In the film The Hunt For Red October, at one point the escaping Russian submarine has to navigate a complicated underwater trench. This unit is built around that idea and brings in the measurement of angle and distance. It is expected at this stage that the students will know how to use protractors.
Specific Learning Outcomes
- Follow directions given in bearings.
- Invent their own maps using bearings.
- Use a protractor to produce maps.
Description of Mathematics
Angle can be seen as and thought of in at least three ways. These are as:
- the spread between two rays
- the corner of a 2-dimensional figure
- an amount of turning
Angle as a measurement of turn underpins the ways. The origin of the unit of measure for angle, the degree, is in the division of a full rotation by 360. It is widely believed that the ancient Babylonians chose 360⁰ for the base because it was divisible into many fractions. We retain that standard international unit, the degree, to this day.
Broadly the development of understanding of angle in The New Zealand Curriculum progressed though the levels as follows:
Level 1: quarter and half turns as angles
Level 2: quarter and half turns in either a clockwise or anti-clockwise direction
angle as an amount of turning
Level 3: sharp (acute) angles and blunt (obtuse) angles
degrees applied to simple angles – 90°, 180°, 360°, 45°, 30°, 60°
Level 4: degrees applied to all acute angles
degrees applied to all angles
angles applied in simple practical situations
Level 5: angles applied in more complex practical situations
At secondary school angle is used extensively in trigonometry (sine, cosine, tangent, etc.) to measure unknown or inaccessible distances. This deals with situations where only right-angled triangles are present in 2-dimensional situations through to more complicated triangles in 3-dimensional applications.
Opportunities for Adaptation and Differentiation
The learning opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. Ways to differentiate include:
- Using grid/graph paper to create co-ordinate systems and label the axes before expecting students to image the system.
- Providing students with physical and digital experiences that require moving in directions with distance, e.g., Turtle maths, Scratch.
- Pair students in mixed ability to groups for mutual support.
- Provide digital tutorials by looking online for videos on using scale maps and compass directions.
The context for this unit is around a classic film about submarines. You might choose to base the story on a different navigation chase, such as Tamatea’s navigation of Aotearoa, or on a story, such as exploring the Marlborough Sounds. Video games frequently involve navigation. That context may be motivating for some students.
Required Resource Materials
- Tell the story of The Hunt for Red October (1990). You will find several clips online including a trailer and highlight sections of the film. (If the students have seen the film then they can be asked to give the main points.) A Russian submarine captain decides to defect to the West with a new submarine. At one point in the film the captain navigates through a natural trench in the Atlantic Ocean. He can’t see where he is going and relies on his navigator to give him directions. We are going to recreate this part of the film in this unit.
- Today you are going to draw a course through the Norwegian Ravine in the Atlantic Ocean from the instructions that I am about to give you (Copymaster 1). Just like the Captain you cannot see the ravine, only obey the navigator’s instructions. The important thing here is that you should end up at the right place and not hit the sides of the Ravine. I will give you compass directions and distances so that you can construct a course through the ravine.
- In groups of two, let the students draw the course in the empty part of Copymaster 1. A scale of 1cm:1km is needed. You may need to discuss that 1cm on the map represents 1km in the real ravine. The co-ordinates of the various points are
B = (0, 5); C = (8, 5); D = (8, 11); E = (4, 11); F = (4, 8); G = (2, 8).
- When students have finished that exercise gather the class. Spend some time discussing how the students worked out each co-ordinate.
Did you need to put a grid system in place first?
Can the coordinates be found without drawing a grid? How?
- You might show students a video of how the cartesian coordinate was developed by Rene Descartes (Search: Descartes and the Cartesian coordinate system). The story of observing the fly in bed makes an amusing story and reinforces the idea that the directions, horizontal then vertical, must be consistently interpreted. For example, the point (3, 2) is three units to the right and two units up from the origin (0,0).
- Let students devise their own Ravine map using only the four points of the compass and distance. They should draw their maps and then list the instructions needed to recreate the maps. For example, instructions might be North for 3km, East for 6km, etc. Pairs of students then swap instructions so that a different pair draws the map. Get the two pairs to check each other’s work. You may need to support some pairs.
- Imagine you are in training to be a navigator at submarine school. Let’s consider the instructions you need to be able to give and receive.
What language might you use to tell the Captain where to take the submarine?
Students might suggest directions and distances. Both aspects are important in plotting a course. Take the class outside to an open area.
- Ask students to act out a sequence of instructions involving compass directions and distances.
Face North and walk for 2 metres (Students might use their mobile phone or a compass to do so. The sun is about North at midday).
Turn to face East and walk 3 metres.
Turn to face South and walk 1 metre.
If you started at (0, 0) what point are you at now?
- Ask students to come up with sets of instructions for others and ask for the coordinate of the finishing point.
- Return inside for some navigator training.
We have drawn the Newfoundland Net which is a navigation path needed to navigate a trench in the ocean off the East of Canada (see Copymaster 2).
Use a protractor and a ruler to draw the route to scale so that 1 km is 1 cm on your drawing. When you have finished, work out the co-ordinates of the points A, B, C, D, E and F. You should also measure the straight-line distance from the start to the finish of the Net.
- After a suitable time ask the students to pair up to see that they have created the same path. Agreement means they pass navigator school.
The Russian Graveyard
- In this session students use the map in Copymaster 3. The students work in pairs to navigate the Russian Graveyard by plotting the course. One student is the navigator and has the map from the copymaster. The other student is the pilot who steers the Red October. The navigator tells the pilot instructions to move the submarine.
The coxswain marks the turning points on graph paper and joins them up. The co-ordinates of the turning points should be read end to see if the pilot has got the Red October safely through the Russian Graveyard.
Interchange the navigator and the pilot. The navigator must sketch the path as well as give instructions. At the completion of the route the navigator and pilot compare notes.
The Scandinavian Sweep
- Use Copymaster 4 to play a game where there is a navigator and a pilot in each of two opposing submarines, Red October and Konovalov. The Konovalov is chasing the Red October. Can the Konovalov catch the Red October?
- Begin at (0,0) and use a scale of 1cm:1km. In each turn the navigator rolls a standard dice. The number that comes up is the distance the submarine can travel in that turn. The distance can be broken up into as many smaller distances as the navigator needs. However, each command must involve a compass direction and a distance, e.g., 2 km East, 3 km North for a dice roll of five. The pilot draws the route.
- Red October gets a head start of three dice rolls before the Konovalov starts. A win for Red October is reaching the open sea. A win for Konovalov is catching Red October.
- The pairs can swap submarines and can start moving through the Sweep from either end (Rotate the Copymaster by a half turn). Check co-ordinates at the end to make sure that no submarine has collided with the wall of the Sweep. That is considered a loss.
In the final lesson of this unit allow the students to produce their own maps with instructions for navigating the maps. The channels should be such that the submarines are never moving in the direction of any of the major points of the compass. You might encourage students to use intercardinal directs like Southeast and Northwest as well as the cardinal directions.
Groups should interchange their instructions and try to recreate each other’s maps.
Dear family and whānau,
This week we have been looking at the construction of routes on a map. We would like you to work together with your child to choose two points on a map and then make a list of instructions on how to go along that route from beginning to end using a protractor and compass.
We would like to see these instructions to see if we can follow them in class. Try to find routes where the sections are straight lines.