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Instructional Accommodations and Curricular Modifications Bringing Learning Within the Reach of Every Student

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Geometry The Shape of Our World

Participant Materials

Geometry Course Participant Materials


Introduction

Component 1: Geometric Thinking Geometric thinking and logic that is the foundation of the study of other topics

Component 2: Use of Formulas Describe and compute area, surface area and volume of two and three-dimensional shapes

Component 3: Geometric Relationships Relationships between parallel and perpendicular lines

Component 4: Right Triangles Geometry of right triangles

Component 1: Geometric Thinking

What else might you see in a geometry classroom?

Vocabulary Pre-Test

Perimeter Cubes Sides Ratio

Radius Points Square Line Segment

Angles Circle



Geometry Word Wall angle

The union of two rays that have the same endpoint; measured in degrees or radians. The five types of angles are zero, acute, right, obtuse and straight.

area The amount of space taken up in a plane by a figure. It is the total number of same size square units that cover a shape without gaps or overlaps.

circle The set of points on a plane that are equidistant (radius) from a certain point (center). A polygon with infinite sides.

circumference The distance around a circle. Circumference is to a circle as perimeter is to a rectangle.

congruent

Identical in form or coinciding exactly when superimposed. An example shows that congruent line segments are the same length, congruent angles have the same measure, and congruent figures are the same shape and size. The symbol for congruent is a wavy line above an equal sign.

cube A cube is a three-‐dimensional figure with six square faces. diameter A straight line going through the center of a circle connecting two points on the circumference.

line In geometry a line is: straight; has not thickness; extends in both directions without end (infinitely)

line segment Part of a line connecting two points. It has definite end points. The word “segment” is important because a line normally extends in both directions without end.

midpoint The point that divides a line segment into two equal parts. The midpoint of a diameter divides a circle into two equal parts.

parallel Always the same distance apart and never touching.

parallelogram A four-‐sided figure with opposite sides that are parallel. A square is one kind of parallelogram.

perimeter The distance around a two-‐dimensional shape. The perimeter of a circle is called the circumference.

perpendicular Perpendicular means “at right angles.” A line is perpendicular to another if it meets or crosses it at right angles.

plane A plane is a flat surface with no thickness. It extends forever. We often draw a plane with edges, but it really has no edges.

point An exact location. It has no size, only position.

proportional When two quantities always have the same relative size; having the same ratio. protractor A semicircular tool used to measure and draw angles.

radius Circle: The distance from the center to a point on the edge of a circle; half the circle’s diameter. Polygon: The distance from the center to a vertex of a regular polygon.

ratio A comparison of two quantities; can be shown using “:” to separate example values, or as a single number dividing one value by the total.

right triangle A triangle that has a right angle (90 side One of the lines that make a flat (2-‐dimensional) shape.

similar Two figures are similar if they have the same shape. They may or may not be the same size. Similar figures have corresponding angles that are congruent and corresponding sides that are in proportion.

slope How steep a straight line is.

square A 4-‐sided flat shape with straight sides where: all sides have equal length; every angle is a right angle; it is a Quadrilateral and a Regular Polygon

surface area The surface area of a 3-‐dimensional figure is the sum of the areas of the faces of the figure. The Pythagorean Theorem

In a right-‐angled triangle the square of the long side (the “hypotenuse”) is equal to the sum of the squares of the other two sides.

volume Volume is the total number of same size cubes that will fit in the interior of a 3-‐dimensional shape.



Critical Thinking

Skillful teachers provide the instruction in this most critical life skill and manage their classrooms to accommodate argument, defense, proof, and precise communication.

Forming a Parallelogram Conclusion

1 2

3



Component 2: The Geometry of Size

*Formula Chart located at the end of this packet.

George Polya’s 4-Step Plan A medium-‐sized orange has a radius of 1.5 inches.

A quart contains 57.75 inches.

The juice in an orange occupies 1/3 of its volume. In your own words, what is the problem asking?

Now make a plan! Approximate the number of medium-sized oranges needed to make a quart of Orange Juice. a use volume of sphere formula b Take 1/3 of that c Divide 57.75 by answer to B

OR… R I D D

Read the problem carefully

Imagine to make a visual,

auditory, and kinesthetic image of the problem

Decide what to do to solve the problem

Do it!

~F.B. Jackson



Successful Teachers… Imbed instruction in the context of solving problems that have meaning to students.

Hook the student’s interest by relating the lesson to its practical uses.

Surface Area

* The surface area of a 3-dimensional figure is the sum of the areas of the faces of the figure. Use Polya’s Plan to solve the quiz question!

Linear Measure

1-‐Dimensional

Examples: inches meters cm

Square Measure

2-‐Dimensional

Examples:

square inches m2 cm2

Cubic Measure

3-‐Dimensional

Examples: cubic inches

m3 ft3

Polya ’s Four-Step Prob lem Solv ing P lan ! !

! !

!

1 2

4 3



Which equation would you use? A. 16x + 12x =560

B. 2(16x) + 2(12x) =560

C. 2(16x) + 2(12x) + 2(10x) =560

Farmer Brown needs to build a fence around a trapezoid-shaped vegetable garden. If the area of the garden is 40 square feet, what length of fence does he need?

Students who systematically a use problem-solving format can:

Avoid errors Build models and make and label sketches to plan their work Work carefully in a step-by-step manner using correct formulas

Conclusion

16’ 12’

x

8’

10’

2’



Component 3: Geometric Relationships

The study of generally usually begins with definitions of lines, points, angles and shapes.

As students master basic definitions that enable them to communicate clearly about their work, teachers ask students to consider relationships possible with multiple lines and shapes.

*Refer to the Word Wall on page three if you missed any of the vocabulary!

LINES

Lines in the same plane that never intersect are parallel lines. Perpendicular lines are lines in the same plane that intersect at a 90• angle.

Prove that ABCD is a parallelogram: Prove that A, B, C, and D are all right angles:

Two non-vertical lines are parallel if

and only if they have the same

slope.



1. Use the slope formula, slope=y2-y1 divided by x2-x1 to calculate the slope of line

segment BC. Which is the correct slope of BC?

2. Calculate the slopes of line segments AB and CD to learn if they are also parallel. What is the slop of line segments AB and CD?

When working with students who are visual learners, some teachers rely on colored markings to make the distinction of corresponding sides even more clear.

Strategies from

Successful Teachers



If corresponding angles of triangles ABC and DEF are congruent and corresponding sides are proportional, what can we say about the relationship between the two triangles? They are:

congruent

corresponding

similar Scale Factor In similar figures, the scale factor is the ratio of the corresponding linear measures.

A proportion that compares width to height shows that 8 compares to 10 as 2 and ½ compares to x.

Review __________ lines have the same slope.

__________ lines have slopes whose product is -1.

The symbol means two shapes are __________.

The symbol ~ means that two shapes are __________.

If two figures are congruent, they are the same size and __________.

If two figures are similar, corresponding angles are congruent and corresponding sides are in __________.

Talking Math!

Strategies from

Successful Teachers

Students who have difficulty solving proportions benefit from writing words that describe the ratio before they use the numbers from the problem.



Component 4: Right Triangles Right triangles form rigid structures that are the

foundation of modern architecture.

Trigonometry extends the study of right triangles.

A right triangle is: __________ A right angle measures: __________ The side opposite the right angle is the: __________ The other two sides are the: __________ Activity Successful Teachers Ask Students To…

1. Reflect on their discovery 2. Use vocabulary to explain discovery 3. Record discovery in a journal

Strategies from

Successful Teachers

Guide students to construct their own understanding. Only learning that has meaning and importance is stored in long-term memory for future use.

What statement would you write to explain your discovery of this method?



The door to the Grand Hotel ballroom is 6 feet wide and 8 feet high. The wedding planner wants to use a round table whose diameter is 10 ½ feet in the center of the room. Will it fit through the door? Practice!

You are standing 500 ft. from the base of a cliff you plan to climb. Using an electronic measuring device, you learn the distance from where you are standing to the top of the cliff is 825 feet. If the cliff is close to a vertical climb, approximately how far will you climb?

a2 + b2 = c2



Post Assessment 1. The value of is the ratio of any circle’s:

diameter to its radius radius to its circumference circumference to its diameter

2. If you multiply times a circle’s radius squared, you will find the ____ of the circle:

Matches a number of objects to a card showing the numeral Places objects from different groups side-by-side Touches one object for each number that is said

3. George Polya’s problem solving format includes all of the following steps except:

Copy the problem onto your paper Restate the question asked Make a plan

4. The sum of the areas of the faces of a three-dimensional figure is known as the ____ of the figure: A commutative operation A negative operation An inverse operation

5. The area of a trapezoid could be _____: 932 meters 932 square meters 932 cubic meters

6. Lines that are parallel have: Slopes whose product is -1 The same slope Slopes with a common factor

7. Figures with the exact same shape and size are:

Congruent Similar Corresponding

8. Figures with the same shape but different sizes are: Congruent Similar Corresponding

9. Rectangle ABCD has a length of 6 and a height of 10. Rectangle EFGH has a length of 3 and height of 5. The scale factor of ABCD to EFGH is _____. 1 to 2 3 to 5 2 to 1

10. The Pythagorean Theorem states that in any right triangle, the sum of the squares of the lengths of the legs is equal to: The square root of the hypotenuse Product of the hypotenuse and the right angle The square of the hyptenuse



Grades 9, 10, and Exit Level Mathematics Chart Perimeter rectangle P = 2l + 2w or P = 2(l + w)

Circumference circle C = 2!r or C = !d

Area rectangle A = lw or A = bh triangle A = bh or A =

trapezoid A = (b1 + b2)h or A =

regular polygon A = aP

circle A = !r 2

1 2

bh 2

1 2

(b1 + b2)h 2

1 2

P represents the Perimeter of the Base of a three-dimensional figure. B represents the Area of the Base of a three-dimensional figure.

Surface Area cube (total) S = 6s 2

prism (lateral) S = Ph prism (total) S = Ph + 2B

pyramid (lateral) S = Pl

pyramid (total) S = Pl + B

cylinder (lateral) S = 2!rh cylinder (total) S = 2!rh + 2!r 2 or S = 2!r(h + r) cone (lateral) S = !rl cone (total) S = !rl + !r 2 or S = !r (l + r) sphere S = 4!r 2

1 2

1 2

Volume prism or cylinder V = Bh

pyramid or cone V = Bh

sphere V = !r 3

1 3 4 3

Special Right Triangles 30°, 60°, 90° x, x"3, 2x

45°, 45°, 90° x, x, x"2

__

__

Pythagorean Theorem a 2 + b 2 = c 2

Distance Formula d = " (x2 # x1) 2 + (y2 # y1) 2

Slope of a Line m = y2 # y1

x2 # x1

Midpoint Formula M = ( , )x1 + x2

2 y1 + y2

2

Quadratic Formula x = # b ± "b 2 # 4ac 2a

Slope-Intercept Form of an Equation y = mx + b

Point-Slope Form of an Equation y # y1 = m(x # x1)

Standard Form of an Equation Ax + By = C

Simple Interest Formula I = prt TX-03300140



Bibliography Geometry the Shape of Our World

Grades 8-‐10 Ashlock, Robert B, Error Patterns in Computation. Pearson. 2006

Beigie, Darin. “The Leap from Patterns to Formulas”, Mathematics Teaching in the Middle

School, Vol. 16, No. 6, February 2011.

Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics. National Council of

Teachers of Mathematics. 2006

Developing Effective Fractions Instruction for Kindergarten Through 8th Grade. National Center

for Educational Evaluation and Regional Assistance: Institute of Education Sciences, NCEE

2010-4039, U.S. Department of Education.

Larson, Roland E., Boswell, Laurie, Stiff, Lee. Geometry, An Integrated Approach, McDougal

Littell, 1998.

Jensen, Eric. Teaching with the Brain in Mind, ASCD, 1998.

Sherman, Helene J., Richardson, Lloyd I., Yard, George J. Teaching Children Who Struggle

with Mathematics. Pearson, 2005.

Sousa, David A. How the Brain Learns Mathematics. Corwin Press,2008.

Wall, Edward S, Posamentier, Alfred S., What Successful Math Teachers Do, Grades 6-12.

Corwin Press, 2007.

Documents

Geometry Participant Materialsstetsonassociates.com/Files/Geometry_Course_Demo... · Lines in the same plane that never intersect are parallel lines. Perpendicular lines are lines