Download Chapter 10 Review

Chapter 1: Viewing Mathematics





Mathematics as Problem Solving
The Role of Problem Solving
o Central to the development and application of mathematics
o Used extensively in all branches of mathematics
o The Meaning of a Problem p. 36
o A problem is a situation for which the following conditions exist
 It involves a question that represents a challenge for the individual
 The question cannot be answered immediately by some routine
procedures known to the individual
 The individual accepts the challenge
o “Can every map be colored with only four different colors if regions that
have a border in common must be colored differently?”
o “How can you cut a cake into eight pieces with three straight cuts?”
o The Meaning of Problem Solving p. 37
o Problem solving is a process by which an individual uses previously
learned concepts, facts, and relationships, along with various reasoning
skills and strategies, to answer a question about a situation
o Algorithms are known steps used for solving different types of equations –
the problem solving process CANNOT be made into an algorithm
o Answer vs. Solution
 Answer – final result
 Solution – process used to find the answer
A Problem-Solving Model
o George Polya’s model p. 39
o Understanding the problem
o Making a plan
o Carrying out the plan
o Looking back
o Estimation is the process of determining an answer that is reasonably
close to the exact answer used in different stages of problem solving
Problem-Solving Strategies
o Make a model
o Act it out
o Choose an operation
o Write an equation
o Draw a diagram
o Guess – check – revise
o Simplify the problem
o Make a list
o Look for a pattern
o Make a table
o Use a specific case
o Work backward
o Use reasoning
o Learning when and how to use problem solving strategies is an important
problem solving skill
Importance of Problem Solving
o Mathematics is primarily used to solve problems in mathematics and in the
real world
o Learning to solve problems is the principal reason for studying
mathematics
o Mathematics is MUCH more than algorithms
o Problem solving applies to all aspects of our lives, NOT just mathematics
Chapter 10: Introducing Geometry
10.1 Basic Ideas of Geometry
 Geometry in nature
o Honey combs
o Snow flakes
o Fibonacci sequence
 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, …
 Sunflowers
 Ratio of counterclockwise spirals to clockwise spirals is often
55:34 or 34:21
 pine cone
 Ratio = 13:8 or 8:5
 Golden ratio
 Approximately 1.618
 Ratio of successive Fibonacci numbers
 Starfish
 Snail shell
 Geometry in human endeavors
o Egyptian pyramids
o Pentagon in Washington, D.C.
 Defining basic ideas
o Points, lines, planes, and space
o Segments, rays, angles
o Special angles and perpendicular lines
o Circles and polygons
o Triangles
o Quadrilaterals
10.2 Solving Problems in Geometry
 A traversable network is also considered to be a simple path
 Network Traversability Theorem
o All even vertices = traversable type 1 (start from any vertex)
o Exactly 2 odd vertices = traversable type 2 (start at one odd vertex end at
the other odd vertex)
o >2 odd vertices = NOT traversable
 Concurrency Relationships in Triangles Theorem
o Centroid = intersection of all three triangle medians
 Balance point
 Center of gravity
 Two thirds the distance from each vertex to the opposite side
o Orthocenter = intersection of all three triangle heights
o Circumcenter = intersection of all three triangle perpendicular bisectors
 Center of the circle containing the triangle vertices or
 Center of the circle that circumscribes said triangle
 The triangle would be inscribed in the circle
o Incenter = intersection of all three triangle angle bisectors
 Center of a circle tangent to all three sides of the triangle
 Center of the circle inscribed in the triangle
 Euler’s line
o contains 3 of the four points of concurrency
o Centroid, Orthocenter, and Circumcenter form Euler’s line
o Leonard Euler (1707-1783) Pretty famous guy!
 Tangrams
10.3 More About Angles
 Angles in Intersecting Lines
o transversal – a line cutting through two or more distinct lines
o alternate interior angles – congruent angles formed on opposite sides of
a transversal between the two lines intersected
o alternate exterior angles – congruent angles formed on opposite sides of
a transversal outside the two lines intersected
o corresponding angles – congruent angles formed on the same side of a
transversal where one angle is between the two lines including one line
and the other angle is outside including the other line of the two lines
intersected
o same-side interior angles – same-side interior angles are supplementary
angles
o same-side exterior angles – same-side exterior angles are
supplementary angles
o vertical angles – congruent angles formed by the intersection of any two
distinct lines such that opposite pairs of angles are congruent
 Angles in Polygons
o sum of the interior angles of any polygon – the sum of the measures of the
interior angles of an n-gon is (n – 2) 180
o sum of the exterior angles of any polygon – the sum of the exterior angles
of any polygon is 360
o Interior angle measures for a regular polygon – the measure of each
n  2180
interior angle of a regular n-gon is
n
o exterior angle measures for a regular polygon – the measure of an exterior
360
angle of a regular n-gon is
n
o central angle measure for a regular polygon – the measure of the central
360
angle of a regular n-gon is
n
 Angles in Circles
o arc – portion of a circle cut off by a pair of rays
o relating arc measure to angle measure –
1
 mP =
m(arc s)
2
 angle inside the circle

angle vertex on circle
1
 mP =
[m(arc s) – m(arc r)]
2
 angle outside the circle
1
 mP =
[m(arc s) + m(arc r)]
2
 angle inside the circle
 angle vertex NOT on the circle
10.4 More About Triangles
 Congruent Triangles
o Definition of congruent triangles – Two triangles are congruent if and
only if, for some correspondence between the two triangles, each pair of
corresponding sides are congruent and each pair of corresponding angles
are congruent
o Triangle congruence postulates
 SSS – if all of the corresponding pairs of sides of a triangle are
congruent, then the two triangles are congruent
 SAS – If two sides and the included angle of the corresponding
pairs of sides and angles of a triangle are congruent, then the two
triangles are congruent
 ASA – If two angles and the shared side of the corresponding pairs
of angles and sides of a triangle are congruent, then the two
triangles are congruent
 AAS – If two angles and a non-shared side of the corresponding
pairs of angles and sides of a triangle are congruent, then the two
triangles are congruent
 For Right Triangles ONLY –
 HA – If the hypotenuse and one angle of the corresponding
pairs of angles and sides of a right triangle are congruent,
then the two right triangles are congruent
 HL – If the hypotenuse and one leg of the corresponding
pairs of sides of a right triangle are congruent, then the two
right triangles are congruent
 The Pythagorean Theorem
o a2  b2  c 2
o a and b are legs of a right triangle
o c is ALWAYS the hypotenuse of the right triangle
o Pythagorean triples
 Special Right Triangles
 45, 45, 90
o c= a 2
OR
o c= b 2
 30, 60, 90
o c = 2a
where a is the shorter leg
o b= a 3
10.5 More About Quadrilaterals
 Properties of Quadrilaterals
o parallelogram – quadrilateral with two pairs of parallel sides
 opposite sides are parallel
 opposite sides are congruent
 one pair of opposite sides are parallel and congruent
 opposite angles are congruent
 consecutive angles are supplementary
 diagonals bisect each other
o rectangle – quadrilateral with four right angles
 a parallelogram is a rectangle if and only if
 it has at least one right angle
 its diagonals are congruent
o rhombus – quadrilateral with four congruent sides
 a parallelogram is a rhombus if and only if
 it has four congruent sides
 its diagonals bisect the angles
 its diagonals are perpendicular bisectors of each other
o square – quadrilateral with four right angles and four congruent sides
 a square is a parallelogram if and only if
 it is a rectangle with four congruent sides
 it is a rhombus with a right angle
 its diagonals are congruent and perpendicular bisectors of
each other
 its diagonals are congruent and bisect the angles
Chapter Summary – p. 589
Key Terms, Concepts, and Generalizations – p. 591
Chapter Review – p. 592
 Work on problems 1-22 in your groups
 Questions?