stage 5: mathematics stem advanced pathway flight 5.3
TRANSCRIPT
NSW Education Standards Authority – Sample unit Mathematics Stage 5 - STEM Advanced Pathway: Algebraic Fractions Page 1 of 18
Stage 5: Mathematics STEM Advanced Pathway Flight 5.3 sample program
Overview Duration
In Flight 5.3 students will recall, consolidate and develop the following essential skills:
§ how to describe direction using angles and bearings § use of trigonometry with right-angled triangles § midpoints and gradients of straight lines § use of Pythagoras’ Theorem
This program will focus on:
§ equipping students to select relevant information from a diagram or written description and appropriate degrees of accuracy for practical examples.
In Flight 5.3 students will develop the following essential STEM understandings:
§ how modern flight provides a context for the mathematics of location, gradient and coordinate geometry.
5 weeks
Outcomes
A student:
§ selects relevant information from diagrams and written descriptions and applies appropriate degrees of accuracy to solve practical problems § recognises the science, technology and engineering feats that allow modern flight § uses the gradient-intercept form to interpret and graph linear relationships (MA5.2-9NA) § uses formulas to find midpoint, gradient and distance on the Cartesian plane, and applies standard forms of the equation of a straight line
(MA5.3-8NA) § applies trigonometry, given diagrams, to solve problems, including problems involving angles of elevation and depression (MA5.1-10MG) § applies trigonometry to solve problems, including problems involving bearings (MA5.2-13MG) § applies Pythagoras’ theorem, trigonometric relationships, the sine rule, the cosine rule and the area rule to solve problems, including
problems involving three dimensions (MA5.3-15MG) § interprets mathematical or real-life situations, systematically applying appropriate strategies to solve problems (MA5.2-2WM)
NSW Education Standards Authority – Sample unit Mathematics Stage 5 - STEM Advanced Pathway: Algebraic Fractions Page 2 of 18
Common misconceptions Language/Literacy
Students may:
§ not have practical experience of printed maps and compass orientations (GPS maps orient to direction of travel and provide directions rather than an overview)
§ not have developed the skill of reading values around a curved edge (digital time replacing clock faces)
§ find 2D representations of 3D situations difficult to interpret § be confused by notation of the form (𝑥!, 𝑦!) § lack confidence naming opposite and adjacent sides in right-angled triangles in different
orientations § lack confidence selecting from formulae that look very similar and refer to the same variables
not have developed their understanding of gradient
Maintain correct descriptions with particular care for:
§ SOHCAHTOA as a memory aid – not a topic name
§ Degrees and minutes in angles – not ‘bubble button’
§ ‘Inverse sine’, not ‘shift sine’
NSW Education Standards Authority – Sample unit Mathematics Stage 5 - STEM Advanced Pathway: Algebraic Fractions Page 3 of 18
Content Teaching and Learning STEM Resources and Stimulus
§ interpret three-figure bearings (eg 035°, 225°) and compass bearings (eg SSW)
§ draw diagrams to assist in solving practical problems involving bearings (Communicating, Problem Solving)
Activity: Where am I? (TES): https://www.tes.com/teaching-resource/where-am-i-11417302 is a bearings exercise to be done by students in pairs.
Link to learning:
The teacher leads a class discussion: Which methods of transport are able to travel along exact bearings? What impedes those that cannot travel this way? How does this contribute to the efficiency of different transport modes? Hence, which professions make regular use of bearings?
Consolidation for learning and guided practice:
§ Students demonstrate method for converting between true (three-figure) and compass bearings
§ Students complete the bearings activity: ‘Eagle cat’ (Scootle): http://www.scootle.edu.au/ec/viewing/L10094/index.html could be used in a game format for practice. Teacher displays the compass and hides one of the bearings. Teams race to calculate the hidden value. Students can use the ‘Malabar Island’ tab for their own interactive practice.
§ Students complete the bearings activity: ‘Picture this’ (TES): https://www.tes.com/teaching-resource/where-am-i-11417302 could be run by nominating a student to read each of the ‘stories’. Each class member translates the worded description into a bearings diagram. Ideally this could be done on mini whiteboards which students hold up to compare drawings.
§ Students revise using angle relationships to find unknown angles
§ Students complete the ‘Angles round a point’ worksheet (TES) https://www.tes.com/teaching-resource/angles-round-a-point-worksheet-6317913
STEM: Ideally this activity would be done in cooperation with the Science faculty. Alternatively it could be set as a homework task.
§ Make a Homemade Compass (Scientific American): https://www.scientificamerican.com/article/steering-science-make-a-homemade-compass/
§ Develop the mathematics by adding three-figure and compass bearings to the device
§ Student discussion – Which of these forms of writing bearings makes most sense to them?
Note: For some students familiar with navigation, there may be a need to clarify that in this context the term ‘true bearing’ is not being used as a reference to bearing from ‘true north’ as distinct from magnetic north – particularly if the activity is done in cooperation with the Science Faculty
§ Find the midpoint and gradient of a Activity: Students watch the video ‘World’s Largest Urban Zipline’: STEM: Human flight takes many
NSW Education Standards Authority – Sample unit Mathematics Stage 5 - STEM Advanced Pathway: Algebraic Fractions Page 4 of 18
Content Teaching and Learning STEM Resources and Stimulus
line segment (interval) on the Cartesian plane (ACMNA294)
§ plot and join two points to form an interval on the Cartesian plane and form a right-angled triangle by drawing a vertical side from the higher point and a horizontal side from the lower point
§ interpret and use coordinate
notation of the form (𝑥!, 𝑦!) and (𝑥!, 𝑦!)
§ use Pythagoras' theorem to
establish the formula for the distance, d, between two points (𝑥!, 𝑦!) and (𝑥!, 𝑦!)on the Cartesian plane: 𝑑 = 𝑥! − 𝑥! ! + 𝑦! − 𝑦! !
§ use the formula to find the distance
between two points on the Cartesian plane
§ use the concept of an average to establish the formula for the midpoint, 𝑀, of the interval joining two points (𝑥!, 𝑦!) and (𝑥!, 𝑦!) on the Cartesian plane: 𝑀 𝑥, 𝑦 = !!!!!
!, !!!!!
!
§ use the formula to find the midpoint
of the interval joining two points on
https://www.youtube.com/watch?v=YcwrRA2BIlw&feature=youtu.be
§ Pause video at 1:22 to show the lines in full. § Students sketch the right-angled triangle formed by the
building, ground and zip lines, estimating measurements. § Continue video. § The teacher asks students to mark on their diagrams the
point along the zipline that ‘flyers’ must disengage. Compare and discuss answers. Discuss the implications of getting the calculations wrong.
Link to learning: The start and end point of the zipline are two points on a Cartesian plane. The line forms the hypotenuse of a right-angled triangle.
From this simple diagram the following can be answered:
§ Any one of zipline length, building height or distance along ground as long as two of the three values are known
§ The steepness of the zipline described in a number of ways: § Its gradient or slope § The angle of depression of the line viewed by a person
at the top (video of Urban zipline at 0:47) § The angle of elevation of the line viewed by a person at
the bottom.
Because the shortest distance between two points is a straight line, straight lines are very significant. If the equation of a straight line is known, it is simple to locate points anywhere along that line.
Consolidation for learning:
Differentiation:
§ Structured – Introduction to Geometry: Points, Lines, Planes and Dimensions (Skills you need):
forms.
Students could be set videos to view at home. Some examples are:
§ Video: ‘Sense of Flying’ (Goovinn): https://www.youtube.com/watch?v=ER1PGYe9UZA&feature=youtu.be
§ Video: ‘Meet the world’s only fully functional jet pack’ (Science Channel): https://www.youtube.com/watch?v=cIMWYeP7O1w&feature=youtu.be
§ Video: ‘Flyboard Air Demo at Flyboard World Cup’ (Atlantic Flyboard): https://www.youtube.com/watch?v=deyMNPbaRpA&feature=youtu.be
Together Science, Technology, Engineering and Mathematics have made the dream of human flight possible.
Class discussion about what might be invented in their lifetimes.
STEM extension activity:
Activity: ‘Where am I?’ (2D flight
NSW Education Standards Authority – Sample unit Mathematics Stage 5 - STEM Advanced Pathway: Algebraic Fractions Page 5 of 18
Content Teaching and Learning STEM Resources and Stimulus
the Cartesian plane
§ use the relationship gradient=rise/run to establish the formula for the gradient, 𝑚, of the interval joining two points (𝑥!, 𝑦!) and (𝑥!, 𝑦!) on the Cartesian plane: 𝑚 = !!!!!
!!!!!
§ find the equation of a line passing
through two points
§ find the equation of a straight line in the form 𝑦 = 𝑚𝑥 + 𝑐, given the gradient and the 𝑦-intercept of the line
§ find the gradient and the 𝑦-intercept
of a straight line from its graph and use these to determine the equation of the line
§ find the equation of a line passing
through a point (𝑥!, 𝑦!), with a given gradient m, using point-gradient form: 𝑦 − 𝑦! = 𝑚(𝑥 − 𝑥!), and gradient-intercept form: 𝑦 = 𝑚𝑥 + 𝑐
§ describe the equation of a line as the relationship between the 𝑥- and 𝑦-coordinates of any point on the
http://www.skillsyouneed.com/num/geometry.html . A very simple fact sheet for students who are yet to develop a sense of the three dimensions
§ Extension – students can develop this learning through the STEM extension activity.
The teacher explicitly teachers coordinate notation of the form (𝑥!, 𝑦!) and (𝑥!, 𝑦!) and the use of diagrams to solve coordinate geometry problems, including:
§ how to find the distance by Pythagoras’ theorem from diagram and distance formula
§ how to find the midpoint by averaging points from diagram and midpoint formula
§ how to find the gradient by rise/run from diagram and gradient formula
§ how to find the equation of a straight line between points by rise/run and 𝑦-intercept from diagram and the point-gradient ‘formula’
Guided practice: School-based and online worksheets could be used as resources.
path) (TES): https://www.tes.com/teaching-resource/where-am-i-2d-flight-path-11423688
Learn by doing:
An online 3D plotter is used to compare driving to flying across the Hay Plains in NSW.
Activity: ‘Where am I?’ (3D extension) (TES): https://www.tes.com/teaching-resource/where-am-i-3d-extension-of-bearings-11423666
Students calculate distances using Pythagoras in three dimensions and compass directions.
STEM extension: a real-world application:
This discussion on Geographic Information Systems: https://gis.stackexchange.com/questions/4690/determine-angle-down-to-horizon-from-different-flight-altitudes shows responses to a question from a pilot who has clients wanting to see a ‘double sunrise’ in one flight:. He wants to know whether a change in altitude will make this possible.
NSW Education Standards Authority – Sample unit Mathematics Stage 5 - STEM Advanced Pathway: Algebraic Fractions Page 6 of 18
Content Teaching and Learning STEM Resources and Stimulus
line Other pilots explain using Pythagoras’ theorem and other formulae as well as pointing out some of the practicalities to be considered.
STEM: Compare distances of road and air travel between two locations.
As a class, choose a location some distance from the school that can only be reached by a circuitous route, perhaps due to landscape features.
Use a tool such as Google Maps to determine the distance and travel time by road and compare this to distance ‘as the crow flies’.
Extension: Quick research will reveal the running costs of a car, helicopter and small plane. Students can calculate an estimated cost of travelling to the location by these different modes of transport and evaluate cost in dollars against cost in time and convenience.
NSW Education Standards Authority – Sample unit Mathematics Stage 5 - STEM Advanced Pathway: Algebraic Fractions Page 7 of 18
Content Teaching and Learning STEM Resources and Stimulus
§ connect the alternate angles formed when parallel lines are cut by a transversal with angles of elevation and depression (Reasoning)
§ determine the angle of inclination, 𝜃 , of a line on the Cartesian plane by establishing and using the relationship 𝑚 = tan 𝜃 where 𝑚 is the gradient of the line
Note to teachers: sometimes ‘angles of elevation and depression’ are presented to students as a ‘new’ part of Trigonometry after they have mastered solving right-angled triangles, and students who had developed confidence lose it when faced with long descriptions of boats, kites and cliffs. This loss of confidence might be avoided if the terms ‘angle of elevation’ and ‘angle of depression’ are introduced as nothing more than clarification of where objects are compared to one another. The following activity introduces the terminology independent of learning about the trigonometric ratios.
Activity: ‘I see you, you see me’
Students work in groups of four.
§ The teacher finds an environment where students can be safely situated at different heights, eg staircase, sloping landscape, playground equipment, upper floor windows.
§ Two group members (objects) position themselves at different heights. The other members (surveyors) sketch the situation, estimate angles and measure one accessible distance.
§ The teacher explains how students should issue instructions to ‘objects’ once they are in position – first both look and point straight ahead with left arm, then without moving the rest of their bodies, look and point right arm at the other ‘object’. ‘Surveyors sketch the situation, estimate the angle made by each ‘object’s’ arms and label their diagrams. (link to samples note: these include measurements for trigonometric calculations which are not required at this time)
Link to learning: Students discuss:
§ How can we describe the angles that were estimated in such a way that someone reading our description would know exactly where those angles were?
§ Will the angles of elevation and depression between two objects always be the same as each other? Why? Are students
STEM: ‘flipped learning’
Suggested as ‘at home’ tasks:
View ‘Exploring Trigonometry’ (Learn Alberta) : http://www.learnalberta.ca/content/mejhm/index.html?l=0&ID1=AB.MATH.JR.SHAP&ID2=AB.MATH.JR.SHAP.TRI&lesson=html/video_interactives/trigonometry/trigonometrySmall.html
Read the article ‘The physics of off-roading’ (How stuff works): http://adventure.howstuffworks.com/outdoor-activities/off-roading/off-roading1.htm and decide whether an approach angle is an angle of depression, elevation or neither:
NSW Education Standards Authority – Sample unit Mathematics Stage 5 - STEM Advanced Pathway: Algebraic Fractions Page 8 of 18
Content Teaching and Learning STEM Resources and Stimulus
reminded of any previous Maths topics about angles? (Angles in parallel lines.)
Consolidation for skill development and guided practice:
If the need for revision is indicated, school-based and online worksheets could be used such as: ‘Angles formed by a transversal’ worksheet (Maths Worksheets 4 Kids): https://www.mathworksheets4kids.com/transversal.php
§ Students list the common words that describe gradient, for example ‘steepness’ or ‘slope’
§ Students recall that gradient is rise/run, ie how much you step up (or down) for each step forward.
§ Students practise calculating gradient as rise/run for a number of right-angled triangles, ideally ‘real-world’ slopes.
§ Students use the worksheet, ‘Introducing tan as adding meaning to gradient’ (TES): https://www.tes.com/teaching-resource/introducing-tan-as-adding-meaning-to-gradient-11427366
§ Students introduce the tangent ratio as a way to convert rise/run values into degrees. (Note to teachers: this is not the typical way that tan is introduced. At this stage the focus is on rise/run in a right-angled triangle where angle of elevation or depression is required. Sides are not named opposite and adjacent.)
Differentiation:
Extension - challenge students to justify the percentages shown in the worksheet graphic in terms of rise/run of a ski slope. Where do the percentages come from?
§ identify the hypotenuse, adjacent sides and opposite sides with
Consolidation for skill development and guided practice:
§ The teacher refers to the previous learning about tan and
STEM: Tutorial on air navigation: ‘The effect of wind’ (Fly Safe):
NSW Education Standards Authority – Sample unit Mathematics Stage 5 - STEM Advanced Pathway: Algebraic Fractions Page 9 of 18
Content Teaching and Learning STEM Resources and Stimulus
respect to a given angle in a right-angled triangle in any orientation
§ label the side lengths of a right-angled triangle in relation to a given angle, eg side 𝑐 is opposite angle 𝐶
§ define the sine, cosine and tangent ratios for angles in right-angled triangles
§ select and use appropriate
trigonometric ratios in right-angled triangles to find unknown sides, including the hypotenuse and including where the given angle is measured in degrees and minutes
§ select and use appropriate trigonometric ratios in right-angled triangles to find unknown angles correct to the nearest degree and in degrees and minutes
§ solve right-angled triangle problems,
including those involving angles of elevation and depression (ACMMG245)
redescribe rise/run as opposite/adjacent side lengths. § The teacher presents labelled diagrams in different orientations
and tangent ratio. § The teacher defines sine and cosine ratios with labelled
diagrams. § The teacher presents the convention for labelling side lengths
of a right-angled triangle in relation to a given angle, eg side 𝑐 is opposite angle 𝐶
Students follow each step of the lesson, creating the diagram and making measurements and performing calculations for themselves, comparing these to the progression of slides.
School-based and online worksheets could be used as resources.
https://www.recreationalflying.com/tutorials/navigation/wind.html
Show this page to students and let them know that while it might look confusing right now, by the end of this topic they will be able to understand and use the maths in this article.
Guide a class discussion about wind speed and direction and aircraft navigation.
§ solve a variety of practical problems, including those involving angles of
Activity: The teacher challenges students to calculate a height or distance they could not measure, and make the necessary tools
STEM extension: Read the article ‘How far away is that
NSW Education Standards Authority – Sample unit Mathematics Stage 5 - STEM Advanced Pathway: Algebraic Fractions Page 10 of 18
Content Teaching and Learning STEM Resources and Stimulus
elevation and depression, when given a diagram and problems for which a diagram is not provided
§ draw diagrams to assist in solving practical problems involving angles of elevation and depression (Communicating, Problem Solving)
themselves.
Making a Clinometer (University of Cambridge and Centre for Mathematical Sciences enrichment program (NRICH)): https://nrich.maths.org/5382
Differentiation:
Extension – Students use their own understanding of symmetry and dividing of 360° to create a protractor from scratch (hint: they might fold paper to ensure symmetry without access to any measuring tools)
contrail?’ (Contrail Science): http://contrailscience.com/how-far-away-is-that-contrail/
As a US site the discussion is in miles, but kilometres are also provided in the table to be used in calculations.
If the school is in an area where contrails are visible students could make these observations and calculations themselves.
If contrails are not visible then the size of, or distance to, clouds might be estimated using the instructions on this site.
§ apply Pythagoras' theorem and trigonometry to solve three-dimensional problems in right-angled triangles (ACMMG276)
§ draw diagrams and use them to solve word problems involving right-angled triangles in three dimensions, including using bearings and angles of elevation or depression, eg 'From a point, 𝐴, due south of a flagpole 100 metres tall on level ground, the angle of elevation of the top of the flagpole is 35°. The top of the same flagpole is observed with an angle of elevation
Activity: Students practise sketching 3D objects
(Note to teachers: The emphasis here is on drawing by hand. This activity might be done in cooperation with the Technology faculty)
§ Students sketch a variety of geometric solids from actual objects
§ Students sketch a range of rectangular and triangular prisms and identify right-angled triangles on each surface and through the centre of the prism. Example sketches of 3D prisms: https://www.tes.com/teaching-resource/quick-sketches-of-right-angled-triangles-in-prisms-11428684
§ Students develop this to a 3D representation of a landscape with a compass in the centre. Students design features in the landscape and describe their bearings from the centre compass . Example sketch of 3D landscape:
STEM: View the video: ‘Navy SEALS' Insane Parachute Jump into Football Stadium’ (Public domain TV): https://www.youtube.com/watch?v=x_7tsRrgA2Q&feature=youtu.be
This video is 5 minutes long and sections of the middle can be skipped.
Before playing, inform students that they will sketch a diagram showing the path of travel between the plane and the stadium.
NSW Education Standards Authority – Sample unit Mathematics Stage 5 - STEM Advanced Pathway: Algebraic Fractions Page 11 of 18
Content Teaching and Learning STEM Resources and Stimulus
22° from a point, B, due east of the flagpole. What is the distance from A to B?'
§ check the reasonableness of
solutions to problems involving bearings and right-angled triangle problems in three dimensions (Problem Solving)
§ use appropriate trigonometric ratios
and formulae to solve two-dimensional problems that require the use of more than one triangle, where the diagram is provided and where a verbal description is given (Problem Solving)
https://www.tes.com/teaching-resource/quick-sketch-3d-landscape-bearings-11428690
Differentiation:
Extension - show distance from centre and altitude of landscape features
Link to learning: It is not uncommon for students to be able to solve trigonometric problems in two dimensions but difficulty visualising a three-dimensional setting keeps them from being able to apply their skills. Persisting with their drawings should help students interpret and sketch 3D problems.
Consolidation for skill development and guided practice:
The teacher leads students to a systematic method sequence, such as:
§ Read the problem § Identify key information § Sketch and label the situation § Highlight the required length or angle § Identify right-angled triangle within the 3D sketch and redraw it
in isolation § Select most appropriate formula, substitute and solve § Check for reasonableness:
§ Students should be able to generalise about building, flagpole and streetlight heights, have some knowledge of cruising altitudes for aircraft, altitudes at which birds tend to fly and distances at which a yacht at sea would be visible from a cliff.
School-based and online worksheets could be used as resources.
Ask students to let you know when they first spot the stadium and pause the video.
Use ground features to estimate horizontal distance and angle of depression to the stadium at that time.
Ask: Can the altitude of the camera at that time be calculated?
Extension: students draw a 3D rectangular prism with the plane and stadium at diametrically opposed corners.
Standard: students draw a 2D rectangle with the plane and stadium at opposing corners.
Further considerations: What else can be calculated from the information in the video? (eg, speed of descent)
§ solve problems involving parallel Activity: Students look at the google maps illustration of all Sydney STEM: Students interact with the
NSW Education Standards Authority – Sample unit Mathematics Stage 5 - STEM Advanced Pathway: Algebraic Fractions Page 12 of 18
Content Teaching and Learning STEM Resources and Stimulus
and perpendicular lines (ACMNA238)
§ graph a variety of straight lines, including perpendicular lines, using digital technologies and compare their gradients to establish the condition for lines to be perpendicular (Communicating, Reasoning)
§ determine that straight lines are
perpendicular if the product of their gradients is −1
§ find the equation of a straight line
parallel or perpendicular to another given line using 𝑦 = 𝑚𝑥 + 𝑐
§ rearrange an equation of a straight line in the form 𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0 ('general form') to gradient-intercept form to determine the gradient and the 𝑦-intercept of the line
Airport flight paths: https://www.google.com/maps/d/u/0/viewer?mid=16LrBie9zbz4M9JJq1a4d5vepruU&hl=en_US&ll=-33.90333742803796%2C150.96519&z=10
The teacher displays this map for students and ask what they think the different colours indicate, then have them brainstorm single words that describe what they see. If they do not come up with it themselves, prompt students towards “parallel”. Discuss why flight paths would be parallel.
Link to learning:
Mathematics allows us to describe our location in space and time with the accuracy to allow busy airports to operate safely. With their ability to work out the gradient of a line segment joining two points, students already have the capacity to predict whether aircraft flying in straight lines are travelling parallel to one another or not. Students might think this is ‘all done by computers’ and if so, a reminder that computers only do what humans instruct them to
Live flight tracker (Flightradar24.com): https://www.flightradar24.com/-33.86,151.2/7
Note to teachers: after 15 seconds you can eliminate advertising from this display page.
Allow students time to digest this real-time image and imagine themselves with the responsibility of an air-traffic controller.
Can they effectively compare the googlemap of flight paths to this live flight tracker?
NSW Education Standards Authority – Sample unit Mathematics Stage 5 - STEM Advanced Pathway: Algebraic Fractions Page 13 of 18
Content Teaching and Learning STEM Resources and Stimulus
might add meaning to their learning.
Consolidation for skill development:
§ The teacher defines parallel and perpendicular and the symbols indicating each.
§ Students recall earlier gradient work and the location of gradient information in the equation of a straight line.
§ Students practise rearranging equations expressed in general form (𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0) to gradient-intercept form
§ Students use digital tools to graph a series of parallel and then perpendicular lines. Students to observe similarities and differences between equations. § Note to teachers: many online lessons and activities exist
for this learning. For example the activity: ‘Parallelendicular’ (via Scootle) : https://schoolsequella.det.nsw.edu.au/file/72bfb0f7-a8c6-453b-968f-e42e7ae94580/1/8881.zip/index.htm
§ The teacher introduces the notation 𝑚! and 𝑚! to use in the formula 𝑚!×𝑚! = −1
Guided practice: School-based and online worksheets could be used as resources.
§ use graphing software to graph a variety of equations of straight lines, and describe the similarities and differences between them, eg 𝑦 = −3𝑥,
𝑦 = −3𝑥 + 2, 𝑦 = −3𝑥 − 2,
𝑦 =12𝑥, 𝑦 = −2𝑥,
𝑦 = 3𝑥, 𝑥 = 2, 𝑦 = 2 (Communicating)
Activity: Design an aircraft hangar (see STEM stimulus column).
Link to learning:
Students in Stage 5 may not yet have considered shapes as being constructed from line segments and hence, not be aware that every side has its own equation if the shape has a location in space (ie, if it is real).
Consolidation for skill development:
STEM: Activity: ‘Design an aircraft hangar’ (TES): https://www.tes.com/teaching-resource/design-an-aircraft-hanger-11429791
Potential assessment activity:
Students are given a design brief for an aircraft hangar and aircraft
NSW Education Standards Authority – Sample unit Mathematics Stage 5 - STEM Advanced Pathway: Algebraic Fractions Page 14 of 18
Content Teaching and Learning STEM Resources and Stimulus
§ interpret and graph linear
relationships using the gradient-intercept form of the equation of a straight line, 𝑦 = 𝑚𝑥 + 𝑐
§ sketch the graph of a line by using its equation to find the 𝑥- and 𝑦-intercepts
Students should complete the STEM activity without the aid of digital graphing tools. They can then evaluate their finished work by creating a digital graph of their equations and comparing this to their hand-drawn hangar. This self-evaluation should reveal any misconceptions or errors.
Differentiation:
Extension - the teacher asks students to consider all parallel lines as transformations of a line through (0, 0) with a particular gradient. Hence, the y-intercept describes the transformation.
Students explore the conditions of 𝑥- and 𝑦-intercepts by considering:
§ the value of 𝑥 at the 𝑦-intercept § the value of 𝑦 at the 𝑥-intercept
§ the teacher asks students whether this is a rule to memorise or a thing they can see will always be true.
§ students practise substituting zero for 𝑥 and 𝑦 to find intercepts.
§ the teacher leads a class discussion: 𝑥- and 𝑦-intercept method vs gradient-intercept method vs plotting points from a table of values for drawing lines from linear equations.
Guided practice: School-based and online worksheets could be used as resources.
specifications are provided in a graphic.
By plotting the extremities of the aircraft on a Cartesian plane, students provide the equations that define the front-on edges of their walls and roof sections.
§ use the sine rule to find unknown sides and angles of a triangle, !
!"#!= !
!"#!= !
!"#!
§ use the cosine rule to find unknown
sides and angles of a triangle,
Activity: Take a walking tour of the school or neighbourhood and observe triangles in the built environment. Note whether they are right-angled, isosceles, equilateral or other.
Link to learning:
Point out that all the formulae they currently have for triangles
STEM:
Guided class discussion – What does architecture have to do with flight?
Suggest students search online
NSW Education Standards Authority – Sample unit Mathematics Stage 5 - STEM Advanced Pathway: Algebraic Fractions Page 15 of 18
Content Teaching and Learning STEM Resources and Stimulus
𝑎! = 𝑏! + 𝑐! − 2𝑏𝑐 cos𝐴
cos𝐴 =𝑏! + 𝑐! − 𝑎!
2𝑏𝑐
§ select and apply the appropriate rule
to find unknowns in non-right-angled triangles
§ solve a variety of practical problems that involve non-right-angled triangles, including problems where a diagram is not provided
require a pair of lengths at right-angles to one another.
Introduce the sine and cosine rules as useful tools for non right-angled triangles.
Consolidation for skill development:
The teacher explicitly teaches:
§ formulae with diagrams in different orientations. § how to decide which formulae are required.
Guided practice: School-based and online worksheets could be used as resources.
for “amazing architecture + airports”. Have any students visited any of these airports? Did they notice the architecture while there?
§ solve a variety of problems by applying coordinate geometry formulas, eg: prove that a particular triangle drawn on the Cartesian plane is right-angled (Communicating, Reasoning)
Challenge:
Students on the Mathematics STEM Pre-calculus Pathway and who have been engaging with the suggested Scope and Sequence will, by this stage in the Pathway, have significant problem-solving and some modelling experience.
In conclusion to the Flight 5.3 Program, students could work in small groups to create posters identifying where they recognise links between the different formulae, where one formula can be used to prove another, or connect this topic’s learning to formulae from other areas of Mathematics and to other subjects.
STEM: combining skills and recognising common properties, features and processes.
Assessment strategies
Activity: Students nominate a write-up or demonstration of one of the activities they completed to be evaluated against criteria created by class discussion.
Topic test: Short answer and multiple choice test.
NSW Education Standards Authority – Sample unit Mathematics Stage 5 - STEM Advanced Pathway: Algebraic Fractions Page 16 of 18
Assessment strategies
Student self-evaluation: Students rate their own development through this unit - their understanding & skills, their application to learning and working mathematically. Students discuss these with one another and then with teacher 'For Learning' in order to identify their readiness to move on to the next topic and personal learning objectives they might set themselves for the next topic (eg: participation in class, completion of homework, developing skills).
Resources overview
Additional Teacher Resources URLs:
§ Examples of elevation and depression (TES): https://www.tes.com/teaching-resource/angles-of-elevation-and-depression-in-the-field-and-in-the-classroom-11305036
§ Video: ‘How far can Legolas see?’ (Minute Physics): https://youtu.be/Rk2izv-c_ts § Video: ‘Vision in aviation’ (Federal Aviation Administration): https://www.youtube.com/watch?v=oWOFE-r8B6g § Video: ‘Helicopter cockpit view take off’ … with instructor explanations (youtube): https://youtu.be/LYXJgcoBqI8 § Video: ‘Student pilot has near miss’ (youtube): https://youtu.be/1mAf6NJBLWA § Interactive trigonometry (downloadable PDF version also available), Learn Alberta:
http://www.learnalberta.ca/content/mejhm/html/video_interactives/trigonometry/trigonometryInteractive.html § ‘Linear graphs’ (Scootle): http://www.scootle.edu.au/ec/viewing/L10090/html/index.html § Diagram of Ski field slopes: https://commons.wikimedia.org/w/index.php?curid=51854778
Teaching & Learning URLs of linked resources:
§ Bearings: ‘Where am I?’ (TES): https://www.tes.com/teaching-resource/where-am-i-11417302 § Bearings: ‘Eagle cat’ (Scootle): http://www.scootle.edu.au/ec/viewing/L10094/index.html § Bearings: ‘Picture this’ (TES): https://www.tes.com/teaching-resource/where-am-i-11417302 § ‘Angles round a point’ worksheet (TES): https://www.tes.com/teaching-resource/angles-round-a-point-worksheet-6317913 § Video: ‘World’s Largest Urban Zipline’: https://www.youtube.com/watch?v=YcwrRA2BIlw&feature=youtu.be § ‘Introduction to Geometry: Points, Lines, Planes and Dimensions’ (Skills you need): http://www.skillsyouneed.com/num/geometry.html § ‘Angles formed by a transversal’ worksheet (Maths Worksheets 4 Kids): http://www.mathworksheets4kids.com/transversal.php § ‘Introducing tan as adding meaning to gradient’ worksheet (TES): https://www.tes.com/teaching-resource/introducing-tan-as-adding-
meaning-to-gradient-11427366 § Making a Clinometer (University of Cambridge and Centre for Mathematical Sciences enrichment program (NRICH)):
https://nrich.maths.org/5382
NSW Education Standards Authority – Sample unit Mathematics Stage 5 - STEM Advanced Pathway: Algebraic Fractions Page 17 of 18
Resources overview
§ Example sketches of 3D prisms: https://www.tes.com/teaching-resource/quick-sketches-of-right-angled-triangles-in-prisms-11428684 § Example sketch of 3D landscape: https://www.tes.com/teaching-resource/quick-sketch-3d-landscape-bearings-11428690 § Google maps illustration of all Sydney Airport flight paths:
https://www.google.com/maps/d/u/0/viewer?mid=16LrBie9zbz4M9JJq1a4d5vepruU&hl=en_US&ll=-33.90333742803796%2C150.96519&z=10
§ Parallel lines: ‘Parallelendicular’ (Scootle): https://schoolsequella.det.nsw.edu.au/file/72bfb0f7-a8c6-453b-968f-e42e7ae94580/1/8881.zip/index.htm
STEM Resources & Stimulus URLs of linked resources:
§ Video: ‘Simple way to make a parachute’: https://www.youtube.com/watch?v=ZlFiTlkjNtc § Make a Homemade Compass (Scientific American): https://www.scientificamerican.com/article/steering-science-make-a-homemade-
compass/ § Video: ‘Sense of Flying’ (Goovinn): https://www.youtube.com/watch?v=ER1PGYe9UZA&feature=youtu.be § Video: ‘Meet the world’s only fully functional jetpack’ (Science Channel):
https://www.youtube.com/watch?v=cIMWYeP7O1w&feature=youtu.be § Video: ‘Flyboard Air Demo at Flyboard World Cup’ (Atlantic Flyboard): https://www.youtube.com/watch?v=deyMNPbaRpA&feature=youtu.be § ‘Where am I?’ (2D flight path) (TES): https://www.tes.com/teaching-resource/where-am-i-2d-flight-path-11423688 § ‘Where am I?’ (3D extension) (TES): https://www.tes.com/teaching-resource/where-am-i-3d-extension-of-bearings-11423666 § Discussion on Geographic Information Systems: http://gis.stackexchange.com/questions/4690/determine-angle-down-to-horizon-from-
different-flight-altitudes § ‘Exploring Trigonometry’ (Learn Alberta):
http://www.learnalberta.ca/content/mejhm/index.html?l=0&ID1=AB.MATH.JR.SHAP&ID2=AB.MATH.JR.SHAP.TRI&lesson=html/video_interactives/trigonometry/trigonometrySmall.html
§ 'The physics of off-roading' (How stuff works): http://adventure.howstuffworks.com/outdoor-activities/off-roading/off-roading1.htm § Air navigation: ‘The effect of wind’ (Fly Safe): https://www.recreationalflying.com/tutorials/navigation/wind.html § ‘How far away is that contrail?’ (Contrail Science): http://contrailscience.com/how-far-away-is-that-contrail/ § Video: ‘Navy SEALS' Insane Parachute Jump into Football Stadium’ (Public domain TV): https://www.youtube.com/watch?v=x_7tsRrgA2Q § Live flight tracker (Flightradar24.com): https://www.flightradar24.com/-33.86,151.2/7 § ‘Design an aircraft hangar’ (TES): https://www.tes.com/teaching-resource/design-an-aircraft-hanger-11429791
NSW Education Standards Authority – Sample unit Mathematics Stage 5 - STEM Advanced Pathway: Algebraic Fractions Page 18 of 18
Teacher Evaluation of Unit