C I R C L E S P H E R E
SHARING IS CARING2
C O N G R U E N T
C Y L I N D E R
YOU WILL SOON PICK WHAT YOU LIKE TO DO AND SPECIALIZE IN THAT
C I R C L E S P H E R E C O N G R U E N T
C Y L I N D E R
,Y X A figure can represent several types simultaneously Lines Adjacent Vertex Ray Straight line 180 degrees Y Longitude Lines Meridians Vertical 0-180 Up down N north S south X Latitude Lines parallels horizontal N north S south Left/ Right N/S 0-90 east west
TABLE OF CONTENTS POLYGON PRISM POLYHEDRAL QUADRILATERAL CYLINDER CONE TRIANGLE POLYHEDRON PLANE PERPENDICULAR VECTOR MAGNITUDE DIRECTION RAY SLOPE LINE POINT
INTERSECT SEGMENT PARALLEL ADJACENT TRAVERSE LINEAR 90 DEGREES 180 DEGREE DIAGONAL CIRCLE SPHERE MULTIPLE
apex Peak
DIMENSION
puff puff little polygon prism train that could can
1D
algebra works
Torus (1/4) * pi2 * (r1 + r2) * (r1 r2)2
CONGRUENT = SAME Two circles are congruent if they have the same size. The size can be measured as the radius, diameter or circumference. They can overlap. Congruent polygons have an equal number of sides, and all the corresponding sides and angles are congruent. However, they can be in a different locaKon, rotated or flipped over. Two angles are congruent if they have the same measure.
NOTE: Similar polygons which be in the same proporKons but different sizes.
2D
polygon(n-gon) pol·y·gon [ póllee gòn ] many-sided Flat Surface Like Paper Figure: a two-dimensional geometric figure formed of three or more straight sides Triangle (3-sided) quadrilateral (4-sided) pentagon (5-sided), hexagon (6-sided) and so on. The word quadrilateral is made of the words quad (meaning "four") and lateral (meaning "of sides")
e l g n a t l Rec a r e t a l i r d a Qu
circle
5
rectangular
6
7
Definitions of cylinder (n) cyl·in·der [ síllindər ] object shaped like tube: an object or shape with straight sides and circular ends cylinder of equal size
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C Y L I N D E R
Non-‐ Polyhe dra: NOT FLAT (if any surface is not flat IT IS A NON Poly He dra)
An ellipse IS the geometric shape that results from cutting a circular conical or cylindrical surface with an oblique plane (the two unbounded cases being the parabola and the hyperbola).
THESE PLANES are Flat Like Paper 2-D
Pentagon Five 5 C Y L I N D E R
circle
In Euclidean (Greek) plane geometry, a quadrilateral is a polygon with four sides (or 'edges') and four vertices or corners. Sometimes, the term quadrangle is used, by analogy with triangle, and sometimes tetragon for consistency with pentagon (5-sided), hexagon (6-sided) and so on. The word quadrilateral is made of the words quad (meaning "four") and lateral (meaning "of sides").
trapezoid
Diamond
Parallelogram
In Euclidean geometry, a rhombus or rhomb is a convex quadrilateral whose four sides all have the same length. The rhombus is often called a diamond
quadrilateral |ˌkwädrə ˈlatərəl| noun a four-sided figure. adjective having four straight sides.
QUADS
RIGHT TRIANGLE ONE ANGLE IS 90 THE SIDE OPPOSITE THE 90 ANGLE IS THE LONGEST SIDE EQUILATERAL TRIANGLE ALL ANGLES ARE EQUAL ALL SIDES ARE EQUAL ISOSCELES TRIANGLES TWO ANGLES ARE EQUAL TWO SIDES ARE EQUAL
1D Planes Perimeter
MEASURING SIDES OF TRIANGLES
Interior Angles The sum of the measures of the interior angles of a triangle is 180 degrees. The interior angles of an equilateral triangle are all 60 degrees. HOW MANY TRIANGLES Square? TWO TRIANGLES
Interior Angles
The relation between the sides and angles of a right triangle is the basis for MEASURING ANGLES IN TRIANGLES
SCALENE TRIANGLES NO ANGLES ARE EQUAL NO SIDES ARE EQUAL ISOSCELES TRIANGLES TWO ANGLES ARE EQUAL TWO SIDES ARE EQUAL EQUILATERAL TRIANGLE ALL ANGLES ARE EQUAL ALL SIDES ARE EQUAL RIGHT TRIANGLE ONE ANGLE IS 90 THE SIDE OPPOSITE THE 90 ANGLE IS THE LONGEST SIDE ACUTE TRIANGLE ALL ANGLES ARE LESS THAN 90 DEGREES OBTUSE TRIANGLE ONE ANGLE IS GREATER THAN 90 DEGREES
OBTUSE TRIANGLE ONE ANGLE IS GREATER THAN 90 DEGREES
ACUTE TRIANGLE ALL ANGLES ARE LESS THAN 90 DEGREES
SIDES
I began by asking the child to construct three triangles with the geometric sQck material. The first triangle was made from 3 pieces of the same size, we spoke about how all the sides are equal. Then I asked the child to construct a triangle with 2 sides the same and one side different. 3. Thirdly she made a triangle with all the sides different. We then did a three-‐period -‐lesson and aZer the final period I quesQoned her 'Why is this an isosceles?', 'Why is this a scalene?', 'Why is this an equilateral?'. We then labelled them. To ensure that she fully understood, I got her to draw the three triangles in her book, write the names and a descripQon for each.
SEE TRIANGLES
SCALENE TRIANGLES NO ANGLES ARE EQUAL NO SIDES ARE EQUAL
EQUILATERAL TRIANGLE ALL ANGLES ARE EQUAL ALL SIDES ARE EQUAL
RIGHT ISOSCELES TRIANGLE ONE ANGLE IS 90 TRIANGLES TWO ANGLES THE SIDE OPPOSITE THE 90 ARE EQUAL ANGLES IS THE TWO SIDES LONGEST SIDE ARE EQUAL
ACUTE TRIANGLE ALL ANGLES ARE LESS THAN 90 DEGREES
OBTUSE TRIANGLE ONE ANGLE IS GREATER THAN 90 DEGREES
POLYNOMIALS trapezoid
Quad means four
parallelogram
rectangle
Poly means many A Octagon + 2(Rectangle) + 2 (Square) +1 (Rectangle) =TOTAL (HOUSE)
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POLYNOMIALS
Torus
FORMULA (1/4) * pi2 * (r1 + r2) * (r1 - r2)2
WHEEL DONUT TORUS
Perimeter formula Square 4 * side Rectangle 2 * (length + width) Parallelogram 2 * (side1+ side2) Triangle side1 + side2 + side3 Regular n-polygon n * side Trapezoid height * (base1 + base2) / 2 Trapezoid base1 + base2 + height * [csc(theta1) + csc(theta2)] Circle 2 * pi * radius Ellipse 4 * radius1 * E(k,pi/2) E(k,pi/2) is the Complete Elliptic Integral of the Second Kind k = (1/radius1) * sqrt(radius12 - radius22)
Area formula Square side2 Rectangle length * width Parallelogram base * height Triangle base * height / 2 Regular n-polygon (1/4) * n * side2 * cot(pi/n) Trapezoid height * (base1 + base2) / 2 Circle pi * radius2 Ellipse pi * radius1 * radius2 Cube (surface) 6 * side2 Â
S U R F A C E A R E A
P L A N E D E M E N S I O n
SQUARE AREA2
2d
T
R
Sphere (surface) 4 * pi * radius2 Cylinder (surface of side) perimeter of circle * height2 * pi * radius * height Cylinder (whole surface) Areas of top and bottom circles + Area of the side2(pi * radius2) + 2 * pi * radius * height Cone (surface) pi * radius * side Torus (surface) pi2 * (radius2)2 - radius1)2
Volume formula Cube side3 Rectangular Prism side1 * side2 * side3 Sphere (4/3) * pi * radius3 Ellipsoid (4/3) * pi * radius1 * radius2 * radius3 Cylinder pi * radius2 * height Cone (1/3) * pi * radius2 * height Pyramid (1/3) * (base area) * height Torus (1/4) * pi2 * (r1 + r2) * (r1 - r2)2 Regular n-polygon (1/4) * n * side2 * cot(pi/n) Trapezoid height * (base1 + base2) / 2 Circle pi * radius2 Ellipse pi * radius1 * radius2 Cube (surface) 6 * side2
3D V o L U M e
#Demensions
S U R F A C E A R E A
P L A N E D E M E N S I O n
SQUARE
2 AREA
2d
Sphere (surface) 4 * pi * radius2 Cylinder (surface of side) perimeter of circle * height2 * pi * radius * height Cylinder (whole surface) Areas of top and bottom circles + Area of the side2(pi * radius2) + 2 * pi * radius * height Cone (surface)pi * radius * side Torus (surface) pi2 * (radius2)2 - radius1)2
Volume formula
Cube side3 Rectangular Prism side1 * side2 * side3 Sphere (4/3) * pi * radius3 Ellipsoid (4/3) * pi * radius1 * radius2 * radius3 Cylinder pi * radius2 * height Cone (1/3) * pi * radius2 * height Pyramid (1/3) * (base area) * height Torus (1/4) * pi2 * (r1 + r2) * (r1 - r2)2 Regular n-polygon (1/4) * n * side2 * cot(pi/n) Trapezoid height * (base1 + base2) / 2 Circle pi * radius2 Ellipse pi * radius1 * radius2 Cube (surface)6 * side2
1d  Prisms(volume) 3d Planes (flat) 2d Circles (flat) Oblique Spheres 3d
Ordered Pair
An ordered pair is a set of two numbers in the form: (x, y) Example: (2, -3) Ordered pairs are used in Cartesian coordinates.
Origin
The origin is the point (0, 0) on the Cartesian plane. It's called the origin because it's the starting place when you plot a point.
Y X
Coordinate Plane. Coordinate
HICH W WAY DO WE GO?
Plane is a plane (FLAT SURFACE) formed by the intersection of a horizontal number line with a verKcal number line. COORDINATE’S X HORIZONTAL Y VERTICAL
I PO
S NT
Horizontal Let X be the points east and west (right and left) ROWS LaQtude (φ) Angular distance on the earth's surface, measured east or west north south Vertical COLUMNS Let Y be the points Vertical Longitude (λ) North South UP DOWN
Symbol The symbol for congruence is  Â
LINES/ ANGLES Two angles are congruent if they have the same measure.
same
In geometry, adjacent angles, often shortened as adj. ∠s, are angles that have a common ray coming out of the vertex going between two other rays.( line with direction to infinit). In other words ... In geometry, two lines or planes (or a line and a plane) are considered perpendicular (or orthogonal) to each other if they form congruent adjacent angles (a T-shape). In geometry, two figures are congruent if they have the same shape and size. Two line segments are congruent if they have the same length. But they need not lie at the same angle or position on the plane
Definitions of parallel (adj) par·al·lel [ párrə lèl ] always same distance apart: relating to or being lines, planes, or curved surfaces that are always the same distance apart and therefore never meet
par·al·lel·o·gram [ pàrrə léllə gràm ] four-sided geometrical figure: a two-dimensional geometric figure formed of four sides in which both pairs of opposite sides are parallel and of equal length, and the opposite angles are equal Synonyms: rhomboid, diamond, lozenge
Par Al Lel ogram
plane
CONE Parallel lines s e l g n a i r t
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Cone Cylinder
Prism Have Flat Sides A prism is a polyhedron that is formed with two parallel polygons (the bases - top and bottom) that are connected at the
edges with rectangles.
t a Fl s e d Si rism
3d volume
P
Polyhedra : have flat faces A polyhedron (plural polyhedral or polyhedron s) is often defined as a geometric solid with flat faces and straight edges
Measure me like this!
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PRISM : a polyhedron with two polygonal faces lying in parallel planes and with the other faces parallelograms
FLAT SIDES
Polyhedron Definition pol·y·he·dron [ pòllee h!drən ] pol·y·he·drons Plural NOUN manysided solid figure: a threedimensional geometric figure formed of many faces
Triangles DefiniQons of pyramid solid triangular shape: a solid shape or structure that has triangular sides that slope to meet in a point and a base that is o`en, but not necessarily, a square.
SPHERE
Congruent angles Two angles are congruent if they have the same measure.
ANGLES INSIDE
Alternate Interior Angles: All equal angles make parallel Suppose that L, M, and T are distinct lines. Then L and M are parallel if and only if alternate interior angles of the intersection of L and T and M and T are equal.
Two parallel lines that are cut by a non-perpendicular transversal. Perpendicular deďŹ nition: Perpendicular simply means
'at right angles .A line is perpendicular to another if they meet at 90 degrees.
LINES /ANGLES Not oblique not parallel or perpendicular: Oblique
Oblique is either plus or minus 90 degrees An angle, such as an acute or obtuse Right Angle angle, an angle that that is not a right is 90° exactly angle or a mulKple Straight of a right angle. Angle an angle that is 180° exactly
Reflex Angle an angle that is greater than 180° Obtuse Angle an angle that is greater than 90° but less than 180 Right Angle an angle that is 90°exactly Acute Angle IS an angle that is less than 90° Straight Angle an angle that is 180° exactly Right Angle an angle that is 90° exactly
RelaKvity Â
a x b = c c / a= B c / b= a SO, 3 x 25 = 75 75 / 3 = 25 75 / 25 = 3
Oblique Angle IS + or – 90 degrees NOT 90 DEGREES An oblique angle is an angle that is not a right angle. (An oblique angle is an angle whose measure is not 90 degrees.) + or -‐ Tennis can be broken down into a game of angles. Obtuse Angle An obtuse angle is an angle whose measure is greater than 90 degrees.
equality congruent parallel intersec+ons intersec+ons straight angle theorem measures corresponding angles corresponding angles theorem
Beverly Mackie’S CROSS MULTIPLY RATIO PROPORTION PERCENTS PERCENTS RATIOS PROPORTIONS – WORD SEARCH BASE CHANGE CROSS MULTIPLY FIFTY HUNDRED LEFT ORDER OUT PERCENTS PROPORTIONS RATIO TO TRUE UNIT
Fill in the blank (Use the above word bank) 1. Percent means how many ____ of 100. 1. Percent means how many out of 100. 2. Percents can be greater than 100%. True or False 2. Percents can be greater than 100%. True or False True 3. To change a percent to a decimal…drop the percent sign and move the decimal two places to the _______________. 3. To change a percent to a decimal…drop the percent sign and move the decimal two places to the left. 4. Part/base = rate/100 used to work with proportions and ______________ 4. Part/base = rate/100 used to work with proportions and percents. 5. In the above statement, ________ – represents the whole amount. 5. In the above statement, base – represents the whole amount.
6. Two for one is the same as ______________ percent. 6. Two for one is the same as fifty percent. If a calculator costs ten dollars, two calculators would could twenty dollars. However, “two for one” means I can buy two calculators for the price of one calculator. In other words, I pay a total of ten dollars for both calculators. That’s the same as paying half price or five dollars for each one. What a deal! Half price is equivalent to fifty percent. 7. Part / ______ represents a fraction 7. Part / whole represents a fraction. By definition, a fraction represents a part of a whole. Some teachers may say a fraction represents part of a total. Let’s say you ordered a taco pizza. When the pizza arrives it has been cut into eight slices with no slices eaten. At this point, it is a whole pizza. Since your brother fell asleep while waiting on the pizza, you decide to get a head start before waking him up. So, you eat three slices of pizza! Your brother catches you wiping your mouth after finishing the third slice. He accuses you of eating three parts of the whole pizza, or 3/8 of the pizza! You are guilty as charged.
r o w s
12 X 25 300
COLUMN Y ROWS X
X
300-75=225
12 -‐ -‐ -‐
-‐ -‐ -‐ -‐ -‐ -‐
Minus 25 X 3 = 75 25
25
column
3
Order of Operations
Order of operations tells us what order we are supposed to do things in a math problem. For example, what's the answer to this? 4 + 10 x 2 Do we do the 4 + 10 first? Or the 10 x 2 first? Here is the official order of operations: Remember (1) Parenthesis (2) Exponents (3) Multiplication & Division (4) Addition & Subtraction So, for 4 + 10 x 2, we do multiplication, then addition... 4 + 10 x 2 = 4 + 20 = 24.
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A common phrase to remember the order is: Please Excuse My Dear Aunt
 8. 3:4 is the same as 3 to 4 8. The above ratio can be expressed as 3:4, 3 to 4, or ž. 9. The denominator for percents when represented as a fraction is always ______. 9. The denominator for percents when represented as a fraction is always 100. In order to help students remember this, I ask the following question. What percentage of the problems do you want to get correct on the next test? Of course, the answer is 100%. However, if you grade was an 83. It means used answered 83 out of 100 or 83/100 or 83% correct. 11. _______________ - comparison of two amounts 11. Ratio - comparison of two amounts 12. _______________ rate 12. Unit rate Unit rate â&#x20AC;&#x201C; the amount compared to one; in a ratio, the second term would be one. For instance,a box of 12 pens cost $24.00. One pen or one unit cost $2. How did I get $2? Divide 24 by 12.
Finding Polygons
Things come apart and go together
PIE
CHORD
Midpoint
The midpoint of the segment (x1, y1) to (x2, y2) The midpoint (also known as class mark in relation to histogram) is the middle point of a line segment. It is equidistant from both endpoints. Formulas The formula for determining the midpoint of a segment in the plane, with endpoints (x1) and (x2) is: The formula for determining the midpoint of a segment in the plane, with endpoints (x1, y1) and (x2, y2) is: The formula for determining the midpoint of a segment in the space, with endpoints (x1, y1, z1) and (x2, y2 z2) is: More generally, for an n-dimensional space with axes , the midpoint of an interval is given by: Construction The midpoint of a line segment can be located by first constructing a lens using circular arcs, then connecting the cusps of the lens. The point where the cusp-connecting line intersects the segment is then the midpoint. It is more challenging to locate the midpoint using only a compass, but it is still possible.
SIMILAR AND
Congruent
Examining Quads Parallel sides/vectors
Line segments Shape QUADS
13. Equal ratios are called ________________. 13. Equal ratios are called proportions. Example 3/8 = 6/24 14. __________ _____________ to solve for a missing number in a proportion. 14. Cross multiply to solve for a missing number in a proportion. Example: 4/5 = 8/? 4x?=5x8 4 x ? = 40 4 times what = 40? Or divide 40 by 4 = 10 In algebraic terms: 4x = 40 4x/4 = 40/4 x=10 15. In a proportion, write the terms of ratios in the same ________________. 15. In a proportion, write the terms of ratios in the same order. Compare the same units: boys/ girls = boy/girls Itâ&#x20AC;&#x2122;s wrong to compare boys/girls = girls/boys
10. Percent of ________ = (original amount – new amount) / original amount 10. Percent of change = (original amount – new amount) / original amount Percent of change means the amount a rate has changed over a period of time.
/ = DIVIDED BY ORIGINAL AMOUNT
KEY WORDS Lessons HAVE NO Grade Level, Early Childhood Lessons, Elementary Lessons, Jr. High/Middle School, High School Lessons, Undergraduate Lessons, Elementary Substitute, Middle School â&#x20AC;&#x201C; Substitute, Lesson Idea Pages, Drama and Art, Increased student performance in mathematics, as well as greater numbers of students enrolling in college preparatory mathematics classes, is a well documented outcome of the project's work.