BASIC EDUCATION ASSISTANCE FOR MINDANAO LEARNING GUIDE

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THIRD YEAR - MATHEMATICS

Plane Coordinate Geometry

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Basic Education Assistance for Mindanao (BEAM) project. Prior approval must be

given by the author(s) or the BEAM Project Management Unit and the source must

be clearly acknowledged.

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MODULE 21: PROPERTIES OF TRIANGLES AND QUADRILATERALS USING COORDINATE PROOF

Mind Map

The Mind Map displays the organization and relationship between the concepts and activities in this Learning Guide in a visual form. It is included to provide visual clues on the structure of the guide and to provide an opportunity for you, the teacher, to reorganize the guide to suit your particular context.

Stages of Learning

The following stages have been identified as optimal in this unit. It should be noted that the stages do not represent individual lessons. Rather, they are a series of stages over one or more lessons and indicate the suggested steps in the development of the targeted competencies and in the achievement of the stated objectives.

Assessment

All six Stages of Learning in this Learning Guide may include some advice on possible formative assessment ideas to assist you in determining the effectiveness of that stage on student learning. It can also provide information about whether the learning goals set for that stage have been achieved. Where possible, and if needed, teachers can use the formative assessment tasks for summative assessment purposes i.e as measures of student performance. It is important that your students know what they will be assessed on.

1. Activating Prior Learning

This stage aims to engage or focus the learners by asking them to call to mind what they know about the topic and connect it with their past learning. Activities could involve making personal connections.

Background or purpose

Students will plot points on the Cartesian Plane.

Strategy

THINK PAIR SHARE. This strategy allows students to think individually about an issue, question or problem and record response. Discuss ideas with a partner and record what they have shared. Share with the whole group or join another pair to reach consensus.

Materials

• manila paper

• masking tape

• graphing paper

Activity 1: “Plot and Tell”

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2. Pair the students.

3. Let each student work individually then share and discuss the answers with his/her partner to reach a consensus.

4. Then, let them compare outputs with another pair.

5. Ask at least two volunteers to present their outputs to the class. 6. Process the activity through an interactive discussion.

7. Let the students keep their outputs for the next activity.

Formative Assessment

Monitor the performance of the students as they perform the activity. Check their outputs. Refer to the answer key on page 14.

Roundup

Students would have plotted points on the Cartesian Plane.

2. Setting the Context

This stage introduces the students to what will happen in the lessons. The teacher sets the objectives/expectations for the learning experience and an overview how the learning experience will fit into the larger scheme.

Background or purpose

Students will define and illustrate the different triangles and quadrilaterals.

Strategy

ROUND TABLE DISCUSSION. This strategy encourages group discussion. In groups, students write down their thoughts, solutions and ideas. A sheet of paper will be passed around the group. As each person reads, he/she initials if he/she agrees or leaves blank if he/she does not. When the paper returns to the owner, he/she reads ideas and reviews own thoughts to present an argument using new and different ideas.

Materials

• activity sheets (refer to Student Activity 2 on pages 15-16)

• crayons

Activity 2: “The Cutting Edge”

Instructions:

1. Organize the class into groups of 5 or as desired. 2. Provide each group the materials.

3. Tell each group to assign a problem to each member and write his/her answers/solutions on a sheet of paper.

4. After which, ask them to pass the paper around their group. Let them initial it if they agree on the answers/solutions or leave blank if they do not.

5. Ask the owner of each paper to review his/her answer/solution if there is no initial found on it.

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Formative Assessment

Monitor students' involvement in the activity and make sure that everyone interacts and shares ideas in accomplishing the task.

Check the groups' outputs. The answer key is provided on page 17.

Roundup

Students would have defined and illustrated the different triangles and quadrilaterals.

3. Learning Activity Sequence

This stage provides the information about the topic and the activities for the students. Students should be encouraged to discover their own information.

Background or purpose

In this stage, students will verify properties of triangles and quadrilaterals using coordinate proof.

Strategies

INTERACTIVE LECTURE. This strategy provides the students with the general outline to give them the framework for thinking about a subject and to structure their note-taking. This type of lecture involves students by focusing their attention on key words. It

emphasizes information transfer at the knowledge, recall and comprehension levels of learning.

MATH TRAIL. As groups move from one station to another, they will solve challenging problems the soonest possible time. This strategy encourages teamwork and cooperation. This also enhances their mathematical/logical, bodily/kinesthetic and interpersonal intelligences.

Materials

• station number and problem strips (refer to Teacher Resource Sheet 4 on pages 18-21)

• thumbtacks

Teacher's Input

Begin this stage by asking students what properties of triangles and quadrilaterals they have listed in Stage 2 and ask the following questions.

1. When can you say that a given triangle is isosceles? 2. How will you prove that it is really an isosceles triangle?

Verifying Properties of Triangles and Quadrilaterals using

Coordinate Proof

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Illustrative Example 1

To prove that the given figure in the Cartesian Plane is an isosceles triangle, we need to show that its two sides have equal lengths. To do that, we have to determine the lengths of the three sides of the triangle. This can be done using the distance formula.

Distance Formula

Given two points, P(x1, y1) and Q(x2, y2), the distance between P and Q is given by

PQ=



x2−x1 2

y2−y1 2 _.

➢Solving the Length of AB : A(-a, 0), B(a, 0) AB=



[a−−a]20−02

AB=



[aa]202

AB=



[2a]2

AB=a

➢Solving the Length of AC : A(-a, 0), C(0, b) AC=



[0−−a]2b−02

AC=



a2

b2

➢Solving the Length of BC : B(a, 0), B(0, b) BC=



[0−−a]2b−02

BC=



a2__b2

Notice that AC = BC, AC

≅

BC .

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Illustrative Example 2

To prove that the given figure is a square, we need to show that all its sides are equal in lengths and all the interior angles measure 900_{; that is, the two consecutive sides are}

perpendicular to each other. Remember that a square is a parallelogram where the opposite sides are parallel and having equal slopes.

Statement Reason

1. Lengths of the sides

OR=



a−020−02=



a2₌_a

OP=



0−02a−02=



a2₌_a

PQ=



a−02a−a2=



a2₌_a

PR=



a−a2a−02=



a2₌_a

1. Distance Formula

Given two points, P(x1, y1) and Q(x2, y2),

the distance between P and Q is given by

PQ=



x2−x1 2

y2−y1 2 _.

2. Slopes of the line segments

Slope of OR=0−0

a−0 =0

Slope of OP=a−0

0−0=undefined

Slope of PQ=a−a

a−0=0

Slope of QR=a−0

a−a =undefined

2. Definition of Slope: m=y2−y1

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3. OR∥PQ OP∥QR

3. Lines/Segments that have equal slopes are parallel.

4. OR⊥OP

OR⊥PQ PQ⊥QR QR⊥OR

4. Horizontal lines have slope zero. Vertical lines have undefined slope. A horizontal line and a vertical line are perpendicular.

5. Therefore, OPQR is a square 5. All the four sides are equal and all the interior angles are right angles.

Now, present and discuss the proof of this next theorem.

The median of a trapezoid is parallel to the bases and has length equal to half the sum of the lengths of the bases.

Given: Trapezoid KLMN with median AB. Prove: AB∥KN∥LM and AB=1

2KNLM

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1. Coordinates of the endpoints of the median, AB

Ac0

2 , b

0 2  =A

c 2, b2

c

2, b2 ⇒coordinates of A

Bda

2 , b

0 2  =B

da 2 , b2

da

2 , b2 ⇒coordinatesof B

1. The endpoints of a median are midpoints of the legs of a trapezoid.

2. Slopes of the median and the bases

Slope of AB=

b 2−

b 2 da

2 − c 2

=0

Slope of KN=0−0

a−0=0

Slope of LM=b−b

d−c =0

x₂−x₁

3. Therefore, AB∥KN∥LM 3. Lines/Segments that have equal slopes are parallel.

4. Lengths of the median and the bases

AB=



da

2 − c 2 2 b 2− b 2 2

AB=



da−c

2 

2

AB=da−c

2

LM=



d−c2b−b2

LM=



d−c2

LM=d−c KN=a

4. Distance Formula

Given two points, P(x1, y1) and Q(x2, y2),

the distance between P and Q is given by

PQ=



x2−x1 2

y2−y1 2 _.

5. Therefore, AB=1

2KNLM 5. AB=

da−c

2 , LM=d−c, KN=a

AB=1

2KNLM da−c

2 =

1

2d−ca da−c

2 =

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Activity 3: “Prove Me Right”

Instructions:

1. Divide the class into 6 groups.

2. Read and explain the mechanics of the game on page 18. 3. Let them perform the task at a given time.

Formative Assessment

Monitor students' performance in the activity.

Check the groups' outputs using the answer key on pages 22-23.

Roundup

Students would have verified properties of triangles and quadrilaterals using coordinate proof.

4. Check for Understanding of the Topic or Skill

This stage is for teachers to find out how much students have understood before they apply it to other learning experiences.

Background or purpose

The students will enhance their knowledge and skills in verifying the properties of triangles and quadrilaterals using coordinate proof.

Strategy

JIGSAW. Students are formed into groups and each student is given an aspect of a topic to discuss or research. They research their aspect and prepare to report back to their home group. They take turns to report on their aspect of the topic.

Material

• activity sheets (refer to Student Activity 4 on pages 24-26)

Activity 4: “A Clear Proof”

Instructions:

1. Divide the class into groups of 6. 2. Distribute the activity sheets.

3. Tell each group to divide the task among the members. 4. Set a time allotment for them to perform the activity.

5. Instruct students with the same task to group together and share ideas about the problems.

6. After which, ask them to go back to their original group and share. 7. Let them deliberate and finalize all their answers.

8. Process the activity through an interactive discussion.

Formative Assessment

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Roundup

Students would have enhanced their knowledge and skills in verifying the properties of triangles and quadrilaterals using coordinate proof.

5. Practice and Application

In this stage, students consolidate their learning through independent or guided practice and transfer their learning to new or different situations.

Background or purpose

Students will apply their learning on verifying properties of triangles and quadrilaterals using coordinate proof in solving problems related to real-life experiences.

Strategy

TASK CARDS. This strategy specifies a task or activity for students to complete

individually, in pairs or as a small group. This can be used across the curriculum areas. Making tasks real-life tasks make them more meaningful.

Material

• task cards (refer to Teacher Resource Sheet 7 on page 30)

Activity 4: “The Real Proof Applied”

Instructions:

1. Prior to the activity, prepare the task cards. 2. Organize the class into 8 groups.

3. Distribute to each group the task cards.

4. Set a time allotment for them to perform the task. 5. Let two groups compare their outputs to reach consensus.

6. Ask a volunteer to present their output for each task to the class for discussion.

Formative Assessment

Check the groups' outputs using the answer key on page 31.

Roundup

Students would have applied their learning on verifying properties of triangles and quadrilaterals using coordinate proof in solving problems related to real-life experiences.

6. Closure

This stage brings the series of lessons to a formal conclusion. Teachers may refocus the objectives and summarize the learning gained. Teachers can also foreshadow the next set of learning experiences and make the relevant links.

Background or purpose

In this stage, students will consolidate their learning about verifying properties of triangle and quadrilaterals using coordinate proof by doing an activity which requires their

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Strategy

BUZZ SESSIONS. This strategy aims to maximize students' engagement and to tap their verbal/linguistic and mathematical intelligences through answering questions or solving problems and come up to some kind of conclusion.

Material

• activity sheet (refer to Student Activity 6 on page 32)

Activity 6: “Are You Sure?”

Instructions:

1. Organize the class into groups of 6

2. Distribute to each group the activity sheet.

3. Set a time allotment for them to perform the activity.

4. When they are done, let them exchange their outputs with other groups for comparison.

5. Ask volunteers to present their outputs to the class for discussion.

Formative Assessment

Ensure that each student works cooperatively with his/her group. Check the groups' outputs using the answer key on page 33.

Roundup

Students would have consolidated their learning about verifying properties of triangle and quadrilaterals using coordinate proof.

Teacher Evaluation

(To be completed by the teacher using this Teacher’s Guide) The ways I will evaluate the success of my teaching this unit are: 7.

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MODULE 21: PROPERTIES OF TRIANGLES AND QUADRILATERALS USING COORDINATE PROOF

Teacher Resource Sheet 1

Plot and Tell

Directions:

1. Plot the following points on the Cartesian Plane using a graphing paper.

2. Connect them from A to R and describe the figure formed.

3. Finally, write in the box shown below the appropriate letter being described by

the statement to answer the trivia.

A dog was the first living creature to orbit the earth aboard the

Soviet's Sputnik 2. What was the name of this dog?

1. A (1, 6)

2. B (4, 3)

3. C (4, 2)

4. D (4, 0)

5. E (4, -9)

6. F (4, -11)

7. G (-4, -11)

8. H (-2, -9)

9. I (-2, -7)

10. J (-2, -2)

11. K (-6, -2)

12. L (-6, 2)

13. M (-7, 3)

14. N (-2, 3)

15. O (3, 8)

16. P (-1, 10)

17. Q (-4, 7)

18. R (0, 5)

The point 4 units above K

The point between R and O

The point 5 units below J

The point 4 units to the left of J

The point 15 units above the midpoint

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Teacher Resource Sheet 2

Plot and Tell

Answer Key

Note:

The figure seems to be a human carrying something on his/her hands.

Students may give different answers based on their own individual perceptions.

A dog was the first living creature to orbit the earth aboard the Soviet's

Sputnik 2. What was the name of this dog?

1 -1 -2 -3 -4 -5

1

-7 2 3 5 6 7

-1 -2 -3 -4 -5 -6 -7 -8 -9 -10 2 3 6 7 8 9 11

The point 4 units above K

The point between R and O

The point 5 units below J

The point 4 units to the left of J

The point 15 units above the midpoint

between H and E

(15)

Student Activity 2

“The Cutting Edge”

Worksheet 1

Objectives:

Define and illustrate the different triangles and quadrilaterals.

Directions:

1. Divide the figure on the Cartesian Plane to form the following polygons and

shade them with the indicated color.

• 1 isosceles triangle ⇒ Red • 1 square ⇒ Green • 2 trapezoids ⇒ Blue

• 2 right triangles ⇒ Orange • 1 rectangle ⇒ Violet

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Student Activity 2

The Cutting Edge

Worksheet 2

1. Define and illustrate each of the following polygons.

Polygons Definition Illustration

Isosceles Triangle

Right Triangle

Parallelogram

Square

Rectangle

Rhombus

Trapezoid

2. State some properties of the following polygons.

Polygons Properties

Triangles

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Teacher Resource Sheet 3

“The Cutting Edge”

Answer Key

Worksheet 1.

Possible Answer

Worksheet 2

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Teacher Resource Sheet 4

Prove Me Right

Directions:

1. Reproduce each station number strip and cut.

2. Reproduce also six copies of each problem for each station.

Note:

You can conduct this activity inside the classroom and use the three

corners of the room as the stations.

Mechanics:

1. Choose a group leader who will facilitate the group's discussion.

2. The leader will pick a station number strip that indicate the order of the

stations that the group will follow.

3. You will be given 5 minutes for each station. After you went through all

stations, discuss and finalize your answer.

4. Then, post your outputs on the walls.

STATION NUMBER STRIP

Stations: 3, 1, 2

Stations: 2, 3, 1

Stations: 1, 2, 3

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PROBLEM STRIP

Given: A(1,1), B(5,1), C(5,3), D(1,3)

Prove: ABCD is a rectangle

x y

1 -2 -1 -3 -4 -5

-2 -1

-3 -4 -5

2

1 3 4 5

2 1 3 4 5

A B

C D

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Given: A(0,0), B(4,2), C(3,3), D(1,2)

Prove: ABCD is a trapezoid that is not isosceles

x y

1 -2 -1 -3 -4 -5

-2 -1

-3 -4 -5

2

1 3 4 5

2 1 3 4 5

A

B C

D

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Given: ∆MON is a right triangle with right angle O. P is the midpoint of segment MN.

Prove: OP = ½ MN

x y 1 -2 -1 -3 -4 -5 -2 -1 -3 -4 -5 2 1 3 4 5

2 1 3 4 5 O M N P

Given: ∆MON is a right triangle with right angle O. P is the midpoint of segment MN.

Prove: OP = ½ MN

x y 1 -2 -1 -3 -4 -5 -2 -1 -3 -4 -5 2 1 3 4 5

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Teacher Resource Sheet 5

Prove Me Right

Answer Key

STATION 1 PROBLEM

1. Slope of AB=1−1

5−1 =0

Slope of DC=3−3

5−1 =0

Slope of AD=3−1

1−1=undefined

Slope of AD=3−1

5−5=undefined

x₂−x₁

2. AB∥DC

AD∥BC

2. Lines/Segments having equal slopes are parallel.

3. AB⊥BC AD⊥DC

3. A horizontal and vertical lines have slopes zero and undefined respectively, and they are perpendicular.

4. AB=



1−125−12=4 CD=



3−325−32=4 AD=



3−121−12=2

BC=



3−125−52=2

4. Distance Formula

Given two points, P(x1, y1) and Q(x2, y2), the

distance between P and Q is given by PQ=



x2−x1

2

y2−y1 2 _.

5. AB

≅

CD AD

≅

BC

5. Segments having the same lengths are congruent.

6. Therefore, ABCD is a rectangle. 6. The opposite sides are congruent and parallel, and the two consecutive sides are

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STATION 2 PROBLEM

1. Slope of AB= 2−0

4−0= 1 2 Slope of CD=3−2

3−1= 1 2 Slope of AD=2−0

1−0 =2 Slope of BC=3−2

3−4 = −1

x₂−x₁

2. AB∥CD 2. Lines/Segments having equal slopes are

parallel. 3. AD and BC are neither parallel nor

perpendicular.

3. Definition of slopes of parallel and perpendicular lines.

• Parallel lines have equal slopes.

• Perpendicular lines have slopes which are negative reciprocal of each other.

4. AD=



2−021−02=



5 BC=



2−324−32=



2

4. Distance Formula



x2−x1

2

y2−y1 2 _.

5. AD is not congruent to BC 5. Segments that are not equal in lengths are not congruent.

6. Therefore, ABCD is not an isosceles trapezoid 6. One pair of sides are parallel legs are not congruent.

STATION 3 PROBLEM

1. MN=



0−224−02=2



5 1. Distance Formula



x2−x1

2

y2−y1 2 _.

2. P02

2 , 4

0

2  =P1, 2 (1, 2) are coordinates of P.

2. Midpoint Formula

Given two points P(x1, y1) and Q(x2, y2), the

coordinates of the point half-way between them is x2x1

2 , y2y1

2  .

3. OP=



1−022−02=



5 3. Distance Formula

4. Therefore, OP=1

2MN

4. OP=



5 , MN=2



5

OP=1

2MN



5=1

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Student Activity 4

A Clear Proof

Objective:

Verify properties of triangles and quadrilaterals using coordinate proof.

Directions:

1. Illustrate the figure described by the given conditions on the Cartesian Plane.

2. Then, show your poof of the properties stated.

The diagonals of an isosceles trapezoid are congruent.

Given:

Trapezoid EFGH with FE

≅

GH

Prove:

EG

≅

HF

Statement

Reason

x y

(25)

The diagonals of a parallelogram bisects each other.

Given:

Parallelogram ABCD

Prove:

AC bisects BD , and BD bisects AC

Statement

Reason

x y

(26)

The line segment connecting the midpoints of the two sides of a triangle is

parallel to the third side. Its length is one-half of the length of the third side.

Given:

∆POQ. R is the midpoint of PO . S is the midpoint of PQ .

Prove:

RS

∥

OQ and RS = ½ OQ

Statement

Reason

x y

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Teacher Resource Sheet 6

A Clear Proof

Answer Key

1. MN=



0−224−02=2



5

EG=



[b−−a]2c−02=



ba2c

1. Distance Formula



x2−x1

2

y2−y1 2 _.

2. EG

≅

HF 2. Two segments that have equal lengths are

congruent.

(28)

To prove that AC bisects BD , and BD bisects AC , we need to show that E is the

midpoint of AC and BD.

1. Midpoint between A(0, 0) and C(a+b, c): Eab0

2 , c

0 2  =E

ab 2,

c 2 Midpoint between B(b, c) and D(a, 0):

Eab

2 , 0

c 2  =E

ab 2,

c 2

2. Midpoint Formula

coordinates of the point half-way between them is x2x1

2 , y2y1

2  .

2. Therefore, AC bisects BD, and BD bisects AC

. 2. The midpoint of

AC and BD

is E.

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1. Rb0

2 , c

0 2  =R

b

2, c2 Midpoint of PO Sba

2 , c

0 2  =S

ba

a , c2 Midpoint of PQ

1. Midpoint Formula

coordinates of the point half-way between

them is x2x1

2 , y2y1

2  .

2. Slope of RS=

c 2−

c 2 ba

2 − b 2

=0

Slope of OQ =0−0

a−0=0

x₂−x₁

3. RS∥OQ 3. Segments that have equal slopes are parallel

4. RS=



ba

2 − b 2 2 c 2− c 2 2 =a 2 OQ=



a−020−02=a

4. Distance Formula

5. Therefore, RS=1

2OQ 5. RS=

a

2, OQ=a RS=1

2OQ a ₌1__a_

⇒ a =a

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Teacher Resource Sheet 7

The Real Proof Applied

TASK CARDS

Directions:

Reproduce two copies of each card and cut.

Roy is designing a kite to look like

the one on the left. Its diagonals

should measure 64 cm and 90

cm. He will use a ribbon to

connect the midpoints of its side.

How much ribbon will Roy need.

Arnold wants to paddle his banca

across the lake. To determine

how far he must paddle, he paced

out a triangle, counting the

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Teacher Resource Sheet 8

The Real Proof Applied

Answer Key

TASK 1

A and B are midpoints of PS and PQ respectively of ∆PQS. Recall that the line segment connecting the midpoints of the two sides of a triangle is one-half the length of the third side. Therefore,

AB=1

2QS

AB=1

264 cm, where QS=64cm AB=32 cm

B and C are midpoints of PQ and QR , respectively of ∆PQR. Therefore,

BC=1

2PR

BC=1

290 cm, where PR=90 cm BC=45 cm

The midpoints of the sides of the kite form a rectangle. Hence, CD=AB=32 cm and DA=BC=45cm . Total length of the ribbon = AB + BC + CD + DA

= 32 cm + 45 cm + 32 cm + 45 cm = 154 cm

TASK 2

Let AB be the distance Juana will paddle across the lake. CD= 2501.5ft =375 ft

AB=1

2CD

AB=1

(32)

Student Activity 6

Are You Sure?

Directions:

1. Determine whether each statement is

always true

,

sometimes true

, or

never

true

.

2. Encircle the icon which represents your answer.

3. If the statement is never true, replace the underlined word or phrase to make it

always true.

1. If a quadrilateral has a pair of sides that have equal slopes and another pair of sides that have equal lengths, then it is an isosceles trapezoid.

___________________________________________________________

2. The median of a trapezoid has a length equal to half the difference of the lengths of the bases.

____________________________________________________________

3. The diagonals of a rhombus have equal lengths.

___________________________________________________________

4. The midpoint of the hypotenuse of a right triangle is equidistant from the vertices.

___________________________________________________________

5. The diagonals of a rectangle intersect at their midpoints.

___________________________________________________________

6. The length of the median to the hypotenuse of a right triangle is equal to the length of the hypotenuse.

___________________________________________________________

7. The opposite sides of a parallelogram have equal slopes.

___________________________________________________________

8. The segment joining the the midpoints of two sides of a triangle has slope equal to the third side.

___________________________________________________________

⇒

(33)

Teacher Resource Sheet 9

Are You Sure?

Answer Key

1. If a quadrilateral has a pair of sides that have equal slopes and another pair of sides that have equal lengths, then it is an isosceles trapezoid.

___________________________________________________________

2. The median of a trapezoid has a length equal to half the difference of the lengths of the bases.

The median of a trapezoid has a length equal to half the sum of the lengths of the bases.

3. The diagonals of a rhombus have equal lengths.

___________________________________________________________

4. The midpoint of the hypotenuse of a right triangle is equidistant from the vertices.

____________________________________________________________

5. The diagonals of a rectangle intersect at their midpoints.

___________________________________________________________

6. The length of the median to the hypotenuse of a right triangle is equal to the length of the hypotenuse.

The length of the median to the hypotenuse of a right triangle is half the length of the hypotenuse.

7. The opposite sides of a parallelogram have equal slopes.

___________________________________________________________

8. The segment joining the the midpoints of two sides of a triangle has slope equal to the third side.

___________________________________________________________

⇒

(34)

For the Teacher:

Translate the information in this Learning Guide into the following matrix to help you prepare your lesson plans.

Stage

_1.

Activating Prior Learning

2.

Setting the

Context

3.

Learning

Activity Sequence

4.

Check for

Understanding

5.

Practice and

Application

6.

Closure

Strategies

Activities from the Learning Guide

Extra activities you may wish to include

Materials and planning needed

Estimated time for this Stage