Grade 7 Mathematics Unit 8 Geometry -...

Grade 7 Mathematics Curriculum Outcomes 233

Outcomes with Achievement Indicators

Unit 8

Grade 7 Mathematics

Unit 8

Geometry

Estimated Time: 18 Hours

[C] Communication [PS] Problem Solving [CN] Connections [R] Reasoning

[ME] Mental Mathematics [T] Technology

and Estimation [V] Visualization



Unit 8

Unit 8: Geometry

Grade 7 Math Curriculum Guide 235

Unit 8 Overview

Introduction

Students will develop a vocabulary of geometric terms and learn to create some simple constructions.

Several Explore investigations will be performed to enable the learning of concepts and techniques.

Students will be introduced to the Cartesian plane (or coordinate plane), and will advance their knowledge

of transformations. The big ideas in this unit are:

• There are a variety of methods used to create parallel and perpendicular line segments.

• There are a variety of methods used to create line segment bisectors and angle bisectors

• Locations on a Cartesian plane are indentified using ordered pairs.

• Under translations, rotations, and reflections the objects and the images are congruent to each

other.

• Transformations can be applied in sequence, one after the other. Objects will be denoted with

capital letters (like A) and images will use the prime notation

( AA ′′′ or mtion transforasingle afor for a combination of two transformations).

Context The students will begin by identifying parallel and perpendicular line segments that are found in the

environment. They will explore a variety of methods to construct their own parallel and perpendicular

segments, all the while learning the correct language and definitions. The concept of a bisector, both line

and angle, will be introduced and the construction of bisectors will be learned. The Cartesian plane will be presented and the four quadrants will be explained. Naming of locations, or

points on the plane, will be done with ordered pairs.

The students will learn the mathematical terms for slide, flip, and turn. A discussion of congruence and

orientation will be made with each transformation. The use of the prime notation will be introduced, and

the students will explore the properties of combined transformations.

Why are these concepts important?

Developing a good understanding of geometry will permit students to:

• Be active participants in many of today’s fields that require a strong knowledge of geometric

concepts. Fields like engineering, carpentry, surveying, interior decorating, architecture and

more.

• Be good logical thinkers. Geometry requires the ability to reason in a linear and coherent

manner; the ability to make statements and support them until a logical conclusion is reached.

Whatever future path a student may walk the ability to analyze and reason logically will serve

them well.

• Be better prepared for the higher order geometry that they will study in high school and beyond.

“And since geometry is the right foundation of all painting, I have decided to teach its rudiments and

principles to all youngsters eager for art.”

Albrecht Durer (1471 – 1528)

Strand: Shape and Space (3-D Objects and 2-D Shapes)



Unit 8

General Outcome: Describe the characteristics of 3-D objects

and 2-D shapes, and analyze the relationships among them.

Specific Outcome

It is expected that students will:

7SS3. Perform geometric

constructions, including:

• perpendicular line

segments

• parallel line segments

• perpendicular bisectors

• angle bisectors.

[CN, R, V]

Achievement Indicators

Elaborations: Suggested Learning and Teaching Strategies

“What makes shapes alike and different can be determined by

an array of geometric properties. For example, shapes have

sides that are parallel, perpendicular, or neither.” (Van de

Walle and Lovin 2006, p. 179)

Students have been introduced to the concept of:

• lines (or line segments) that are parallel (never

intersect) or that are perpendicular (meet at right

angles) in familiar shapes and in the real world.

• identifying the parallel sides of squares, rectangles,

hexagons, trapezoids, and parallelograms.

• identifying pairs of adjacent sides that are

perpendicular.

This achievement indicator is discussed in Lesson 8.1 (Parallel

Lines) and Lesson 8.2 (Perpendicular Lines).

Examples of parallel lines in the environment.

• Opposite sides of picture frames

• Railroad/Roller Coaster tracks

• Lines on loose-leaf paper

• Rows of siding on a house

• Lines of latitude

• Guitar strings

The Explore activity on page 300 in the textbook allows

students to discover their own methods of creating parallel

lines using a wide variety of tools. It is open ended and most

likely will lead to many novel approaches.

The textbook has a host of methods (pp.300-301) that illustrate

how to draw parallel line segments. They are all effective and

offer their own perspectives on the concepts involved.

7SS3.2 Describe examples

of parallel line segments

in the environment.

7SS3.1 Identify line

segments on given

diagrams that are either

parallel or perpendicular.

7SS3.3 Draw a line

segment parallel to

another line segment, and

explain why they are

parallel.




Unit 8



Suggested Assessment Strategies

Activity

Ask the students to make a list of as many pairs of parallel lines as

they can find in the classroom in two minutes. After the two

minutes are up have the students pass their list to another student.

Then ask students, one at a time, to read an entry from the list in

front of them. Everybody who has that entry on their list will cross

it off. At the end, the list with the most remaining entries will be the

winner. (Students can be paired up for this activity.)

Paper and Pencil

1. Have students draw a line that is neither vertical nor horizontal.

Then, using a method of their choice, draw a second line that is

parallel to the first.

2. Where might you see this pattern and what is the purpose of

these parallel lines?

Journal

Have students think of 2-D shapes (excluding quadrilaterals) that

have parallel sides. Ask the students to include diagrams to

illustrate their thinking.

Resources/Notes

Math Makes Sense 7

Lesson 8.1

Unit 8: Data Analysis

TR: ProGuide, pp. 4–6

Master 8.8, 8.24, 8.15

CD-ROM Unit 8 Masters

ST: pp. 300–302

Practice and HW Book

pp. 178–181




Unit 8



Specific Outcome





segments




[CN, R, V]

(Cont’d)



Examples of perpendicular lines in the environment.

• Crosses

• Railway tracks and railway ties

• Fence posts and fence rails

• Four way stops

• Lines of latitude and longitude

• A wall and a shelf


students to discover their own methods of creating

perpendicular lines using a wide variety of tools. It is open

ended and most likely will lead to many novel approaches.

The textbook has a host of methods (pp.303-304) that illustrate

how to draw perpendicular line segments. They are all

effective and offer their own perspectives on the concepts

involved.

7SS3.5 Draw a line

segment perpendicular to

another line segment, and

explain why they are

perpendicular.


of perpendicular line

segments in the

environment.




Unit 8




Activity

Ask the students to make a list of as many pairs of perpendicular

lines as they can find in the classroom in two minutes. After the two

minutes are up have the students pass their list to another student.

Then ask students, one at a time, to read an entry from the list in

front of them. Everybody who has that entry on their list will cross

it off. At the end, the list with the most remaining entries will be the

winner. (Students can be paired up for this activity)

Paper and Pencil

Have students draw a line that is neither vertical nor horizontal.

Then, using a method of their choice, draw a second line that is

perpendicular to the first.

Journal

1. Can two lines be both parallel and perpendicular?

2. Can a line have more than one other line that is perpendicular to

it? Explain your reasoning. Can you think of an example?

Resources/Notes

Math Makes Sense 7

Lesson 8.2



Master 8.9, 8.25, 8.16


ST: pp. 303–305


pp. 182–185




Unit 8



Specific Outcome





segments




[CN, R, V]

(Cont’d)



A perpendicular bisector is a line or segment that intersects

another segment at a right angle and divides it into two equal

parts.

Examples of perpendicular bisectors segments in the

environment.

The Explore activity on page 306

in the textbook allows students to

discover their own methods of

creating perpendicular bisectors

using a wide variety of tools. It is

open ended and most likely will

lead to many novel approaches.

The textbook has three strategies (pp.307-308) that illustrate

how to draw perpendicular bisectors. They are all effective and

offer their own perspectives on the concepts involved.


of perpendicular bisectors

in the environment.

7SS3.7 Draw the

perpendicular bisector of a

line segment, using more

than one method, and

verify the construction.

MBAM

CDAB

=

⊥




Unit 8




Journal

The Reflect activity on page 309 in the textbook is suggested as an

exercise.

Paper and Pencil

For the Perpendicular Bisector worksheet, refer to Appendix 8–A.

Resources/Notes

Math Makes Sense 7

Lesson 8.3 Unit 8: Data Analysis


Master 8.10, 8.26


ST: pp. 306–309


pp. 186–189




Unit 8



Specific Outcome





segments




[CN, R, V]

(Cont’d)



For every angle, there exists a line that divides the angle into

two equal parts. This line is known as the angle bisector.

Example in the environment:


students to discover their own methods of creating angle

bisectors using a wide variety of tools. It is open ended and

most likely will lead to many novel approaches.

The textbook has three strategies (pp.310-311) that illustrate

how to draw angle bisectors. They are all effective and offer

their own perspectives on the concepts involved.

7SS3.9 Draw the bisector

of a given angle, using

more than one method,

and verify that the

resulting angles are equal.


of angle bisectors in the

environment.




Unit 8




Pencil & Paper

For the Angle Bisector worksheet, refer to Appendix 8–B.

Informal Observation

Mix-up Match-up: Create a set of cards. On one half write the terms

in the list below; on the second half write definitions that match the

terms on the first half. Distribute these cards to the students and

have them circulate the room trying to match up the correct cards.

Instruct them to sit together when they find their match.

parallel lines right angle

perpendicular lines straight angle

angle bisector reflex angle

perpendicular bisector vertex

line segment radius

obtuse angle diameter

acute angle circumference

Resources/Notes

Math Makes Sense 7



Master 8.11, 8.27, 8.17,

8.18a,b


ST: pp. 310–313


pp. 190–192

Unit Problem P 338, 339

Strand: Shape and Space (Transformations)



Unit 8

General Outcome: Describe and analyze position and motion

of objects and shapes.

Specific Outcome


7SS4. Identify and plot

points in the four quadrants

of a Cartesian plane, using

integral ordered pairs.

[C, CN, V]



Key Terms

coordinate plane the plane containing the x- and y-axes

coordinate system the grid formed by the intersection of two

perpendicular number lines that meet at their zero points

ordered pair a pair of numbers used to locate any point on a

coordinate plane

quadrant one of the four regions into which the x- and y-axes

separate the coordinate plane

x-axis the horizontal number line on a coordinate plane

y-axis the vertical number line on a coordinate plane

x-coordinate the first number in a coordinate pair

y-coordinate the second number in a coordinate pair

Identify the ordered pair that names point A. Step 1 Move left on the x-axis to find the x-coordinate of point

A, which is -3.

Step 2 Move up the y-axis to find the y-coordinate, which is 4.

Point A is named by (-3, 4).

7SS4.1 Label the axes of a

four quadrant coordinate

plane (or Cartesian plane),

and identify the origin.

7SS4.2 Identify the

location of a given point

in any quadrant of a

Cartesian plane, using an

integral ordered pair.

(5, 4)

Origin

(0, 0)

7SS4.3 Plot the point

corresponding to a given

integral ordered pair on a

Cartesian plane with units

of 1, 2, 5 or 10 on its axes.




Unit 8




Games

Have students play the commercial board game Battleship or create

a game similar to battleship using the coordinate plane. Creating a

game would probably be better because you can utilize all four of

the quadrants. The game can also be played online at the link

indicated below:

http://www.learn4good.com/games/board/battleship.htm

For the website, refer to: www.learn4good.com

Paper and Pencil

For the Coordinate Plane worksheet, refer to Appendix 8–C.

Graphic Organizer (Foldable)

A four-door booklet can be created to organize students’ notes on a

Cartesian plane. (Refer to the Appendix 8–D for the Coordinate

Plane foldable.)

Journal

Using the internet research why coordinate planes are often called

Cartesian planes. Write a brief paragraph explaining your findings.

Resources/Notes

Math Makes Sense 7



Master 8.20, 8.12, 8.28

PM 22


ST: pp. 315–319


pp. 193–195




Unit 8



Specific Outcome


7SS4. Identify and plot

points in the four quadrants

of a Cartesian plane, using

integral ordered pairs.

[C, CN, V]

(Cont’d)



The following achievement indicators are addressed together.

Students would be expected to perform both of these tasks:

1. Given sets of points as ordered pairs, plot them on a

coordinate plane and join those points to create a shape.

i.e.: given point A(1, 3), point B……., plot the points

and connect to form a shape.

2. Using shapes drawn on a coordinate plane, identify the

locations of the vertices.

i.e. Given the shape in the diagram identify the ordered

pair that describes each vertex.

7SS4.4 Draw shapes and

designs in a Cartesian

plane, using given integral

ordered pairs.

7SS4.5 Create shapes and

designs, and identify the

points used to produce the

shapes and designs, in any

quadrant of a Cartesian

plane.

A




Unit 8




Paper and Pencil

1. Natasha is creating an X-pattern for her needlepoint project in

home economics. She has plotted the X on a coordinate plane

using these ordered pairs.

A(3, 0) B(2, -1) C(1, -2)

D(-3, -2) E(-1, -4) F(-1, 0)

G(0, -1) H(2, -3) I(3, -4)

Will Natasha make an X? If not, what ordered pair will she need to

change to fix it?

2. Refer to Appendix 8–E for the worksheet entitled Coordinate

Plane Worksheet: Newfoundland Flag.

Resources/Notes

Math Makes Sense 7

Lesson 8.5

(continued)




Unit 8



Specific Outcome

It is expected that students will: 7SS5. Perform and describe

transformations

(translations, rotations or

reflections) of a 2-D shape in

all four quadrants of a

Cartesian plane (limited to

integral number vertices).

[C, CN, PS, T, V]



Students have been exposed to transformational geometry in

previous grades. An emphasis at this level should be the use of

the formal language of transformations, such as translation,

reflection, and rotation, instead of slides, flips, and turns.

Students will be working with transformations and

combinations of transformations in all four quadrants of the

Cartesian plane.

With respect to describing transformations, students should be

able to recognize a given transformation as one of the

following: a reflection, a translation, a rotation, or some

combination of these. In addition, when given an object and its

image students should be able to describe:

• a translation, using words and notation describing the

translation (E.g. ∆A′B′C′ is the translation image of ∆ABC, or

D′(5, 8) is the translation image of D (-5, 8). Students will be

taught that when describing translations they have to describe

the horizontal change first and the vertical change second.

• a reflection, by determining the location of the line of

reflection

• a rotation, using degree or fraction-of-turn measures, both

clockwise and counterclockwise, and identify the location of

the centre of a rotation. A centre of rotation may be located

on the shape (such as a vertex of the original image) or off the

shape.

Note: A′ is read as “A prime”. It is used to label the point that

matches point A after the transformation has been applied.

When investigating properties of transformations, students

should consider the concepts of congruence, which were

developed informally in previous grades. In discussing the

properties of transformations, students should consider if the

transformation of the image:

• has side lengths and angle measures the same as the given

image;

• is both similar to and congruent to the given image;

• has the same orientation as the given image;

• appears to have remained stationary with respect to the given

image.

7SS5.2 Describe the

horizontal and vertical

movement required to

move from a given point

to another point on a

Cartesian plane.

7SS5.1 (It is intended that

the original shape and its

image have vertices with

integral coordinates.)

Identify the coordinates of

the vertices of a given 2-D

shape on a Cartesian

plane.

7SS5.3 Determine the

distance between points

along horizontal and

vertical lines in a

Cartesian plane.




Unit 8




Group Activity(Discussion)

Ask students to identify identical objects in the classroom (desks,

books, posters, etc.). Discuss how these objects could be regarded

as objects and images, and describe the transformations that relate

them to each other.

i.e. Desks are arranged in 5 rows with 6 desks in each row. What

transformation could be applied to relate the first desk in the first

row to the fourth desk in the third row?

Paper and Pencil

Resources/Notes

Math Makes Sense 7

Lesson 8.6

Lesson 8.7



& pp. 29–33

Master 8.20, 8.13, 8.29

Master 8.20, 8.14, 8.30

PM 22


ST: pp. 320–324

ST: pp. 325–329


pp. 196–198

pp. 199–201

The transformation is

a translation of two

desks to the right,

and three desks back.

O is the object.

A, B, C, and D

are images of O.

1) Identify the

coordinate pairs

of the vertices

for the object O

and its images.

2) Describe the

movement

required to

move from any

point on O to

each of its

image points.




Unit 8



Specific Outcome


7SS5. Perform and describe

transformations

(translations, rotations or

reflections) of a 2-D shape in

all four quadrants of a

Cartesian plane (limited to

integral number vertices).

[C, CN, PS, T, V]

(Cont’d)



Transformational geometry is another way to investigate and

interpret geometric figures by moving every point in a plane

figure to a new location. To help students form images of

shapes through different transformations, students can use

concrete objects such as cardboard cut-outs or geometry sets,

figures drawn on graph paper, mirrors or other reflective

surfaces, or appropriate technology. Some transformations,

like translations, reflections, and rotations, do not change the

figure itself.

Remind students what has been learned in previous lessons

that will be pertinent to this lesson and/or have them begin to

think about the words and ideas of this lesson:

Can someone tell me where you might see a reflection in

everyday life?

Can anyone tell me what it means to rotate an object?

Can anyone guess what it might mean to translate an object?

Successive transformations are defined as more than one

transformation being applied to an object. For example, a

translation is applied to A and then a second transformation is

applied to A′ creating A ′′ .

The textbook addresses successive transformations in Lessons

8.6 and 8.7. There is no elaboration in the text on this topic but

there are exercises. However, students have had prior

experience with successive transformations in Grade 6.

7SS5.4 Describe the

positional change of the

vertices of a given 2-D

shape to the corresponding

vertices of its image as a

result of a transformation,

or successive

transformations, on a

Cartesian plane.

7SS5.5 Perform a

transformation or

consecutive

transformations on a given

2-D shape, and identify

coordinates of the vertices

of the image.

7SS5.6 Describe the image

resulting from the

transformation of a given

2-D shape on a Cartesian

plane by identifying the

coordinates of the vertices

of the image.




Unit 8




Paper and Pencil

For the worksheets entitled Describing Transformations Worksheet

and Successive Transformations Worksheet, refer to the Appendix

8–F and 8–G.

Journal

If a shape undergoes two transformations, one after the other, does

it matter in what order they are applied? Will you get the same final

image either way?

Technology/Web Resources

1. A neat game addressing successive transformations aimed at

ages 9-13 is available at www.mathsonline.co.uk

2. For a golf game:

http://www.mathsonline.co.uk/nonmembers/gamesroom/transfo

rm/golftrans.html

This website was found at: www.mathsonline.co.uk

Examples of Islamic art, easily found on the Internet, would be

useful for illustrating successive transformations. Some examples

are found below.

Resources/Notes

Math Makes Sense 7

Lesson 8.6

Lesson 8.7

(continued)




Unit 8

Grade 7 Mathematics Unit 8 Geometry -...

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