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 interior angle of a regular n-gon is
o exterior angle measures for a regular polygon – the measure of an exterior angle of a regular n-gon is
o central angle measure for a regular polygon – the measure of the central angle of a regular n-gon is
· Angles in Circles
o arc – portion of a circle cut off by a pair of rays
o relating arc measure to angle measure –
§ mÐP = m(arc s)
· angle inside the circle
· angle vertex on circle
§ mÐP = [m(arc s) – m(arc r)]
· angle outside the circle
§ mÐP = [m(arc s) + m(arc r)]
· 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
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 = OR
o c =
· 30°, 60°, 90°
o c = 2a where a is the shorter leg
o b =
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?