Teaching and learning plan on introducing angles

Teaching and learning plan on introducing angles
Teaching & Learning Plans
Plan 7: Introduction to Angles
Junior Certificate Syllabus
The Teaching & Learning Plans
are structured as follows:
Aims outline what the lesson, or series of lessons, hopes to achieve.
Prior Knowledge points to relevant knowledge students may already have and also
to knowledge which may be necessary in order to support them in accessing this new
topic.
Learning Outcomes outline what a student will be able to do, know and understand
having completed the topic.
Relationship to Syllabus refers to the relevant section of either the Junior and/or
Leaving Certificate Syllabus.
Resources Required lists the resources which will be needed in the teaching and
learning of a particular topic.
Introducing the topic (in some plans only) outlines an approach to introducing the
topic.
Lesson Interaction is set out under four sub-headings:
i.
Student Learning Tasks – Teacher Input: This section focuses on teacher input
and gives details of the key student tasks and teacher questions which move the
lesson forward.
ii.
Student Activities – Possible and Expected Responses: Gives details of
possible student reactions and responses and possible misconceptions students
may have.
iii. Teacher’s Support and Actions: Gives details of teacher actions designed to
support and scaffold student learning.
iv. Checking Understanding: Suggests questions a teacher might ask to evaluate
whether the goals/learning outcomes are being/have been achieved. This
evaluation will inform and direct the teaching and learning activities of the next
class(es).
Student Activities linked to the lesson(s) are provided at the end of each plan.
Teaching & Learning Plan 7:
Introduction to Angles
Aims
• To introduce students to the concept of an angle as a rotation and naming
an angle
• To help students to learn about various angle types and the measurement
of angles (In degrees)
• To help students to recall and/or deepen their understanding of the
concepts of parallel and perpendicular lines and horizontal and vertical
lines
Prior Knowledge
Students should understand and recall the concepts of planes, points, etc. introduced in
Teaching and Learning Plan 6. They should be familiar with the circle as a shape.
Learning Outcomes
As a result of studying this topic, students will be able to
• recognise angles in terms of rotation
• classify and describe angles
• describe the various angle types in terms of degrees
• relate angles to shape
• name an angle
• identify and name parallel and perpendicular lines, vertical and horizontal
lines
Relationship to Junior Certificate Syllabus
Sub-topics
Ordinary Level
2.1 Synthetic geometry
The geometrical results should be
first encountered through discovery
and investigation.
(Refer to the appendix on geometry)
Terms: Angle; Rotate; acute, right angle,
obtuse, straight angle, reflex angle,
ordinary angle; circle, perpendicular lines
Concepts (introduced as required)
Constructions: 1, 2, 3, 4, 5, 6, 8, 9, 10,
11
Theorems: non-rigorous treatment of
theorems 1 – 6.
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Teaching & Learning Plan 7: Introduction to Angles
Resources Required
Geostrips and geometry sets for each student
Introducing the Topic
Students may well have some misconceptions about angles, which will need to be
explored and overcome:
• the size of an angle varies with the length of the arms
• the size of an angle varies with the size of arc made with the angle vertex
as centre
• different orientations of an angle as a source of confusion
• that a right angle only exists between vertical and horizontal lines
© Project Maths Development Team 2009
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2
© Project Maths Development Team 2009
»» Demonstrate the concept
of angle by using two (long
red) Geostrips fastened
together at one end. The
strips will be on top of each
other at first, then the top
one is slowly rotated to
reveal an angle. Put major
emphasis on the ROTATION
part of this motion.
»» Start a “new words”
section on the board. Insert
the word “ANGLES”.
Teacher’s Support and
Actions
• student answer/response
»» Are students moving the
Geostrips in a way that
shows they understand the
term ‘rotate’?
Checking Understanding
KEY: » next step
»» Give each student two
»» Students fasten Geostrips
Geostrips: two of the same
(of same colour and length)
together and rotate.
colour and length and an
additional two strips of a
different colour and length
to the previous two.
Student Activities: Possible
and Expected Responses
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»» Today’s class is all about
angles. You meet angles
everywhere you go – all
sorts and sizes of angles in
buildings and rooms and
even in natural things. Let
us start by talking about
how an angle is made.
When we have two rays in
the same place in the plane
(“on top of each other”),
and we rotate (or turn) one
about some common point,
while the other stays where
it is, we get an angle, just
like these two geo-strips
here. As you can see when
we rotate one strip, we get
an angle. The angle is the
amount of rotation that is
done.
Student Learning Tasks:
Teacher Input
Lesson Interaction
Teaching & Learning Plan 7: Introduction to Angles
3
© Project Maths Development Team 2009
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»» The two rays which make
up the angle are called the
arms of the angle.
»» The key thing here is that
the angle is the rotation of
one ray about another ray
through a common point.
The angle changes as we
change the amount of
rotation.
• student answer/response
»» Are students sure what it
means for something to
rotate?
Checking Understanding
KEY: » next step
»» Draw an angle on the
board and indicate the rays
as arms of the angle, and
adds “ARMS of the ANGLE”
to the “new words” list.
»» Rotate the top Geostrip
further to make a bigger
angle, then vary the
amount of rotation to
emphasise the point.
»» Now let us rotate the strip
further; we get a bigger
angle. If we rotate more,
we get a bigger angle still.
»» Students rotate Geostrips.
»» Go to the classroom door
and open it. An angle
forms between the door
and some wall.
»» Write the word ROTATE
on the board in the “new
words” list.
Teacher’s Support and
Actions
»» What about when we open
the door? The door rotates
around the hinges, does
it not? See how the door
makes an angle with the
wall as I rotate it.
»» Can you give examples of
things that you have seen
rotating?
• Bottle top, steering wheel
of car…your arm…your
leg.
• Turn…twist.
»» Do you understand the
meaning of “rotate”?
»» Can you give me another
word meaning the same?
Student Activities: Possible
and Expected Responses
Student Learning Tasks:
Teacher Input
Teaching & Learning Plan 7: Introduction to Angles
4
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• They are the same size
because they both have
the same degree of
rotation.
»» Now have a look at these
two angles on the board.
They have the same degree
of rotation, but the arcs are
different sizes. Are these
angles the same size or is one
bigger than the other?
© Project Maths Development Team 2009
»» Students mirror with
their geostrips what the
teacher has demonstrated.
»» Can you see that the amount
of rotation of the larger strip
is the same as the amount of
rotation of the longer strip?
• student answer/response
»» Can the students explain
clearly why the two
angles are the same size
even though the arcs
are different sizes?
»» Are you satisfied that
the explanations
students give you, show
that the length of the
arms of the angle do
not dictate the size of
the angle?
KEY: » next step
»» Ask the class and give them
a moment to think before
asking one student.
»» Draw two equal angles
on the board, but insert
different sized arcs on each.
»» Invite students to agree or
disagree, but explain they
must have a valid reason for
so doing.
»» Do you agree or disagree with
this?
Teacher’s Support and Actions Checking Understanding
»» Use two shorter (yellow
for contrast) strips. Rotate
one of these and one
of the longer red strips
simultaneously to show
that the amount of rotation
and therefore angle size, is
unaffected by the length of
strip used.
Student Activities: Possible
and Expected Responses
»» Now let us see what happens • No
if we change the length of
the strip we use to make
the angle. Let us rotate the
shorter strip at the same time
as the larger. Does this change
the size of the angle?
Student Learning Tasks:
Teacher Input
Teaching & Learning Plan 7: Introduction to Angles
5
© Project Maths Development Team 2009
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• student answer/response
»» Are students actively
participating in the
discussion?
Checking Understanding
KEY: » next step
»» Write the words “RIGHT
ANGLE” on the board in
the “new words” list.
»» Brief class discussion of the
meaning of the word right
in the English language.
• Students may say because it
is…”upright”?
»» Why do you think this is
called a right angle?
»» The word “right” has many
meanings in English.
»» Ask the class and give them
a moment to think before
asking one student.
• A right angle (some may
say 90°).
»» Do you know what we call
this angle?
»» Write the word VERTEX on
the board.
»» Indicate on Geostrips
where the vertex is.
»» Move the strip to the
vertical position to show a
right angle.
• Acute appendicitis, acute
shortage etc…
»» Write the words ACUTE
ANGLE on the board.
»» Ask the class and give them
a moment to think before
asking one student.
»» Use the red (longer) strip to
show an acute angle.
Teacher’s Support and
Actions
»» Now let us rotate the strip
further.
»» Where the two arms meet
is called the VERTEX of the
angle.
»» Have you met the word
“acute” before? Where?
»» Yes. Notice how sharp the
point of the angle looks.
“acute” means “sharp”.
• Acute angle.
»» Students mirror with their
geostrips what the teacher
has demonstrated.
»» Now let us rotate the strip
further.
»» Do you know what we call
this angle?
Student Activities: Possible
and Expected Responses
Student Learning Tasks:
Teacher Input
Teaching & Learning Plan 7: Introduction to Angles
6
• No
»» Now if I were to turn the
same size angle the other
way, would I call it a “left
angle”?
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»» A reflex angle
»» Do you know what we call
this angle?
© Project Maths Development Team 2009
»» Students mirror with their
geostrips what the teacher
has demonstrated.
»» Now let us rotate the strip
further.
»» Students may not have met
this word before in other
conversation?
»» Notice how the point of
the angle looks less sharp.
“Obtuse” means “blunt”.
Have you met the word
“acute” before?
• student answer/response
»» Are any students having
difficulty with the new
words?
KEY: » next step
»» Write the words REFLEX
ANGLE on the board.
»» Ask the class; then select an
individual student to answer.
»» Move the strip to the
position to show a reflex
angle.
»» Write the words OBTUSE
ANGLE on the board.
»» Ask the class; then select an
individual student to answer.
»» Do you know what we call
this angle?
• An obtuse angle.
»» Move the strip to the
position to show an obtuse
angle.
»» Ask the class to explain why.
»» Turn the angle round so that
the right angle is on the left
as seen by the students.
Teacher’s Support and Actions Checking Understanding
»» Now let us rotate the strip
even further.
»» Students mirror with their
geostrips what the teacher
has demonstrated.
Student Activities: Possible
and Expected Responses
Student Learning Tasks:
Teacher Input
Teaching & Learning Plan 7: Introduction to Angles
7
© Project Maths Development Team 2009
• 360°
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»» This is a funny number to
have for a full circle is it
not? Well, the reason for
this goes back to the ancient
Babylonians of 2000 B.C. and
their way of measuring the
time in a day.
»» How many degrees in a full
circle?
»» Does anyone know what unit • Degrees
we use to measure angles?
»» Now we must talk of
measuring angles.
»» Look around the classroom
and identify where angles
are formed. What kind of
angles are they? Which is
the most common angle and
which is the least common
angle?
»» Students look for angles
in the room and write a
number of them into their
copies.
»» Students mirror with their
geostrips what the teacher
has demonstrated.
»» Now let us rotate the strip
further until we come back
to where we started.
»» Now we are back where we
started and we have done a
full revolution which means
that the end of the strip has
completed a full circle.
Student Activities: Possible
and Expected Responses
Student Learning Tasks:
Teacher Input
Checking Understanding
• student answer/response
»» Are students actively
participating?
KEY: » next step
»» Ask the class; then select
an individual student to
answer.
»» Ask the class; then select
an individual student to
answer.
»» Walk around to see what
»» Has everyone completed
students are writing down.
the task?
Some students may have
difficulty. As you walk
around identify and guide
those students.
»» Move the strip to the
position where it started.
Teacher’s Support and
Actions
Teaching & Learning Plan 7: Introduction to Angles
8
»» Move the strip to the
position to show a straight
angle.
»» Ask the class; then select
an individual student to
answer.
»» Students mirror with their
geostrips what the teacher
has demonstrated.
• Perpendicular lines
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»» Remember you must have at least two
lines to compare if you want to use the
word “perpendicular”.
»» Yes they are called perpendicular lines.
KEY: » next step
»» Write the words
PERPENDICULAR LINES
on the board in the “new
words” list.
»» Ask the class; then select
an individual student to
answer.
• Because at this point we
are half-way to a full circle
and 180°= ½ of 360°.
»» Why?
»» There is a name for two lines which
are at an angle of 90° to each other.
Do you know this name?
»» Move the strip to the
position to show a straight
angle.
• Because at this point we
are one quarter of the way
to a full circle and 90° =1/4
of 360°.
»» Why?
»» Ask the class; then select
an individual student to
answer.
»» Prepare to demonstrate
various angle sizes using
board and geo-strips. Move
the strip to the vertical
position to show a right
angle.
Teacher’s Support and
Actions
»» How many degrees in a straight angle? • 180°
• 90°
Student Activities: Possible
and Expected Responses
»» How many degrees in a right angle?
»» Let us look again at the strips as we
make the various angles.
Student Learning Tasks: Teacher Input
Teaching & Learning Plan 7: Introduction to Angles
• student answer/response
Checking
Understanding
9
• A vertical line
• A horizontal line
»» What is the name for a
line which goes straight up
and is perpendicular to the
ground?
»» What is the name for a
line which is level and runs
parallel to the ground?
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»» Students look for parallel
»» Walk around to see what
lines in the room and write
students are writing down.
a number of them into
Some students may have
difficulty. As you walk around
their copies.
you identify and guide those
students.
»» Can you see any lines
parallel to each other in
this room?
© Project Maths Development Team 2009
• Parallel lines.
»» What do you call two lines
which are always the same
distance apart?
• student answer/response
»» Do students understand
these words?
»» Has everyone
completed the task?
»» Has everyone
completed the task?
Checking Understanding
KEY: » next step
»» Write the word HORIZONTAL
on the board in the “new
words” list.
»» Write the word VERTICAL on
the board in the “new words”
list.
»» Ask the class; then select an
individual student to answer.
»» Write the words PARALLEL
LINES on the board in the
“new words” list.
»» Ask the class; then select an
individual student to answer.
»» Move the strips to the position
to show parallel lines.
»» Walk around to see what
students are writing down.
Some students may have
difficulty. As you walk around
identify and guide those
students.
»» Students look for
perpendicular lines in the
room and write a number
of them into their copies.
»» Can you see any lines
perpendicular to each
other in this room?
Teacher’s Support and Actions
Student Activities: Possible
and Expected Responses
Student Learning Tasks:
Teacher Input
Teaching & Learning Plan 7: Introduction to Angles
10
© Project Maths Development Team 2009
»» Complete Student Activity
2 and 3.
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»» Students fill out Student
Activity 2 and 3.
»» Students copy different ideas
to their own from the board.
• student answer/response
»» Do students understand
the words, phrases and
diagrams?
»» Did students have lots of
different suggestions?
Checking Understanding
KEY: » next step
»» Circulate and support
students who have
difficulty with the task.
»» Distribute Student Activity
2 and 3.
»» Take selections from each
group and record them on
the board/flipchart/laptop.
»» Walk around to see what
students are writing down.
If anyone is struggling, ask
questions which will give
them a hint of an example.
»» Distribute Student Activity
1.
»» Students fill out Student
Activity 1.
»» Working in pairs, complete
Student Activity 1.
Teacher’s Support and
Actions
»» Demonstrate naming an
angle on the board using
three letters in two ways,
for example <ABC or <CBA.
Student Activities: Possible
and Expected Responses
»» Finally we want to name
an angle. We name an
angle by the three letters
which give the two rays
that make up the angle, so
that the point denoting the
vertex is in the middle.
Student Learning Tasks:
Teacher Input
Teaching & Learning Plan 7: Introduction to Angles
11
© Project Maths Development Team 2009
»» Write down any question
you may have.
»» Write down anything you
found difficult.
»» Write down 3 items you
learned about angles and
lines today.
Reflection
Student Learning Tasks:
Teacher Input
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7. A line which is level and runs parallel to
the ground is called a horizontal line.
6. A line which goes straight up and is
parallel to the ground is called a vertical
line.
5. Lines which are always the same distance
apart are called parallel lines.
4. Angles are named using three letters with
the vertex in the middle.
3. Two lines that meet at 90 degrees to each
other are called perpendicular lines.
2. A full circle has 360 degrees.
1. There are many sizes of angles: acute,
right, obtuse, straight and reflex.
Student Activities: Possible and Expected
Responses
KEY: » next step
»» Circulate and take
note particularly of
any questions students
have and help them to
answer these.
Teacher’s Support and
Actions
Teaching & Learning Plan 7: Introduction to Angles
• student answer/response
Checking
Understanding
12
Teaching & Learning Plan 7: Introduction to Angles
Student Activity 1
Students work in pairs.
1. Find and list 5 surfaces in the classroom which are horizontal and 5 which are
vertical._______________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
2. See if you can find lines or surfaces which are parallel.___________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
3. See if you can find lines or surfaces which are perpendicular to each other.
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
4. Can you find perpendicular lines in the classroom where the lines are not vertical or
horizontal?_ __________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
5. What type of lines are used in the pages of your copy?___________________________
______________________________________________________________________________
______________________________________________________________________________
6. Why is it important to have walls vertical?______________________________________
______________________________________________________________________________
______________________________________________________________________________
7. Is it important to have floors horizontal? Why?__________________________________
______________________________________________________________________________
______________________________________________________________________________
8. Do you know what a builder uses to ensure that he builds a wall vertically?
______________________________________________________________________________
______________________________________________________________________________
9. What building in Italy is famous for not being vertical?___________________________
______________________________________________________________________________
______________________________________________________________________________
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13
Teaching & Learning Plan 7: Introduction to Angles
Student Activity 2
Look at the mathematical words/phrases in the middle column below and rewrite each into the
correct description box on the left and diagram box on the right.
Description
Mathematical words/
phrases
Line going directly upwards
Horizontal Line
Angle greater than 90° but
less than 180°
Acute angle
Line parallel to the ground
Right Angle
Angle of 180°
Obtuse Angle
Angle greater than 180°
Straight Angle
Angle of 90°
Reflex Angle
Angle less than 180°
Ordinary Angle (more than one
diagram)
Lines which never meet
Perpendicular Lines
Lines at an angle of 90° to
each other
Parallel lines
Angle less than 90°
Vertical line
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Diagram
14
Teaching & Learning Plan 7: Introduction to Angles
Student Activity 3
1. Name each of these angles using three letters in two ways:
Angle
Name one
Name two
2. What is a wide-angle lens in a camera? What does it do? What type of picture does it
take?_________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
3. What does it mean when we say that in football that “the goalkeeper narrowed the
angle”? What angle did the goalkeeper narrow?_________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
4. Draw out the last five letters of the alphabet in capitals. What kinds of angles are
formed in them?_ _____________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
5. What kinds of angle are formed by the hands of a clock?
(a) At 9.00p.m?________________________________________________________________
(b) At 6.00 a.m.?_______________________________________________________________
(c) At 3.00 a.m.?_______________________________________________________________
6. Through how many degrees will the minute hand of a clock rotate in:
(a) 30 minutes?________________________________________________________________
(b) 20 minutes?________________________________________________________________
(c) 40 minutes?________________________________________________________________
(d) 45 minutes?________________________________________________________________
© Project Maths Development Team 2009
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