course: grade 9 applied mathematics (mfm1p) unit … · unit 2 plane geometry section activity page...

Course: Grade 9 Applied Mathematics (MFM1P)

Unit 2: Plane Geometry

Unit 2

Plane Geometry

Section Activity .1.1 I Remember 3 2.1.2 Plane Geometry Record Sheet 6 2.1.4 Parallel Lines Exploration – Optional 9 2.1.P Practice 10 2.5.1 What’s So Special Guide Sheet 17 2.5.2 What’s So Special Record Sheet 18 2.6.1 Learn the Lingo 20 2.7.1 Exterior and Interior Angles of a Polygon 23 2.7.2 Interior Angle Sums 25 2.7.J Journal Activity 27 2.7.P Practice 28 2.S Unit Summary Page 30 2.R Reflecting on My Learning (3, 2, 1) 31 2.RLS Reflecting on Learning Skills 32

MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008) 2-2

2.1.1: I remember…. Work with a partner to complete the definitions below that you know. Leave the ones you are unsure of and come back to them throughout this unit as you learn more about them.

Word/Term Definition Diagram

Supplementary Angles

Complimentary Angles

Opposite Angles

Corresponding Angles

Alternate Angles

Co-Interior Angles

Parallel Lines

Transversal


2.1.1: I remember….(continued)


Triangle

Isosceles Triangle

Equilateral Triangle

Right Triangle

Acute Triangle

Obtuse Triangle

Scalene Triangle


2.1.1: I remember….(continued)


Quadrilateral

Parallelogram

Rhombus

Trapezoid

Square

Rectangle

Hexagon

Polygon


2.1.2: Plane Geometry Record Sheet Use this page to record your observations and conclusions from the Plane Geometry GSP®4 file. Determine the unknown angle in the right column. Give reasons for your answer.


2.1.2: Plane Geometry Record Sheet (continued)


2.1.4: Parallel Lines Exploration - Optional

Explore and Reflect Sketch 1. How did you know when Line 1 and Line 2 were

parallel?

Angle Relationships 2. Find one pair of equal angles. Explain how you

know they are equal.

3. Find another pair of equal angles. Explain how you know they are equal.

4. Find as many pairs of angles that are supplementary (add to 180°) as you can. Explain how you know.

Summary (to be completed as a whole class)


2.1.P: Practice 1. Look at this diagram. (a) Name two parallel line segments.

(b) Name two transversals.

(c) Name two corresponding angles.

(d) Name two alternate angles.

(e) Find the measures of ∠ ECD, ∠ ACE, and ∠ BCA.

A

C B

E

50º

65º

D

•

•


2.1.P: Practice (continued) 2. (a) Draw parallelogram ABCD with ∠ A = 51º.

(b) How can you use what you know about parallel line segments and a transversal

to find the measures of the other 3 angles in the parallelogram? Explain your work.

(c) When is a quadrilateral a parallelogram? Explain.


2.1.P: Practice Sheet (continued) Define each principle and determine the unknown angles. 1.

x° 85°

Reason:

Reason:

Reason:

Reason:

x ==

o

2.

r° 70°

r ==

o

3.

m° 30°

m ==

o

4.

q°

50° 55°

q ==

o


2.1.P: Practice Sheet (continued)

5.

Reason:

Reason:

Reason:

b° 45°

60° b =

=

o

6.

7.

a° 75°

x°

a ==

o

x ==

o


2.1.P: Practice Sheet (continued) 8.

x°

72°

Reason: x ==

o

m°

68°

Reason:

Reason: m =o

w° 83°

w =o


2.1.P: Practice (continued) 9. Find the measure of a. 5. Find the value of x. Give reasons. Give reasons.

xº

49º

68º

71º

aº

46º

10. Find x. Give reasons. 7. Find the values of the missing angles. Give reasons.

89º

44º

xº

yº

42º

xº

96º

42º 11. The diagram shows two parallel lines cut by a transversal. The measure of a + b is

_____. Give reasons.

aº

bº


2.1.P: Practice (continued) 12. For the following diagram, list as many examples of each Angle Theorem as possible.

h°

e°

g°

f°

b° a° d°

c° Z – pattern C – pattern F - pattern

e.g. ∠c = ∠ g

13. Solve for x and y a) b)


2.5.1: “What’s So Special?” Guide Sheet Explore! Drag each vertex in the figure. As you drag vertices, look for some of the following: • measurements that always seem to be equal to each other • measurements that never seem to change • measurements that might have a constant ratio (proportional) • lines that always seem to be parallel or perpendicular • line segments that always seem to be bisected • figures that always seem to be congruent • objects that don’t seem to be connected, yet they move together when something is

dragged Make an Hypothesis Decide which measurements you need to test your hypothesis. Drag each vertex again while you pay close attention to the way the object moves and to the way the measurements change. Test Your Hypothesis Collect and record evidence to test your hypothesis.

What can you measure? • angles • lengths • areas • perimeters • slopes • •

What can you calculate? • sums • ratios • formulas • • • •


2.5.2: What’s So Special? Record sheet

Investigation 1. Special Triangles

Explore: Move the vertices of the triangles around and examine how the angles and side lengths change. Hypothesis: Make a hypothesis about what type of triangle each figure is and record it in the chart below. Test your Hypothesis: Make any measurements that will help test your hypothesis.

Hypothesis:

Type of Triangle Conclusion:

Type of Triangle Evidence: (what measurements support your conclusion)

ΔABC

ΔDEF

ΔGHI

ΔKLM

Investigation 2. Parallel or Perpendicular?

Explore: Drag the endpoints of the line segments. Hypothesis: Make a hypothesis about which lines are parallel, which are perpendicular, and which are neither. Test your Hypothesis: Make any measurements that will help test your hypothesis. Conclusions: Make a statement about which lines are parallel and which are perpendicular and provide evidence (which measurements support your claim)


2.5.2: What’s So Special? Record sheet Investigation 5: Special Quadrilaterals?

Explore: Drag each vertex of each figure. Hypothesis: Make a hypothesis about what type of quadrilateral each figure is and record your hypothesis in the chart below. Test your Hypothesis: Make any measurements that will help test your hypothesis. Conclusions:

Quadrilateral Hypothesis Conclusions Evidence

(What measurements prove your conclusions?)

A

B

C

D

E

F

G


2.6.1: Learn the Lingo 1. Part a) shows an example of how to complete a word chart.

Complete the remaining word charts.

a) b) Term: Term:

Equilateral Triangle

Visual Representation:

Triangle


Definition: An equilateral triangle is a triangle for which all sides have the same length.

Association: A Yield sign

Definition: Association:

c) d)

Term: Term:

Exterior Angle


Interior Angle


Definition: Association: Definition: Association:


2.6.1: Learn the Lingo (continued)

e) f) Term: Term:

Parallel Lines


Transversal



g) h)

Term: Term:

Perpendicular Bisector


Diagonal




2.6.1: Learn the Lingo (continued) 2. Determine the unknown angle. Give reasons for your answer.

a)

∠AB = AC = BC

ACB = ________

b) c)

∠∠

o

TZ UYTWX = 75UVW = __

∠DEG = ____

T

d)

∠∠∠

oCOD = 64FOE = ___COF = ___

e)

∠∠

MP NONQR = 115MRQ = ____

o

f)

∠∠∠

oBOC = 43COE = ____EOD = ____

g)

∠

∠∠

o

o

VRW = 42RT = RU

SRT = 19RSZ =

h)

∠∠

o

WX = WYYWX = 118WXZ = _____

i) Create your own question!

A

B C

42 ° R

Z A

W V

S T U

X WXY Z

O

D

C

B

ER

P

M

OS

N

Q

O

D

F C

E

V W Z

AX

Y

UD

108°

FEG


2.7.1: Exterior and Interior Angles of a Polygon Part A – Exterior Angles of a Triangle Using the GSP files: Angles Triangles. gsp

1. Click the “Show Measurements” Tab. 2. Drag vertices A, B, and C. 3. What do you notice? 4. Click the “Reset the triangle” Tab. 5. Click the “Make the triangle smaller” Tab. 6. If we decrease the size of the triangle, what do you notice about the sum of the exterior angles?

Part B – Exterior Angles of a Quadrilateral Similarly with the quadrilateral,

1. Click the “Make the quadrilateral smaller” Tab. 2. Describe what just happened. 3. If we decrease the size of the quadrilateral, what do you notice about the sum of the exterior angles?


http://www.curriculum.org/lms/files/tips4rm/TIPS4RManglestriangle.gsp

2.7.1: Exterior and Interior Angles of a Polygon (continued) Part C – Exterior Angles of Any Polygon

Compare the conclusions you reached in Part A and Part B. Write your final conclusion about the sum of the exterior angles of any polygon.

Similarly with any polygon, 1. Click on the tab “Show Sum”. 2. Drag vertices A, B, and C. 3. What do you notice? 4. Click the “Reset” Tab. 5. Click the “Shrink polygon” Tab. 6. If we decrease the size of the polygon, what do you notice about the sum of the exterior angles? 7. Click the “One less side” Tab. What shape do you have now? What is the sum of the exterior angles? 8. Click the “Another side less” Tab. What shape do you have now? What is the sum of the exterior angles?


2.7.2: Interior Angle Sums 1. Complete the chart.

Diagram Number of sides

Sum of interior angles Understanding

3 180°

The sum of the angles in any triangle is 180o.

4

5

n

2. a) Determine the sum of the interior angles in a polygon with 15 sides. Show your work. b) Determine the number of sides in a polygon if the sum of the interior angles is 5400°.

Show your work.


2.7.2: Interior Angle Sums (continued) 3. Derek is building a deck for his summer job in the shape of a regular octagon. a) Define: regular octagon ? b) Determine the measure of the interior angles of the deck.

Show your work. 4. A Canadian $1 coin, known as a loonie, is a regular polygon with 11 sides, called an

undecagon. a) Define a regular polygon with 11 sides. b) Determine the sum of the interior angles of the loonie. c) What is the size of one of the interior angles?


2.7.J: Journal Activity Write a letter to Abe, who missed Math class, explaining how he can determine the sum of the interior and exterior angles in a decagon (10-sided polygon).


2.7.P: Practice 1. Determine the measure of each indicated angle and state reasons. a) b) c)

49º yº

107º

104º100º

xº

41º

xº 70º

2. Determine the values of x, y, and z. Give reasons. a) b) c)

96º

yº zº

47º

zº yº57º

64º

xº

84xº

º

zº yº 132º

108º xº

3. Determine the measures of a and b. Give reasons. a) b)

83º

bº aº

105º

55º

25º

aº

115º

bº


2.7.P: Practice (continued) 4. Find the measure of x in the following pentagon. Give reasons.

100º 100º

100º 100º

xº

5. Find the measures of a, b, and c. Give reasons.

cº

aº

bº

135º


2.S: Unit Summary Page Complete the following concept map to relate all the terms from this unit.

PLANE GEOMETRY


2.R: Reflecting on My Learning (3, 2, 1)

3 Things I know well from this unit

2 Things I need explained more

1 Question I still have


2.RLS: Reflecting on Learning Skills

Students should be aware of the importance that these skills have on your performance. After receiving your marked assessment, answer the following questions. Be honest with yourself. Good Learning Skills will help you now, in other courses and in the future.

• E – Always • G – Sometimes • S – Need Improvement • N – Never

Organization • E G S N I came prepared for class with all materials • E G S N My work is submitted on time • E G S N I keep my notebook organized.

Work Habits • E G S N I attempt all of my homework • E G S N I use my class time efficiently • E G S N I limit my talking to the math topic on hand • E G S N I am on time • E G S N If I am away, I ask someone what I missed, • E G S N I complete the work from the day that I missed.

Team Work • E G S N I am an active participant in pairs/group work • E G S N I co-operate with others within my group • E G S N I respect the opinions of others

Initiative • E G S N I participate in class discussion/lessons • E G S N When I have difficulty I seek extra help • E G S N After I resolve my difficulties, I reattempt the problem • E G S N I review the daily lesson/ideas/concepts

Works Independently • E G S N I attempt the work on my own • E G S N I try before seeking help • E G S N If I have difficulties I ask others but I stay on task • E G S N I am committed to tasks at hand

Yes No I know all the different ways available in my school, where I can seek extra help. Yes No I tried my best. What will I do differently in the next unit to improve? ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________


course: grade 9 applied mathematics (mfm1p) unit … · unit 2 plane geometry section activity page...

Documents