Chapter 14 and 16
Coordinate Geometry
(x,y) the x is the horizontal axis and the y is the vertical axis. The x always comes first, so the y always comes last.
Obtuse Triangle
1 angle is greater than 90 degrees.
Right Triangle
1 angle of the triangle = 90 degrees
7 teaching strategies for research projects
1. Brainstorm Questions 2. Choose a question 3. Predict the outcome 4. Create a plan to test the outcome 5. Implement the plan 6. Analyze the data 7. Reflect
4 graphing stages for children
1. Concrete 2. Concrete-Pictorial 3. Pictorial-Abstract 4. Abstract
5 reasons statistics should be included in math programs
1. make math more meaningful 2. help us deal with uncertainty 3. help us understand statistical arguments 4. help us decide what we should and should not purchase 5. interesting and motivating for children
Dodecahedron
12 regular pentagons
Isoceles Triangle
2 out of 3 sides are of equal length
Icosohedron
20 equilateral triangles
Polygons
2D figures with closed geometric curves (may be straight or curved), curves don't cross. They are either convex or concave.
Polyhedra
3D shapes with faces consisting of polygons (plane figures 2d) with 3 or more straight lines
Tetrahedron
4 equilateral triangles
How many types of triangles are there?
6
Hexahedron
6 squares
Octahedron
8 equilateral triangles
Isosceles Trapezoid
A trapezoid with two nonparallel sides equal
Concrete
Actual objects are used to graph
Concrete-Pictorial
Actual objects as well as pictures are used.
Van Hiele Level 2: Informal Deduction
Children not only think about properties of shapes but are also able to notice relationships within and between figures. It is at this level of thinking that children are first able to understand inclusion relationships, using words such as all, some and none. ex. all squares are rectangles, but not all rectangles are squares. Children at this level are able to formulate meaningful definitions. Before this level, they could memorize definitions, but not formulate their own or understand them.
Van Hiele Level 1: Analysis
Children think in terms of properties, they don't see relationships. They learn everything in isolation, for example, they can't see a square as a rectangle because they have learned the properties of each in isolation.
What is Euler's Rule?
Formula that describes the relationship between faces(F), vertices(V), and edges(E), of polyhedra. If given 2 of the variables, the 3rd one can be figured out using the following formula. F+V-2=E.
Statistics that help us Interpret
Frequency, Measures of Central Tendency (mode, median, mean), Variation
Line Plots
Graphic representations in which marks above a number line indicate the frequency of each value. Important parts of the definition are number line and frequency.
Teachers Relationship to Van Hiele levels
Most of our k-5 students will be on levels 0-1. Levels are not age related, levels are sequential.
Which of the following is false concerning traversable networks
One odd vertex makes a network traversable.
Probability Algorithm
P=desired outcomes/total possible outcomes
Van Hiele Level 3: Formal Deduction
Students can create proofs because they can understand relationships between axioms, definitions, theorems, corollaries and postulates.
Van Hiele Level 4: Rigor
Students can think abstractly. Math majors and mathematicians are able to think at this level.
Van Hiele Level 0: Visualization
The child thinks about shapes in terms of what they resemble. ex. a child thinks a triangle resembles a mountain. If the triangle is turned differently, they no longer recognize it. Children are able to sort shapes into groups that look alike bases on their perceptions. They can sort, identify, match and describe objects in a limited capacity. They need lots of examples and nonexamples to learn.
Which shapes tessellate by themselves?
Triangles, squares, and hexagons are the only regular shapes that tessellate by themselves. Other regular shapes can tessellate if more than one shape is used.
Circle graphs
Usually not used until grade 6. Appropriate for sources of revenue, breakdown of expenditures, time a child devotes to various activities, family budgets.
Does a parallogram have line and rotational symmetry?
Yes, both
Graph
a diagram that shows a relationship between two or more things.
Rhombus
a parallelogram with all sides equal
Square
a polygon with 2 pairs of parallel sides, with all right angles. Can be classified as a Quadrilateral, a Parallelogram, a Rectangle and a Rhombus
Axiom
a proposition that is assumed without proof for the sake of studying the consequences that follow from it. Also called a postulate
Theorem
a proposition, statement, or formula embodying something to be proved from other propositions and formulas
Trapezoid
a quadrilateral with at least one pair of parallel sides
Kite
a quadrilateral with two pairs of adjacent sides equal.
Convex
all angles are less than 180 degrees
Equilateral Triangle
all sides of triangle are of equal length.
Acute Triangle
angles all less than 90 degrees
Quadrilaterals
any simple, closed figure consisting of 4 straight lines.
Concave
at least one angle is greater than 180 degrees
One Mobius strip property is that is can
become two strips
John Tukey
first used a stem and leaf plot in 1977 in a study of volcanos, John Tukey was a mathematician and statistician who is credited with first using the term software to describe the programs that run on computers.
Place and Order
knowing concepts like inside, outside, between, will help students explain mathematical concepts and properties.
Network
lines connecting two or more points.
Translations
movement along a straight line that has distance and direction. The direction can be horizontal, vertical or oblique(diagonal). Normally called a slide by a child.
Reflections
movement of a figure about a line outside the figure, one a side of the figure or intersecting with a vertex. Normally called a flip by a child.
Rotations
movement of a figure around a point. The turning point may be inside the figure, on the figure, or outside the figure.
Transformational Geometry
movements of an object where the object itself is not distorted or changed in any way are called rigid transformations. Also called motion geometry.
Does a Scalene Triangle have line and rotational symmetry?
no, neither one
Scalene Triangle
none of the sides are the same length
Stem and Leaf Plots
normally 3 digits, the stem is usually the first 2 digits, the leaf is usually the last digit.
Abstract
one to many correspondence is introduced, rectangular bar graphs can be introduced, in addition to picture graphs, line graphs can be used.
Regular Polyhedra
one whose faces consist of the same kind of regular congruent polygons (all squares or all equilateral triangles) with the same number of edges meeting at each vertex of the figure. There are only 5 regular polyhedra. These are also known as platonic solids.
Rectangle
polygon with 2 pairs of parallel sides but has all right angles.
Parallelogram
polygon with 2 pairs of parallel sides.
Corollary
proposition that is incidentally proved in proving another proposition. ( if 1 + 3=4, then 4-1= 3)
Tessellations
repetition of a geometric shape with no overlaps and no gaps.
Histograms
represent continuous data. Bar graphs represent categorical data.
Similar Figures
same shape but not same size
Semiregular Polyhedra
semiregular shapes composed of more than one kind of regular polygon. 13 shapes classified as semiregular solids or Archimedian solids
Tangram
seven piece puzzle consisting of various shapes that can be put together to create various figures. The shapes are 5 triangles of various sizes, one square, and one parallelogram.
Rotational Symmetry
shape can be rotated around a point less than 360 degrees.
What concepts contribute to organizing and analyzing data?
sorting,measuring, geometric attributes, various types of graphs and tables, proportions, measures of central tendency and range
Truncated and Stellated Polyhedra
stellating: process of building onto solids by cutting off sections in a systematic way. truncating: cutting off solids in a systematic way.
Probability
the area of mathematics that analyzes the chance of something occurring. Probability requires children to collect and organize data which further develops their problem solving skills.
Mean
the average, divide the sum of a set of numbers by the number of numbers in the set.
Symmetry
the identical reflection of two sides. In order to be symmetrical, the right side of the pattern or object is exactly the same as the left side of the pattern or object. Folding can test symmetry.
Edges of a Polyhedra
the line made when two faces of a polyhedra join together.
Median
the midpoint. To find the median, put the numbers in order. Either the number in the middle or an average of the numbers is the median.
Traversible Networks
the network can be traced in one continuous motion without retracing over a line. In order to be traversible, it needs to have all of the vertices even, or just two odd vertices.
Frequency
the number of times something happens, from a set of numbers related to frequency , a frequency distribution table can be made.
Vertices of a Polyhedra
the points where the edges of a polyhedra meet
A student who cannot distinguish a three sided from a four sided figure would be at
the prerecognition state
Faces of a Polyhedra
the sides of the shape, ex. a cube has 6 faces.
Topology
the study of properties of figures that stay the same even under distortions , except tearing or cutting.
Mode
the value that occurs the most. Most is preferred for use with children as opposed to mode. Mode can be found by creating a frequency distribution table.
Variation
the word we use for variation is range, the range is the difference between the highest value and the lowest value.
Congruent Figures
those that have the same size and shape, all corresponding angles and the length of corresponding sides are equal. Superimposing figures is a good test for congruency. Another way to test for congruency is to measure all sides and angles to identify that each shape has corresponding parts.
Postulate
to assume without proof. ex. all right angles are congruent
Topological Transformations
transformations that change the shape and size of the object. ex. stretching
How can a teacher increase success in Van Hiele levels?
use language on a childs level, accept informal terms such as six-a-gon instead of hexagon at the elementary level. Homogenous grouping may work better than heterogenous grouping in this subject.
Bar Graphs
used mainly in primary grades, graph drawn using rectangular bars to show how large each value is. The bars can be horizontal or vertical.
Things that Change and Things that do not Change
when children understand place and order, they can understand these concepts. Think of drawing a face on a balloon. The eyes are always going to be inside something (the face). There will always be something not closed (the nose), There is order ( the nose will always be above the mouth). Something intersects (the eyelashes with the eyes) and something connected and something not connected.(any of the features on the face). Things that could change when the balloon is blown up are that one eye might be larger than the other and the mouth may not be symmetrical.
Does a regular polygon have line and rotational symmetry?
yes, both
The focal points of the NCTM standards match the _________ of the CCGPS
Standards for Mathematical Content
Pictorial-Abstract
Start with the pictorial and transition into the abstract. Pictographs using a 1 to 1 correspondence can be used.
How are Mobius Strips used in real life?
Conveyor Belts, Typewriter Ribbons, Music, Physics, Chemistry
Van Hiele Levels of Geometric Thinking
Dina and Pierre van Hiele were Dutch educators who identified 5 levels of geometric thinking.
When students are discussing shapes, they must use mathematically correct terms at all times.
False
Distortions
Figures can be distorted by stretching, twisting, bending and shrinking
Why do we need to teach Geometry
Helps students describe the world in which they live, stepping stone to other topics of mathematics such as area and perimeter, Increases their spatial abilities, tends to be exciting to children and therefore contributes to their eagerness to learn mathematical concepts, research indicates that children's knowledge of geometry needs improvement.
3 facts about tessellations
If a basic shape tessellates, then its corners or vertices should all be able to meet at one single point. For this reason, regular pentagons will not tessellate. If a = the number of degrees in a vertex of the regular polygon in question. If 360/a = a whole number, then the shape will tessellate.
Prerecognition
In 1992 Clements and Battista hypothesized that a level exists that is below visualization called prerecognition. Al this level, a child couldn't tell a 3 sided figure from a 4 sided figure.
Properties of a Mobius Strip
It is one sides, if you cut down the middle of the strip, it becomes one long strip, if you cut 1/3 of the way across the strip you create 2 strips.
Does an isoceles triangle have line and rotational symmetry?
Line symmetry, but not rotational symmetry
How can a teacher increase success in Van Hiele levels?
Providing many experiences at a students current level for him to be successful at a higher level. Teachers should identify a goal and design activities to meet that goal. Teachers should connect Geometry to the real world.