of an iceberg is visible—the other 
lies unseen below the surface of the
water, and is a danger to ships sailing nearby.
Have you ever seen an iceberg? When the sea is calm, you can see both the top of the iceberg and its
reflection in the water. But, beauty can be deceiving! Only of an iceberg is visible—the other lies unseen below the surface of the
water, and is a danger to ships sailing nearby.
The reflection in the photograph is an example of one of the transformations in motion geometry that you will explore in this lesson. Reflections are often called flips. You will study slides, flips, and turns, and how these transformations can be used to describe relationships among geometric figures. You will also examine how these transformations are used in everyday life.
Now you will explore the meaning of slides, turns, and flips.
For centuries, navigation at sea has involved mathematics. A ship's crew must be able to calculate the position of their ship, the direction it is headed, the distance it should travel along this course, and when to change direction.
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The map below shows three ships that are heading towards an island with buried treasure. Imagine that you are standing at the wheel of one of the ships and are responsible for guiding the ship to the treasure island.
1.
a. Describe the course that each ship follows. In your description, include directions and distances.
b. Which ship's course was easiest to describe? Explain.
c. Compare your descriptions of the courses for Ship 1 and Ship 3. What do you notice? Explain.
d. If all three ships sail at the same speed, give the order in which the ships will arrive at the island. Explain.
Look at the path of Ship 2 on page 16. It is a slide.
A slide is a motion in which an object moves from a starting position to a finishing position in a straight line. The new position of the object after the slide is the slide image .
An example of a slide in everyday life is an object moving on a straight conveyor belt.
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Look at the paths of Ships 1 and 3 on page 16. If you fold the grid along Ship 2's path, Ship 3's path falls on Ship 1's path. This means that Ship 1's path is the flip image of Ship 3's path. Ship 2's path is the flip line.
A flip is a motion in which an object is flipped or reflected over a fixed line, called the flip line. The flip image is the mirror image of the original position.
An example of a flip in everyday life is a reflection in water or a pancake being flipped.
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2.
a. The following diagram shows a shape that has been moved in a slide motion. What will the shapes in the next two boxes look like?
b. The following diagram shows a shape that has been moved in a flip motion. What will the shapes in the next two boxes look like?
Look at the ships again. The paths that Ship 1 and Ship 3 took included several turns.
A turn is a motion in which an object is turned about a fixed point, called the turn centre. The new position of the object after the turn is called the turn image.
An example of a turn in everyday life is a ferris wheel or a carousel.
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3. The following diagram shows a shape that has been moved in a turn motion three times. What will the shapes in the next two boxes look like?
The direction of a turn can be described as clockwise or counterclockwise. Remember, the hands of a clock move in a clockwise direction. The opposite direction is called counterclockwise.
The angle of a turn can be expressed in degrees or as a fraction of a turn. Remember, a full turn is
360°, a turn is 90°, a 
turn is 180°, and a 
turn is 270°.
4. Write 
turn, 
turn,
turn, or full
turn to describe the clockwise turn of each of the following letters. Each letter began in the
upright position.
a. 
b. 
c. 
d. 
5. Write 
turn, 
turn,
turn, or full
turn to describe the counter-clockwise turn of each of the following letters. Each letter began
in the upright position.
a. 
b. 
c. 
d. 
You will now discover how symmetry is related to flips and turns.
Have you ever made designs like these by folding and cutting paper?
Each of these designs has line symmetry.
Line symmetry is the property in which one-half of the shape can be flipped onto the other half so the two halves match exactly. The line that divides the shape into two matching parts is called the line of symmetry.
You can test for line symmetry using tracing paper.
Example
Does this star have line symmetry?
To test for line symmetry, do the following steps:
Step 1: Trace the figure and cut out the tracing.
Step 2: If the star has line symmetry, you can fold the cutting so that the left side matches the right side (a flip).
Note: Test to see how many different lines of symmetry there are. In other words, check how many ways the cutting can be folded so the left side matches the right side.
1. With a page of shapes, use tracing paper to test each shape for line symmetry. Which shapes have line symmetry? Make a sketch of each shape and show all lines of symmetry.
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Have you ever played with a pinwheel? A pinwheel does not have line symmetry, but it can be turned around its centre in such a way that it is in the same position more than once in a full turn.
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The pinwheel has point symmetry.
Point symmetry is the property in which a shape matches with its original position more than once in a full turn.
The turn centre about which a figure can be turned is called the point of symmetry.
You can use thin paper and pins to test for point symmetry.
Example
Does this star have point symmetry?
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To test for point symmetry, do the following steps:
Step 1: Cover the figure with the thin paper (airmail paper or tracing paper), put a pin through the turn centre, and trace the figure. To keep track of the positions as the star turns, mark one point on the tracing of the star.
Step 2: Test to see if the star has point symmetry. Does the figure match the original more than once in a full turn?
Yes, the star has point symmetry. In one full turn, the star matches the original position five times.
2. With the page of shapes, use tracing paper and a pin to test each shape for point symmetry. Which shapes have point symmetry?
Now you will explore line symmetry and point symmetry.
Hold this page up to a large wall mirror. Compare the message and its reflection.
1. Look at each word of the message and the reflection.
a. In which direction is each word spelled in the message?
b. In which direction is each word spelled in its reflection?
2. Look at each letter of the message and the reflection.
a. Which letters look the same as their reflections?
b. Which letters have reflections that appear to be formed backward?
The letters of the alphabet that look the same in the reflected message have a vertical line of symmetry. The vertical line of symmetry cuts the letter in half. When you place a mirror on this line, you can see the whole letter by looking at either the right half or the left half of the letter, along with its reflection.
Example
The letter A has a vertical line of symmetry.
3. With a sheet of letters and numerals, use a small rectangular mirror to discover which letters have a vertical line of symmetry.
a. Name all the letters that have a vertical line of symmetry.
b. For each letter that has a vertical line of symmetry, draw a line to show where you placed the mirror.
Sometimes a letter has a horizontal line of symmetry that cuts the letter in half. When you place a small rectangular mirror on this line, you can see the whole letter by looking at either the top half or the bottom half of the letter, along with its reflection.
Example
The letter E has a horizontal line of symmetry.
4. With a sheet of letters and numerals, use a small rectangular mirror to discover which letters have a horizontal line of symmetry.
a. Name all the letters that have a horizontal line of symmetry.
b. For each letter that has a horizontal line of symmetry, draw a line to show where you placed the mirror.
5. Name all the letters that have both vertical and horizontal lines of symmetry.
Some letters have point symmetry. These letters look the same upside down as they do right-side up.
Example
The letter S has point symmetry.
6. With a sheet of letters and numerals, use it to discover which letters have point symmetry. Name all the letters that have point symmetry.
7. Name all the letters that have both line symmetry and point symmetry.
8. Name all the letters that have neither line symmetry nor point symmetry.
Now it's time to show your home instructor what you have been learning.
With a sheet of letters and numerals, discuss with your home instructor which numerals on the sheet have line symmetry and which have point symmetry.
You will now use your knowledge of turns to describe signals at sea.
Earlier in the course you learned about some flags that ships use to communicate. Another way that flags are used to send messages at sea is using a special pattern called semaphore. The semaphore flag is square and divided diagonally. The upper part of the flag is red and the lower part of the flag is yellow. One flag is held in the right hand, and a second flag is held in the left hand.
For the semaphore alphabet, the signaller stands with arms extended so that the top of the arm at the shoulder is a turn centre. Each hand turns like a clock hand and stops at one of the following positions: up, high, out, low, or down.
Notice how the flag positions are related to each other. Answer the following questions as if you are facing the signaller.
1.
a. Tell what kind of motion you see when the signaller moves any of the right-hand flag positions to a different right-hand flag position.
b. Tell what kind of motion you see when the signaller moves any of the left-hand flag positions to a different left-hand flag position.
2. Tell what turn direction (clockwise or counterclockwise) and amount of turn 
you see when the signaller makes the
following changes with his or her right hand.
a. from an up position to an out position
b. from an out position to an up position
c. from a down position to an up position
d. from an up position to a down position
e. from a high position to a low position
f. from a low position to a high position
g. from a down position to an out position
h. from an out position to a down position
3. Tell what turn direction (clockwise or counterclockwise) and amount of turn 
you see when the signaller makes the
following changes with his or her left hand.
a. from an up position to an out position
b. from an out position to an up position
c. from a down position to an up position
d. from an up position to a down position
e. from a high position to a low position
f. from a low position to a high position
g. from a down position to an out position
h. from an out position to a down position
4. How does the answer to each part of question 2 compare to each part of question 3?
5. Tell what kind of motion relates the two flags in a signal if the signaller holds the flags in the following manner.
a. the left-hand flag low and the right-hand flag high
b. the left-hand flag high and the right-hand flag low
6. The following picture shows a sailor signalling the letter A.
a. Describe the hand positions for the letter A.
b. If you face the sailor and "mirror" his signal, you will signal the letter G. Describe how you would hold the flags to signal the letter G.
c. Draw a picture showing the flags for the letter G.
7. The following picture shows a sailor signalling the letter R.
a. Describe the hand positions for the letter R.
b. What kind of motion relates the two flags for the letter R?
c. The letter D is the only letter that can be signalled two ways. To signal the letter D, the signaller can do whichever of the following is more convenient:
- left hand down and right hand up
- left hand up and right hand down
Describe two different ways a signaller can move his or her hands to signal the letter D immediately after signalling the letter R.
8. The following picture shows a sailor signalling the letter U.
a. Describe the hand positions for the letter U.
b. What kind of motion relates the two flags for the letter U?
c. To signal the letter N immediately after signalling the letter U, a sailor can make a quarter turn clockwise with his left hand and a quarter turn counterclockwise with his right hand. Describe the signal for the letter N.
d. Draw a picture that shows the signal for the letter N.
For additional information on semaphore, do some research on the Internet. You may find the following website helpful!
http://www.anbg.gov.au/flags/semaphore.html
Move only three of the bowling pins in the bottom picture so that they form the mirror image of the pins in the top picture.
In this lesson you were introduced to motion geometry by investigating slides, turns, and flips. You saw how these motions are used in the world around you.
Look for slides, turns, and flips in the designs of a variety of everyday objects such as the patterns on a bedspread, your clothing, wallpaper, or tiles in your kitchen.
Removed due to copyright.
1.
a. Ship 1: 2 km west, 5 km north, 3 km west, 2 km north, 2 km east, 6 km north, 1 km west, 3 km north, 3 km east Ship 2: 15 km north. Ship 3: 2 km east, 5 km north, 3 km east, 2 km north, 2 km west, 6 km north, 1 km east, 3 km north, 3 km west
b. The course for Ship 2 was easiest to describe. The ship travelled north in a straight line without making any turns.
c. Both ships travelled north, but whenever one of the ships travelled west, the other ship travelled east. All of the distances were the same.
d. Ship 2 would arrive first because it travelled the shortest distance. Ships 1 and 3 would arrive later than Ship 2, but at the same time as each other. They travelled farther than Ship 2, but they travelled the same distance as each other.
2.
a. 
b. 
3.
4.
a. 
b. 
c. 
d. 
a full turn
5.
a. 
b. 
c. 
d. 
a full turn
| 1. | Figure A has two lines of symmetry. | Figure B doesn't have line symmetry. | ||
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| Figure C doesn't have line symmetry. | Figure D has three lines of symmetry. | |||
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| Figure E has six lines of symmetry. | Figure F has one line of symmetry. | |||
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2.
Figure A has point symmetry. The shape will fit onto itself twice in a full turn.
1 full turn Figure B has point symmetry. The shape will fit onto itself twice in a full turn.
1 full turn Figure C doesn't have point symmetry.
Figure D has point symmetry. The shape will fit onto itself three times in a full turn.
1 full turn Figure E has point symmetry. The shape will fit onto itself six times in a full turn.
Figure F doesn't have point symmetry.
1.
a. Each word in the message is spelled from left to right.b. Each word in its reflection is spelled backward, from right to left.
2.
a. The letters W, H, A, T, O, Y, U, I, and M look the same as their reflections.b. The letters D, S, E, N, and R have reflections that are backward.
3.
a. The letters that have a vertical line of symmetry are A, H, I, M, O, T, U, V, W, X, and Y.
b.
4.
a. The letters that have a horizontal line of symmetry are B, C, D, E, H, I, K, O, and X.
b.
5. The letters that have both vertical and horizontal lines of symmetry are H, I, O, and X.
6. The letters that have point symmetry are H, I, N, O, S, X, and Z.
7. The letters that have both line symmetry and point symmetry are H, I, O, and X.
8. The letters that have neither line nor point symmetry are F, G, J, L, P, Q, and R.
1.
a. I see a turn motion.b. I see a turn motion.
2.
a. quarter turn, counterclockwise b. quarter turn, clockwise c. half turn, clockwise d. half turn, counterclockwise e. quarter turn, counterclockwise f. quarter turn, clockwise g. quarter turn, clockwise h. quarter turn, counterclockwise
3.
a. quarter turn, clockwise b. quarter turn, counterclockwise c. half turn, counterclockwise d. half turn, clockwise e. quarter turn, clockwise f. quarter turn, counterclockwise g. quarter turn, counterclockwise h. quarter turn, clockwise
4. The turn changes described in each part of questions 3 and 4 are the same, but the turn motion is in the opposite direction.
5.
a. a flip with a horizontal flip line through the signaller's waistb. a flip with a horizontal flip line through the signaller's waist
6.
a. The signal for the letter A is left hand down, right hand low.
b. The signal for the letter G is left hand low, right hand down.
c.
7.
a. The signal for the letter R is left hand out, right hand out.
b. a flip (with a vertical flip line through the signaller)
c. To signal the letter D immediately after signalling the letter R, the signaller can turn both hands a quarter turn clockwise or both hands a quarter turn counterclockwise.
8.
a. The signal for the letter U is left hand high, right hand high.
b. a flip (with a vertical flip line through the signaller)
c. The signal for the letter N is left hand low, right hand low.
d.
Pins A, B, and C are moved as shown.
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