
https://goo.gl/CixWM4
Writing equations of circles centered at the
origin
https://goo.gl/1GNfdh
8.2 – Complete the square to find the center
and radius of a circle given by an equation
Completing the square
https://goo.gl/ew413g
Writing equations of circles in standard form
https://goo.gl/CgqJqc
8.3 – Writing the equation of a circle given
various information
Image- http://images.slideplayer.com/10/2817405/slides/slide_26.jpg
Vertex (8.4) – the point where a parabola
crosses its axis of symmetry, or the two points
where an ellipse and hyperbola meet their major
axes.
Using the distance formula
https://goo.gl/lBWd5B
8.4 – Derive the equation of a parabola given
a focus and directrix
Graphing quadratics, quadratic transformations
https://goo.gl/Ns0cRa
PUSD Math News – Mathematics 2
Module 8: Circles and Other Conics
8.4- cont’d
Graphing a parabola given a focus and directrix
https://goo.gl/bU8GmR
Using completing the square to rewrite the
equation of a circle in standard form
https://goo.gl/6LbG2p
8.5 – Connecting the equations of parabolas
to prior work with quadratic functions
Maximum or minimum values of a quadratic
https://goo.gl/awXZT8
Writing the equation of an ellipse in standard
form using completing the square
https://goo.gl/0Z8ic0
Linear equations on point-slope form
https://goo.gl/DcrjAI
8.8H – Develop the definition of a hyperbola
as the set of all points in a plane such that the
difference between the distances from the
point to each of the two foci is constant
Identifying a conic section from its equation
https://goo.gl/Z0uOYy
https://goo.gl/j47L4X
8.6 – Writing the equation of a parabola with
a vertical directrix, and constructing an
argument that all parabolas are similar
Graphing a hyperbola
https://goo.gl/1CmEx3
Parabolas with a vertical directrix
https://goo.gl/T850YC
Writing the equation for a hyperbola
https://goo.gl/Ex0sL0
Writing the equation of a circle given its center
and a point on the circle
https://goo.gl/E0VIgP
Key features of quadratics
https://goo.gl/DzaOsa
8.7H – Build understanding of the definition
of an ellipse as the set of all points whose sum
of distances from two distinct points (foci) is
constant.
Solving radical equations
https://goo.gl/sKtG6Y
Introduction to ellipses
https://goo.gl/EepfMw
8.7- cont’d