Maths: Geometry and measures KS3. Sketching and drawing lines and angles. Pythagoras Theorem.

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1 TEACHERS NOTES About this activity This activity helps pupils practice accurately drawing lines and angles, and using Pythagoras Theorem, with a real life scenario set at Manchester Airport. Taking off can be the noisiest part of a flight and so aircraft follow Preferred Noise Routes (PNRs) that guide them away from built-up areas wherever possible. Pupils use distances, angles and triangles to plot PNRs on a map. They then extend their use of Pythagoras Theorem to consider how aircraft increase altitude along one PNR, solving a problem to find the aircraft s altitude at the release point of the PNR. Learning outcomes Pupils can: Accurately draw lines and angles. Solve problems involving Pythagoras Theorem. Curriculum links Maths: Geometry and measures KS3. Sketching and drawing lines and angles. Pythagoras Theorem. Resources and preparation The student resources for this lesson comprise Pupil Sheet One and Pupil Maps One and Two. Print enough pupil sheets and maps for the class. Review the presentation slides. You will need a whiteboard and data projector for the activity. Pupils will need pencils, protractors, rulers and pens. Timings The activities can be delivered in a single lesson of approximately one hour, but you may wish to extend over two lessons. 1

2 LESSON PLAN Starter activity ideas Before you begin you may wish to: Review how to use a ruler and protractor to accurately measure and draw lines and angles up to 180 o using mm and degrees. Main activities Introduce the activity and explain that pupils are going to use lines, angles and triangles to plot some important routes aircraft follow after taking off from Manchester Airport. These routes have been chosen so whenever possible aircraft don t cause too much noise for people living nearby and avoid flying low over places like the nearby Peak District National Park. Show slide 1 and talk about what pupils think Manchester Airport might be like. Show slide 2. Briefly discuss why take off may be the noisiest part of a flight. (It is because the aircraft is at its heaviest as it is carrying all its fuel for the journey, and is working against gravity to gain height.) Ensure pupils understand that a Preferred Noise Route (PNR) is a pathway through the sky. Aircraft follow a track on the ground while also gaining height as they climb (this will be elaborated more towards the end of the lesson). Show the first part of slide 3 (don t click through for now) and ask pupils to look at their copies of the Pupil Map One. Explain that aircraft must always take off and land into the wind, and on about 4 days out of 5, this comes from the southwest and the Airport is in westerly operation on these days. Show the direction aircraft should take off and land from the Airport (from the top right corner towards the bottom left corner. Explain that the rest of the time, the wind tends to come from the northeast and the Airport is in easterly operation on these days. Explain that this lesson will include first Westerly and then both Westerly and Easterly Operation. Task 1: distances, angles and PNRs Establish that PNRs leaving Runway 1 (used for Westerly Operation) begin at the southwest end of that runway and initially head southwest. PNRs leaving Runway 2 (during Easterly Operation) begin at the northeast end of that runway and initially head northeast. Ensure pupils can identify these two points on their Pupil Map One. Review PNR table 1 on Pupil Sheet One. Each line in the table shows the distance and angle data to complete the red PNRs on the Pupil Map One. Each pair of angle + scale distance information represents one leg of the PNR. Now click through the remaining elements of Screen 3.4 and explore how PNR LIS2R can be drawn using distance and angle data. This example is shown on the Pupil Map already. 2

3 LESSON PLAN Task 1: distances, angles and PNRs (cont) All: Using Pupil Map One, ask pupils to draw each PNR in table 1 and label it with its codename. Explain that all the PNRs curve down to the south pupils need this information to know which direction to measure their angles. Each PNRs has its first leg drawn already. The ends of these legs are labelled with a letter for the first angle to use. Some legs may require more than one angle. Less able: Assign one or more PNRs according to ability or allow pupils to work with a partner. More able: Pupils complete all PNRs in table 1. Challenge pupils to achieve 1mm and 1 degree accuracy. Review answers on slide 4. Task 2: Pythagoras Theorem and PNRs Show the first part of slide 5 (don t click through for now) and briefly review Pythagoras Theorem. Explain that this provides another way to help plot PNRs on the map. Work through the rest of slide 5 and explore how PNR DES1S can be drawn using Pythagoras Theorem. This example is shown on the screens already. (Note that the dotted lines form part of the triangles used to create these PNRs, but are not part of the PNR itself.) Show slide 6 All: Ask pupils to calculate the missing distance in each PNR using the information on Pupil Map Two and complete table 2 on their Pupil Sheet. Less able: Complete one PNR together as a class before pupils complete the other two. Review answers, which are at the bottom of this lesson plan and on slide 6.1. Show slide 7. Explain that in real life PNRs curve gently to reflect how aircraft turn in the air (they can t turn a sharp corner), so the PNRs pupils plotted are just rough approximations. More able: discuss briefly how more accurate PNRs could be developed using more lines and angles to represent each one and introduce approximations to a curve. There are also more PNRs eight for Westerly Operations and six for Easterly Operations. PNRs are also very wide, 3km, or 3000m. This gives aircraft plenty of space as for safety, it s vital that aircraft stay well apart from one another. 3

4 LESSON PLAN Task 3: what s the release altitude for PNR SAN1R? Show slide 8. Help pupils identify that the lines they have been constructing on their maps are here shown by the dotted lines they are the tracks along the ground. In reality, the aircraft follow the solid line as they climb steadily. Explain that pupils need to apply their ideas to solve one final question: how high should aircraft following PNR SAN1R be flying in order to be released from the PNR. That is, they are high enough for their engine noise to no longer be a potential issue. Explain that for the purpose of this task, aircraft should reach their release altitude when they reach the end of the PNR at the edge of the map. Review the information on the Pupil Sheet together. Help pupils identify that they need to complete the diagram consisting of a series of triangles that join together like the ones on slide 8 using: the remaining along the ground distances of each leg of PNR SAN1R, (which pupils should identify they know from table 1) the knowledge that the aircraft will fly a distance of 101.2mm, to scale, (which pupils should identify is the hypotenuse of the triangle) Ensure pupils understand that each triangle represents one stage of SAN1R, starting with when an aircraft leaves the end of Runway 1. The aircraft climbs at a constant rate so each angle (see angle a on slide 8) is identical. All: Ask pupils to fill in the missing information on their diagrams and then use the information to find the release altitude for SAN1R, using Pythagoras Theorem. (Pupils should remember that 1mm = 250m.) Share the answer on slide 10. Explain that this is represents a really shallow angle aircraft only have to climb at an angle of 3.5 o to achieve this. In fact, many aircraft will reach their release altitude earlier than this as they can climb much faster. Plenary Remind pupils that PNRs are the routes aircraft follow after taking off from Manchester Airport, so whenever possible they don t cause too much noise for people living nearby and avoid places like the nearby Peak District National Park. Extension ideas Pupils can consider helicopter flights from the airport to local towns. Working in pairs or small groups, each pupil can develop a 1, 2 or 3 leg route on their map, measuring the distance and bearing of each leg and remembering that 1mm on the map = 250m. Pupils then share their journey plan with their partner, who must plot the route and identify the destination. 4

5 LESSON PLAN Answers Task 2: Pythagoras Theorem and PNRs Distance in mm Distance in nautical miles SON1R 85mm 11.5 nautical miles ASM1S 20mm 2.7 nautical miles LIS1S 69mm 9.3 nautical miles 5

6 PUPIL SHEET ONE Manchester Airport is a busy place. Each year, there are nearly 170,000 flights in and out of the Airport. Taking off can be the noisiest part of a flight. Preferred Noise Routes (PNRs) guide aircraft away from built-up areas wherever possible. Use these angles and distances to plot these Westerly Operation PNRs on Pupil Map One. Label each PNR with its code, eg EKL1R. Each PNR has its first leg drawn already. Table 1 LIS1Y a=118 o and 20mm, then 157 o and 25mm SAN1R b=135 o and 23mm, then 106 o and 31mm EKL1R c=149 o and 57mm Use the information on your map to find the three missing distances using Pythagoras Theorem, to complete the remaining PNRs Table 2 Calculate the missing distance shown with a? on Pupil Map Two. Distance in mm Distance in nautical miles SON1R ASM1S LIS1S 6

7 PUPIL SHEET ONE Convert to real world distances Pilots measure distances in nautical miles. The scale on the map is 1mm = 250m. Can you convert each distance into nautical miles? One nautical mile = 1852m. What s the release altitude for PNR HON1R? PNR SAN1R consists of four legs, beginning at the end of runway 1. By the time it reaches release altitude an aircraft has flown 101.2mm (to scale) along its flightpath. What is the release altitude in metres? mm Release altitude 17mm 30mm mm mm 7

8 Pupil Map One KS3 Maths: Lines, Angles and Triangles Distance, angles and PNRs We can use angles and distance to plot simplified PNRs. N W E S N W E S 8

9 Pupil Map Two KS3 Maths: Lines, Angles and Triangles Distance, angles and PNRs We can use angles and distance to plot simplified PNRs. N W E S 9