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L1 – Foundations Name ______________________________ 1.1 – Introduction Per _____ Date _______________________

Geometry Q1: L1 – Foundations Handouts

The field of study referred to as Geometry was originally formalized during ancient Greek times, approximately 2,300 years ago. Geometry was formalized to answer questions dealing with 2 and 3-dimesional objects, such as:

• Measurement: How long is each side of an object, what is its total perimeter length, what is the area or volume enclosed within the boundaries of an object, etc.

• Angles: What is the angle between two adjacent sides or faces of an object? • Equivalence: Under what conditions are two objects essentially the “same?” For example,

how do we determine that two triangles have three angles with identical measures, and/or three sides with identical length?

• Proof: How can we formally justify that our conjectures are correct?

Consider, for example, the obstacles the engineers in ancient Egypt needed to overcome in order to build the pyramids. How could they determine the angles at which the faces needed to meet in order for the four pyramid faces to meet perfectly at the desired height? How did they know how large to make the base and the building blocks? Whatever methods they used, they were not able to use formal Geometry, since it was non-existent for another 1500 years. In more recent times, how did the Central Pacific Railroad engineers building the Summit tunnel in the Sierra Nevada Mountains in 1865 know, without the benefit of a bird’s-eye view, which direction the blasting crews on opposite sides of the mountain should proceed in order for the two sides to meet in the middle?1

As you may have noticed, Geometry was originally developed to answer questions relating to construction. With this in mind we loosely framed this Geometry course around a construction theme; helping Grandma Ruthie build her new house and place accessories around her land. A general plot plan and simple building plans for Grandma’s house are included below, and we encourage you to refer to them when appropriate throughout the course.

Before getting started with the deeper concepts we need to first introduce some of the terminology and notation necessary to identify different parts of Geometric objects, and to distinguish those parts themselves from their measurements. This will allow you to quickly identify and refer to the various parts of each Geometric Object, and to efficiently use the notation in your work. When building a whole area of study such as Geometry, you first need to begin with something; that is, you cannot build something from nothing. As such, we begin in the next section with some basic Geometry vocabulary, and we assume we already know some previously learned mathematics such as how to add and multiply numbers.

Good luck! J

1 The engineers actually sank a shaft down into the middle of the tunnel so they could work on four rock faces simultaneously, which complicated even further the meeting of both sets of ends. When the four sub-‐tunnels met, they were within 2 inches of each other, an amazing engineering feat!

Geometry Q1: L1 – Foundations Handouts

Bird House

Fire Pit

Grandma Ruthie

’s

House

Water

Tank

Roses

Hibiscus Gate

Pool and Jacuzzi

Mauna Young Mt.

Veg. Garden

Lava Tube

Chicken

Coop

Corn

Field

Radio Tower

Playground

Geometry Q1: Lesson 1 – Foundations 1 Handouts

Front Elevation

Grandmas’ House Plans

Master Bath Dining

Room

Bath Kitchen

Short Hallway

Storage

Master

Bedroom

Main Hallway

Living

Room Bedroom

Porch

Ramp

L1 – Foundations Name ______________________________ 1.2 – In the Beginning Per _____ Date _______________________


In the Beginning

In the beginning, there was nothing but empty space. Then BANG! Out of the nothingness, some basic building blocks appeared, A, !" and C.

From these building blocks, the world began to take shape; that is, the world of Geometry. J

One day, a geometer came along and attempted to define these 3 building blocks. He found he was having trouble and turned to some friends for help. Try to help him out by attempting to fill in the table below. Discuss your results with a partner.

CE

BA

A BE C

Name/Term

Descriptors



1. What are some problems you and your classmates ran into trying to define these terms?

Geometers had similar problems for many years and so they finally agreed to call these three “building blocks” the undefined terms. They realized that if they were going to “build” plane Geometry they needed to start with something, and instead of rigorously defining them, as is the usual case for mathematical objects, they agreed to begin with descriptions of each of the undefined terms.

2. Using the table your class just created, attempt to come up with descriptions for each of the three undefined terms.

- Point (#VOC):

- Line (#VOC):

- Plane (#VOC):



3. Now that we have descriptions of each term/object, we can agree upon common notations for each. We will consistently use this notation throughout the course, and your ability to recognize each of these (and subsequent objects) from its notation will be essential to your success. It is also important to know the proper notation for each geometric object so that others can recognize what you are referring to when you use it. Guided notes: - Point: The notation for a point is _____________________

- Example:

- Line: There are two different ways to notate a line. - Notation for a line containing two points A and B is ______ or ______ and their

order does not matter. - Another notation for a line is _________________________

- Example:

- Plane: The notation for a plane is ___________________. (Note that this is the same

notation as that used for a _____________ and it is the context that distinguishes a point

from a plane.)

- Example:

L1 – Foundations Name ______________________________ 1.3 – Vocabulary Booklet Per _____ Date _______________________


Vocabulary Booklet

For this activity, you will create a booklet to keep your vocabulary words in. Throughout the year, every time you learn a new vocabulary word you can add it to this booklet and use it as a study guide or a reference booklet. The style of booklet you will use for your Vocabulary Booklet depends on your teacher. The instructions below are for building your own booklet. If your teacher uses a different type of booklet, such as a composition tablet, you should adjust accordingly, but you should include all the important information on the cover and within.

1. Your teacher will provide you with paper to create this booklet. Fold the paper in half to make a booklet.

2. On the front of your booklet, include the following: a. “Vocabulary Booklet” b. “Geometry” c. Your Name d. Period

3. On the inside of your booklet, start a table for your vocabulary words.

4. Put your three undefined terms and descriptions your class created as your first three vocabulary words.

5. Draw lines in between to separate vocabulary words. Include examples and/or drawings, and/or whatever you think will help you study in the future.

6. Remember, you will be adding to this booklet, and referencing it frequently throughout the entire course, so make every effort to keep it neat and well-organized.

Vocabulary Word Notation(s) Definition/ Description/Example

L1 – Foundations Name ______________________________ 1.4 – Undefined Terms (Homework/Practice) Per _____ Date _______________________


E

M

F

Z

M

K

Y

Q

T

O

LW

Practice:

For each figure, state what it is, and denote it using proper notation.

1)

2)

3)

4) 5) 6)

7) 8) 9)

For each of the following, draw an example that fits the description.

10) Point L 11) Plane F 12) !"

13) Plane J 14) !" 15) Point K

A

L1 – Foundations Name ______________________________ 1.5 – Grandma’s Ramp Per _____ Date _______________________


Grandma Ruthie is coming home from the hospital with a brand new wheel chair. She needs you to build her a ramp up to her front porch, which is 4 feet higher than her front yard.

1. Draw 2 different possible ramps that could be built. Note: the ramps will differ depending on where the ramp begins in the front yard.

2. What is one essential piece of information about the ramps that distinguishes one from the other?

In order to have _________________, we have to look at just a section of a line. We call this a line _____________.

Porch

4 ft. Front yard

Help me get up to my porch, please.



3. What are some other examples of line segments on Grandma’s house?

4. Work with your classmates to use your undefined terms to come up with a definition for line segment.

Line Segment (#VOC).- Notation:

The line segment with endpoints A and B, can be written as ____________________ or

_____ or_____.

The length of a line segment is denoted by just _______ and it indicates the

__________________ between the two endpoints.

What if the line goes on forever in one direction, but not the other? This is known as a _______.

5. What are some examples of this type of Geometric object in everyday life?

6. Work with your class to use your undefined terms to come up with the definition for a ray. Ray (#VOC)-

Notation: For a ray with endpoint A that goes in the direction of B, the proper notation is

___________ or _____________.



B

A

C

Now that we’ve defined line segment and ray, let’s get back to Grandma’s ramp.

7. What other important piece of information would distinguish Grandma’s ramp from other ramps? (Hint: how do you describe how steep the ramp is? It can’t be too steep or Grandma won’t be able to wheel herself up.)

8. Work with your classmates to use your existing terms to come up with a definition for an ______________. (#VOC)

Notation: There are several ways to notate an angle. One way is by its points: _____ or _________ or __________. It can also be named by a number, such as ________.

The notation for the measure of angle ABC is !∠! !" !∠!"# !" !∠!"#.



When we think about angles we usually think of them in terms of their measure. There are

actually ________ measurements for each angle, one less than or equal to _______ and one

between ________ and _________. We are usually interested in the measurement less than or

equal to 180° that can be measured with a protractor. The second angle determined by the same

two rays is known as the reflex angle. (#VOC)

An angle whose measure is equal to 180° is called a ____________________ and is made of two

opposite rays. (#VOC)

9. What is the measure of the reflex angle in the diagram below?

10. An analog clock is given below. If you just landed in a new time zone and need to set your clock back 3 hours, what is the measure of the angle you would need to move the short hand?

11. For the previous problem most people would say 90˚ counterclockwise, but some people are superstitious and will only turn their hands clockwise. How many degrees clockwise would you need to turn your short hand in this case?

12. The angle from the previous problem is known as the _________________ angle.

m∠ABC = 40°

reflex angle

B

A

C



13. Now that you’ve defined line segment and angle, decide what a realistic plan for Grandma’s ramp would look like. Include drawings and/or descriptions below. Try to include as many details as possible.

14. In order to prevent Grandma from falling off the edge of the ramp you will probably want to include railings on both sides. Assuming the rectangle below represents a top view of the ramp, place points indicating where you would approximately attach the railings.

15. Did you put these points in a line? When multiple points lie on the same line, they are known as ________________. (#VOC)

16. Likewise, when multiple points (e.g. rail points) lie on the same plane (the ramp), they are known as _______________. (#VOC) Draw and label 3 such points on the plane below.



Q SR

17. Another way to designate a plane is by using 3 non-collinear points that lie on the plane. Why do you think it is important that these points are non-collinear?

Going back to collinear points, when a point lies on a line segment, there is a postulate that relates the lengths of the resulting line segments.

18. Segment Addition Postulate: If point C lies on the line segment AB, such that C is between A and B, then AC + CB = ______ (#THM – See section 1.7 for building your Theorems Booklet)

***Remember, when there is no line above the letter (AC), we are referring to the length of the line segment.

Example: Find QS, if QR= 5 and RS= 3.

______ + ______ = QS

______ + ______ = QS

_______ = QS

19. Find RS, if QR= 7 and QS=12.

20. Note: A postulate is a ________ statement that does not require proving. It can be used to derive the logical statements to solve a problem.

A BC

L1 – Foundations Name ______________________________ 1.6 – Practice/Homework Per _____ Date _______________________


U

VW

Foundations 1 Practice

Determine if each figure gives an example of a line segment, a ray, an angle, collinear points, or coplanar points. Then, if possible, name it.

1. 2.

3. 4.

5. 6.

7. 8.

G

M

L

T

P S

Q

R

M

J

E

F

X

YZ



Use the figure below for Exercises 9–16. Note that !" pierces the plane at N, and R is not a point on plane V.

9. Name two line segments shown in the figure.

10. What is the intersection of !" and !"?

11. Name three collinear points. 12. What are two ways to name plane V?

13. Are points R, N, M, and X coplanar? 14. Name two rays shown in the figure.Explain

15. Name a pair of opposite rays with endpoint N.

16. How many lines are shown in the drawing?

For Exercises 17-22, determine whether each statement is always, sometimes, or never true.

17. !" and !" are the same ray. 18. !" and !" are opposite rays.

19. A plane contains only three points. 20. Three non-collinear points are contained in only one plane.

21. If !" lies in plane X, point G lies in plane X.

22. If three points are coplanar, they are collinear.



Corner B:

Corner C:

Corner D:

Corner E:

23. Is it possible for one ray to be shorter in length than another? Explain.

Given the angle measure, find its reflex angle measure.

24. 82˚ 25. 155˚ 26. 26˚ 27. 178˚

Given the reflex angle, find its angle measure.

28. 284˚ 29. 190˚ 30. 300˚ 31. 279˚

32. A boat is traveling North up the coast. The lookout spots a reef ahead and tells the captain they need to change their course 43˚ to the right (i.e. clockwise or starboard). Unfortunately, their rudder is broken and they can only turn left (i.e. counter-clockwise or port). What can the captain do to still correctly change the ship’s course and avoid the reef?

33. Peter Perfect is walking a fashion show runway in the shape of a pentagon. The models are walking counterclockwise, but Peter is eccentric and only turns to his right (clockwise). Help him figure out how many degrees he has to turn to the right at each corner so he doesn’t fall off the runway.

Backstage

m∠EAB = 89°m∠DEA = 111°m∠CDE = 114°m∠BCD = 114°

m∠ABC = 111°

B

A

E

DC



Use the Segment Addition Postulate to solve for the missing lengths. Use the diagram below for reference.

34. EF= 5, FG= 2, EG= ? 35. FG= 26, EG= 35, EF= ?

36. EF= 15, EG= 21, FG= ? 37. FG= 101, EF= 127, EG= ?

38. Grandma Ruthie is walking back to her car at the other end of the mall. She passes a sign that says the mall is 143 yds long. If she has already walked 85 yds, how much longer does she have to walk to get to her car?

39. Isaac, Grandma’s grandson, is on a road trip from Boston to Seattle. On a particular straight stretch of freeway, he has driven 291 miles since his last pit stop at a gas station when he sees a sign that says the next two gas stations are located in 22 miles and 54 miles. If his car can go approximately 340 miles on one tank of gas, should he stop at the first gas station or will he make it to the second one?

E GF

L1 – Foundations Name ______________________________ 1.7 – Theorems Booklet Per _____ Date _______________________


Theorems Booklet

For this activity, you will be creating a booklet in which to record your theorems and postulates. Throughout the year, every time you learn a new theorem, corollary or postulate, you can add it to this booklet and use it as a reference and study guide. It will be very similar to your vocabulary booklet.

1. Your teacher will provide you with paper to create this booklet. Fold the paper in half to make a booklet.

2. On the front of your booklet, include the following: a. “Theorems & Postulates” b. “Geometry” c. Your Name d. Period 3. On the inside of your booklet, start a table for your Theorems.

4. Add the Segment Addition Postulate from this section as your first postulate. 5. Draw lines in between for separation. Include examples and/or drawings, whatever you think

will help you in the future.

Remember, you will be adding to this booklet, and referencing it frequently throughout the entire course, so make every effort to keep it neat and well-organized.

Theorem, Corollary or Postulate

Statement & Examples

In this section, make sure to specify if it is a Theorem, Corollary or Postulate first, then write the name of it. Ex: Postulate Segment Addition Postulate

If point C lies on the line segment AB, such that C is between A and B, then AC + CB = AB

L1 – Foundations Name ______________________________ 1.8 – Project Instructions Per _____ Date __________________________


Project Instructions

You are going to be working in groups to create mini posters for the vocabulary words you learned this lesson. Be prepared to present your completed poster to the class.

Posters should include: - Word - Definition or description - Notation - Image or diagram - Example from real life

Your poster should be:

- Colorful and eye-catching - Neat and legible - ACCURATE. Make sure to use your notes.

Write down your vocabulary word here: _____________________________

l1 foundations handouts - weeblymrchowmath.weebly.com/.../l1_foundations_handouts.pdf · l1 –...

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