Download Unit 4 Geometry

Unit 4 Geometry
In this unit, students will use protractors, rulers, set squares, and various
strategies to
• construct related lines using angle properties
• investigate geometric properties related to symmetry, angles, and sides
• sort and classify triangles and quadrilaterals by their geometric
properties
Protractors
If you need additional protractors for individual students, you can photocopy
a protractor (or BLM Protractors, p E-50) onto a transparency and cut it out.
Such protractors are also convenient to use on an overhead projector.
Paper Folding
Many Activities in these lessons involve paper folding. Unless otherwise
noted, the starting shape is a regular 8 1/2” × 11” sheet of paper.
Sometimes the starting shape is an oval or a cloud, to make sure there are
no angles for students to start with or refer to.
Technology: Geometer’s Sketchpad
Students are expected to investigate geometric properties of lines, angles,
and triangles using dynamic geometry software. Many activities in this unit
use a program called Geometer’s Sketchpad, and some are instructional—
they help you teach students how to use the program. If you are not
familiar with Geometer’s Sketchpad, the built-in Help Centre provides
explicit instructions for many constructions. Search the Index using phrases
such as “How to construct congruent angles” or “How to construct a line
segment of given length.”
Please note: Dynamic geometry software is not required to complete the unit.
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Summary BLMs
Step-by-step instructions for constructions used in the unit are summarized
on BLMs, for easy reference. This chart lists the summary BLMs available
and the lesson(s) they relate to.
Teacher’s Guide for Workbook 7.1
Summary BLM
Lesson(s) Constructions
Measuring and
Drawing Angles
and Triangles
(p E-42)
G7–3
Measuring an angle
Drawing an angle
Drawing lines that intersect at an angle
Drawing a triangle
Drawing
G7–4
Perpendicular Lines G7–5
and Bisectors
(p E-43)
Drawing a line segment perpendicular
to AB through point P (using a set
square, using a protractor)
Drawing
Parallel Lines
(p E-44)
Drawing a line parallel to AB through
point P (using a set square, using
a protractor)
G7–6
Drawing the perpendicular bisector
of line segment AB
E-1
G7-1 Points and Lines
Pages 97–99
Curriculum
Expectations
Ontario: conceptual review; 7m2, 7m4
WNCP: conceptual review,
[R, T]
Vocabulary
point
line segment
intersection point
ray
endpoint
intersect
Goals
Students will review the basic geometric concepts of (and notation for)
points, lines, line segments, and rays. They will also review intersecting
lines and lines segments.
PRIOR KNOWLEDGE REQUIRED
Materials
Can use a ruler to draw lines
Dynamic geometry software
(optional)
Review the concepts of a point, line, line segment, and ray. A dot
represents a point. A point is an exact location. It has no size—no length,
width, or height. The dot has size, or you couldn’t see it, but real points
do not.
A line extends in a straight path forever in two directions. It has no ends.
Lines that are drawn have a thickness, but real lines do not.
line
A line segment is the part of a line between two points, called endpoints.
It has a length that can be measured.
line segment
A ray is part of a line that has one endpoint and extends forever in
one direction.
Review naming a point, line, ray, and line segment.
ray
A
B
Line AB or BA
A point is
named with
a capital
letter.
To name a line,
give the names of
any two points on
the line.
A
B
ray AB (not BA)
To name a ray, give
the name of the
endpoint and any
point on the line.
F
G
Line segment FG or GF
To name a line
segment, give the
names of the
endpoints.
Point out that you don’t need to draw arrows and/or dots at the ends
of lines, rays, or line segments unless you especially want to show that
something is a line, ray, or line segment.
Introduce the intersection point—a point that lines, line segments, or
rays have in common. Draw several examples of intersecting line segments,
including the following:
E-2
Teacher’s Guide for Workbook 7.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
A
Invite volunteers to identify the intersection point in each picture.
B
A
C
D
Review intersecting lines and line segments. Draw the picture at left
on the board and ASK: Does AB intersect CD? The answer depends on
whether AB and CD are lines, line segments, or rays. Check all possible
combinations of lines and line segments, extending the lines to show
intersection.
ANSWERS: If either one of these is a line segment, they do not intersect.
Lines AB and CD intersect.
process Expectation
Revisiting conjectures that
were true in one context
If students are engaged, ask them to check all four possible rays, as well
as all combinations of rays and lines or rays and line segments.
ANSWERS: Out of four possible rays, only BA and DC intersect. Also, line
AB intersects with ray DC and ray BA intersects with line CD.
EXAMPLES:
A
C
B
A
D
rays AB and DC
do not intersect
C
line AB and ray DC
intersect
B
B
A
D
C
D
line AB and ray CD
do not intersect
ACTIVITY
process Expectation
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Technology
Teach students to draw lines, line segments, and rays using the line
tool of Geometer’s Sketchpad. Then ask them to draw a line and an
independent point. Ask them to move the point so that it looks like it
is on the line. Then have them modify the line. Does the point stay on
the line? (no) Now show students how to construct a line through two
given points and a point on the line, so that modifying the points keep
the line and the points together.
Geometry 7-1
E-3
G7-2 Angles and Shapes
Pages 100–101
Goals
Curriculum
Expectations
Students will review the naming of angles and shapes.
Ontario: review, 7m4, 7m7
WNCP: 6SS1, review, [C, T]
Vocabulary
angle
arms
vertex, vertices
line segment
PRIOR KNOWLEDGE REQUIRED
Materials
Knows what an angle is
Can identify polygons
Knows the terms clockwise,
counter-clockwise
Dynamic geometry software
(optional)
Review the concept of an angle and how to name it by following the
progression on the worksheet.
An angle is formed by two rays with the same endpoint.
The endpoint is the vertex of the angle. The two rays are the arms of
the angle.
vertex
arms
Point out that the letter for the point at the vertex has to be in the middle
of the name for an angle.
Extra practice:
X
1. Name the angle in all possible ways.
Y
A
K
Q
Z
∠XYZ or ∠ZYX
(not ∠XZY or ∠YXZ)
B
E
2. a) Which of the following are possible names for this angle?
∠CAT, ∠CTP, ∠PTA, ∠UTA, ∠UTC, ∠TCP
C
B
C
A
D
E-4
T
U
P
b) Write three more different names for the angle.
Point out that sometimes, when there is no chance of confusion, only
the vertex letter is used to name an angle. For example, there is only one
possible ∠D in the picture at left, but there are three possibilities for
∠A: ∠BAD or ∠BAC or ∠DAC.
Teacher’s Guide for Workbook 7.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
A
Polygons
vertex
Not Polygons
Review the concept of a polygon. A polygon
• is a closed 2 D (flat) shape
• has sides that are straight line segments
• has each side touching exactly two other line segments, one at
each of its endpoints
The point where two sides of a polygon meet is called a vertex. (The plural
of vertex is vertices.)
Have students explain (using the definition above) why each of the shapes
below is not a polygon.
Review naming polygons.
B
A
C
A
1. Start at
any vertex.
Choose a
letter to name
the vertex.
D
D
C
A
S
B
F
M
Q
3. You can choose
2. Go around the polygon clockwise or
any sequence of
counter-clockwise, labelling the other
letters.
vertices. To name the polygon, write the
letters at the vertices in order. For example,
you could name this rhombus ABCD or
BADC or DCBA, but not DBAC.
Extra practice:
a) Circle the correct names for this polygon.
ABKLXY
ALYXBK
BKALYX
BKXAYL
XYLAKB
b) Write another correct name for this polygon.
B
K
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
X
Y
A
L
ACTIVITY
process Expectation
Technology
Geometry 7-2
Teach students to construct polygons using Geometer’s Sketchpad.
Have them measure the lengths of the sides of the polygons.
E-5
G7-3 M
easuring and Drawing Angles
and Triangles
Pages 102–104
Goals
Curriculum
Expectations
Students will measure and draw angles.
Ontario: 5m51, 5m52, 5m54, 6m48, 6m49, 7m3, 7m4,
7m46
WNCP: 6SS1, review,
[T, R, V]
PRIOR KNOWLEDGE
REQUIRED
Materials
Knows what an angle is
Can name an angle and identify a named angle
dice
geoboards and elastics
grid paper
Dynamic geometry software
(optional)
Vocabulary
angle
vertex
arms
acute
obtuse
Review the concept of an angle’s size. Draw two angles:
ASK: Which angle is smaller? Which corner is sharper? The diagram on the
left is larger, but its corner is sharper, and mathematicians say that this angle
is smaller. The distance between the ends of the arms in both diagrams
is the same, but this does not matter; angles are made of rays and these
can be extended. What matters is the “sharpness.” The sharper the corner
on the “outside” of the angle, the narrower the space between the angle’s
arms. Explain that the size of an angle is the amount of rotation between the
angle’s arms. The smallest angle is closed; both arms are together. Draw the
following picture to illustrate what you mean by smaller and larger angles.
Smaller
Larger
You might also illustrate what the size of an angle means by opening a book
to different angles. Draw some angles and ask your students to order them
from smallest to largest.
A
E-6
B
C
D
E
Teacher’s Guide for Workbook 7.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
You can show how much an angle’s arm rotates with a piece of chalk. Draw
a line on the board then rest the chalk along the line’s length. Fix the chalk
to one of the line’s endpoints and rotate the free end around the endpoint to
any desired position.
Define acute and obtuse angles in relation to right angles. Obtuse angles
are larger than a right angle; acute angles are smaller than a right angle.
ExTRa PRaCTICE:
1. Copy the shapes onto grid paper and mark any right, acute, and
obtuse angles. Which shape has one internal right angle? What did
you use to check?
2. Which figures at left have
a) all acute angles?
b) all obtuse angles?
c) some acute and some obtuse angles?
Introduce protractors. On the board, draw two angles that are close to
each other—say, 50° and 55°—without writing the measurements and in
a way that makes it impossible to compare the angles visually. aSK: How
can you tell which angle is larger? Invite volunteers to try different strategies
they suggest (such as copying one of the angles onto tracing paper and
comparing the tracing to the other angle, or creating a copy of the angle
by folding paper). Lead students to the idea of using a measurement tool.
ExaMPLE:
Explain that to measure an angle, people use a protractor. A protractor
has 180 subdivisions around its curved edge. These subdivisions are
called degrees ( °). Degrees are a unit of measurement, so just as we write
cm or m when writing a measurement for length, it is important to write
the ° symbol for angles.
origin
base line
Using protractors. Show your students how to use a protractor on the
board or on the overhead projector. Identify the origin (the point at which
all the degree lines meet) and the base line (the line that goes through
the origin and is parallel to the straight edge). When using a protractor,
students must
• place the vertex of the angle at the origin;
• position the base line along one of the arms of the angle.
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
You could draw pictures (see below) to illustrate incorrect protractor use,
or demonstrate it using an overhead projector and a transparent protractor.
Geometry 7-3
E-7
Introduce the degree measures for right angles (90°), acute angles (less than
90°), and obtuse angles (between 90° and 180°). Point out that there are
two scales on a protractor because the amount of rotation can be measured
clockwise or counter-clockwise. Students should practise choosing the
correct scale by deciding whether the angle is acute or obtuse, then saying
whether the measurement should be more or less than 90°. (You may want
to do some examples as a class first.)
Then have students practise measuring angles with protractors. Include
some cases where the arms of the angles have to be extended first.
Introduce angles in polygons, then have students measure the angles in
several polygons. Students can draw polygons (with both obtuse and acute
angles) and have partners measure the angles in the polygons.
Drawing angles. Model drawing angles step by step (see Workbook p.
104 or BLM Measuring and Drawing Angles and Triangles, p E-42).
Emphasize the correct position of the protractor. Have students practise
drawing angles. You could also use Activity 2 for that purpose.
Have students practise drawing lines that intersect at a given angle.
Another way to practise drawing angles is to construct triangles with given
angle measures.
ACTIVITIES 1–4
1. Students can use geoboards and elastics to make right, acute,
and obtuse angles. When students are comfortable doing that, they
can create figures with different numbers of given angles. EXAMPLES:
d)
e)
12°
9°
4°
process Expectation
Selecting tools and
strategies, Technology
E-8
2. Students will need a die, a protractor, and a sheet of paper. Draw a
starting line on the paper. Roll the die and draw an angle of the measure
given by the die; use the starting line as your base line and draw the
angle counter-clockwise. Label your angle with its degree measure.
For each next roll, draw an angle in the counter-clockwise direction so
that the base line of your angle is the arm drawn at the previous roll.
The measure of the new angle is the sum of the result of the die and the
measure of the angle in the previous roll. Stop when there is no room to
draw an angle of the size given by the roll. For example, if the first three
rolls are 4, 5, and 3, the picture will be as shown.
3. Teach students to draw and measure angles using Geometer’s
Sketchpad. Then ask them to try moving different points (on the arms
of the angle, or its vertex) so that the size of the angle becomes, say,
50°. Is it easy or hard to do? When you move the line segments, does
Teacher’s Guide for Workbook 7.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
a) a triangle with 3 acute angles
b) a quadrilateral with 0, 2, or 4 right angles
c) a quadrilateral with 1 right angle
d) a shape with 3 right angles
e) a quadrilateral with 3 acute angles
Sample ANSWERS:
the angle change? (yes) When you move the vertex or other point on
the arms, does the angle change? (yes) Show students how to draw
an angle of fixed measure (using menu options). Will moving the
endpoints change the size of the angle now? (no) Show students how
to draw angles equal to a given angle.
4. Have students draw polygons in Geometer’s Sketchpad and
measure the size of the angles and the length of the sides of these
polygons. Have students check that the angle measures they obtain
make sense. For example, clicking on three vertices of a quadrilateral
and then using menu options to measure the angle might produce
different angles, depending on the order in which the vertices were
selected. Also, the software sometimes measures angles in the wrong
direction, producing an answer more than 180°.
process Expectation
Technology, Reflecting
on the reasonableness
of the answer
Extensions
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1. Angles on an analogue clock.
What is the angle between the hands at 12:24? 13:36? 15:48? (Draw the
hands first!) To guide students to the answers, draw an analogue clock
that shows 3:00 on the board. Ask your students what angle the hands
create. What is the measure of that angle? If the time is 1:00, what is the
measure of the angle between the hands? Do you need a protractor to
tell? Have students write the angle measures for each hour from 1:00 to
6:00. Which number do they skip count by?
An hour is 60 minutes and a whole circle is 360°. What angle does the
minute hand cover every minute? (6°) How long does it take the hour
hand to cover that many degrees? How do you know? (12 minutes,
because the hour hand covers only one twelfth of the full circle in an
hour, moving 12 times slower than the minute hand)
If the time is 12:12, where do the hour hand and the minute hand point?
What angle does each hand make with a vertical line? What is the angle
between the hands? (ANSWER: The hour hand points at one minute
and the angle that it makes with the vertical line is 6°. The minute hand
points at 12 minutes and the angle that it makes with the vertical line is
12 × 6 = 72°. The angle between the hands is 72° − 6° = 64°.)
2. Some scientists think that moths travel at a 30° angle to the sun when
they leave home at sunrise. Note that the sun is far away, so all the rays
it sends to us seem parallel.
sun’s rays
flower
N
30°
evening
Geometry 7-3
moth home
W
morning
E
S
E-9
a) What angle do the moths need to travel at to find their way back at sunset? Hint: Where is the sun in the evening?
b) A moth sees the light from the candle flame and thinks it’s the sun. The candle is very near to us, and the rays it sends to us go out in all directions. Where does the moth end up? Draw the moth’s path.
30°
30°
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
E-10
Teacher’s Guide for Workbook 7.1
G7-4 Perpendicular Lines
Pages 105–106
Goals
Curriculum
Expectations
Students will identify and draw perpendicular lines.
Ontario: 7m4, 7m5, 7m46
WNCP: 6SS1, 7SS3,
[CN, V, T]
PRIOR KNOWLEDGE REQUIRED
Materials
Knows what a right angle is
Can name an angle and identify a
named angle
Dynamic geometry software
(optional)
protractors
set squares
Vocabulary
angle
perpendicular
slant line
right angle
arms
Introduce perpendicular lines (lines that meet at 90°) and show how
to mark perpendicular lines with a square corner. Draw several pairs of
intersecting lines on the board and have students identify the perpendicular
lines. Include pairs of lines that are not horizontal and line segments that
intersect in different places and at different angles (see examples below).
Invite volunteers to check whether the lines are perpendicular using a corner
of a page, a protractor, and/or a set square.
Ask students where they see perpendicular lines or line segments—also
called perpendiculars—in the environment (sides of windows and desks,
intersections of streets, etc.).
Extra practice:
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Which lines look like they are perpendicular?
K
C
P
B
F
a)
b)
c) J
d) O
L
G
Q
D
A
H
M
I
R
N
E
A perpendicular through a point. Explain that sometimes we are interested
in a line that is perpendicular to a given line, but we need an additional
condition—the perpendicular should pass through a given point. In each
diagram below, have students identify first the lines that are perpendicular to
the segment AB, then the lines that pass through point P, and finally the one
line that satisfies both conditions.
G
D
K
a)
b)
E
G
B
P
C
F
P
A
C
B
F
A
D
H
E
H
Constructing a perpendicular through a point. Model using a set square
(and then a protractor) to construct a perpendicular through a point that
is not on the line. Emphasize the correct position of the set square (one
Geometry 7-4
E-11
side coinciding with the given line, the other touching the given point) and
the protractor (the given line should pass through the origin and through
the 90° mark). Have students practise the construction. Circulate among
the students to ensure that they are using the tools correctly. Then invite
volunteers to model the construction of a perpendicular through a point that
is on the line. (Emphasize the difference in the position of the set square:
the square corner is now at the point, though one arm still coincides with
the given line.) Then have students practise this construction as well.
Extra practice:
Draw a pair of perpendicular slant lines (i.e., lines that are neither vertical
nor horizontal) and a point not on the lines. Draw perpendiculars to the slant
lines through the point. What quadrilateral have you constructed? (rectangle)
Bonus
Draw a slant line and a point not on the line. Using a protractor
and a ruler or set square, draw a square that has one side on the slant
line and one of the vertices at the point you drew. (ANSWER: Draw a
perpendicular through the point to the given line. Measure the distance from
the point to the line along the perpendicular. Then mark a point on the given
line that is the same distance from the intersection as the given point. Now
draw a perpendicular to the given line through this point as well. Finally,
draw a perpendicular to the last line through the given point.)
connection
Science
Why perpendiculars are important. Discuss with students why
perpendiculars are important and where are they used in real life. For
example, you can explain that it is easy to determine a vertical line (using
gravity—just hang a stone on a rope and trace the rope), but you need
a right angle to make sure that the floor of a room is horizontal.
process Expectation
1. Have students use Geometer’s Sketchpad to:
Technology
a) Draw a line. Label it m.
b) Mark a point A on the line m.
c) Draw another line through point A.
d) Measure the angle between the two lines.
e) Try to make the angle a right angle by moving the points
around. Is it easy or hard to do?
f) Check whether the lines stay perpendicular when you move any
of the points in the picture.
g) Repeat parts a) through f) with a point not on the line. Note
that when point A is not on the line, the second line might be
modified so that it does not intersect the line m, and the angle
you measure disappears.
Explain that you need a method to draw perpendicular lines that
will keep them perpendicular even if the points are moved around.
E-12
Teacher’s Guide for Workbook 7.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
ACTIVITIES 1–2
Teach students to draw perpendicular lines using the perpendicular
line command from the menu. Do these lines stay perpendicular
to the given line even if points are moved around? (yes)
2. Have students draw a triangle using the polygon tool in
Geometer’s Sketchpad. Ask them to move the points around to
make it look like a right triangle. Then ask them to measure the
angles of the triangle and to check whether it is indeed a right
triangle. Is it easy to draw a perfect right triangle this way? (no)
If you move the points around, does the triangle remain a right
triangle? (no) Ask students to think about how they draw a right
triangle on paper. What tools do they use and why? (a protractor
or a set square) What tools could we use instead of protractors
and set square in Geometer’s Sketchpad? (perpendicular lines)
Then have students draw a right triangle in Geometer’s Sketchpad.
Ask them to check that the triangle remains a right triangle even if
points are moved around.
process Expectation
Technology
process Expectation
Selecting tools
and strategies
Students can also add the measures of the acute angles in the
right triangle they created, and check that the sum remains 90°
even when the triangle is modified.
Extension
process Expectation
Visualizing
Discuss with your students whether there can be more than one line
perpendicular to a given line through a given point, and whether such
a perpendicular always exists. You can use the diagrams below to help
students visualize the answers.
P
P
B
A
B
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
A
Geometry 7-4
E-13
G7-5 Perpendicular Bisectors
Pages 107–108
Goals
Curriculum
Expectations
Students will identify and draw perpendicular bisectors.
Ontario: 7m1, 7m3, 7m4,
7m48
WNCP: 6SS1, 7SS3,
[C, R, V, T]
PRIOR KNOWLEDGE REQUIRED
Can identify and construct a perpendicular using a set square
or a protractor
Can identify and mark right angles
Can name line segments and identify named line segments
Can draw and measure with a ruler
Vocabulary
line segment
midpoint
right angle
perpendicular
bisector
Example:
Materials
paper circles or BLM Circles (p E-51)
Dynamic geometry software
(optional)
Introduce the notation for equal line segments. Explain that when we
want to show that line segments are equal, we add the same number of
marks across each line segment. This is particularly useful for sketches,
when you are not drawing everything exactly to scale.
Introduce the midpoint. Model finding the midpoint of a segment using a
ruler, and mark the halves of the line segment as equal, then have students
practise this skill.
Introduce bisectors. Explain that a bisector of a line segment is a line
(or ray or line segment) that divides the line segment into two equal parts.
There can be many bisectors of a line segment. Ask students to draw a
line segment with several bisectors.
ASK: Can a line have a bisector? What about a ray? (no, because lines
and rays have no midpoints)
Draw a scalene triangle. Choose a side and draw a bisector to that side
that passes through the vertex of the triangle that is opposite that side.
Repeat with the other sides. What do you notice? (ANSWER: All three
bisectors, called medians, pass through the same point.)
Introduce perpendicular bisectors. Of all the bisectors of a line segment
only one is perpendicular to the line segment, and it is called the
perpendicular bisector. The perpendicular bisector of a line segment
• divides the line segment into two equal parts AND
• intersects the line segment at right angles (90°).
Point out that there are two parts in the definition, and both must be true.
ASK: How can we draw a perpendicular bisector? How is that problem
E-14
Teacher’s Guide for Workbook 7.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Extra practice:
similar to constructing a bisector? Constructing a perpendicular? Lead
students to the idea that they should first find the midpoint of the line
segment, then construct a perpendicular through that point. Have students
practise drawing perpendicular bisectors using set squares and protractors.
process Expectation
Changing into a
known problem
E
D
F
J
G
C
H
B
Present several diagrams that combine intersecting line segments,
perpendicular lines, and bisectors, such as the one at left, and have students
identify equal segments, then find perpendicular lines and bisectors.
(Example: Find a bisector of EG. Is it a perpendicular bisector?) Ask other
questions about the diagram, such as:
• J is the midpoint of what segment? (CG) Why not AD? How do you
know? (We do not have any information about the lengths of DJ and
AJ. They look equal, but might have different lengths.)
• Name three line segments GJ is perpendicular to.
A
Finally, have students identify all the perpendicular bisectors and the line
segments they bisect. (ANSWERS: AD, JA, and JD bisect line segment CG;
CJ, JG, and CG bisect line segment FH)
process assessment
Workbook Question 8f) –
[C,], 7m3
Ask students to find examples of equal segments, midpoints, and
perpendicular bisectors in the classroom or elsewhere, such as in letters
of the alphabet, in pictures or photographs, and so on.
ACTIVITIES 1–3
process Expectation
Technology, Selecting
tools and strategies
2. Paper folding and line segments
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
A
C
B
1. Show students how to construct the midpoint of a line segment
in Geometer’s Sketchpad. Explain that there is no command that
will construct a perpendicular bisector of a line segment. Challenge
students to find a way to construct a perpendicular bisector of a
line segment using the tools they are familiar with in Geometer’s
Sketchpad. (Find the midpoint of the line segment, then construct a
line through the midpoint and perpendicular to the given line segment)
PROMPT: Think of the way you construct a perpendicular line segment
on paper. Which tools can you use? (protractor or set square and
ruler) Which tool replaces a protractor or a set square in Geometer’s
Sketchpad? (“Perpendicular Line” in the “Construct” menu)
Draw a line segment AB dark enough that you can see through the
paper. Fold the paper so that A meets B. What line has your crease
made? (ANSWER: a perpendicular bisector) Use a ruler and protractor
to check your answer.
3. Paper folding and circles
Give each student a circle (you can use BLM Circles) and ask students
to draw and label a triangle on their circle.
a) Fold the circle in half so that A meets B.
Geometry 7-5
E-15
Look at the line that the crease in your fold makes. Is it a bisector of
angle C? Is it a perpendicular bisector of line segment AB?
b) Fold the circle in half again, this time making A meet C. What two
properties will the crease fold have?
c) Repeat, making B meet C.
At what point in the circle will all three perpendicular bisectors meet?
Extensions
1. Given a triangle ABC, how can you use perpendicular bisectors to help
you draw the circle going through the points A, B, and C? Draw an acute
scalene triangle. Draw the perpendicular bisectors by hand—do not cut
and fold the triangle.
2. Start with a paper circle. Choose a point C on the circle and draw a right
angle so that its arms intersect the circle. Label the points where the
arms intersect the circle A and B and draw the line segment AB. Repeat
with several circles to produce different right triangles. (You can use
BLM Circles for this Extension.)
A
Fold the circle in two across the side AC so that A falls on C (creating a
perpendicular bisector of AC). Mark the point where B falls on the circle.
Repeat with the side BC, marking the point where A falls on the circle.
What do you notice? (The image of A is the same as the image of B.)
What type of special quadrilateral have you created? (a rectangle)
Repeat the exercise starting with an obtuse or an acute angle C. Do the
images of A and B coincide?
B
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
C
E-16
Teacher’s Guide for Workbook 7.1
G7-6 Parallel Lines
Pages 109–111
Goals
Curriculum
Expectations
Students will identify and draw parallel lines.
Ontario: 7m1, 7m2, 7m46
WNCP: 5SS5, 7SS3,
[C, R, PS]
PRIOR KNOWLEDGE REQUIRED
Can identify and construct a perpendicular using a set square
or a protractor
Can identify and mark right angles
Can name a line segment and identify a named line segment
Can draw and measure with a ruler
Vocabulary
line segment
parallel
right angle
perpendicular
Materials
BLM Distance Between Parallel Lines (p E-46)
BLM Drawing Parallel Lines (p E-44)
Introduce parallel lines—straight lines that never intersect, no matter how
much they are extended. Show how to mark parallel lines with the same
number of arrows.
Have students identify parallel lines in several diagrams. Then have students
identify and mark parallel sides of polygons. Include polygons that have
pairs of parallel sides that are neither horizontal nor vertical.
Introduce the symbol || for parallel lines, label the vertices of the polygons
used above, and have students state which sides are parallel using the new
notation (EXAMPLE: AB || CD).
Ask students to think about where they see parallel lines. Some examples of
parallel lines in the real world are a double centerline on a highway and the
edges of construction beams.
p
?
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
m
process assessment
Workbook Question 8
[R, C], 7m2
n
Determining if two lines are parallel. Have students draw a triangle on grid
paper. Then ask them to draw line segments that are parallel to the sides
of the triangle. For each pair of parallel lines (say, m and n) ask students
to draw a perpendicular (say, p) to one of the lines (m) so that it intersects
both lines. Ask students to predict what the angle between n and p is. Ask
students to explain their prediction. Then have them measure the angle
between the lines. Was the prediction correct?
Have students check the prediction using the other two pairs of parallel lines
they drew. Explain that this property—If one line in a pair of parallel lines
interests a third line at a right angle, the other parallel line also makes a right
angle with the same line—allows us to check whether two lines are parallel
and to construct parallel lines.
Drawing a line segment parallel to a given line segment. Have students
problem-solve how to construct parallel lines using what they’ve just learned
about a perpendicular to parallel lines. As a prompt, you could use the
Geometry 7-6
E-17
process assessment
[PS], 7m1
rectangle and square problems from G7-4 (Extra practice and Bonus,
p E-12—students are required to draw a rectangle or a square using a point
and a pair of parallel lines). ASK: What do you know about the sides of a
rectangle? How does constructing a rectangle mean that you constructed
a pair of parallel lines?
Model drawing the line parallel to a given line through a point using a
protractor (see p 111 in the Workbook or BLM Drawing Parallel Lines), and
then model doing the same thing using a set square. Have students practise
drawing parallel lines using both tools.
ACTIVITIES 1–2
1. Paper folding. Draw a line dark enough so you can see it through
the page. Fold the paper so that you can find a line perpendicular to
AB that does not bisect AB. How can you use this crease to find a line
parallel to AB? How can you use a ruler or any right angle to find a line
parallel to AB?
ANSWER: Fold the paper so that the perpendicular to AB falls onto itself.
The crease is perpendicular to the perpendicular, so it is parallel to AB.
2. Students can investigate distances between parallel lines with BLM
Distance Between Parallel Lines.
Extensions
process assessment
7m1, [PS]
2. Draw:
a) a hexagon with three parallel sides
b) an octagon with four parallel sides
c) a heptagon with three pairs of parallel sides
d) a heptagon with two sets of three parallel sides
e) a polygon with three sets of four parallel sides
f) a polygon with four sets of three parallel sides
Sample ANSWERS:
a)
E-18
b)
c)
Teacher’s Guide for Workbook 7.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
1. A plane is a flat surface. It has length and width, but no thickness.
It extends forever along its length and width. Parallel lines in a plane
will never meet, no matter how far they are extended in either direction.
Can you find a pair of lines not in a plane that never meet and do
not intersect?
d)
e) f)
3. Lines are parallel if they point in the same direction—that’s why we use
arrows to show parallel lines! We can regard direction as an angle with a
horizontal line. For example, if two lines are both vertical, they both make
a right angle with a horizontal line, and they are parallel. The choice of
a horizontal line as a benchmark is arbitrary—it is just a convention; any
line could be used for that purpose. Indeed, any two lines perpendicular
to a third line are parallel.
Students can complete the Investigation on BLM Properties of Parallel
Lines (p E-45) to learn what happens when parallel lines meet a third
line at different angles.
4. Ask students if they can draw a parallelogram that’s not a rectangle
using only a ruler and a set square. SOLUTION:
Step 2: Draw two of the
parallel sides using the triangle.
Step 3: Use the
ruler to complete
the figure.
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Step 1: Draw
one side of the
parallelogram.
Geometry 7-6
E-19
G7-7 Angle Relationships
Pages 112–113
Goals
Curriculum
Expectations
Ontario: 7m3, 7m7, 7m46,
7m47
WNCP: 6SS2, 7SS1,
[C, R, V]
Students will find angles in triangles, discover the sum of the angles
in a triangle, and use this sum to solve problems.
PRIOR KNOWLEDGE REQUIRED
Can use a protractor to measure angles
Materials
BLM Sum of the Angles in a Triangle (pp E-47–E-48)
Vocabulary
straight angle
acute, obtuse, right angle
adjacent angles
intersecting lines
linear pair
A straight angle is formed when the arms of the angle point in exactly
opposite directions and form a straight line through the vertex of the angle.
Adjacent angles share an arm
and a vertex.
a
c
a
b
f
d
a
b
Two adjacent angles are a
linear pair if, together, they form
a straight angle.
Have students name all the pairs of adjacent angles in the picture at left.
b
ASK: Which angles make a linear pair? Which two combinations of angles
make a straight angle?
1 cup
3
=
1 cup
2
5 cup
6
Point out that the same procedure applies to capacities, volumes, and areas.
4 cm3 + 2 cm3 = 6 cm3
45°
30°
E-20
What happens with angles? Their measures are also added: the measure
of the large angle at left is 30° + 45° = 75°.
Sum of the angles around a point. Ask students to draw a pair of
intersecting lines, measure the angles, and write the measures on the
picture. Then ask them to add up the measures. What is the sum of the
angles around the point? Did they all get the same answer? Show the
Teacher’s Guide for Workbook 7.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
+
Angle measures in adjacent angles add to the measure of the large
angle. Draw a line segment divided into two smaller segments. Mark the
lengths of the smaller line segments. ASK: What is the length of the whole
line segment? How do you know? What do you do with the lengths of the
smaller line segments to obtain the length of the whole line segment?
53°
127°
53°
128°
picture at left (mention that these angles were made by two intersecting
lines) and ask students how they can tell that whoever measured the angles
made a mistake. (Any two adjacent angles in this picture make a linear pair,
so their sums should all be 180°).
Sum of the angles in a triangle. Have students complete the Investigation
on BLM Sum of the Angles in a Triangle. What is the sum of the angles in
a triangle? Then have students draw several triangles, measure their angles,
and add the measures. Did they get the same sum every time? Explain
that the sums might not add to 180° due to mistakes in measurement. The
protractor, though a convenient tool, is imprecise.
Finding the measure of the angles using the sum of the angles in a
triangle. Draw a triangle on the board and write the measure of two of
the angles in the triangle. ASK: How can I find the measure of the third
angle? (180° minus the sum of the other two angles) Have students find the
measures of the angles in several problems of this sort, then proceed to
more complicated questions, such as the following:
process Expectation
Reflecting on the
reasonableness of
the answer
• All the angles in a triangle are equal. What is the size of each angle?
•O
ne angle of a triangle is 30°. The other two angles are equal. What is
the size of these angles?
• A triangle has two equal angles. One of the angles in this triangle is
90°. What are the sizes of the other two angles?
With the last question, ask whether the equal angles can be 90° each.
Have students explain why this is not possible. (The two angles would
already add to 180°, leaving no room for the third angle.) Invite a volunteer to
draw the correct triangle on the board and mark the measures of the angles.
Then present a similar problem:
• A triangle has two equal angles. One of the angles in this triangle
is 50°. What are the sizes of the other two angles?
ASK: How is this problem different from the previous problem? (The given
process Expectation
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Reflecting on other ways
to solve the problem
process assessment
Workbook, Question 11
[C, V], 7m7
angle is an acute angle, not a right angle.) Can a triangle have two angles of
50°? What is the third angle then? (80°) Draw an acute isosceles triangle on
the board and ask volunteers to mark the angles on the picture. Then draw
another acute isosceles triangle, mark the base angles as equal, and mark
the unequal angle as 50°. Can this situation happen? What are the measures
of the other two angles?
Extra practice:
a) If half an angle is 20°, the whole b) If one-third of an angle is 30°, angle is ______°.
the whole angle is ______°.
20°
Geometry 7-7
30°
E-21
c) What is half of 90°? ______°
d) If one-quarter of an angle is 25°, the whole angle is ______°.
25°
?
Bonus
What is one-third of 120°? ______°
ACTIVITIES 1–2
1. Create right angles by paper folding
Give your students a piece of paper that has no corners (e.g., an oval
or a cloud shape). Ask students to create a right angle on the paper by
folding. (POSSIBLE ANSWER: Fold once, unfold to see your straightline crease, then fold again across the crease so that the sides of the
crease coincide.)
ASK: When you folded the paper the first time, what is the size of the
process assessment
7m7, [C]
angle you created? (180°, a straight angle) What fraction of the straight
angle is the right angle? (one half) Use your answers to these two
questions to explain why your angle is a right angle. (ANSWER: A right
angle is 90°, so two right angles are 180°, or a straight angle. A right
angle is exactly half of a straight angle. In other words, when you fold
the paper the second time and the halves of the first crease coincide,
you know that the two angles you have created are equal. Together,
the angles make a straight angle, so they have to be right angles.)
2. Start with a cloud- or oval-shaped piece of paper. How can you fold
the paper to create the following?
Students can work independently or in pairs to try and create the angle
and shapes listed. Invite students to share strategies and solutions
with the class, then, if necessary, give students the correct sequence of
steps to follow. Do not unfold the paper unless it is mentioned as part
of instructions. The instructions might be easier to understand if you
actually perform the folding as you go along.
ANSWERS:
a) Make a 45° angle:
Create a right angle by the method of Activity 1. Fold the right angle
through the vertex so that its arms fall onto each other. This divides the
angle into two equal parts, so the angle is a 45° angle.
E-22
Teacher’s Guide for Workbook 7.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
a) a 45° angle
b) a square
c) a rectangle
b) Make a square:
i) Fold the paper to make
a line segment. Fold the line
segment onto itself to create
a right angle; the vertex of
the right angle will be one of
the vertices of the square.
ii) Fold the right angle in
two to make a 45° angle.
Sides of square
iii) Fold the corner of the
45° angle onto one of the
arms, so that parts of that
arm meet each other. Trace
the fold you created with
a pencil.
iv) Unfold once and trace
the crease that is revealed
with a pencil.
v) Unfold once again and flip
the paper over. The lines you
drew form two more sides of
the square.
c) Make a rectangle:
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Geometry 7-7
i) Repeat step i) from part b).
Sides of rectangle
E-23
ii) Choose a point on one of
the arms of your right angle
to be the second vertex of
the rectangle. Fold the paper
through the chosen point so
that the arm of the right angle
you created folds onto itself.
This will be the third side of the
rectangle.
Second corner
of the rectangle
Third side of
the rectangle
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
iii) Flip the paper over. Fold
the bottom part of the strip
backwards so that both creases
forming the sides of the strip
fold onto themselves.
E-24
Teacher’s Guide for Workbook 7.1
G7-8 Triangle Properties
Pages 114–116
Curriculum
Expectations
Ontario: 7m3, 7m7, 7m47,
7m48
WNCP: 6SS4, review,
[C, CN, R, V]
Goals
Students will classify triangles by side and angle measures.
PRIOR KNOWLEDGE REQUIRED
Knows the sum of the angles in a triangle
Can draw triangle using given sides and angles
Materials
Vocabulary
straight angle
acute, obtuse, right angle/triangle
scalene, isosceles,
equilateral triangle
dice, spinners, rulers, and protractors (see Activity 1)
rulers, scissors, straws (see Activity 3)
Do Investigations 1 and 2 as a class.
After the second investigation, have students work in groups of three.
Each student has to draw three triangles using a protractor:
Student 1
process assessment
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Worksheet, Question 7
[C, R], 7m3, 7m7
Student 2
Student 3
Triangle 1
two angles of 60° two angles of 60° two angles of 60°
Triangle 2
two angles of 30° two angles of 40° two angles of 25°
Triangle 3
two angles of 55° two angles of 75° two angles of 70°
Ask students to find the measure of the third angle in each triangle and
to classify the triangles they created by their angle measures. Then ask
students to cut the triangles out and fold them, so as to check whether
they have any equal sides. Compare the findings in each group. Students
should understand that when two angles are equal, there will be two equal
sides, regardless of the measure of the third angle. If all three angles are
What should the measure
equal, there will be three equal sides. Bonus
of the two equal angles should be in order for the triangle to be an isosceles
right triangle?
ACTIVITIES 1–2
1. Each student will need a die, a ruler, a protractor, and a spinner
divided into three parts and labelled with the types of triangles they
have been studying (acute triangle, obtuse triangle, right triangle).
The object of the game is to build a set of triangles around any vertex
that fills all 360°.
Students start with a horizontal line of 5 cm. They roll the die and
spin the spinner each turn. The spinner gives the type of triangle to
be constructed. The result of the die multiplied by 10 gives the size of
one angle of the triangle. (The other angles should also be multiples
of 10°.) Students have to draw the first triangle using the 5 cm line as
the base. Each new triangle should use one of the sides of an existing
Geometry 7-8
E-25
triangle as a side. The position of the angles is up to the student.
The triangles cannot overlap. Students may find it convenient to write
the sizes of the angles on the triangles they draw.
Let students know that there is one combination of the results of the
spinner and the die that makes it impossible to draw a triangle with
angles that are all multiples of 10°. Ask students to figure out what
combination that is. (1 and acute) When this combination is rolled,
students have to roll again.
60°
20°
2
80°
60°
3
30° 4
80°
1
50°
50°
70°
60°
5
70°
SAMPLE GAME: This game started with 5 and acute triangle, which
gave Triangle 1. The next roll was 2 and the next spin was acute
triangle, so Triangle 2 had to be isosceles with angles of 80° at the
base. The student decided to put the 80° angle next to the 50° angle.
The next two turns were 6, right triangle (Triangle 3) and 1, right
triangle (Triangle 4), and the game ended with 6, acute triangle
(Triangle 5). In the last turn, the 60° angle did not fit in the remaining
50° gap, but a triangle with 50° and 60° angles is still acute, so the
student used the 50° angle to fill the gap and end the game. If the final
spin had given obtuse triangle instead of acute triangle, the student
would have a choice of drawing the next triangle either around a
different vertex or place the smallest angle of the triangle (which could
be only 10° or 20°) in the remaining gap. In any case the game would
have continued.
2. Paper folding
First, create a square from a rectangular sheet of paper:
a) Fold the short side of the paper onto the long side (to create a
right trapezoid with a 45° angle and a triangle with a 45° angle).
b) Fold the extra part of the page—a rectangle—over the triangle.
c) Unfold the paper and cut off the rectangle.
unfold
a) Fold the square in half (vertically, not diagonally) so that a crease
divides the square into two rectangles.
b) Fold the top right corner of the square down so that the top right
vertex meets the crease. The vertex should be slightly above the
bottom edge.
c) Mark the point where the corner meets the crease and trace a line
along the folded side of the square with a pencil.
d) Unfold.
E-26
Teacher’s Guide for Workbook 7.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Now, create a triangle:
e) Repeat steps b) – d) with the top left corner of the square. The
corner will meet the crease at the same point, which is the vertex of
your triangle.
f) Cut the triangle out along the traced lines. Which triangle have your
created? Explain.
30˚
30˚
30˚
60˚
ANSWER: You started with a square and ensured in b) that the sides
of the triangle are equal. It is an equilateral triangle.
3. The triangle inequality. Give students a ruler, scissors, and some
straws. Have students measure and cut a set of 10 straws:
• one straw of each of the following lengths: 10 cm, 9 cm, 8 cm,
6 cm, 3 cm
• two straws of 4 cm
• three straws of 5 cm
Questions:
a) How many (distinct) right triangles can you make using the straws?
ANSWER: 2 distinct triangles with sides of length 3, 4, 5 and 6, 8, 10.
b) How many isosceles triangles can you make?
ANSWER: 9 triangles with the following side lengths:
3, 4, 4
8, 5, 5
3, 5, 5
9, 5, 5
4, 5, 5
5, 5, 5
5, 4, 4
6, 4, 4
6, 5, 5
Note that equilateral triangles are also isosceles – this is a
special case.
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
c) Why can you not make triangles with side lengths 8, 4, 4; 9, 4, 4;
10, 4, 4; and 10, 5, 5? If you had two straws of length 3 cm, could you
make a triangle with sides 3 cm, 3 cm, and 6 cm? If you had two straws
of length 6 cm, could you make a triangle with sides 3 cm, 6 cm, and
6 cm?
(This question provides a hint for Question 7i) on the worksheet. If
students have trouble producing the triangle in 7i), they should create
the triangle with straws and trace it.)
Triangle inequality
a
b
c
a+b>c
Geometry 7-8
Finish the activity by discussing this question: What could be a rule
for determining the sets of straws that will make a triangle and those
that won’t?
The rule is known as the triangle inequality. It says that for three lengths
to make a triangle, the sum of the lengths of any two sides must be
greater than the length of the third side.
E-27
G7-9 Angle Bisectors
Page 117
Curriculum
Expectations
Ontario: 7m2, 7m5, 7m47,
7m48
WNCP: 6SS4, 7SS3,
[C, CN, PS, R, V]
Goals
Students will identify and draw angle bisectors.
PRIOR KNOWLEDGE REQUIRED
Can divide 2-digit and 3-digit numbers by 2
Can double 2-digit numbers
Can use a protractor for measurement and construction
Vocabulary
straight angle
acute, obtuse,
right angle/triangle
scalene, isosceles, equilateral triangle
Materials
dice
Introduce angle bisectors. An angle bisector is a ray that cuts an angle
exactly in half, making two equal angles. Point out that an angle only has
one bisector. Ask students to think about where they see angle bisectors
in the real world. For example, angle bisectors are often seen in corners on
furniture and picture frames.
Explain that we can usually use an equal number of small arcs or lines
to show that the angles in a diagram are equal. Show the diagrams below
and ASK: Which diagrams look like they show an angle bisector? Invite
volunteers to mark the angles that look like equal angles.
Draw several bisected angles on the board and give the measure of one
angle (as in Question 2 on the worksheets). Have students determine the
measure of the second angle and of the whole (unbisected) angle. Then do
the reverse—provide the measure for the whole angle and have students
determine the measures of the parts.
a)
Find the size of each of the equal angles.
b)
80°
60°
Constructing angle bisectors. Draw an angle on the board, then have a
volunteer measure it and write the measurement. ASK: If you were to draw
an angle bisector, what would be the degree measure of each half? Model
drawing the bisector. Emphasize that the bisector lies between the arms
of the angle. Have students practise bisecting different angles, including
E-28
Teacher’s Guide for Workbook 7.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Bonus
process assessment
Worksheet, Question 6
7m2, [R,C]
Worksheet, Question 7
[V]
obtuse, right, and acute angles. (You might use Activity 1 at this point.)
Ask students to identify both the angles they drew and the halves as acute
or obtuse angles. Can they get two obtuse angles after bisecting an angle?
(no) Why not? (Because double the obtuse angle is more than 180°. You
might point out that such angles are called reflexive angles.) If students need
a prompt, have them bisect a straight angle or draw two equal obtuse angles
with a common arm.
Extra practice:
Draw a scalene triangle and bisect each of the angles. If the bisection is
performed correctly, the bisectors should meet at the same point.
ACTIVITIES 1–2
1. Students will each need two dice of different colours. If two dice are
unavailable, students can roll one die twice, taking the first roll as the
result on the red die (r) and the second roll as the result on the blue die
(b). Students roll the dice and draw an angle with the degree measure
equal to 25r + b. (EXAMPLE: If you roll 2 on the red die and 6 on the
blue die, you draw an angle of 56°.) Then they bisect the angle.
process Expectation
Connecting (algebra)
Bonus
What is the largest angle you can create with this rule?
(156°) What is the smallest angle? (26°) How many ways can you create
an angle that is a multiple of 5? (6 ways: 25r is a multiple of 5 for any r,
so to get the whole sum to divide by 5 you need b to be 5.)
2. Paper folding. Make an angle using folds. To bisect your angle, fold
the paper through the point of intersection so that the creases that form
the two arms of the angle fall one on top of the other.
Extensions
process assessment
[CN, PS], 7m5
1. You are given an angle and a transparent mirror (a Mira). How can you
find the angle bisector? Hint: An angle bisector is a line of symmetry.
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
2. The incentre of a triangle
Geometry 7-9
a) Draw a triangle.
b) Draw the angle bisectors for all the angles of the triangle. What do you notice? (The bisectors all pass through the same point, called the incentre of the triangle.)
c)
Label the point where the bisectors intersect O. Draw perpendiculars through O to each of the sides of the triangle. Measure the distances along the perpendicular lines from O to the sides of the triangle. What do you notice? (the distances are all the same)
E-29
G7-10 Quadrilateral Properties
Pages 118–119
Curriculum
Expectations
Ontario: 7m2, 7m47, 7m48
WNCP: 5SS6, 6SS5, [R, V]
Goals
Students will investigate properties of quadrilaterals related to angles
and sides and sort quadrilaterals according to these properties.
PRIOR KNOWLEDGE REQUIRED
Vocabulary
acute, obtuse, right angle
equilateral
special quadrilaterals:
trapezoid, parallelogram,
rectangle, rhombus,
square, kite, right
trapezoid, isosceles
trapezoid
opposite sides
Is familiar with special quadrilaterals
Can identify equal sides and angles
Can identify and mark parallel lines
Can identify and mark right angles
Materials
BLM Quadrilaterals (pp E-52–E-53)
BLM Straw Quadrilaterals (p E-49)
straws
Introduce quadrilaterals (polygons with four sides). Explain that some
quadrilaterals have special properties. This means that they have certain
attributes that apply to all of the quadrilaterals of that type. Ask students
which special quadrilaterals they know and what special properties these
quadrilaterals have. Introduce any special quadrilaterals that are not
mentioned (see Vocabulary). Make sure students understand the meaning
of opposite sides and adjacent sides.
Properties of sides Shapes with Properties of angles Shapes with
the property
the property
E-30
No equal sides
No equal angles
One pair of equal
sides
One pair of equal
angles
Two pairs of equal
sides
Two pairs of equal
angles
Equal sides are
adjacent
Equal angles are
adjacent
Teacher’s Guide for Workbook 7.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Sort quadrilaterals by properties of sides and angles. Give your
students some paper parallelograms, rhombuses, rectangles, squares, and
trapezoids (see BLM Quadrilaterals for a sample of shapes). Discuss with
students how they can check for equal sides or angles using folding, and
check whether angles add to 180° using rolling or folding (the example
at right shows how to check that for a parallelogram). Review with students
how they can check whether sides are parallel by drawing (or folding) a
perpendicular to one of the sides. If the perpendicular makes a right angle
with the second side as well, the sides are parallel. Ask students to sort the
shapes into this table:
Equal sides are
opposite
Equal angles are
opposite
Four equal sides
Four equal angles
One pair of parallel
sides
No pairs of angles
add to 180°
Two pairs of parallel
sides
One pair of angles
add to 180°
Two pairs of angles
add to 180°
Four pairs of angles
add to 180°
Process expectation
Making and investigating
conjectures
Ask students to look at the table closely. Are there any properties that go
together? For example, all quadrilaterals that have two pairs of parallel sides
also have four pairs of angles that add to 180°. Ask students to pick at least
one such pair of properties and to try to draw (using protractors and rulers)
a quadrilateral that shares one of these properties but not the other. For
some properties, such as the pair in the example above, this is impossible,
but in some cases students might be able to come up with a shape. Debrief
as a class.
EXAMPLE: “One pair of parallel sides” might seem to go “with two pairs of
angles add to 180°;” however, a kite with two right angles as shown has two
pairs of angles adding to 180° and no parallel sides.
ACTIVITY
Students can create quadrilaterals from straws and investigate them
using BLM Straw Quadrilaterals.
Extensions
process Expectation
Visualizing
1. You can build a diagram that shows relationships between different
special quadrilaterals. If a shape has all the properties of some other
shape, it is drawn inside a more general shape.
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Start with a rectangle. All squares are rectangles, so we can draw a
square as part of a rectangle:
or even
All squares are also rhombuses. Let’s draw a square as part of
a rhombus:
Geometry 7-10
E-31
Can you find a shape that is a rectangle and a rhombus at the same time
but is NOT a square? No. Let’s show that—a square is the intersection of
the rhombus and the rectangle:
All rectangles and rhombuses are also parallelograms, so we can draw
the shape from the last step inside a parallelogram:
A rhombus is a parallelogram that is also a kite. Can you find a
parallelogram that is also a kite but NOT a rhombus? No, so the
rhombus is the intersection of a kite and a parallelogram.
2. For 3 lengths to make a triangle, the sum of any 2 lengths must be longer than the third length.
Process expectation
Making and investigating
conjectures
A. Predict the rule that describes when 4 lengths will make a
quadrilateral and when they will not.
B. Test your prediction. Try to make a quadrilateral using 3 small
paperclips and a long pencil as the sides. Did it work? Why or why not?
D. Check off the correct ending for the statement.
For 4 lengths to make a quadrilateral,

2 sides must be equal in length.

any 3 sides must together be longer than the fourth side.

each of 2 sides must be longer than the other 2 sides.

the sum of 2 sides must be longer than the sum of the
other 2 sides.
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Teacher’s Guide for Workbook 7.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
C. Cut various lengths from straws and do some more tests. Sketch
the results of your tests.
Process expectation
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Connecting
3. Draw a parallelogram. Bisect each angle of the parallelogram and
extend each bisector so that it intersects two other bisectors. What
geometric shape can you see in the middle of the parallelogram? (a
rectangle) Use the sum of the angles in the shaded triangle in the picture
to explain why this is so. (ANSWER: The acute angles of the triangle
are both half of the angles of a parallelogram. The adjacent angles of a
parallelogram add to 180°, so their halves add to 90°. By the sum of the
angles in a triangle, the third angle is a right angle. Since there are three
other triangles like the shaded triangle in the parallelogram, the shape
in the middle has four right angles and must be a rectangle.)
Geometry 7-10
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G7-11 Symmetry
Pages 120–121
Curriculum
Expectations
Ontario: 6m46, 6m47, 7m47; 7m1, 7m3, 7m6, 7m7
WNCP: 6SS4, 7SS3
[C, CN, R, V]
Vocabulary
line of symmetry
straight angle
acute, obtuse, right angle/triangle
scalene, isosceles, equilateral triangle
Goals
Students will identify lines of symmetry in polygons and sort
quadrilaterals by symmetry properties.
PRIOR KNOWLEDGE REQUIRED
Can identify congruent shapes
Can identify lines of symmetry
Can use a protractor for measurement and construction
Materials
BLM Regular Polygons (p E-56)
BLM Quadrilaterals (pp E-52–E-53)
BLM 2-D Shapes Sorting Game (pp E-54–E-55)
MIRAs (optional)
Line symmetry and line of symmetry. A polygon has line symmetry,
or reflection symmetry, if you can fold it in half along a line so that the two
halves match exactly. The folding line is called a line of symmetry. A line
of symmetry divides the shape into two equal halves. Students might be
familiar with the definition from earlier grades.
Show a parallelogram with a diagonal drawn in and ask whether this is a
line of symmetry. Why or why not? Repeat with the diagonal of a rectangle.
Regular polygons. Explain to your students that a regular polygon has
all sides and all angles equal. ASK: Which triangle is a regular triangle?
(equilateral) A rhombus has all sides equal. Draw a rhombus that is not
a square. Is it a regular polygon? (no) Why not? (angles are not equal)
Can a polygon have equal angles but be not regular? (yes: a rectangle,
for example, has equal angles but not equal sides) Which quadrilateral is
regular? (square)
Lines of symmetry in regular polygons. As an alternative to Investigation
1 on the worksheet, give each student a set of paper regular polygons
(triangle, square, pentagon, and hexagon—see BLM Regular Polygons)
and have students fold the shapes to find the lines of symmetry in each.
How many lines of symmetry does each shape have? Is there a relationship
between the number of edges and the number of lines of symmetry in a
regular polygon? (yes, they are equal) As an alternative, students can use
Miras to check for lines of symmetry.
E-34
Teacher’s Guide for Workbook 7.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Ask students to draw a line of symmetry in a rectangle. ASK: Can you draw
a line of symmetry for a parallelogram? (only if it is a rhombus or a rectangle,
or both)
Types of lines of symmetry. Every line of symmetry in a polygon is one
of these:
• a bisector of two
opposite angles
• a perpendicular
bisector of two
opposite sides
• an angle bisector,
and a perpendicular
bisector of the opposite
side
Some polygons have one line of symmetry, some have none, and some
have more than one.
Process expectation
Looking for a pattern
process assessment
Worksheets, Investigation
1 C – [C], 7m7
Have students identify the types of lines of symmetry in regular polygons.
Can students see a pattern? Ask them to try to explain why there is such a
pattern. ANSWER: When the number of sides is even, there are opposite
sides and opposite vertices. A line of symmetry then passes through
opposite sides (bisecting them perpendicularly) or through opposite
vertices, bisecting the angles. When the number of sides is odd, there
are no opposite sides or vertices; each vertex has a side opposite to it,
and the line of symmetry bisects both.
Ask students to draw a number of different triangles and to classify them
by the number of line of symmetry. What do they notice?
ANSWER:
• Equilateral triangles have 3 lines of symmetry.
• Isosceles triangles have 1 line of symmetry.
• Scalene triangles have no lines of symmetry.
Give students the quadrilaterals from BLM Quadrilaterals. Ask them to find
the lines of symmetry by folding (or using Miras). Then ask students to sort
the quadrilaterals by the number of lines of symmetry. They can also sort the
quadrilaterals by the types of symmetry lines using Venn diagrams:
1. Symmetry lines that bisect angles
2. Symmetry lines that perpendicularly bisect sides
ANSWER: The only quadrilaterals in the central zone of the Venn diagram
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
will be squares. Shapes in 1: kites and rhombuses. Shapes in 2: rectangles
and isosceles trapezoids.
ACTIVITIES 1–3
For each Activity, students will need a set of quadrilaterals (they can use
some shapes from Activity 1 and the quadrilaterals from BLM Quadril­
aterals and the property cards from BLM 2-D Shapes Sorting Game
1. Each student flips over a property card and then sorts his or her
shape cards into two piles: those that have that property and those that
Geometry 7-11
E-35
do not. If you prefer, you could choose a property and have everyone
sort their shapes using that property. Students can also choose two
property cards and make a Venn diagram using the two properties.
2. Students can play this analogue of Solitaire:
Process expectation
Connecting, Visualizing
Shuffle the shape and property cards together. Deal 5 piles of cards
on the table, face down, so that each pile has one more card than the
last pile—1 card in the leftmost pile, 2 cards in the next pile, and so on.
Leave the rest of the cards in a spare pile. Turn over the topmost card
in each pile.
The object of the game is to get all the cards into columns. The shape
and property cards will alternate: each shape will have to satisfy the
properties above and below it, and each property will have to be found
in the shapes above and below it. You can move parts of any column
to any other column.
If you cannot move any of the cards that are face up (as in the
illustration of the game below), take a card from the spare pile and try
to add it to a column. It the card doesn’t fit anywhere, take another card
from the spare pile, place it on top of the card that you could not play,
and try again. You can use only the topmost card of the new spare pile
in front of you at any time. When you have turned over all the cards
in the original spare pile, turn the new spare pile face down and start
again. The game ends when you have placed all the cards or when you
cannot place any more cards.
13
14
10
No obtuse
angles
2
Not equilateral
18
11
9
No right
angles
12
All angles
equal
8
Spare pile
12
At least one
reflexive angle
Cards that
could not be
played
3. Turn over a property card. Sort the shape cards into those that
have that property and those that do not (discard pile). Then turn over
a second property card and sort the shapes from the discard pile
according to the property on the second card. Repeat until the discard
pile disappears (in which case you win) or you run out of property
cards (in which case you lose).
Play again using two property cards at a time: a card goes into the
discard pile unless it satisfies both properties. Play a third round using
three property cards together at a time.
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Teacher’s Guide for Workbook 7.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
16
Equilateral
Discuss with the students when it is easiest and when is it hardest to
win. ANSWER: It is easiest to win with one property card and hardest
with three cards; the discard pile is larger the more property cards you
have at a time. Some properties might even be contradictory, in which
case all cards go into the discard pile, e.g., no quadrilateral can have
all angles equal (so be a rectangle) and have no lines of symmetry at
the same time.
Process expectation
Reflecting on what made
the problem easy or hard
Extension
Lines of symmetry and reflections
process assessment
[C], 7m7
60°
60°
a) Can you draw a triangle that is isosceles but not equilateral and has
an angle of 60°? Explain. ANSWER: No.
EXPLANATION: There are two cases: The 60° angle is either between
the equal sides or at the base. If it is between the equal sides, the other
two angles are equal. Since all three angles add to 180°, the equal
angles together make 120°, which means they are 60° each, so the
triangle is equilateral.
If the 60° angle is at the base, the other angle at the base is also 60°,
so together they add to 120°. Since the sum of the angles in a triangle
is 180°, the third angle must also be 60°.
b) Draw a triangle with an angle of 60° and each of the following properties:
i) equilateral
ii) acute but not equilateral
iii) right-angled
iv) obtuse
c) In each triangle from b), extend one of the sides that is adjacent to the
60° angle. Reflect the triangle through that side. Look at the shape the
triangle and the reflection form together. Which special quadrilateral or
triangle do you see? ANSWER:
i) ii)
60°
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
60°
iii)
60°
60°
60°
60°
iv)
60° 60°
60°
60°
60°
60°
Geometry 7-11
E-37
d) In each picture from c), reflect the original triangle through the other side
adjacent to the 60° angle. Look at the shape produced by the triangle
and both reflections. How many lines of symmetry does it have? What is
special about the triangle that produced a symmetrical shape?
ANSWER: Only one of these figures has a line of symmetry, and the
equilateral triangle that produced it was the only one that had a line of
symmetry from the beginning.
e) Add a third mirror line, so that there are three lines dividing 360° into
six equal angles. Reflect the shapes in this mirror line as well. How many
lines of symmetry does each figure have now?
ANSWER and EXPLANATION: Only the figure that was produced using
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
an equilateral triangle has 6 lines of symmetry, because each line of
symmetry of the figure (that passes through the same vertex as the
mirror lines used in its construction) when reflected in the mirror lines,
produced more lines of symmetry. The other figures have 3 lines of
symmetry—the mirror lines used in their construction.
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Teacher’s Guide for Workbook 7.1
G7-12 Making a Sketch
Pages 122–126
Goals
Curriculum
Expectations
Students will use logic, symbols, and all of the geometric properties and
relationships they have learned to make quick and accurate sketches.
Ontario: 7m1, 7m2, 7m3, 7m5, 7m6, 7m7
WNCP: [PS, C, R, V, CN]
PRIOR KNOWLEDGE REQUIRED
Can identify equal sides, angles, and right angles
Understands properties of triangles and quadrilaterals based on
number of sides, angles, symmetry, and parallel sides
Can name angles and polygons; can identify a named angle or polygon
Vocabulary
acute, obtuse,
right angle/triangle
scalene, isosceles, equilateral triangle
special quadrilaterals: trapezoid, parallelogram, rectangle, rhombus, square, kite, right trapezoid, isosceles trapezoid
Explain to your students that a sketch is a quick drawing made without
using tools such as a ruler or protractor. Knowing how to make a sketch
is an important math skill. Sketches can help us organize information,
see relationships, and solve problems.
Have students complete the worksheets one page at a time. Stop, if and
when necessary, to discuss or take up specific questions as a class. You
can use the questions below as extra practice for students who work more
quickly than others (to keep the class working through the questions at
approximately the same pace) or for students who are struggling with a
particular concept or step.
process assessment
Extra practice for page 122:
Worksheets:
Question 10 – [R, V], 7m3
Question 11 – [CN], 7m5
Questions 14, 15 – [R, V], 7m1, 7m6
Question 16 – [C], 7m7
1. Sketch a square and all its lines of symmetry.
2. Circle the better sketch.
a)
b)
c)
d)
60°
60°
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
3.
Sketch the figures.
a) a line segment AC with midpoint B
b) a rectangle 2 cm by 6 cm
c) a rhombus with angles 20° and 160°
d) DKLM with sides 5 cm, 3 cm, and 3 cm
Extra practice for page 123:
1. Add the information that is not on the sketch.
a) BD is the line of symmetry in triangle DABC with ∠A = 50°.
B
A
Geometry 7-12
C
E-39
b) E is the midpoint of AD. BE is perpendicular to the side AD of a parallelogram ABCD. BE = 3 cm.
B
C
3 cm
A
E
D
2. Make a sketch for each problem. Ignore the unnecessary information.
a) A spinner is made of six triangles, three red and three blue,
sharing a common vertex. The red and blue triangles alternate.
All triangles are equilateral with sides of 3 cm each. What is the
shape of the spinner?
b) A traffic island has the shape of a right trapezoid with one of
the angles 65°. The island contains three shrubs and a circular
flowerbed 1 m wide. What are the sizes of the angles of the
traffic island?
Bonus
John climbs a ladder to the attic window. The ladder is propped
against the wall, and the foot of the ladder is 1 m from the wall. The ladder
is made of two pieces and is 4.5 m long in total. The window is 7 m from the
ground. Can John reach the window from the ladder?
Extra practice for page 124:
1. An isosceles triangle has sides of 5 cm and 3 cm. What is the perimeter of the triangle? Make two different sketches to show that the relative position of sides in the triangle matters.
B
D
Tower base
35 m
C
a) BD is the line of symmetry in isosceles triangle ABC with ∠A = 50°.
What is the size of CBD?
b) A city tower is a rectangular prism completely covered with glass panels. The base of the building is a rectangle 35 m by 24 m. To
clean the glass panels, workers use a platform that is 6 m long. How
many times will the workers have to move the scaffold up and down
to clean the entire building? NOTE: Explain to the students that the
platform can be shifted sideways at any height without moving the
platform up or down, but not around the corner.
Extra practice for page 125:
1. Sketch each quadrilateral. Add all necessary labels and side and
angle markings.
a) ABCD is a parallelogram with sides of length 3 m and 5 m and
∠A = 40°.
b) KLMN is an isosceles trapezoid with three sides that are 3 m long and the fourth side 5 m long.
E-40
Teacher’s Guide for Workbook 7.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
A
2. Add other information you can deduce to the sketch. Then solve
the problem.
c) PQRS is a kite with sides of length 2 cm and 5 cm and two
right angles.
d) WXYZ is a rectangle with ∠ZXY = 30°.
process assessment
Solve each problem by making a sketch.
[PS], 7m2
2. The shortest side of a parallelogram is 5 cm. The longest side is 2 cm longer than the shortest side. What is the perimeter of the parallelogram?
3. ABC is an isosceles triangle with one of the angles 100°. What are the
sizes of the other angles?
4. A square is cut into two identical parts and rearranged to make a
rectangle. The short side of the rectangle is 6 cm. How long is the long
side of the rectangle?
5. A square is cut into two identical parts and rearranged to make a
triangle. What are the angles of the triangle?
6. A rectangle is cut into two identical triangles and rearranged to make
a parallelogram. Sarah thinks the parallelogram is a rhombus. Steven
thinks it cannot be a rhombus. Who is correct—Sarah or Steven?
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
7. In DKLM, KL = LM and LP is the perpendicular bisector of KM.
∠KLP = 50°. What are the sizes of the angles of DKLM?
Geometry 7-12
E-41